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\hypersetup{pdftitle={Likelihood inference in an autoregression with f\mbox{}ixed ef\mbox{}fects: Supplementary material}, pdfauthor={G.~Dhaene and K.~Jochmans}, pdftex,linkcolor=blue,citecolor=blue,urlcolor=black}


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\year 2013
\received{March 15, 2007}
\accepted{June 10, 2013}
\volume{10}
\setcounter{page}{1}


\title[Likelihood inference in an autoregression with f\mbox{}ixed ef\mbox{}fects: Supplementary material]{{\normalsize LIKELIHOOD INFERENCE IN AN AUTOREGRESSION WITH FIXED EFFECTS: SUPPLEMENTARY MATERIAL}}



\author[G.~Dhaene and K.~Jochmans]{Geert Dhaene}

\address{{\normalsize University of Leuven, Department of Economics, Naamsestraat 69, B-3000 Leuven, Belgium}}
\email{geert.dhaene@kuleuven.be}

\author{Koen Jochmans}

\address{{\normalsize Sciences Po, Department of Economics, 28 rue des Saints-P\`eres, 75007 Paris, France}}
\email{koen.jochmans@sciencespo.fr}

\begin{document}


\vspace{2cm}
\noindent
Tables 1 to 3 below present simulation results for the same design as in the
main text, but for a more extensive range of parameter values and sample
sizes. We ran a full factorial design with%
\begin{eqnarray*}
N &=&100,250,500,1000,2500,5000,10000; \\
T &=&2,4,6,8,16,24;
\end{eqnarray*}%
and

\begin{itemize}
\item in the f\mbox{}irst-order autoregression (Table 1),%
\[
\rho _{0}=.8,.9,.99;\qquad \psi =0,1,2;
\]

\item in the second-order autoregression (Table 2),%
\[
\rho _{0}=\left(
\begin{array}{c}
.6 \\
.2%
\end{array}%
\right) ,\left(
\begin{array}{c}
1 \\
-.2%
\end{array}%
\right) ;\qquad \psi =.3,1,2;
\]

\item in the f\mbox{}irst-order autoregression with a covariate  (Table 3),%
\[
\theta _{0}=\left(
\begin{array}{c}
\rho _{0} \\
\beta _{0}%
\end{array}%
\right) =\left(
\begin{array}{c}
.5 \\
.5%
\end{array}%
\right) ,\left(
\begin{array}{c}
.9 \\
.1%
\end{array}%
\right) ,\left(
\begin{array}{c}
.99 \\
.01%
\end{array}%
\right) ;\qquad \gamma =.5,.99;\qquad \psi =0,1,2.
\]
\end{itemize}

\noindent
We ran 10,000 replications at each design point.


\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.5$ & $-.142$ & --- & $-.747$ & $.267$ & --- & $.153$ &
$.819$ & $.921$ & $.000$ \\
$100$ & $2$ & $1$ & $.5$ & $.027$ & --- & $-.373$ & $.266$ & --- & $.141$ & $%
.903$ & $.934$ & $.090$ \\
$100$ & $2$ & $2$ & $.5$ & $.019$ & --- & $.111$ & $.166$ & --- & $.113$ & $%
.946$ & $.945$ & $.880$ \\
$100$ & $4$ & $0$ & $.5$ & $.008$ & $-.039$ & $-.295$ & $.141$ & $.148$ & $%
.066$ & $.924$ & $.926$ & $.004$ \\
$100$ & $4$ & $1$ & $.5$ & $.016$ & $-.053$ & $-.139$ & $.124$ & $.164$ & $%
.067$ & $.945$ & $.928$ & $.327$ \\
$100$ & $4$ & $2$ & $.5$ & $.001$ & $-.016$ & $.071$ & $.064$ & $.082$ & $%
.056$ & $.946$ & $.936$ & $.684$ \\
$100$ & $6$ & $0$ & $.5$ & $.008$ & $-.032$ & $-.147$ & $.091$ & $.082$ & $%
.051$ & $.952$ & $.920$ & $.131$ \\
$100$ & $6$ & $1$ & $.5$ & $.002$ & $-.048$ & $-.073$ & $.068$ & $.096$ & $%
.049$ & $.952$ & $.914$ & $.580$ \\
$100$ & $6$ & $2$ & $.5$ & $-.001$ & $-.021$ & $.043$ & $.044$ & $.063$ & $%
.041$ & $.945$ & $.930$ & $.744$ \\
$100$ & $8$ & $0$ & $.5$ & $.001$ & $-.026$ & $-.085$ & $.056$ & $.057$ & $%
.042$ & $.953$ & $.918$ & $.400$ \\
$100$ & $8$ & $1$ & $.5$ & $-.001$ & $-.040$ & $-.045$ & $.048$ & $.070$ & $%
.040$ & $.943$ & $.907$ & $.730$ \\
$100$ & $8$ & $2$ & $.5$ & $-.001$ & $-.023$ & $.028$ & $.036$ & $.051$ & $%
.034$ & $.946$ & $.930$ & $.812$ \\
$100$ & $16$ & $0$ & $.5$ & $.000$ & $-.019$ & $-.021$ & $.028$ & $.030$ & $%
.026$ & $.944$ & $.902$ & $.841$ \\
$100$ & $16$ & $1$ & $.5$ & $-.001$ & $-.027$ & $-.013$ & $.027$ & $.035$ & $%
.025$ & $.947$ & $.879$ & $.899$ \\
$100$ & $16$ & $2$ & $.5$ & $-.001$ & $-.023$ & $.009$ & $.023$ & $.032$ & $%
.023$ & $.944$ & $.893$ & $.902$ \\
$100$ & $24$ & $0$ & $.5$ & $.000$ & $-.018$ & $-.010$ & $.021$ & $.022$ & $%
.020$ & $.943$ & $.878$ & $.907$ \\
$100$ & $24$ & $1$ & $.5$ & $-.001$ & $-.023$ & $-.006$ & $.020$ & $.025$ & $%
.020$ & $.948$ & $.852$ & $.926$ \\
$100$ & $24$ & $2$ & $.5$ & $.000$ & $-.020$ & $.004$ & $.019$ & $.024$ & $%
.018$ & $.943$ & $.869$ & $.925$%
\\ \hline \T
$100$ & $2$ & $0$ & $.9$ & $-.144$ & --- & $-.551$ & $.265$ & --- & $.151$ &
$.821$ & $.925$ & $.014$ \\
$100$ & $2$ & $1$ & $.9$ & $-.094$ & --- & $-.472$ & $.267$ & --- & $.153$ &
$.845$ & $.923$ & $.050$ \\
$100$ & $2$ & $2$ & $.9$ & $-.006$ & --- & $-.288$ & $.270$ & --- & $.145$ &
$.884$ & $.931$ & $.323$ \\
$100$ & $4$ & $0$ & $.9$ & $-.083$ & $-.483$ & $-.294$ & $.127$ & $.413$ & $%
.073$ & $.843$ & $.730$ & $.006$ \\
$100$ & $4$ & $1$ & $.9$ & $-.044$ & $-.537$ & $-.227$ & $.126$ & $.455$ & $%
.072$ & $.879$ & $.748$ & $.055$ \\
$100$ & $4$ & $2$ & $.9$ & $.013$ & $-.150$ & $-.085$ & $.126$ & $.266$ & $%
.066$ & $.922$ & $.900$ & $.648$ \\
$100$ & $6$ & $0$ & $.9$ & $-.051$ & $-.283$ & $-.203$ & $.086$ & $.210$ & $%
.051$ & $.858$ & $.691$ & $.006$ \\
$100$ & $6$ & $1$ & $.9$ & $-.020$ & $-.383$ & $-.145$ & $.085$ & $.245$ & $%
.049$ & $.904$ & $.645$ & $.080$ \\
$100$ & $6$ & $2$ & $.9$ & $.011$ & $-.128$ & $-.029$ & $.082$ & $.151$ & $%
.044$ & $.944$ & $.864$ & $.837$ \\
$100$ & $8$ & $0$ & $.9$ & $-.033$ & $-.191$ & $-.154$ & $.066$ & $.127$ & $%
.039$ & $.876$ & $.669$ & $.008$ \\
$100$ & $8$ & $1$ & $.9$ & $-.009$ & $-.283$ & $-.104$ & $.067$ & $.166$ & $%
.038$ & $.914$ & $.576$ & $.124$ \\
$100$ & $8$ & $2$ & $.9$ & $.008$ & $-.113$ & $-.008$ & $.059$ & $.106$ & $%
.033$ & $.949$ & $.815$ & $.890$ \\
$100$ & $16$ & $0$ & $.9$ & $-.003$ & $-.077$ & $-.072$ & $.038$ & $.043$ & $%
.022$ & $.925$ & $.572$ & $.041$ \\
$100$ & $16$ & $1$ & $.9$ & $.002$ & $-.130$ & $-.043$ & $.037$ & $.061$ & $%
.021$ & $.944$ & $.381$ & $.324$ \\
$100$ & $16$ & $2$ & $.9$ & $.000$ & $-.082$ & $.008$ & $.023$ & $.050$ & $%
.017$ & $.958$ & $.583$ & $.869$ \\
$100$ & $24$ & $0$ & $.9$ & $.001$ & $-.050$ & $-.041$ & $.027$ & $.025$ & $%
.015$ & $.951$ & $.478$ & $.157$ \\
$100$ & $24$ & $1$ & $.9$ & $.001$ & $-.083$ & $-.024$ & $.023$ & $.034$ & $%
.015$ & $.952$ & $.274$ & $.514$ \\
$100$ & $24$ & $2$ & $.9$ & $.000$ & $-.068$ & $.007$ & $.015$ & $.033$ & $%
.012$ & $.943$ & $.365$ & $.854$%
\\ \hline \T
$100$ & $2$ & $0$ & $.99$ & $-.144$ & --- & $-.506$ & $.265$ & --- & $.151$
& $.821$ & $.925$ & $.034$ \\
$100$ & $2$ & $1$ & $.99$ & $-.135$ & --- & $-.495$ & $.267$ & --- & $.153$
& $.821$ & $.918$ & $.043$ \\
$100$ & $2$ & $2$ & $.99$ & $-.125$ & --- & $-.475$ & $.266$ & --- & $.150$
& $.827$ & $.929$ & $.053$ \\
$100$ & $4$ & $0$ & $.99$ & $-.087$ & $-.773$ & $-.258$ & $.123$ & $.474$ & $%
.073$ & $.835$ & $.651$ & $.023$ \\
$100$ & $4$ & $1$ & $.99$ & $-.082$ & $-.771$ & $-.249$ & $.123$ & $.475$ & $%
.072$ & $.839$ & $.656$ & $.027$ \\
$100$ & $4$ & $2$ & $.99$ & $-.068$ & $-.737$ & $-.229$ & $.123$ & $.472$ & $%
.072$ & $.849$ & $.675$ & $.049$ \\
$100$ & $6$ & $0$ & $.99$ & $-.060$ & $-.587$ & $-.174$ & $.082$ & $.276$ & $%
.050$ & $.839$ & $.447$ & $.022$ \\
$100$ & $6$ & $1$ & $.99$ & $-.056$ & $-.584$ & $-.167$ & $.080$ & $.272$ & $%
.049$ & $.852$ & $.448$ & $.029$ \\
$100$ & $6$ & $2$ & $.99$ & $-.042$ & $-.550$ & $-.145$ & $.080$ & $.279$ & $%
.049$ & $.871$ & $.493$ & $.075$ \\
$100$ & $8$ & $0$ & $.99$ & $-.046$ & $-.472$ & $-.132$ & $.062$ & $.198$ & $%
.038$ & $.847$ & $.280$ & $.022$ \\
$100$ & $8$ & $1$ & $.99$ & $-.043$ & $-.469$ & $-.125$ & $.061$ & $.193$ & $%
.038$ & $.850$ & $.281$ & $.033$ \\
$100$ & $8$ & $2$ & $.99$ & $-.028$ & $-.434$ & $-.104$ & $.060$ & $.198$ & $%
.037$ & $.881$ & $.337$ & $.098$ \\
$100$ & $16$ & $0$ & $.99$ & $-.025$ & $-.255$ & $-.068$ & $.031$ & $.081$ &
$.020$ & $.843$ & $.034$ & $.020$ \\
$100$ & $16$ & $1$ & $.99$ & $-.020$ & $-.254$ & $-.061$ & $.031$ & $.080$ &
$.020$ & $.867$ & $.033$ & $.045$ \\
$100$ & $16$ & $2$ & $.99$ & $-.009$ & $-.227$ & $-.043$ & $.030$ & $.080$ &
$.019$ & $.910$ & $.057$ & $.213$ \\
$100$ & $24$ & $0$ & $.99$ & $-.016$ & $-.172$ & $-.047$ & $.021$ & $.047$ &
$.014$ & $.857$ & $.003$ & $.019$ \\
$100$ & $24$ & $1$ & $.99$ & $-.012$ & $-.173$ & $-.040$ & $.021$ & $.047$ &
$.013$ & $.873$ & $.004$ & $.055$ \\
$100$ & $24$ & $2$ & $.99$ & $-.003$ & $-.150$ & $-.024$ & $.021$ & $.046$ &
$.012$ & $.920$ & $.009$ & $.358$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}

\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.5$ & $-.125$ & --- & $-.750$ & $.203$ & --- & $.095$ &
$.819$ & $.926$ & $.000$ \\
$250$ & $2$ & $1$ & $.5$ & $.036$ & --- & $-.375$ & $.204$ & --- & $.089$ & $%
.924$ & $.942$ & $.003$ \\
$250$ & $2$ & $2$ & $.5$ & $.008$ & --- & $.109$ & $.096$ & --- & $.071$ & $%
.956$ & $.951$ & $.672$ \\
$250$ & $4$ & $0$ & $.5$ & $.013$ & $-.017$ & $-.295$ & $.110$ & $.097$ & $%
.043$ & $.932$ & $.937$ & $.000$ \\
$250$ & $4$ & $1$ & $.5$ & $.005$ & $-.023$ & $-.140$ & $.078$ & $.106$ & $%
.042$ & $.954$ & $.938$ & $.051$ \\
$250$ & $4$ & $2$ & $.5$ & $.001$ & $-.006$ & $.072$ & $.040$ & $.053$ & $%
.036$ & $.945$ & $.945$ & $.385$ \\
$250$ & $6$ & $0$ & $.5$ & $.002$ & $-.013$ & $-.147$ & $.055$ & $.051$ & $%
.033$ & $.960$ & $.937$ & $.004$ \\
$250$ & $6$ & $1$ & $.5$ & $.000$ & $-.019$ & $-.072$ & $.041$ & $.063$ & $%
.031$ & $.951$ & $.940$ & $.261$ \\
$250$ & $6$ & $2$ & $.5$ & $.000$ & $-.008$ & $.044$ & $.028$ & $.040$ & $%
.026$ & $.947$ & $.941$ & $.508$ \\
$250$ & $8$ & $0$ & $.5$ & $.001$ & $-.011$ & $-.085$ & $.035$ & $.037$ & $%
.027$ & $.950$ & $.939$ & $.078$ \\
$250$ & $8$ & $1$ & $.5$ & $.000$ & $-.017$ & $-.043$ & $.030$ & $.045$ & $%
.026$ & $.945$ & $.931$ & $.515$ \\
$250$ & $8$ & $2$ & $.5$ & $.000$ & $-.009$ & $.029$ & $.023$ & $.033$ & $%
.022$ & $.947$ & $.943$ & $.644$ \\
$250$ & $16$ & $0$ & $.5$ & $.000$ & $-.008$ & $-.021$ & $.018$ & $.019$ & $%
.017$ & $.948$ & $.933$ & $.708$ \\
$250$ & $16$ & $1$ & $.5$ & $.000$ & $-.011$ & $-.012$ & $.017$ & $.022$ & $%
.016$ & $.951$ & $.924$ & $.856$ \\
$250$ & $16$ & $2$ & $.5$ & $.000$ & $-.009$ & $.009$ & $.015$ & $.021$ & $%
.015$ & $.945$ & $.923$ & $.864$ \\
$250$ & $24$ & $0$ & $.5$ & $.000$ & $-.007$ & $-.009$ & $.013$ & $.014$ & $%
.013$ & $.948$ & $.920$ & $.866$ \\
$250$ & $24$ & $1$ & $.5$ & $.000$ & $-.009$ & $-.005$ & $.013$ & $.016$ & $%
.013$ & $.947$ & $.909$ & $.906$ \\
$250$ & $24$ & $2$ & $.5$ & $.000$ & $-.009$ & $.004$ & $.012$ & $.015$ & $%
.012$ & $.947$ & $.915$ & $.906$%
\\ \hline \T
$250$ & $2$ & $0$ & $.9$ & $-.123$ & --- & $-.549$ & $.200$ & --- & $.094$ &
$.823$ & $.926$ & $.000$ \\
$250$ & $2$ & $1$ & $.9$ & $-.075$ & --- & $-.474$ & $.200$ & --- & $.094$ &
$.856$ & $.928$ & $.000$ \\
$250$ & $2$ & $2$ & $.9$ & $.009$ & --- & $-.289$ & $.204$ & --- & $.091$ & $%
.909$ & $.934$ & $.056$ \\
$250$ & $4$ & $0$ & $.9$ & $-.062$ & $-.276$ & $-.292$ & $.096$ & $.329$ & $%
.046$ & $.850$ & $.810$ & $.000$ \\
$250$ & $4$ & $1$ & $.9$ & $-.026$ & $-.369$ & $-.227$ & $.097$ & $.398$ & $%
.046$ & $.892$ & $.804$ & $.000$ \\
$250$ & $4$ & $2$ & $.9$ & $.016$ & $-.056$ & $-.082$ & $.096$ & $.164$ & $%
.042$ & $.940$ & $.927$ & $.384$ \\
$250$ & $6$ & $0$ & $.9$ & $-.037$ & $-.150$ & $-.202$ & $.064$ & $.148$ & $%
.031$ & $.874$ & $.808$ & $.000$ \\
$250$ & $6$ & $1$ & $.9$ & $-.008$ & $-.242$ & $-.144$ & $.067$ & $.200$ & $%
.031$ & $.906$ & $.759$ & $.001$ \\
$250$ & $6$ & $2$ & $.9$ & $.008$ & $-.054$ & $-.029$ & $.058$ & $.094$ & $%
.027$ & $.952$ & $.913$ & $.739$ \\
$250$ & $8$ & $0$ & $.9$ & $-.022$ & $-.096$ & $-.153$ & $.050$ & $.089$ & $%
.025$ & $.886$ & $.803$ & $.000$ \\
$250$ & $8$ & $1$ & $.9$ & $-.001$ & $-.172$ & $-.103$ & $.051$ & $.126$ & $%
.024$ & $.929$ & $.716$ & $.003$ \\
$250$ & $8$ & $2$ & $.9$ & $.004$ & $-.051$ & $-.008$ & $.038$ & $.067$ & $%
.021$ & $.955$ & $.881$ & $.879$ \\
$250$ & $16$ & $0$ & $.9$ & $.001$ & $-.036$ & $-.071$ & $.029$ & $.029$ & $%
.014$ & $.939$ & $.765$ & $.000$ \\
$250$ & $16$ & $1$ & $.9$ & $.003$ & $-.074$ & $-.043$ & $.026$ & $.042$ & $%
.013$ & $.955$ & $.589$ & $.051$ \\
$250$ & $16$ & $2$ & $.9$ & $.000$ & $-.041$ & $.008$ & $.014$ & $.032$ & $%
.011$ & $.954$ & $.759$ & $.817$ \\
$250$ & $24$ & $0$ & $.9$ & $.001$ & $-.023$ & $-.041$ & $.018$ & $.016$ & $%
.010$ & $.958$ & $.716$ & $.005$ \\
$250$ & $24$ & $1$ & $.9$ & $.000$ & $-.045$ & $-.024$ & $.014$ & $.024$ & $%
.009$ & $.957$ & $.517$ & $.191$ \\
$250$ & $24$ & $2$ & $.9$ & $.000$ & $-.037$ & $.007$ & $.009$ & $.023$ & $%
.008$ & $.950$ & $.597$ & $.777$%
\\ \hline \T
$250$ & $2$ & $0$ & $.99$ & $-.123$ & --- & $-.504$ & $.200$ & --- & $.094$
& $.823$ & $.926$ & $.000$ \\
$250$ & $2$ & $1$ & $.99$ & $-.116$ & --- & $-.496$ & $.200$ & --- & $.095$
& $.827$ & $.924$ & $.000$ \\
$250$ & $2$ & $2$ & $.99$ & $-.102$ & --- & $-.476$ & $.199$ & --- & $.095$
& $.839$ & $.932$ & $.000$ \\
$250$ & $4$ & $0$ & $.99$ & $-.067$ & $-.764$ & $-.256$ & $.094$ & $.466$ & $%
.046$ & $.840$ & $.665$ & $.000$ \\
$250$ & $4$ & $1$ & $.99$ & $-.063$ & $-.758$ & $-.249$ & $.093$ & $.470$ & $%
.046$ & $.845$ & $.673$ & $.000$ \\
$250$ & $4$ & $2$ & $.99$ & $-.049$ & $-.695$ & $-.227$ & $.092$ & $.465$ & $%
.046$ & $.866$ & $.701$ & $.000$ \\
$250$ & $6$ & $0$ & $.99$ & $-.047$ & $-.582$ & $-.173$ & $.061$ & $.276$ & $%
.031$ & $.848$ & $.460$ & $.000$ \\
$250$ & $6$ & $1$ & $.99$ & $-.043$ & $-.580$ & $-.166$ & $.063$ & $.278$ & $%
.031$ & $.849$ & $.467$ & $.000$ \\
$250$ & $6$ & $2$ & $.99$ & $-.030$ & $-.514$ & $-.145$ & $.062$ & $.277$ & $%
.031$ & $.873$ & $.539$ & $.001$ \\
$250$ & $8$ & $0$ & $.99$ & $-.036$ & $-.458$ & $-.131$ & $.046$ & $.196$ & $%
.024$ & $.848$ & $.296$ & $.000$ \\
$250$ & $8$ & $1$ & $.99$ & $-.031$ & $-.463$ & $-.123$ & $.046$ & $.195$ & $%
.024$ & $.862$ & $.297$ & $.000$ \\
$250$ & $8$ & $2$ & $.99$ & $-.019$ & $-.407$ & $-.103$ & $.046$ & $.194$ & $%
.024$ & $.887$ & $.374$ & $.002$ \\
$250$ & $16$ & $0$ & $.99$ & $-.018$ & $-.240$ & $-.067$ & $.023$ & $.080$ &
$.012$ & $.857$ & $.041$ & $.000$ \\
$250$ & $16$ & $1$ & $.99$ & $-.014$ & $-.251$ & $-.061$ & $.023$ & $.081$ &
$.012$ & $.876$ & $.031$ & $.000$ \\
$250$ & $16$ & $2$ & $.99$ & $-.004$ & $-.208$ & $-.042$ & $.023$ & $.080$ &
$.012$ & $.913$ & $.084$ & $.015$ \\
$250$ & $24$ & $0$ & $.99$ & $-.012$ & $-.157$ & $-.046$ & $.016$ & $.044$ &
$.008$ & $.858$ & $.005$ & $.000$ \\
$250$ & $24$ & $1$ & $.99$ & $-.008$ & $-.168$ & $-.039$ & $.016$ & $.047$ &
$.008$ & $.881$ & $.004$ & $.000$ \\
$250$ & $24$ & $2$ & $.99$ & $-.001$ & $-.136$ & $-.024$ & $.016$ & $.044$ &
$.008$ & $.932$ & $.017$ & $.063$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}



\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.5$ & $-.106$ & --- & $-.750$ & $.162$ & --- & $.067$ &
$.833$ & $.927$ & $.000$ \\
$500$ & $2$ & $1$ & $.5$ & $.033$ & --- & $-.375$ & $.168$ & --- & $.063$ & $%
.931$ & $.953$ & $.000$ \\
$500$ & $2$ & $2$ & $.5$ & $.004$ & --- & $.108$ & $.067$ & --- & $.051$ & $%
.952$ & $.950$ & $.400$ \\
$500$ & $4$ & $0$ & $.5$ & $.012$ & $-.008$ & $-.295$ & $.088$ & $.069$ & $%
.030$ & $.946$ & $.942$ & $.000$ \\
$500$ & $4$ & $1$ & $.5$ & $.003$ & $-.010$ & $-.139$ & $.053$ & $.076$ & $%
.030$ & $.958$ & $.943$ & $.002$ \\
$500$ & $4$ & $2$ & $.5$ & $.000$ & $-.003$ & $.072$ & $.028$ & $.038$ & $%
.025$ & $.949$ & $.946$ & $.125$ \\
$500$ & $6$ & $0$ & $.5$ & $.001$ & $-.006$ & $-.146$ & $.038$ & $.037$ & $%
.023$ & $.955$ & $.943$ & $.000$ \\
$500$ & $6$ & $1$ & $.5$ & $.000$ & $-.010$ & $-.072$ & $.029$ & $.045$ & $%
.022$ & $.954$ & $.943$ & $.056$ \\
$500$ & $6$ & $2$ & $.5$ & $.000$ & $-.004$ & $.044$ & $.020$ & $.028$ & $%
.018$ & $.947$ & $.946$ & $.243$ \\
$500$ & $8$ & $0$ & $.5$ & $.000$ & $-.006$ & $-.085$ & $.025$ & $.026$ & $%
.019$ & $.946$ & $.941$ & $.002$ \\
$500$ & $8$ & $1$ & $.5$ & $.000$ & $-.008$ & $-.043$ & $.021$ & $.032$ & $%
.018$ & $.948$ & $.939$ & $.246$ \\
$500$ & $8$ & $2$ & $.5$ & $.000$ & $-.005$ & $.029$ & $.016$ & $.023$ & $%
.015$ & $.951$ & $.949$ & $.431$ \\
$500$ & $16$ & $0$ & $.5$ & $.000$ & $-.004$ & $-.021$ & $.012$ & $.014$ & $%
.012$ & $.951$ & $.939$ & $.509$ \\
$500$ & $16$ & $1$ & $.5$ & $.000$ & $-.006$ & $-.011$ & $.012$ & $.016$ & $%
.011$ & $.949$ & $.936$ & $.781$ \\
$500$ & $16$ & $2$ & $.5$ & $.000$ & $-.004$ & $.009$ & $.010$ & $.015$ & $%
.010$ & $.948$ & $.940$ & $.807$ \\
$500$ & $24$ & $0$ & $.5$ & $.000$ & $-.004$ & $-.009$ & $.009$ & $.010$ & $%
.009$ & $.947$ & $.933$ & $.791$ \\
$500$ & $24$ & $1$ & $.5$ & $.000$ & $-.005$ & $-.005$ & $.009$ & $.011$ & $%
.009$ & $.949$ & $.929$ & $.887$ \\
$500$ & $24$ & $2$ & $.5$ & $.000$ & $-.004$ & $.004$ & $.008$ & $.011$ & $%
.008$ & $.947$ & $.929$ & $.887$%
\\ \hline \T
$500$ & $2$ & $0$ & $.9$ & $-.107$ & --- & $-.550$ & $.164$ & --- & $.067$ &
$.826$ & $.931$ & $.000$ \\
$500$ & $2$ & $1$ & $.9$ & $-.060$ & --- & $-.475$ & $.164$ & --- & $.067$ &
$.864$ & $.930$ & $.000$ \\
$500$ & $2$ & $2$ & $.9$ & $.012$ & --- & $-.290$ & $.169$ & --- & $.065$ & $%
.913$ & $.945$ & $.002$ \\
$500$ & $4$ & $0$ & $.9$ & $-.052$ & $-.157$ & $-.292$ & $.078$ & $.258$ & $%
.033$ & $.851$ & $.861$ & $.000$ \\
$500$ & $4$ & $1$ & $.9$ & $-.018$ & $-.227$ & $-.226$ & $.080$ & $.320$ & $%
.032$ & $.900$ & $.854$ & $.000$ \\
$500$ & $4$ & $2$ & $.9$ & $.013$ & $-.026$ & $-.082$ & $.076$ & $.115$ & $%
.030$ & $.951$ & $.937$ & $.135$ \\
$500$ & $6$ & $0$ & $.9$ & $-.030$ & $-.082$ & $-.201$ & $.053$ & $.112$ & $%
.023$ & $.872$ & $.870$ & $.000$ \\
$500$ & $6$ & $1$ & $.9$ & $-.001$ & $-.147$ & $-.143$ & $.055$ & $.153$ & $%
.022$ & $.924$ & $.834$ & $.000$ \\
$500$ & $6$ & $2$ & $.9$ & $.005$ & $-.028$ & $-.028$ & $.041$ & $.066$ & $%
.020$ & $.957$ & $.933$ & $.601$ \\
$500$ & $8$ & $0$ & $.9$ & $-.016$ & $-.052$ & $-.153$ & $.041$ & $.065$ & $%
.018$ & $.892$ & $.868$ & $.000$ \\
$500$ & $8$ & $1$ & $.9$ & $.003$ & $-.104$ & $-.102$ & $.043$ & $.094$ & $%
.017$ & $.933$ & $.802$ & $.000$ \\
$500$ & $8$ & $2$ & $.9$ & $.001$ & $-.027$ & $-.008$ & $.025$ & $.047$ & $%
.015$ & $.961$ & $.915$ & $.864$ \\
$500$ & $16$ & $0$ & $.9$ & $.002$ & $-.019$ & $-.071$ & $.023$ & $.020$ & $%
.010$ & $.944$ & $.852$ & $.000$ \\
$500$ & $16$ & $1$ & $.9$ & $.002$ & $-.042$ & $-.042$ & $.019$ & $.031$ & $%
.009$ & $.957$ & $.740$ & $.001$ \\
$500$ & $16$ & $2$ & $.9$ & $.000$ & $-.023$ & $.008$ & $.010$ & $.023$ & $%
.008$ & $.954$ & $.845$ & $.731$ \\
$500$ & $24$ & $0$ & $.9$ & $.001$ & $-.012$ & $-.041$ & $.013$ & $.011$ & $%
.007$ & $.960$ & $.828$ & $.000$ \\
$500$ & $24$ & $1$ & $.9$ & $.000$ & $-.025$ & $-.023$ & $.010$ & $.017$ & $%
.007$ & $.957$ & $.699$ & $.024$ \\
$500$ & $24$ & $2$ & $.9$ & $.000$ & $-.021$ & $.007$ & $.006$ & $.016$ & $%
.005$ & $.950$ & $.756$ & $.638$%
\\ \hline \T
$500$ & $2$ & $0$ & $.99$ & $-.107$ & --- & $-.505$ & $.164$ & --- & $.067$
& $.826$ & $.931$ & $.000$ \\
$500$ & $2$ & $1$ & $.99$ & $-.102$ & --- & $-.497$ & $.163$ & --- & $.067$
& $.831$ & $.928$ & $.000$ \\
$500$ & $2$ & $2$ & $.99$ & $-.090$ & --- & $-.476$ & $.164$ & --- & $.067$
& $.842$ & $.932$ & $.000$ \\
$500$ & $4$ & $0$ & $.99$ & $-.056$ & $-.748$ & $-.256$ & $.076$ & $.474$ & $%
.033$ & $.839$ & $.671$ & $.000$ \\
$500$ & $4$ & $1$ & $.99$ & $-.054$ & $-.756$ & $-.248$ & $.076$ & $.474$ & $%
.032$ & $.844$ & $.681$ & $.000$ \\
$500$ & $4$ & $2$ & $.99$ & $-.039$ & $-.640$ & $-.226$ & $.076$ & $.489$ & $%
.032$ & $.864$ & $.727$ & $.000$ \\
$500$ & $6$ & $0$ & $.99$ & $-.040$ & $-.560$ & $-.172$ & $.050$ & $.273$ & $%
.022$ & $.851$ & $.475$ & $.000$ \\
$500$ & $6$ & $1$ & $.99$ & $-.033$ & $-.579$ & $-.164$ & $.050$ & $.279$ & $%
.022$ & $.857$ & $.474$ & $.000$ \\
$500$ & $6$ & $2$ & $.99$ & $-.022$ & $-.469$ & $-.144$ & $.050$ & $.271$ & $%
.022$ & $.884$ & $.577$ & $.000$ \\
$500$ & $8$ & $0$ & $.99$ & $-.030$ & $-.442$ & $-.130$ & $.038$ & $.192$ & $%
.017$ & $.845$ & $.310$ & $.000$ \\
$500$ & $8$ & $1$ & $.99$ & $-.025$ & $-.459$ & $-.123$ & $.038$ & $.194$ & $%
.017$ & $.860$ & $.296$ & $.000$ \\
$500$ & $8$ & $2$ & $.99$ & $-.014$ & $-.367$ & $-.103$ & $.038$ & $.190$ & $%
.016$ & $.884$ & $.425$ & $.000$ \\
$500$ & $16$ & $0$ & $.99$ & $-.015$ & $-.218$ & $-.067$ & $.019$ & $.077$ &
$.009$ & $.852$ & $.055$ & $.000$ \\
$500$ & $16$ & $1$ & $.99$ & $-.010$ & $-.240$ & $-.060$ & $.019$ & $.080$ &
$.009$ & $.878$ & $.040$ & $.000$ \\
$500$ & $16$ & $2$ & $.99$ & $-.002$ & $-.180$ & $-.042$ & $.019$ & $.074$ &
$.008$ & $.924$ & $.125$ & $.000$ \\
$500$ & $24$ & $0$ & $.99$ & $-.010$ & $-.137$ & $-.046$ & $.013$ & $.042$ &
$.006$ & $.852$ & $.011$ & $.000$ \\
$500$ & $24$ & $1$ & $.99$ & $-.006$ & $-.161$ & $-.039$ & $.013$ & $.046$ &
$.006$ & $.886$ & $.005$ & $.000$ \\
$500$ & $24$ & $2$ & $.99$ & $.001$ & $-.116$ & $-.023$ & $.013$ & $.041$ & $%
.006$ & $.936$ & $.034$ & $.002$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}



\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.5$ & $-.090$ & --- & $-.750$ & $.134$ & --- & $.047$
& $.834$ & $.930$ & $.000$ \\
$1000$ & $2$ & $1$ & $.5$ & $.027$ & --- & $-.374$ & $.135$ & --- & $.044$ &
$.943$ & $.954$ & $.000$ \\
$1000$ & $2$ & $2$ & $.5$ & $.001$ & --- & $.108$ & $.046$ & --- & $.035$ & $%
.950$ & $.951$ & $.116$ \\
$1000$ & $4$ & $0$ & $.5$ & $.011$ & $-.003$ & $-.294$ & $.068$ & $.049$ & $%
.021$ & $.948$ & $.951$ & $.000$ \\
$1000$ & $4$ & $1$ & $.5$ & $.002$ & $-.005$ & $-.139$ & $.036$ & $.054$ & $%
.021$ & $.957$ & $.947$ & $.000$ \\
$1000$ & $4$ & $2$ & $.5$ & $.000$ & $-.002$ & $.072$ & $.020$ & $.026$ & $%
.017$ & $.952$ & $.952$ & $.009$ \\
$1000$ & $6$ & $0$ & $.5$ & $.001$ & $-.003$ & $-.146$ & $.026$ & $.026$ & $%
.016$ & $.952$ & $.951$ & $.000$ \\
$1000$ & $6$ & $1$ & $.5$ & $.000$ & $-.005$ & $-.072$ & $.021$ & $.032$ & $%
.015$ & $.949$ & $.944$ & $.001$ \\
$1000$ & $6$ & $2$ & $.5$ & $.000$ & $-.002$ & $.045$ & $.014$ & $.020$ & $%
.013$ & $.950$ & $.949$ & $.046$ \\
$1000$ & $8$ & $0$ & $.5$ & $.000$ & $-.003$ & $-.085$ & $.017$ & $.019$ & $%
.013$ & $.953$ & $.947$ & $.000$ \\
$1000$ & $8$ & $1$ & $.5$ & $.000$ & $-.004$ & $-.043$ & $.015$ & $.023$ & $%
.013$ & $.948$ & $.946$ & $.047$ \\
$1000$ & $8$ & $2$ & $.5$ & $.000$ & $-.002$ & $.029$ & $.011$ & $.017$ & $%
.011$ & $.949$ & $.946$ & $.164$ \\
$1000$ & $16$ & $0$ & $.5$ & $.000$ & $-.002$ & $-.021$ & $.009$ & $.010$ & $%
.008$ & $.946$ & $.939$ & $.240$ \\
$1000$ & $16$ & $1$ & $.5$ & $.000$ & $-.003$ & $-.012$ & $.008$ & $.011$ & $%
.008$ & $.952$ & $.944$ & $.640$ \\
$1000$ & $16$ & $2$ & $.5$ & $.000$ & $-.002$ & $.009$ & $.007$ & $.010$ & $%
.007$ & $.946$ & $.942$ & $.699$ \\
$1000$ & $24$ & $0$ & $.5$ & $.000$ & $-.002$ & $-.009$ & $.007$ & $.007$ & $%
.006$ & $.950$ & $.942$ & $.659$ \\
$1000$ & $24$ & $1$ & $.5$ & $.000$ & $-.002$ & $-.005$ & $.006$ & $.008$ & $%
.006$ & $.949$ & $.943$ & $.845$ \\
$1000$ & $24$ & $2$ & $.5$ & $.000$ & $-.002$ & $.004$ & $.006$ & $.008$ & $%
.006$ & $.952$ & $.942$ & $.854$%
\\ \hline \T
$1000$ & $2$ & $0$ & $.9$ & $-.090$ & --- & $-.550$ & $.134$ & --- & $.048$
& $.833$ & $.933$ & $.000$ \\
$1000$ & $2$ & $1$ & $.9$ & $-.044$ & --- & $-.475$ & $.135$ & --- & $.047$
& $.874$ & $.930$ & $.000$ \\
$1000$ & $2$ & $2$ & $.9$ & $.021$ & --- & $-.289$ & $.142$ & --- & $.046$ &
$.925$ & $.956$ & $.000$ \\
$1000$ & $4$ & $0$ & $.9$ & $-.043$ & $-.079$ & $-.291$ & $.064$ & $.194$ & $%
.023$ & $.853$ & $.905$ & $.000$ \\
$1000$ & $4$ & $1$ & $.9$ & $-.008$ & $-.119$ & $-.226$ & $.066$ & $.234$ & $%
.023$ & $.909$ & $.892$ & $.000$ \\
$1000$ & $4$ & $2$ & $.9$ & $.009$ & $-.014$ & $-.082$ & $.057$ & $.081$ & $%
.021$ & $.956$ & $.943$ & $.012$ \\
$1000$ & $6$ & $0$ & $.9$ & $-.023$ & $-.043$ & $-.201$ & $.044$ & $.081$ & $%
.016$ & $.873$ & $.909$ & $.000$ \\
$1000$ & $6$ & $1$ & $.9$ & $.002$ & $-.079$ & $-.143$ & $.046$ & $.110$ & $%
.016$ & $.926$ & $.887$ & $.000$ \\
$1000$ & $6$ & $2$ & $.9$ & $.002$ & $-.013$ & $-.028$ & $.028$ & $.046$ & $%
.014$ & $.963$ & $.943$ & $.368$ \\
$1000$ & $8$ & $0$ & $.9$ & $-.011$ & $-.027$ & $-.152$ & $.034$ & $.046$ & $%
.012$ & $.899$ & $.911$ & $.000$ \\
$1000$ & $8$ & $1$ & $.9$ & $.005$ & $-.058$ & $-.102$ & $.036$ & $.069$ & $%
.012$ & $.937$ & $.867$ & $.000$ \\
$1000$ & $8$ & $2$ & $.9$ & $.001$ & $-.013$ & $-.007$ & $.017$ & $.033$ & $%
.010$ & $.961$ & $.930$ & $.827$ \\
$1000$ & $16$ & $0$ & $.9$ & $.002$ & $-.010$ & $-.071$ & $.019$ & $.015$ & $%
.007$ & $.949$ & $.899$ & $.000$ \\
$1000$ & $16$ & $1$ & $.9$ & $.001$ & $-.023$ & $-.042$ & $.013$ & $.022$ & $%
.006$ & $.960$ & $.840$ & $.000$ \\
$1000$ & $16$ & $2$ & $.9$ & $.000$ & $-.012$ & $.008$ & $.007$ & $.016$ & $%
.005$ & $.948$ & $.895$ & $.573$ \\
$1000$ & $24$ & $0$ & $.9$ & $.000$ & $-.006$ & $-.041$ & $.009$ & $.008$ & $%
.005$ & $.961$ & $.882$ & $.000$ \\
$1000$ & $24$ & $1$ & $.9$ & $.000$ & $-.014$ & $-.023$ & $.007$ & $.012$ & $%
.005$ & $.952$ & $.814$ & $.001$ \\
$1000$ & $24$ & $2$ & $.9$ & $.000$ & $-.011$ & $.007$ & $.005$ & $.011$ & $%
.004$ & $.947$ & $.846$ & $.417$%
\\ \hline \T
$1000$ & $2$ & $0$ & $.99$ & $-.090$ & --- & $-.505$ & $.134$ & --- & $.048$
& $.833$ & $.933$ & $.000$ \\
$1000$ & $2$ & $1$ & $.99$ & $-.085$ & --- & $-.497$ & $.134$ & --- & $.047$
& $.835$ & $.931$ & $.000$ \\
$1000$ & $2$ & $2$ & $.99$ & $-.072$ & --- & $-.475$ & $.135$ & --- & $.047$
& $.846$ & $.934$ & $.000$ \\
$1000$ & $4$ & $0$ & $.99$ & $-.048$ & $-.735$ & $-.255$ & $.063$ & $.463$ &
$.023$ & $.841$ & $.689$ & $.000$ \\
$1000$ & $4$ & $1$ & $.99$ & $-.043$ & $-.743$ & $-.248$ & $.063$ & $.468$ &
$.023$ & $.845$ & $.684$ & $.000$ \\
$1000$ & $4$ & $2$ & $.99$ & $-.030$ & $-.546$ & $-.226$ & $.063$ & $.456$ &
$.023$ & $.873$ & $.757$ & $.000$ \\
$1000$ & $6$ & $0$ & $.99$ & $-.032$ & $-.534$ & $-.172$ & $.042$ & $.269$ &
$.016$ & $.840$ & $.498$ & $.000$ \\
$1000$ & $6$ & $1$ & $.99$ & $-.028$ & $-.561$ & $-.164$ & $.042$ & $.274$ &
$.016$ & $.858$ & $.486$ & $.000$ \\
$1000$ & $6$ & $2$ & $.99$ & $-.015$ & $-.391$ & $-.144$ & $.042$ & $.261$ &
$.015$ & $.888$ & $.649$ & $.000$ \\
$1000$ & $8$ & $0$ & $.99$ & $-.025$ & $-.405$ & $-.130$ & $.031$ & $.185$ &
$.012$ & $.848$ & $.346$ & $.000$ \\
$1000$ & $8$ & $1$ & $.99$ & $-.020$ & $-.444$ & $-.123$ & $.031$ & $.192$ &
$.012$ & $.861$ & $.320$ & $.000$ \\
$1000$ & $8$ & $2$ & $.99$ & $-.009$ & $-.297$ & $-.102$ & $.031$ & $.177$ &
$.012$ & $.904$ & $.526$ & $.000$ \\
$1000$ & $16$ & $0$ & $.99$ & $-.012$ & $-.185$ & $-.067$ & $.016$ & $.068$
& $.006$ & $.848$ & $.095$ & $.000$ \\
$1000$ & $16$ & $1$ & $.99$ & $-.008$ & $-.227$ & $-.060$ & $.016$ & $.078$
& $.006$ & $.881$ & $.049$ & $.000$ \\
$1000$ & $16$ & $2$ & $.99$ & $.000$ & $-.144$ & $-.042$ & $.016$ & $.065$ &
$.006$ & $.925$ & $.208$ & $.000$ \\
$1000$ & $24$ & $0$ & $.99$ & $-.008$ & $-.108$ & $-.046$ & $.011$ & $.036$
& $.004$ & $.854$ & $.040$ & $.000$ \\
$1000$ & $24$ & $1$ & $.99$ & $-.004$ & $-.147$ & $-.039$ & $.011$ & $.045$
& $.004$ & $.897$ & $.010$ & $.000$ \\
$1000$ & $24$ & $2$ & $.99$ & $.001$ & $-.090$ & $-.023$ & $.011$ & $.035$ &
$.004$ & $.942$ & $.086$ & $.000$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.5$ & $-.075$ & --- & $-.750$ & $.105$ & --- & $.030$
& $.827$ & $.924$ & $.000$ \\
$2500$ & $2$ & $1$ & $.5$ & $.015$ & --- & $-.375$ & $.091$ & --- & $.028$ &
$.954$ & $.956$ & $.000$ \\
$2500$ & $2$ & $2$ & $.5$ & $.001$ & --- & $.107$ & $.029$ & --- & $.023$ & $%
.950$ & $.949$ & $.001$ \\
$2500$ & $4$ & $0$ & $.5$ & $.005$ & $-.002$ & $-.294$ & $.044$ & $.031$ & $%
.013$ & $.959$ & $.947$ & $.000$ \\
$2500$ & $4$ & $1$ & $.5$ & $.000$ & $-.002$ & $-.139$ & $.023$ & $.034$ & $%
.013$ & $.951$ & $.948$ & $.000$ \\
$2500$ & $4$ & $2$ & $.5$ & $.000$ & $-.001$ & $.072$ & $.012$ & $.016$ & $%
.011$ & $.953$ & $.952$ & $.000$ \\
$2500$ & $6$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.146$ & $.016$ & $.017$ & $%
.010$ & $.950$ & $.947$ & $.000$ \\
$2500$ & $6$ & $1$ & $.5$ & $.000$ & $-.002$ & $-.072$ & $.013$ & $.020$ & $%
.010$ & $.950$ & $.949$ & $.000$ \\
$2500$ & $6$ & $2$ & $.5$ & $.000$ & $-.001$ & $.044$ & $.009$ & $.013$ & $%
.008$ & $.948$ & $.948$ & $.000$ \\
$2500$ & $8$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.084$ & $.011$ & $.012$ & $%
.008$ & $.950$ & $.951$ & $.000$ \\
$2500$ & $8$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.043$ & $.010$ & $.014$ & $%
.008$ & $.950$ & $.946$ & $.000$ \\
$2500$ & $8$ & $2$ & $.5$ & $.000$ & $-.001$ & $.029$ & $.007$ & $.011$ & $%
.007$ & $.951$ & $.948$ & $.006$ \\
$2500$ & $16$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.021$ & $.006$ & $.006$ & $%
.005$ & $.949$ & $.951$ & $.015$ \\
$2500$ & $16$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.012$ & $.005$ & $.007$ & $%
.005$ & $.951$ & $.945$ & $.314$ \\
$2500$ & $16$ & $2$ & $.5$ & $.000$ & $-.001$ & $.009$ & $.005$ & $.007$ & $%
.005$ & $.953$ & $.949$ & $.409$ \\
$2500$ & $24$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.009$ & $.004$ & $.004$ & $%
.004$ & $.953$ & $.951$ & $.347$ \\
$2500$ & $24$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.005$ & $.004$ & $.005$ & $%
.004$ & $.950$ & $.948$ & $.706$ \\
$2500$ & $24$ & $2$ & $.5$ & $.000$ & $-.001$ & $.004$ & $.004$ & $.005$ & $%
.004$ & $.948$ & $.946$ & $.732$%
\\ \hline \T
$2500$ & $2$ & $0$ & $.9$ & $-.074$ & --- & $-.550$ & $.104$ & --- & $.030$
& $.836$ & $.928$ & $.000$ \\
$2500$ & $2$ & $1$ & $.9$ & $-.031$ & --- & $-.475$ & $.106$ & --- & $.030$
& $.877$ & $.938$ & $.000$ \\
$2500$ & $2$ & $2$ & $.9$ & $.021$ & --- & $-.289$ & $.109$ & --- & $.029$ &
$.939$ & $.955$ & $.000$ \\
$2500$ & $4$ & $0$ & $.9$ & $-.033$ & $-.033$ & $-.292$ & $.050$ & $.128$ & $%
.015$ & $.852$ & $.932$ & $.000$ \\
$2500$ & $4$ & $1$ & $.9$ & $.000$ & $-.047$ & $-.226$ & $.053$ & $.150$ & $%
.014$ & $.923$ & $.924$ & $.000$ \\
$2500$ & $4$ & $2$ & $.9$ & $.004$ & $-.004$ & $-.082$ & $.035$ & $.051$ & $%
.013$ & $.961$ & $.951$ & $.000$ \\
$2500$ & $6$ & $0$ & $.9$ & $-.016$ & $-.018$ & $-.200$ & $.034$ & $.053$ & $%
.010$ & $.877$ & $.930$ & $.000$ \\
$2500$ & $6$ & $1$ & $.9$ & $.005$ & $-.034$ & $-.143$ & $.036$ & $.072$ & $%
.010$ & $.937$ & $.923$ & $.000$ \\
$2500$ & $6$ & $2$ & $.9$ & $.000$ & $-.005$ & $-.028$ & $.016$ & $.029$ & $%
.009$ & $.952$ & $.949$ & $.062$ \\
$2500$ & $8$ & $0$ & $.9$ & $-.006$ & $-.011$ & $-.152$ & $.026$ & $.030$ & $%
.008$ & $.903$ & $.933$ & $.000$ \\
$2500$ & $8$ & $1$ & $.9$ & $.004$ & $-.023$ & $-.102$ & $.026$ & $.044$ & $%
.008$ & $.947$ & $.919$ & $.000$ \\
$2500$ & $8$ & $2$ & $.9$ & $.000$ & $-.005$ & $-.007$ & $.011$ & $.021$ & $%
.007$ & $.951$ & $.945$ & $.711$ \\
$2500$ & $16$ & $0$ & $.9$ & $.002$ & $-.004$ & $-.071$ & $.013$ & $.009$ & $%
.004$ & $.956$ & $.933$ & $.000$ \\
$2500$ & $16$ & $1$ & $.9$ & $.000$ & $-.010$ & $-.042$ & $.008$ & $.015$ & $%
.004$ & $.956$ & $.898$ & $.000$ \\
$2500$ & $16$ & $2$ & $.9$ & $.000$ & $-.005$ & $.008$ & $.004$ & $.010$ & $%
.003$ & $.952$ & $.931$ & $.234$ \\
$2500$ & $24$ & $0$ & $.9$ & $.000$ & $-.002$ & $-.040$ & $.005$ & $.005$ & $%
.003$ & $.953$ & $.929$ & $.000$ \\
$2500$ & $24$ & $1$ & $.9$ & $.000$ & $-.006$ & $-.023$ & $.004$ & $.008$ & $%
.003$ & $.952$ & $.894$ & $.000$ \\
$2500$ & $24$ & $2$ & $.9$ & $.000$ & $-.005$ & $.007$ & $.003$ & $.007$ & $%
.002$ & $.949$ & $.909$ & $.099$%
\\ \hline \T
$2500$ & $2$ & $0$ & $.99$ & $-.074$ & --- & $-.505$ & $.104$ & --- & $.03$
& $.835$ & $.930$ & $.000$ \\
$2500$ & $2$ & $1$ & $.99$ & $-.07$ & --- & $-.497$ & $.105$ & --- & $.03$ &
$.836$ & $.934$ & $.000$ \\
$2500$ & $2$ & $2$ & $.99$ & $-.054$ & --- & $-.475$ & $.105$ & --- & $.03$
& $.858$ & $.928$ & $.000$ \\
$2500$ & $4$ & $0$ & $.99$ & $-.038$ & $-.672$ & $-.255$ & $.049$ & $.458$ &
$.014$ & $.841$ & $.696$ & $.000$ \\
$2500$ & $4$ & $1$ & $.99$ & $-.034$ & $-.709$ & $-.248$ & $.049$ & $.476$ &
$.015$ & $.849$ & $.688$ & $.000$ \\
$2500$ & $4$ & $2$ & $.99$ & $-.02$ & $-.359$ & $-.226$ & $.049$ & $.402$ & $%
.014$ & $.881$ & $.818$ & $.000$ \\
$2500$ & $6$ & $0$ & $.99$ & $-.026$ & $-.46$ & $-.172$ & $.032$ & $.253$ & $%
.01$ & $.841$ & $.549$ & $.000$ \\
$2500$ & $6$ & $1$ & $.99$ & $-.021$ & $-.525$ & $-.164$ & $.032$ & $.277$ &
$.01$ & $.858$ & $.515$ & $.000$ \\
$2500$ & $6$ & $2$ & $.99$ & $-.01$ & $-.255$ & $-.144$ & $.033$ & $.211$ & $%
.01$ & $.895$ & $.751$ & $.000$ \\
$2500$ & $8$ & $0$ & $.99$ & $-.019$ & $-.33$ & $-.13$ & $.024$ & $.168$ & $%
.008$ & $.85$ & $.437$ & $.000$ \\
$2500$ & $8$ & $1$ & $.99$ & $-.015$ & $-.402$ & $-.123$ & $.024$ & $.19$ & $%
.008$ & $.865$ & $.364$ & $.000$ \\
$2500$ & $8$ & $2$ & $.99$ & $-.004$ & $-.189$ & $-.103$ & $.025$ & $.136$ &
$.007$ & $.908$ & $.672$ & $.000$ \\
$2500$ & $16$ & $0$ & $.99$ & $-.01$ & $-.124$ & $-.067$ & $.012$ & $.053$ &
$.004$ & $.848$ & $.242$ & $.000$ \\
$2500$ & $16$ & $1$ & $.99$ & $-.005$ & $-.192$ & $-.06$ & $.012$ & $.073$ &
$.004$ & $.887$ & $.098$ & $.000$ \\
$2500$ & $16$ & $2$ & $.99$ & $.001$ & $-.085$ & $-.042$ & $.013$ & $.044$ &
$.004$ & $.935$ & $.421$ & $.000$ \\
$2500$ & $24$ & $0$ & $.99$ & $-.006$ & $-.066$ & $-.046$ & $.008$ & $.025$
& $.003$ & $.856$ & $.175$ & $.000$ \\
$2500$ & $24$ & $1$ & $.99$ & $-.002$ & $-.116$ & $-.039$ & $.008$ & $.039$
& $.003$ & $.899$ & $.032$ & $.000$ \\
$2500$ & $24$ & $2$ & $.99$ & $.001$ & $-.053$ & $-.023$ & $.008$ & $.023$ &
$.002$ & $.951$ & $.275$ & $.000$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.5$ & $-.064$ & --- & $-.751$ & $.086$ & --- & $.021$
& $.831$ & $.933$ & $.000$ \\
$5000$ & $2$ & $1$ & $.5$ & $.008$ & --- & $-.375$ & $.063$ & --- & $.020$ &
$.957$ & $.949$ & $.000$ \\
$5000$ & $2$ & $2$ & $.5$ & $.000$ & --- & $.107$ & $.020$ & --- & $.016$ & $%
.950$ & $.948$ & $.000$ \\
$5000$ & $4$ & $0$ & $.5$ & $.002$ & $-.001$ & $-.294$ & $.029$ & $.022$ & $%
.009$ & $.965$ & $.952$ & $.000$ \\
$5000$ & $4$ & $1$ & $.5$ & $.001$ & $-.001$ & $-.139$ & $.016$ & $.024$ & $%
.010$ & $.951$ & $.948$ & $.000$ \\
$5000$ & $4$ & $2$ & $.5$ & $.000$ & $.000$ & $.072$ & $.009$ & $.012$ & $%
.008$ & $.950$ & $.950$ & $.000$ \\
$5000$ & $6$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.146$ & $.012$ & $.012$ & $%
.007$ & $.951$ & $.954$ & $.000$ \\
$5000$ & $6$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.072$ & $.009$ & $.014$ & $%
.007$ & $.951$ & $.948$ & $.000$ \\
$5000$ & $6$ & $2$ & $.5$ & $.000$ & $.000$ & $.044$ & $.006$ & $.009$ & $%
.006$ & $.950$ & $.951$ & $.000$ \\
$5000$ & $8$ & $0$ & $.5$ & $.000$ & $-.001$ & $-.085$ & $.008$ & $.008$ & $%
.006$ & $.947$ & $.950$ & $.000$ \\
$5000$ & $8$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.043$ & $.007$ & $.010$ & $%
.006$ & $.950$ & $.949$ & $.000$ \\
$5000$ & $8$ & $2$ & $.5$ & $.000$ & $.000$ & $.029$ & $.005$ & $.008$ & $%
.005$ & $.946$ & $.949$ & $.000$ \\
$5000$ & $16$ & $0$ & $.5$ & $.000$ & $.000$ & $-.021$ & $.004$ & $.004$ & $%
.004$ & $.949$ & $.947$ & $.000$ \\
$5000$ & $16$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.011$ & $.004$ & $.005$ & $%
.004$ & $.950$ & $.949$ & $.078$ \\
$5000$ & $16$ & $2$ & $.5$ & $.000$ & $-.001$ & $.009$ & $.003$ & $.005$ & $%
.003$ & $.951$ & $.951$ & $.152$ \\
$5000$ & $24$ & $0$ & $.5$ & $.000$ & $.000$ & $-.009$ & $.003$ & $.003$ & $%
.003$ & $.948$ & $.948$ & $.089$ \\
$5000$ & $24$ & $1$ & $.5$ & $.000$ & $.000$ & $-.005$ & $.003$ & $.004$ & $%
.003$ & $.948$ & $.949$ & $.495$ \\
$5000$ & $24$ & $2$ & $.5$ & $.000$ & $.000$ & $.004$ & $.003$ & $.003$ & $%
.003$ & $.949$ & $.950$ & $.550$%
\\ \hline \T
$5000$ & $2$ & $0$ & $.9$ & $-.064$ & --- & $-.551$ & $.086$ & --- & $.021$
& $.831$ & $.933$ & $.000$ \\
$5000$ & $2$ & $1$ & $.9$ & $-.021$ & --- & $-.475$ & $.089$ & --- & $.021$
& $.887$ & $.939$ & $.000$ \\
$5000$ & $2$ & $2$ & $.9$ & $.016$ & --- & $-.289$ & $.088$ & --- & $.021$ &
$.942$ & $.953$ & $.000$ \\
$5000$ & $4$ & $0$ & $.9$ & $-.027$ & $-.018$ & $-.291$ & $.041$ & $.091$ & $%
.010$ & $.858$ & $.942$ & $.000$ \\
$5000$ & $4$ & $1$ & $.9$ & $.004$ & $-.024$ & $-.225$ & $.044$ & $.106$ & $%
.010$ & $.930$ & $.940$ & $.000$ \\
$5000$ & $4$ & $2$ & $.9$ & $.001$ & $-.003$ & $-.082$ & $.023$ & $.036$ & $%
.009$ & $.962$ & $.948$ & $.000$ \\
$5000$ & $6$ & $0$ & $.9$ & $-.011$ & $-.008$ & $-.200$ & $.028$ & $.037$ & $%
.007$ & $.886$ & $.945$ & $.000$ \\
$5000$ & $6$ & $1$ & $.9$ & $.005$ & $-.017$ & $-.143$ & $.030$ & $.052$ & $%
.007$ & $.945$ & $.938$ & $.000$ \\
$5000$ & $6$ & $2$ & $.9$ & $.000$ & $-.003$ & $-.028$ & $.012$ & $.021$ & $%
.006$ & $.951$ & $.945$ & $.002$ \\
$5000$ & $8$ & $0$ & $.9$ & $-.004$ & $-.006$ & $-.152$ & $.022$ & $.021$ & $%
.006$ & $.904$ & $.939$ & $.000$ \\
$5000$ & $8$ & $1$ & $.9$ & $.003$ & $-.011$ & $-.102$ & $.020$ & $.032$ & $%
.005$ & $.954$ & $.933$ & $.000$ \\
$5000$ & $8$ & $2$ & $.9$ & $.000$ & $-.002$ & $-.007$ & $.008$ & $.015$ & $%
.005$ & $.952$ & $.948$ & $.528$ \\
$5000$ & $16$ & $0$ & $.9$ & $.001$ & $-.002$ & $-.071$ & $.009$ & $.007$ & $%
.003$ & $.960$ & $.940$ & $.000$ \\
$5000$ & $16$ & $1$ & $.9$ & $.000$ & $-.005$ & $-.042$ & $.005$ & $.010$ & $%
.003$ & $.949$ & $.928$ & $.000$ \\
$5000$ & $16$ & $2$ & $.9$ & $.000$ & $-.002$ & $.008$ & $.003$ & $.007$ & $%
.002$ & $.950$ & $.940$ & $.044$ \\
$5000$ & $24$ & $0$ & $.9$ & $.000$ & $-.001$ & $-.041$ & $.004$ & $.004$ & $%
.002$ & $.951$ & $.938$ & $.000$ \\
$5000$ & $24$ & $1$ & $.9$ & $.000$ & $-.003$ & $-.023$ & $.003$ & $.006$ & $%
.002$ & $.948$ & $.919$ & $.000$ \\
$5000$ & $24$ & $2$ & $.9$ & $.000$ & $-.002$ & $.007$ & $.002$ & $.005$ & $%
.002$ & $.946$ & $.931$ & $.006$%
\\ \hline \T
$5000$ & $2$ & $0$ & $.99$ & $-.064$ & --- & $-.506$ & $.086$ & --- & $.021$
& $.831$ & $.933$ & $.000$ \\
$5000$ & $2$ & $1$ & $.99$ & $-.060$ & --- & $-.498$ & $.087$ & --- & $.021$
& $.838$ & $.931$ & $.000$ \\
$5000$ & $2$ & $2$ & $.99$ & $-.045$ & --- & $-.475$ & $.087$ & --- & $.021$
& $.858$ & $.929$ & $.000$ \\
$5000$ & $4$ & $0$ & $.99$ & $-.032$ & $-.591$ & $-.255$ & $.040$ & $.443$ &
$.010$ & $.841$ & $.713$ & $.000$ \\
$5000$ & $4$ & $1$ & $.99$ & $-.027$ & $-.655$ & $-.247$ & $.040$ & $.474$ &
$.010$ & $.861$ & $.718$ & $.000$ \\
$5000$ & $4$ & $2$ & $.99$ & $-.015$ & $-.212$ & $-.226$ & $.041$ & $.316$ &
$.010$ & $.881$ & $.860$ & $.000$ \\
$5000$ & $6$ & $0$ & $.99$ & $-.021$ & $-.368$ & $-.171$ & $.027$ & $.232$ &
$.007$ & $.841$ & $.614$ & $.000$ \\
$5000$ & $6$ & $1$ & $.99$ & $-.017$ & $-.466$ & $-.164$ & $.027$ & $.269$ &
$.007$ & $.862$ & $.564$ & $.000$ \\
$5000$ & $6$ & $2$ & $.99$ & $-.006$ & $-.148$ & $-.144$ & $.027$ & $.151$ &
$.007$ & $.903$ & $.828$ & $.000$ \\
$5000$ & $8$ & $0$ & $.99$ & $-.017$ & $-.246$ & $-.130$ & $.020$ & $.142$ &
$.005$ & $.839$ & $.552$ & $.000$ \\
$5000$ & $8$ & $1$ & $.99$ & $-.011$ & $-.349$ & $-.122$ & $.020$ & $.183$ &
$.005$ & $.869$ & $.438$ & $.000$ \\
$5000$ & $8$ & $2$ & $.99$ & $-.002$ & $-.110$ & $-.102$ & $.021$ & $.096$ &
$.005$ & $.918$ & $.788$ & $.000$ \\
$5000$ & $16$ & $0$ & $.99$ & $-.008$ & $-.078$ & $-.067$ & $.010$ & $.039$
& $.003$ & $.844$ & $.448$ & $.000$ \\
$5000$ & $16$ & $1$ & $.99$ & $-.004$ & $-.151$ & $-.060$ & $.010$ & $.062$
& $.003$ & $.891$ & $.173$ & $.000$ \\
$5000$ & $16$ & $2$ & $.99$ & $.002$ & $-.050$ & $-.042$ & $.011$ & $.031$ &
$.003$ & $.941$ & $.624$ & $.000$ \\
$5000$ & $24$ & $0$ & $.99$ & $-.005$ & $-.040$ & $-.046$ & $.007$ & $.018$
& $.002$ & $.859$ & $.381$ & $.000$ \\
$5000$ & $24$ & $1$ & $.99$ & $-.001$ & $-.086$ & $-.039$ & $.007$ & $.031$
& $.002$ & $.911$ & $.089$ & $.000$ \\
$5000$ & $24$ & $2$ & $.99$ & $.001$ & $-.031$ & $-.023$ & $.006$ & $.016$ &
$.002$ & $.954$ & $.501$ & $.000$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}






\clearpage
\addtocounter{table}{-1}
\begin{table}[h]  \small
\caption{Simulation results for the f\mbox{}irst-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & &    &  bias  &                  &  & std &   &  & $\mathrm{ci}_{.95}$ &   \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.5$ & $-.055$ & --- & $-.750$ & $.072$ & --- & $.015$
& $.831$ & $.931$ & $.000$ \\
$10000$ & $2$ & $1$ & $.5$ & $.003$ & --- & $-.375$ & $.040$ & --- & $.014$
& $.962$ & $.950$ & $.000$ \\
$10000$ & $2$ & $2$ & $.5$ & $.000$ & --- & $.107$ & $.014$ & --- & $.011$ &
$.953$ & $.949$ & $.000$ \\
$10000$ & $4$ & $0$ & $.5$ & $.001$ & $.000$ & $-.294$ & $.020$ & $.015$ & $%
.007$ & $.958$ & $.951$ & $.000$ \\
$10000$ & $4$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.139$ & $.011$ & $.017$ & $%
.007$ & $.949$ & $.948$ & $.000$ \\
$10000$ & $4$ & $2$ & $.5$ & $.000$ & $.000$ & $.072$ & $.006$ & $.008$ & $%
.006$ & $.950$ & $.951$ & $.000$ \\
$10000$ & $6$ & $0$ & $.5$ & $.000$ & $.000$ & $-.146$ & $.008$ & $.008$ & $%
.005$ & $.951$ & $.951$ & $.000$ \\
$10000$ & $6$ & $1$ & $.5$ & $.000$ & $-.001$ & $-.072$ & $.006$ & $.010$ & $%
.005$ & $.948$ & $.952$ & $.000$ \\
$10000$ & $6$ & $2$ & $.5$ & $.000$ & $.000$ & $.044$ & $.004$ & $.006$ & $%
.004$ & $.950$ & $.951$ & $.000$ \\
$10000$ & $8$ & $0$ & $.5$ & $.000$ & $.000$ & $-.084$ & $.005$ & $.006$ & $%
.004$ & $.948$ & $.949$ & $.000$ \\
$10000$ & $8$ & $1$ & $.5$ & $.000$ & $.000$ & $-.043$ & $.005$ & $.007$ & $%
.004$ & $.950$ & $.949$ & $.000$ \\
$10000$ & $8$ & $2$ & $.5$ & $.000$ & $.000$ & $.029$ & $.004$ & $.005$ & $%
.003$ & $.951$ & $.952$ & $.000$ \\
$10000$ & $16$ & $0$ & $.5$ & $.000$ & $.000$ & $-.021$ & $.003$ & $.003$ & $%
.003$ & $.947$ & $.947$ & $.000$ \\
$10000$ & $16$ & $1$ & $.5$ & $.000$ & $.000$ & $-.012$ & $.003$ & $.004$ & $%
.003$ & $.951$ & $.946$ & $.003$ \\
$10000$ & $16$ & $2$ & $.5$ & $.000$ & $.000$ & $.009$ & $.002$ & $.003$ & $%
.002$ & $.951$ & $.952$ & $.015$ \\
$10000$ & $24$ & $0$ & $.5$ & $.000$ & $.000$ & $-.009$ & $.002$ & $.002$ & $%
.002$ & $.953$ & $.951$ & $.003$ \\
$10000$ & $24$ & $1$ & $.5$ & $.000$ & $.000$ & $-.005$ & $.002$ & $.003$ & $%
.002$ & $.955$ & $.952$ & $.214$ \\
$10000$ & $24$ & $2$ & $.5$ & $.000$ & $.000$ & $.004$ & $.002$ & $.002$ & $%
.002$ & $.955$ & $.952$ & $.268$%
\\ \hline \T
$10000$ & $2$ & $0$ & $.9$ & $-.055$ & --- & $-.550$ & $.072$ & --- & $.015$
& $.831$ & $.931$ & $.000$ \\
$10000$ & $2$ & $1$ & $.9$ & $-.013$ & --- & $-.475$ & $.074$ & --- & $.015$
& $.898$ & $.947$ & $.000$ \\
$10000$ & $2$ & $2$ & $.9$ & $.013$ & --- & $-.289$ & $.066$ & --- & $.014$
& $.955$ & $.949$ & $.000$ \\
$10000$ & $4$ & $0$ & $.9$ & $-.021$ & $-.008$ & $-.291$ & $.034$ & $.065$ &
$.007$ & $.864$ & $.945$ & $.000$ \\
$10000$ & $4$ & $1$ & $.9$ & $.005$ & $-.012$ & $-.226$ & $.037$ & $.076$ & $%
.007$ & $.936$ & $.942$ & $.000$ \\
$10000$ & $4$ & $2$ & $.9$ & $.001$ & $-.001$ & $-.082$ & $.016$ & $.025$ & $%
.007$ & $.955$ & $.948$ & $.000$ \\
$10000$ & $6$ & $0$ & $.9$ & $-.008$ & $-.004$ & $-.200$ & $.023$ & $.027$ &
$.005$ & $.892$ & $.947$ & $.000$ \\
$10000$ & $6$ & $1$ & $.9$ & $.004$ & $-.009$ & $-.143$ & $.023$ & $.037$ & $%
.005$ & $.950$ & $.942$ & $.000$ \\
$10000$ & $6$ & $2$ & $.9$ & $.000$ & $-.001$ & $-.028$ & $.008$ & $.015$ & $%
.004$ & $.952$ & $.951$ & $.000$ \\
$10000$ & $8$ & $0$ & $.9$ & $-.001$ & $-.003$ & $-.152$ & $.018$ & $.015$ &
$.004$ & $.919$ & $.946$ & $.000$ \\
$10000$ & $8$ & $1$ & $.9$ & $.001$ & $-.006$ & $-.102$ & $.014$ & $.023$ & $%
.004$ & $.961$ & $.940$ & $.000$ \\
$10000$ & $8$ & $2$ & $.9$ & $.000$ & $-.001$ & $-.007$ & $.005$ & $.011$ & $%
.003$ & $.946$ & $.946$ & $.288$ \\
$10000$ & $16$ & $0$ & $.9$ & $.000$ & $-.001$ & $-.071$ & $.006$ & $.005$ &
$.002$ & $.961$ & $.945$ & $.000$ \\
$10000$ & $16$ & $1$ & $.9$ & $.000$ & $-.002$ & $-.042$ & $.004$ & $.007$ &
$.002$ & $.954$ & $.941$ & $.000$ \\
$10000$ & $16$ & $2$ & $.9$ & $.000$ & $-.001$ & $.008$ & $.002$ & $.005$ & $%
.002$ & $.948$ & $.946$ & $.001$ \\
$10000$ & $24$ & $0$ & $.9$ & $.000$ & $-.001$ & $-.040$ & $.003$ & $.003$ &
$.002$ & $.955$ & $.941$ & $.000$ \\
$10000$ & $24$ & $1$ & $.9$ & $.000$ & $-.001$ & $-.023$ & $.002$ & $.004$ &
$.001$ & $.955$ & $.939$ & $.000$ \\
$10000$ & $24$ & $2$ & $.9$ & $.000$ & $-.001$ & $.007$ & $.001$ & $.004$ & $%
.001$ & $.953$ & $.941$ & $.000$%
\\ \hline \T
$10000$ & $2$ & $0$ & $.99$ & $-.055$ & --- & $-.505$ & $.072$ & --- & $.015$
& $.831$ & $.931$ & $.000$ \\
$10000$ & $2$ & $1$ & $.99$ & $-.050$ & --- & $-.497$ & $.072$ & --- & $.015$
& $.841$ & $.928$ & $.000$ \\
$10000$ & $2$ & $2$ & $.99$ & $-.034$ & --- & $-.476$ & $.072$ & --- & $.015$
& $.868$ & $.929$ & $.000$ \\
$10000$ & $4$ & $0$ & $.99$ & $-.027$ & $-.461$ & $-.255$ & $.034$ & $.406$
& $.007$ & $.842$ & $.754$ & $.000$ \\
$10000$ & $4$ & $1$ & $.99$ & $-.022$ & $-.565$ & $-.248$ & $.034$ & $.459$
& $.007$ & $.856$ & $.731$ & $.000$ \\
$10000$ & $4$ & $2$ & $.99$ & $-.010$ & $-.101$ & $-.226$ & $.034$ & $.215$
& $.007$ & $.891$ & $.908$ & $.000$ \\
$10000$ & $6$ & $0$ & $.99$ & $-.018$ & $-.259$ & $-.171$ & $.022$ & $.189$
& $.005$ & $.847$ & $.703$ & $.000$ \\
$10000$ & $6$ & $1$ & $.99$ & $-.013$ & $-.390$ & $-.164$ & $.022$ & $.251$
& $.005$ & $.869$ & $.614$ & $.000$ \\
$10000$ & $6$ & $2$ & $.99$ & $-.003$ & $-.078$ & $-.143$ & $.023$ & $.107$
& $.005$ & $.910$ & $.879$ & $.000$ \\
$10000$ & $8$ & $0$ & $.99$ & $-.013$ & $-.160$ & $-.130$ & $.017$ & $.110$
& $.004$ & $.846$ & $.683$ & $.000$ \\
$10000$ & $8$ & $1$ & $.99$ & $-.009$ & $-.275$ & $-.122$ & $.017$ & $.161$
& $.004$ & $.873$ & $.526$ & $.000$ \\
$10000$ & $8$ & $2$ & $.99$ & $.000$ & $-.059$ & $-.102$ & $.018$ & $.068$ &
$.004$ & $.927$ & $.852$ & $.000$ \\
$10000$ & $16$ & $0$ & $.99$ & $-.007$ & $-.046$ & $-.067$ & $.008$ & $.028$
& $.002$ & $.848$ & $.635$ & $.000$ \\
$10000$ & $16$ & $1$ & $.99$ & $-.002$ & $-.104$ & $-.060$ & $.009$ & $.048$
& $.002$ & $.899$ & $.326$ & $.000$ \\
$10000$ & $16$ & $2$ & $.99$ & $.001$ & $-.027$ & $-.042$ & $.009$ & $.022$
& $.002$ & $.948$ & $.769$ & $.000$ \\
$10000$ & $24$ & $0$ & $.99$ & $-.004$ & $-.022$ & $-.046$ & $.006$ & $.013$
& $.001$ & $.860$ & $.606$ & $.000$ \\
$10000$ & $24$ & $1$ & $.99$ & $-.001$ & $-.056$ & $-.039$ & $.006$ & $.023$
& $.001$ & $.917$ & $.238$ & $.000$ \\
$10000$ & $24$ & $2$ & $.99$ & $.001$ & $-.017$ & $-.023$ & $.004$ & $.011$
& $.001$ & $.960$ & $.691$ & $.000$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_0 y_{it-1} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $y_{i0}$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}


\clearpage
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $.3$ & $.6$ & $-.143$ & --- & $-.848$ & $.262$ & --- & $.304$
& $.822$ & $.927$ & $.029$ \\
&  &  & $.2$ & $-.349$ & --- & $-.742$ & $.665$ & --- & $.807$ & $.956$ & $%
.945$ & $.479$ \\
$100$ & $2$ & $1$ & $.6$ & $-.146$ & --- & $-.844$ & $.267$ & --- & $.167$ &
$.811$ & $.914$ & $.000$ \\
&  &  & $.2$ & $-.173$ & --- & $-.730$ & $.285$ & --- & $.273$ & $.879$ & $%
.918$ & $.047$ \\
$100$ & $2$ & $2$ & $.6$ & $-.143$ & --- & $-.846$ & $.265$ & --- & $.152$ &
$.819$ & $.923$ & $.000$ \\
&  &  & $.2$ & $-.139$ & --- & $-.733$ & $.244$ & --- & $.180$ & $.838$ & $%
.923$ & $.001$ \\
$100$ & $4$ & $.3$ & $.6$ & $-.069$ & $-.282$ & $-.327$ & $.122$ & $.283$ & $%
.065$ & $.860$ & $.781$ & $.000$ \\
&  &  & $.2$ & $-.030$ & $-.123$ & $-.121$ & $.098$ & $.135$ & $.093$ & $.930
$ & $.810$ & $.528$ \\
$100$ & $4$ & $1$ & $.6$ & $-.001$ & $-.044$ & $-.204$ & $.122$ & $.121$ & $%
.061$ & $.920$ & $.919$ & $.050$ \\
&  &  & $.2$ & $-.001$ & $-.022$ & $-.051$ & $.095$ & $.091$ & $.083$ & $.949
$ & $.927$ & $.786$ \\
$100$ & $4$ & $2$ & $.6$ & $.009$ & $-.011$ & $-.005$ & $.082$ & $.064$ & $%
.048$ & $.954$ & $.941$ & $.953$ \\
&  &  & $.2$ & $.005$ & $-.005$ & $.073$ & $.073$ & $.065$ & $.066$ & $.955$
& $.941$ & $.731$ \\
$100$ & $6$ & $.3$ & $.6$ & $-.035$ & $-.153$ & $-.206$ & $.082$ & $.137$ & $%
.050$ & $.896$ & $.778$ & $.008$ \\
&  &  & $.2$ & $-.017$ & $-.074$ & $-.086$ & $.069$ & $.078$ & $.059$ & $.933
$ & $.822$ & $.552$ \\
$100$ & $6$ & $1$ & $.6$ & $.007$ & $-.037$ & $-.111$ & $.083$ & $.074$ & $%
.046$ & $.945$ & $.905$ & $.287$ \\
&  &  & $.2$ & $.004$ & $-.017$ & $-.013$ & $.066$ & $.058$ & $.056$ & $.959$
& $.932$ & $.889$ \\
$100$ & $6$ & $2$ & $.6$ & $.001$ & $-.012$ & $.026$ & $.047$ & $.045$ & $%
.036$ & $.951$ & $.934$ & $.915$ \\
&  &  & $.2$ & $.001$ & $-.004$ & $.086$ & $.046$ & $.045$ & $.047$ & $.944$
& $.942$ & $.475$ \\
$100$ & $8$ & $.3$ & $.6$ & $-.015$ & $-.101$ & $-.144$ & $.064$ & $.088$ & $%
.043$ & $.923$ & $.778$ & $.056$ \\
&  &  & $.2$ & $-.008$ & $-.052$ & $-.066$ & $.055$ & $.056$ & $.046$ & $.946
$ & $.843$ & $.608$ \\
$100$ & $8$ & $1$ & $.6$ & $.006$ & $-.033$ & $-.071$ & $.063$ & $.055$ & $%
.039$ & $.956$ & $.895$ & $.501$ \\
&  &  & $.2$ & $.003$ & $-.015$ & $-.003$ & $.052$ & $.045$ & $.044$ & $.962$
& $.931$ & $.915$ \\
$100$ & $8$ & $2$ & $.6$ & $.000$ & $-.012$ & $.027$ & $.038$ & $.038$ & $%
.031$ & $.940$ & $.930$ & $.883$ \\
&  &  & $.2$ & $.001$ & $-.002$ & $.075$ & $.036$ & $.036$ & $.038$ & $.944$
& $.947$ & $.432$ \\
$100$ & $16$ & $.3$ & $.6$ & $.003$ & $-.041$ & $-.050$ & $.037$ & $.036$ & $%
.028$ & $.963$ & $.793$ & $.512$ \\
&  &  & $.2$ & $.002$ & $-.025$ & $-.030$ & $.035$ & $.031$ & $.028$ & $.963$
& $.868$ & $.780$ \\
$100$ & $16$ & $1$ & $.6$ & $.000$ & $-.024$ & $-.024$ & $.031$ & $.030$ & $%
.027$ & $.950$ & $.871$ & $.826$ \\
&  &  & $.2$ & $.000$ & $-.011$ & $.001$ & $.029$ & $.028$ & $.027$ & $.949$
& $.926$ & $.933$ \\
$100$ & $16$ & $2$ & $.6$ & $.000$ & $-.011$ & $.015$ & $.025$ & $.026$ & $%
.023$ & $.945$ & $.922$ & $.905$ \\
&  &  & $.2$ & $.000$ & $-.003$ & $.041$ & $.024$ & $.024$ & $.025$ & $.946$
& $.945$ & $.576$ \\
$100$ & $24$ & $.3$ & $.6$ & $.000$ & $-.029$ & $-.024$ & $.024$ & $.025$ & $%
.022$ & $.952$ & $.790$ & $.767$ \\
&  &  & $.2$ & $.000$ & $-.018$ & $-.016$ & $.024$ & $.023$ & $.022$ & $.952$
& $.872$ & $.869$ \\
$100$ & $24$ & $1$ & $.6$ & $.000$ & $-.020$ & $-.012$ & $.022$ & $.023$ & $%
.021$ & $.944$ & $.858$ & $.896$ \\
&  &  & $.2$ & $-.001$ & $-.011$ & $.000$ & $.022$ & $.022$ & $.021$ & $.950$
& $.917$ & $.942$ \\
$100$ & $24$ & $2$ & $.6$ & $.000$ & $-.012$ & $.009$ & $.020$ & $.020$ & $%
.019$ & $.949$ & $.909$ & $.931$ \\
&  &  & $.2$ & $.000$ & $-.003$ & $.025$ & $.019$ & $.020$ & $.020$ & $.943$
& $.942$ & $.723$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}



\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $.3$ & $1$ & $-.141$ & --- & $-.856$ & $.263$ & --- & $.273$ &
$.822$ & $.927$ & $.016$ \\
&  &  & $-.2$ & $-.196$ & --- & $-.370$ & $.572$ & --- & $.721$ & $.965$ & $%
.966$ & $.591$ \\
$100$ & $2$ & $1$ & $1$ & $-.145$ & --- & $-.860$ & $.268$ & --- & $.146$ & $%
.811$ & $.914$ & $.000$ \\
&  &  & $-.2$ & $-.102$ & --- & $-.381$ & $.215$ & --- & $.227$ & $.912$ & $%
.921$ & $.226$ \\
$100$ & $2$ & $2$ & $1$ & $-.142$ & --- & $-.859$ & $.265$ & --- & $.133$ & $%
.819$ & $.923$ & $.000$ \\
&  &  & $-.2$ & $-.082$ & --- & $-.376$ & $.158$ & --- & $.133$ & $.856$ & $%
.924$ & $.035$ \\
$100$ & $4$ & $.3$ & $1$ & $-.069$ & $-.237$ & $-.360$ & $.117$ & $.261$ & $%
.059$ & $.861$ & $.802$ & $.000$ \\
&  &  & $-.2$ & $.007$ & $-.018$ & $.009$ & $.088$ & $.084$ & $.095$ & $.962$
& $.939$ & $.810$ \\
$100$ & $4$ & $1$ & $1$ & $.009$ & $-.030$ & $-.213$ & $.114$ & $.099$ & $%
.053$ & $.929$ & $.929$ & $.012$ \\
&  &  & $-.2$ & $.008$ & $-.004$ & $.013$ & $.081$ & $.079$ & $.086$ & $.963$
& $.939$ & $.823$ \\
$100$ & $4$ & $2$ & $1$ & $.003$ & $-.008$ & $-.022$ & $.059$ & $.051$ & $%
.037$ & $.949$ & $.942$ & $.931$ \\
&  &  & $-.2$ & $.003$ & $-.001$ & $.061$ & $.061$ & $.060$ & $.069$ & $.946$
& $.943$ & $.712$ \\
$100$ & $6$ & $.3$ & $1$ & $-.030$ & $-.112$ & $-.216$ & $.076$ & $.116$ & $%
.047$ & $.905$ & $.815$ & $.001$ \\
&  &  & $-.2$ & $.002$ & $-.018$ & $-.024$ & $.063$ & $.058$ & $.060$ & $.963
$ & $.928$ & $.835$ \\
$100$ & $6$ & $1$ & $1$ & $.008$ & $-.025$ & $-.105$ & $.073$ & $.062$ & $%
.043$ & $.954$ & $.921$ & $.271$ \\
&  &  & $-.2$ & $.004$ & $-.001$ & $.031$ & $.054$ & $.052$ & $.056$ & $.955$
& $.941$ & $.829$ \\
$100$ & $6$ & $2$ & $1$ & $.000$ & $-.009$ & $.017$ & $.038$ & $.039$ & $.031
$ & $.945$ & $.940$ & $.940$ \\
&  &  & $-.2$ & $.001$ & $.002$ & $.088$ & $.041$ & $.042$ & $.045$ & $.943$
& $.945$ & $.405$ \\
$100$ & $8$ & $.3$ & $1$ & $-.008$ & $-.067$ & $-.137$ & $.059$ & $.072$ & $%
.041$ & $.937$ & $.837$ & $.052$ \\
&  &  & $-.2$ & $.003$ & $-.014$ & $-.029$ & $.050$ & $.045$ & $.046$ & $.964
$ & $.935$ & $.830$ \\
$100$ & $8$ & $1$ & $1$ & $.002$ & $-.024$ & $-.064$ & $.051$ & $.048$ & $%
.037$ & $.953$ & $.911$ & $.540$ \\
&  &  & $-.2$ & $.001$ & $-.001$ & $.026$ & $.043$ & $.042$ & $.043$ & $.945$
& $.941$ & $.851$ \\
$100$ & $8$ & $2$ & $1$ & $-.001$ & $-.009$ & $.015$ & $.033$ & $.034$ & $%
.028$ & $.939$ & $.934$ & $.934$ \\
&  &  & $-.2$ & $.001$ & $.003$ & $.077$ & $.033$ & $.034$ & $.037$ & $.946$
& $.943$ & $.365$ \\
$100$ & $16$ & $.3$ & $1$ & $.001$ & $-.026$ & $-.037$ & $.030$ & $.032$ & $%
.027$ & $.950$ & $.868$ & $.672$ \\
&  &  & $-.2$ & $.001$ & $-.009$ & $-.014$ & $.029$ & $.028$ & $.027$ & $.949
$ & $.932$ & $.905$ \\
$100$ & $16$ & $1$ & $1$ & $-.001$ & $-.018$ & $-.019$ & $.027$ & $.028$ & $%
.025$ & $.944$ & $.903$ & $.868$ \\
&  &  & $-.2$ & $.000$ & $-.003$ & $.010$ & $.026$ & $.026$ & $.026$ & $.948$
& $.945$ & $.919$ \\
$100$ & $16$ & $2$ & $1$ & $.000$ & $-.009$ & $.004$ & $.023$ & $.024$ & $%
.022$ & $.944$ & $.928$ & $.951$ \\
&  &  & $-.2$ & $.000$ & $.001$ & $.038$ & $.023$ & $.023$ & $.024$ & $.949$
& $.947$ & $.597$ \\
$100$ & $24$ & $.3$ & $1$ & $-.001$ & $-.019$ & $-.016$ & $.022$ & $.023$ & $%
.021$ & $.948$ & $.867$ & $.862$ \\
&  &  & $-.2$ & $.000$ & $-.007$ & $-.007$ & $.022$ & $.022$ & $.021$ & $.945
$ & $.935$ & $.934$ \\
$100$ & $24$ & $1$ & $1$ & $.000$ & $-.015$ & $-.009$ & $.021$ & $.022$ & $%
.021$ & $.945$ & $.899$ & $.916$ \\
&  &  & $-.2$ & $.000$ & $-.004$ & $.004$ & $.021$ & $.021$ & $.021$ & $.945$
& $.945$ & $.934$ \\
$100$ & $24$ & $2$ & $1$ & $-.001$ & $-.010$ & $.002$ & $.019$ & $.020$ & $%
.019$ & $.948$ & $.918$ & $.946$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.019$ & $.019$ & $.020$ & $.945$
& $.945$ & $.771$%
\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}







\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $.3$ & $.6$ & $-.125$ & --- & $-.840$ & $.199$ & --- & $.190$
& $.822$ & $.925$ & $.000$ \\
&  &  & $.2$ & $-.296$ & --- & $-.721$ & $.437$ & --- & $.504$ & $.955$ & $%
.931$ & $.310$ \\
$250$ & $2$ & $1$ & $.6$ & $-.122$ & --- & $-.844$ & $.198$ & --- & $.106$ &
$.822$ & $.920$ & $.000$ \\
&  &  & $.2$ & $-.137$ & --- & $-.729$ & $.201$ & --- & $.173$ & $.874$ & $%
.922$ & $.001$ \\
$250$ & $2$ & $2$ & $.6$ & $-.123$ & --- & $-.843$ & $.200$ & --- & $.096$ &
$.826$ & $.929$ & $.000$ \\
&  &  & $.2$ & $-.117$ & --- & $-.728$ & $.182$ & --- & $.114$ & $.840$ & $%
.927$ & $.000$ \\
$250$ & $4$ & $.3$ & $.6$ & $-.050$ & $-.131$ & $-.326$ & $.092$ & $.197$ & $%
.041$ & $.868$ & $.864$ & $.000$ \\
&  &  & $.2$ & $-.021$ & $-.057$ & $-.117$ & $.068$ & $.095$ & $.058$ & $.925
$ & $.882$ & $.276$ \\
$250$ & $4$ & $1$ & $.6$ & $.008$ & $-.019$ & $-.202$ & $.094$ & $.077$ & $%
.039$ & $.933$ & $.934$ & $.000$ \\
&  &  & $.2$ & $.004$ & $-.009$ & $-.049$ & $.066$ & $.058$ & $.053$ & $.957$
& $.944$ & $.701$ \\
$250$ & $4$ & $2$ & $.6$ & $.003$ & $-.004$ & $-.006$ & $.048$ & $.040$ & $%
.030$ & $.956$ & $.946$ & $.949$ \\
&  &  & $.2$ & $.002$ & $-.002$ & $.073$ & $.044$ & $.041$ & $.042$ & $.952$
& $.947$ & $.476$ \\
$250$ & $6$ & $.3$ & $.6$ & $-.022$ & $-.072$ & $-.205$ & $.062$ & $.091$ & $%
.032$ & $.900$ & $.871$ & $.000$ \\
&  &  & $.2$ & $-.010$ & $-.034$ & $-.083$ & $.048$ & $.053$ & $.038$ & $.936
$ & $.892$ & $.269$ \\
$250$ & $6$ & $1$ & $.6$ & $.008$ & $-.016$ & $-.109$ & $.061$ & $.047$ & $%
.029$ & $.954$ & $.935$ & $.027$ \\
&  &  & $.2$ & $.005$ & $-.006$ & $-.012$ & $.045$ & $.037$ & $.035$ & $.962$
& $.941$ & $.880$ \\
$250$ & $6$ & $2$ & $.6$ & $.001$ & $-.004$ & $.025$ & $.029$ & $.028$ & $%
.023$ & $.948$ & $.945$ & $.834$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $.086$ & $.029$ & $.028$ & $.030$ & $.951$
& $.948$ & $.129$ \\
$250$ & $8$ & $.3$ & $.6$ & $-.007$ & $-.046$ & $-.143$ & $.047$ & $.058$ & $%
.027$ & $.928$ & $.872$ & $.000$ \\
&  &  & $.2$ & $-.002$ & $-.023$ & $-.065$ & $.039$ & $.038$ & $.029$ & $.951
$ & $.900$ & $.293$ \\
$250$ & $8$ & $1$ & $.6$ & $.004$ & $-.013$ & $-.071$ & $.042$ & $.035$ & $%
.025$ & $.965$ & $.933$ & $.157$ \\
&  &  & $.2$ & $.002$ & $-.006$ & $-.003$ & $.033$ & $.029$ & $.028$ & $.964$
& $.943$ & $.907$ \\
$250$ & $8$ & $2$ & $.6$ & $.001$ & $-.004$ & $.026$ & $.024$ & $.024$ & $%
.020$ & $.944$ & $.942$ & $.761$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.076$ & $.024$ & $.024$ & $.024$ & $.944$
& $.942$ & $.092$ \\
$250$ & $16$ & $.3$ & $.6$ & $.002$ & $-.018$ & $-.050$ & $.024$ & $.023$ & $%
.018$ & $.964$ & $.879$ & $.154$ \\
&  &  & $.2$ & $.001$ & $-.011$ & $-.029$ & $.023$ & $.020$ & $.018$ & $.963$
& $.909$ & $.582$ \\
$250$ & $16$ & $1$ & $.6$ & $.000$ & $-.010$ & $-.023$ & $.019$ & $.019$ & $%
.017$ & $.950$ & $.918$ & $.683$ \\
&  &  & $.2$ & $.000$ & $-.005$ & $.001$ & $.018$ & $.017$ & $.017$ & $.949$
& $.942$ & $.932$ \\
$250$ & $16$ & $2$ & $.6$ & $.000$ & $-.004$ & $.015$ & $.016$ & $.016$ & $%
.014$ & $.949$ & $.940$ & $.830$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.042$ & $.015$ & $.015$ & $.016$ & $.949$
& $.949$ & $.206$ \\
$250$ & $24$ & $.3$ & $.6$ & $.000$ & $-.012$ & $-.024$ & $.015$ & $.016$ & $%
.014$ & $.950$ & $.878$ & $.544$ \\
&  &  & $.2$ & $.000$ & $-.008$ & $-.016$ & $.015$ & $.015$ & $.014$ & $.949$
& $.915$ & $.776$ \\
$250$ & $24$ & $1$ & $.6$ & $.000$ & $-.008$ & $-.012$ & $.014$ & $.015$ & $%
.013$ & $.949$ & $.909$ & $.843$ \\
&  &  & $.2$ & $.000$ & $-.005$ & $.001$ & $.014$ & $.014$ & $.014$ & $.952$
& $.938$ & $.937$ \\
$250$ & $24$ & $2$ & $.6$ & $.000$ & $-.005$ & $.009$ & $.013$ & $.013$ & $%
.012$ & $.949$ & $.938$ & $.890$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $.025$ & $.012$ & $.012$ & $.013$ & $.946$
& $.946$ & $.440$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $.3$ & $1$ & $-.124$ & --- & $-.864$ & $.200$ & --- & $.170$ &
$.822$ & $.925$ & $.000$ \\
&  &  & $-.2$ & $-.169$ & --- & $-.393$ & $.369$ & --- & $.449$ & $.967$ & $%
.948$ & $.491$ \\
$250$ & $2$ & $1$ & $1$ & $-.122$ & --- & $-.859$ & $.199$ & --- & $.092$ & $%
.822$ & $.920$ & $.000$ \\
&  &  & $-.2$ & $-.081$ & --- & $-.376$ & $.146$ & --- & $.143$ & $.905$ & $%
.924$ & $.049$ \\
$250$ & $2$ & $2$ & $1$ & $-.123$ & --- & $-.859$ & $.200$ & --- & $.082$ & $%
.826$ & $.929$ & $.000$ \\
&  &  & $-.2$ & $-.069$ & --- & $-.378$ & $.115$ & --- & $.081$ & $.859$ & $%
.927$ & $.000$ \\
$250$ & $4$ & $.3$ & $1$ & $-.048$ & $-.103$ & $-.360$ & $.088$ & $.176$ & $%
.037$ & $.869$ & $.882$ & $.000$ \\
&  &  & $-.2$ & $.004$ & $-.008$ & $.010$ & $.057$ & $.056$ & $.060$ & $.969$
& $.949$ & $.812$ \\
$250$ & $4$ & $1$ & $1$ & $.012$ & $-.013$ & $-.212$ & $.086$ & $.063$ & $%
.034$ & $.945$ & $.935$ & $.000$ \\
&  &  & $-.2$ & $.005$ & $-.001$ & $.014$ & $.053$ & $.050$ & $.055$ & $.961$
& $.944$ & $.814$ \\
$250$ & $4$ & $2$ & $1$ & $.001$ & $-.003$ & $-.022$ & $.036$ & $.032$ & $%
.023$ & $.951$ & $.946$ & $.881$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $.060$ & $.038$ & $.038$ & $.044$ & $.949$
& $.947$ & $.539$ \\
$250$ & $6$ & $.3$ & $1$ & $-.017$ & $-.050$ & $-.214$ & $.057$ & $.076$ & $%
.030$ & $.910$ & $.892$ & $.000$ \\
&  &  & $-.2$ & $.002$ & $-.007$ & $-.024$ & $.041$ & $.037$ & $.039$ & $.966
$ & $.943$ & $.787$ \\
$250$ & $6$ & $1$ & $1$ & $.005$ & $-.011$ & $-.104$ & $.049$ & $.039$ & $%
.027$ & $.961$ & $.939$ & $.020$ \\
&  &  & $-.2$ & $.002$ & $.000$ & $.031$ & $.034$ & $.033$ & $.035$ & $.952$
& $.947$ & $.753$ \\
$250$ & $6$ & $2$ & $1$ & $.000$ & $-.003$ & $.017$ & $.024$ & $.025$ & $.020
$ & $.948$ & $.944$ & $.897$ \\
&  &  & $-.2$ & $.000$ & $.001$ & $.088$ & $.026$ & $.026$ & $.029$ & $.948$
& $.949$ & $.081$ \\
$250$ & $8$ & $.3$ & $1$ & $-.001$ & $-.030$ & $-.136$ & $.043$ & $.047$ & $%
.026$ & $.943$ & $.897$ & $.000$ \\
&  &  & $-.2$ & $.003$ & $-.006$ & $-.029$ & $.033$ & $.030$ & $.029$ & $.968
$ & $.940$ & $.738$ \\
$250$ & $8$ & $1$ & $1$ & $.001$ & $-.009$ & $-.064$ & $.031$ & $.030$ & $%
.023$ & $.959$ & $.936$ & $.186$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.026$ & $.026$ & $.027$ & $.949$
& $.949$ & $.765$ \\
$250$ & $8$ & $2$ & $1$ & $.000$ & $-.003$ & $.015$ & $.021$ & $.022$ & $.018
$ & $.946$ & $.942$ & $.892$ \\
&  &  & $-.2$ & $.000$ & $.001$ & $.076$ & $.021$ & $.022$ & $.023$ & $.944$
& $.944$ & $.057$ \\
$250$ & $16$ & $.3$ & $1$ & $.000$ & $-.011$ & $-.036$ & $.019$ & $.020$ & $%
.017$ & $.952$ & $.914$ & $.389$ \\
&  &  & $-.2$ & $.000$ & $-.004$ & $-.014$ & $.018$ & $.018$ & $.017$ & $.946
$ & $.942$ & $.846$ \\
$250$ & $16$ & $1$ & $1$ & $.000$ & $-.008$ & $-.019$ & $.017$ & $.018$ & $%
.016$ & $.949$ & $.933$ & $.761$ \\
&  &  & $-.2$ & $.000$ & $-.001$ & $.010$ & $.016$ & $.016$ & $.017$ & $.952$
& $.952$ & $.888$ \\
$250$ & $16$ & $2$ & $1$ & $.000$ & $-.004$ & $.005$ & $.015$ & $.016$ & $%
.014$ & $.950$ & $.941$ & $.935$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.037$ & $.014$ & $.015$ & $.015$ & $.947$
& $.947$ & $.254$ \\
$250$ & $24$ & $.3$ & $1$ & $.000$ & $-.008$ & $-.016$ & $.014$ & $.015$ & $%
.014$ & $.950$ & $.917$ & $.756$ \\
&  &  & $-.2$ & $.000$ & $-.003$ & $-.007$ & $.014$ & $.014$ & $.013$ & $.948
$ & $.943$ & $.911$ \\
$250$ & $24$ & $1$ & $1$ & $.000$ & $-.006$ & $-.009$ & $.013$ & $.014$ & $%
.013$ & $.949$ & $.927$ & $.882$ \\
&  &  & $-.2$ & $.000$ & $-.002$ & $.005$ & $.013$ & $.013$ & $.013$ & $.950$
& $.950$ & $.926$ \\
$250$ & $24$ & $2$ & $1$ & $.000$ & $-.004$ & $.002$ & $.012$ & $.013$ & $%
.012$ & $.949$ & $.942$ & $.944$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.012$ & $.012$ & $.012$ & $.949$
& $.950$ & $.542$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $.3$ & $.6$ & $-.104$ & --- & $-.845$ & $.163$ & --- & $.134$
& $.827$ & $.928$ & $.000$ \\
&  &  & $.2$ & $-.253$ & --- & $-.733$ & $.331$ & --- & $.356$ & $.952$ & $%
.926$ & $.142$ \\
$500$ & $2$ & $1$ & $.6$ & $-.105$ & --- & $-.843$ & $.163$ & --- & $.075$ &
$.830$ & $.925$ & $.000$ \\
&  &  & $.2$ & $-.114$ & --- & $-.730$ & $.160$ & --- & $.122$ & $.864$ & $%
.924$ & $.000$ \\
$500$ & $2$ & $2$ & $.6$ & $-.104$ & --- & $-.843$ & $.162$ & --- & $.067$ &
$.834$ & $.927$ & $.000$ \\
&  &  & $.2$ & $-.098$ & --- & $-.729$ & $.147$ & --- & $.080$ & $.843$ & $%
.927$ & $.000$ \\
$500$ & $4$ & $.3$ & $.6$ & $-.040$ & $-.067$ & $-.325$ & $.076$ & $.144$ & $%
.030$ & $.867$ & $.900$ & $.000$ \\
&  &  & $.2$ & $-.017$ & $-.030$ & $-.116$ & $.051$ & $.071$ & $.042$ & $.922
$ & $.909$ & $.079$ \\
$500$ & $4$ & $1$ & $.6$ & $.012$ & $-.009$ & $-.202$ & $.077$ & $.056$ & $%
.027$ & $.942$ & $.938$ & $.000$ \\
&  &  & $.2$ & $.007$ & $-.003$ & $-.049$ & $.051$ & $.041$ & $.037$ & $.962$
& $.950$ & $.566$ \\
$500$ & $4$ & $2$ & $.6$ & $.001$ & $-.002$ & $-.006$ & $.033$ & $.029$ & $%
.021$ & $.956$ & $.947$ & $.951$ \\
&  &  & $.2$ & $.001$ & $-.001$ & $.074$ & $.032$ & $.029$ & $.030$ & $.946$
& $.946$ & $.206$ \\
$500$ & $6$ & $.3$ & $.6$ & $-.014$ & $-.038$ & $-.205$ & $.050$ & $.066$ & $%
.022$ & $.905$ & $.908$ & $.000$ \\
&  &  & $.2$ & $-.007$ & $-.018$ & $-.083$ & $.037$ & $.038$ & $.026$ & $.940
$ & $.919$ & $.062$ \\
$500$ & $6$ & $1$ & $.6$ & $.006$ & $-.008$ & $-.109$ & $.047$ & $.034$ & $%
.021$ & $.954$ & $.937$ & $.000$ \\
&  &  & $.2$ & $.004$ & $-.003$ & $-.012$ & $.032$ & $.026$ & $.025$ & $.968$
& $.949$ & $.862$ \\
$500$ & $6$ & $2$ & $.6$ & $.000$ & $-.002$ & $.026$ & $.021$ & $.020$ & $%
.016$ & $.947$ & $.948$ & $.681$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.086$ & $.020$ & $.020$ & $.021$ & $.950$
& $.946$ & $.009$ \\
$500$ & $8$ & $.3$ & $.6$ & $-.003$ & $-.023$ & $-.143$ & $.039$ & $.041$ & $%
.019$ & $.927$ & $.912$ & $.000$ \\
&  &  & $.2$ & $-.002$ & $-.012$ & $-.065$ & $.030$ & $.028$ & $.020$ & $.948
$ & $.920$ & $.076$ \\
$500$ & $8$ & $1$ & $.6$ & $.003$ & $-.007$ & $-.070$ & $.030$ & $.025$ & $%
.017$ & $.964$ & $.940$ & $.014$ \\
&  &  & $.2$ & $.001$ & $-.003$ & $-.002$ & $.024$ & $.020$ & $.020$ & $.960$
& $.946$ & $.911$ \\
$500$ & $8$ & $2$ & $.6$ & $.000$ & $-.002$ & $.026$ & $.017$ & $.017$ & $%
.014$ & $.949$ & $.949$ & $.568$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.077$ & $.016$ & $.016$ & $.017$ & $.952$
& $.952$ & $.004$ \\
$500$ & $16$ & $.3$ & $.6$ & $.001$ & $-.010$ & $-.050$ & $.017$ & $.017$ & $%
.013$ & $.962$ & $.909$ & $.016$ \\
&  &  & $.2$ & $.001$ & $-.006$ & $-.029$ & $.016$ & $.014$ & $.012$ & $.961$
& $.932$ & $.316$ \\
$500$ & $16$ & $1$ & $.6$ & $.000$ & $-.005$ & $-.023$ & $.013$ & $.014$ & $%
.012$ & $.953$ & $.933$ & $.459$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $.002$ & $.013$ & $.012$ & $.012$ & $.948$
& $.943$ & $.932$ \\
$500$ & $16$ & $2$ & $.6$ & $.000$ & $-.002$ & $.015$ & $.011$ & $.011$ & $%
.010$ & $.948$ & $.946$ & $.701$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.042$ & $.011$ & $.011$ & $.011$ & $.951$
& $.949$ & $.028$ \\
$500$ & $24$ & $.3$ & $.6$ & $.000$ & $-.006$ & $-.024$ & $.011$ & $.011$ & $%
.010$ & $.949$ & $.914$ & $.274$ \\
&  &  & $.2$ & $.000$ & $-.004$ & $-.016$ & $.011$ & $.011$ & $.010$ & $.949$
& $.930$ & $.618$ \\
$500$ & $24$ & $1$ & $.6$ & $.000$ & $-.004$ & $-.012$ & $.010$ & $.010$ & $%
.010$ & $.949$ & $.931$ & $.741$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $.001$ & $.010$ & $.010$ & $.010$ & $.949$
& $.943$ & $.932$ \\
$500$ & $24$ & $2$ & $.6$ & $.000$ & $-.002$ & $.009$ & $.009$ & $.009$ & $%
.009$ & $.948$ & $.942$ & $.819$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.026$ & $.009$ & $.009$ & $.009$ & $.948$
& $.946$ & $.155$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $.3$ & $1$ & $-.104$ & --- & $-.860$ & $.164$ & --- & $.118$ &
$.827$ & $.928$ & $.000$ \\
&  &  & $-.2$ & $-.145$ & --- & $-.380$ & $.274$ & --- & $.314$ & $.965$ & $%
.932$ & $.379$ \\
$500$ & $2$ & $1$ & $1$ & $-.105$ & --- & $-.859$ & $.163$ & --- & $.064$ & $%
.830$ & $.925$ & $.000$ \\
&  &  & $-.2$ & $-.068$ & --- & $-.377$ & $.113$ & --- & $.101$ & $.892$ & $%
.923$ & $.003$ \\
$500$ & $2$ & $2$ & $1$ & $-.104$ & --- & $-.859$ & $.162$ & --- & $.058$ & $%
.833$ & $.927$ & $.000$ \\
&  &  & $-.2$ & $-.057$ & --- & $-.377$ & $.091$ & --- & $.058$ & $.855$ & $%
.927$ & $.000$ \\
$500$ & $4$ & $.3$ & $1$ & $-.036$ & $-.053$ & $-.359$ & $.072$ & $.127$ & $%
.027$ & $.875$ & $.911$ & $.000$ \\
&  &  & $-.2$ & $.003$ & $-.004$ & $.010$ & $.041$ & $.040$ & $.042$ & $.971$
& $.947$ & $.799$ \\
$500$ & $4$ & $1$ & $1$ & $.011$ & $-.007$ & $-.212$ & $.069$ & $.044$ & $%
.024$ & $.950$ & $.944$ & $.000$ \\
&  &  & $-.2$ & $.003$ & $-.001$ & $.014$ & $.038$ & $.035$ & $.039$ & $.960$
& $.949$ & $.802$ \\
$500$ & $4$ & $2$ & $1$ & $.001$ & $-.002$ & $-.022$ & $.025$ & $.023$ & $%
.017$ & $.951$ & $.947$ & $.792$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $.060$ & $.027$ & $.027$ & $.031$ & $.947$
& $.949$ & $.314$ \\
$500$ & $6$ & $.3$ & $1$ & $-.010$ & $-.025$ & $-.213$ & $.047$ & $.055$ & $%
.021$ & $.911$ & $.920$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.004$ & $-.025$ & $.030$ & $.027$ & $.028$ & $.965
$ & $.946$ & $.708$ \\
$500$ & $6$ & $1$ & $1$ & $.002$ & $-.005$ & $-.103$ & $.033$ & $.027$ & $%
.019$ & $.962$ & $.948$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $.030$ & $.023$ & $.023$ & $.025$ & $.951$
& $.947$ & $.636$ \\
$500$ & $6$ & $2$ & $1$ & $.000$ & $-.002$ & $.017$ & $.017$ & $.017$ & $.014
$ & $.951$ & $.947$ & $.828$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.088$ & $.018$ & $.019$ & $.020$ & $.948$
& $.948$ & $.004$ \\
$500$ & $8$ & $.3$ & $1$ & $.002$ & $-.014$ & $-.136$ & $.036$ & $.033$ & $%
.018$ & $.942$ & $.927$ & $.000$ \\
&  &  & $-.2$ & $.002$ & $-.003$ & $-.029$ & $.024$ & $.021$ & $.020$ & $.969
$ & $.945$ & $.583$ \\
$500$ & $8$ & $1$ & $1$ & $.001$ & $-.005$ & $-.063$ & $.022$ & $.022$ & $%
.016$ & $.952$ & $.943$ & $.022$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.018$ & $.019$ & $.019$ & $.949$
& $.949$ & $.629$ \\
$500$ & $8$ & $2$ & $1$ & $.000$ & $-.002$ & $.015$ & $.014$ & $.015$ & $.013
$ & $.950$ & $.951$ & $.816$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.076$ & $.015$ & $.015$ & $.016$ & $.952$
& $.953$ & $.001$ \\
$500$ & $16$ & $.3$ & $1$ & $.000$ & $-.006$ & $-.036$ & $.013$ & $.014$ & $%
.012$ & $.944$ & $.930$ & $.127$ \\
&  &  & $-.2$ & $.000$ & $-.002$ & $-.014$ & $.013$ & $.013$ & $.012$ & $.952
$ & $.948$ & $.762$ \\
$500$ & $16$ & $1$ & $1$ & $.000$ & $-.004$ & $-.019$ & $.012$ & $.013$ & $%
.012$ & $.949$ & $.943$ & $.586$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.012$ & $.012$ & $.012$ & $.949$
& $.950$ & $.842$ \\
$500$ & $16$ & $2$ & $1$ & $.000$ & $-.002$ & $.005$ & $.011$ & $.011$ & $%
.010$ & $.948$ & $.942$ & $.925$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.037$ & $.010$ & $.010$ & $.011$ & $.949$
& $.949$ & $.047$ \\
$500$ & $24$ & $.3$ & $1$ & $.000$ & $-.004$ & $-.016$ & $.010$ & $.011$ & $%
.010$ & $.944$ & $.928$ & $.591$ \\
&  &  & $-.2$ & $.000$ & $-.002$ & $-.007$ & $.010$ & $.010$ & $.009$ & $.949
$ & $.946$ & $.885$ \\
$500$ & $24$ & $1$ & $1$ & $.000$ & $-.003$ & $-.009$ & $.009$ & $.010$ & $%
.009$ & $.949$ & $.939$ & $.824$ \\
&  &  & $-.2$ & $.000$ & $-.001$ & $.005$ & $.009$ & $.009$ & $.009$ & $.950$
& $.948$ & $.913$ \\
$500$ & $24$ & $2$ & $1$ & $.000$ & $-.002$ & $.002$ & $.009$ & $.009$ & $%
.009$ & $.951$ & $.945$ & $.940$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.008$ & $.009$ & $.009$ & $.949$
& $.947$ & $.269$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $.3$ & $.6$ & $-.093$ & --- & $-.842$ & $.134$ & --- & $.095$
& $.831$ & $.931$ & $.000$ \\
&  &  & $.2$ & $-.207$ & --- & $-.728$ & $.248$ & --- & $.253$ & $.945$ & $%
.933$ & $.029$ \\
$1000$ & $2$ & $1$ & $.6$ & $-.091$ & --- & $-.843$ & $.134$ & --- & $.052$
& $.833$ & $.928$ & $.000$ \\
&  &  & $.2$ & $-.096$ & --- & $-.729$ & $.124$ & --- & $.085$ & $.863$ & $%
.930$ & $.000$ \\
$1000$ & $2$ & $2$ & $.6$ & $-.093$ & --- & $-.844$ & $.136$ & --- & $.047$
& $.825$ & $.932$ & $.000$ \\
&  &  & $.2$ & $-.086$ & --- & $-.730$ & $.122$ & --- & $.056$ & $.834$ & $%
.932$ & $.000$ \\
$1000$ & $4$ & $.3$ & $.6$ & $-.032$ & $-.035$ & $-.325$ & $.063$ & $.105$ &
$.021$ & $.862$ & $.925$ & $.000$ \\
&  &  & $.2$ & $-.014$ & $-.016$ & $-.116$ & $.039$ & $.051$ & $.029$ & $.920
$ & $.928$ & $.005$ \\
$1000$ & $4$ & $1$ & $.6$ & $.013$ & $-.003$ & $-.202$ & $.062$ & $.038$ & $%
.019$ & $.953$ & $.949$ & $.000$ \\
&  &  & $.2$ & $.006$ & $-.002$ & $-.048$ & $.039$ & $.029$ & $.026$ & $.964$
& $.953$ & $.368$ \\
$1000$ & $4$ & $2$ & $.6$ & $.000$ & $-.001$ & $-.005$ & $.023$ & $.020$ & $%
.015$ & $.954$ & $.950$ & $.941$ \\
&  &  & $.2$ & $.001$ & $.000$ & $.074$ & $.022$ & $.021$ & $.021$ & $.950$
& $.950$ & $.030$ \\
$1000$ & $6$ & $.3$ & $.6$ & $-.009$ & $-.019$ & $-.205$ & $.041$ & $.048$ &
$.016$ & $.912$ & $.930$ & $.000$ \\
&  &  & $.2$ & $-.004$ & $-.009$ & $-.083$ & $.028$ & $.028$ & $.019$ & $.939
$ & $.936$ & $.003$ \\
$1000$ & $6$ & $1$ & $.6$ & $.004$ & $-.004$ & $-.109$ & $.033$ & $.023$ & $%
.015$ & $.963$ & $.947$ & $.000$ \\
&  &  & $.2$ & $.002$ & $-.002$ & $-.011$ & $.023$ & $.019$ & $.018$ & $.967$
& $.950$ & $.832$ \\
$1000$ & $6$ & $2$ & $.6$ & $.000$ & $-.001$ & $.026$ & $.015$ & $.014$ & $%
.011$ & $.948$ & $.946$ & $.422$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.086$ & $.015$ & $.014$ & $.015$ & $.946$
& $.946$ & $.000$ \\
$1000$ & $8$ & $.3$ & $.6$ & $.001$ & $-.012$ & $-.143$ & $.032$ & $.030$ & $%
.014$ & $.941$ & $.929$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.007$ & $-.065$ & $.023$ & $.020$ & $.015$ & $.949$
& $.933$ & $.004$ \\
$1000$ & $8$ & $1$ & $.6$ & $.001$ & $-.003$ & $-.070$ & $.020$ & $.017$ & $%
.012$ & $.963$ & $.945$ & $.000$ \\
&  &  & $.2$ & $.001$ & $-.001$ & $-.002$ & $.016$ & $.015$ & $.014$ & $.957$
& $.946$ & $.914$ \\
$1000$ & $8$ & $2$ & $.6$ & $.000$ & $-.001$ & $.027$ & $.012$ & $.012$ & $%
.010$ & $.951$ & $.948$ & $.259$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.076$ & $.012$ & $.012$ & $.012$ & $.943$
& $.946$ & $.000$ \\
$1000$ & $16$ & $.3$ & $.6$ & $.000$ & $-.005$ & $-.050$ & $.012$ & $.012$ &
$.009$ & $.957$ & $.932$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.003$ & $-.029$ & $.011$ & $.010$ & $.009$ & $.952$
& $.936$ & $.075$ \\
$1000$ & $16$ & $1$ & $.6$ & $.000$ & $-.003$ & $-.023$ & $.010$ & $.010$ & $%
.008$ & $.948$ & $.940$ & $.184$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.002$ & $.009$ & $.009$ & $.009$ & $.947$
& $.944$ & $.928$ \\
$1000$ & $16$ & $2$ & $.6$ & $.000$ & $-.001$ & $.015$ & $.008$ & $.008$ & $%
.007$ & $.953$ & $.949$ & $.481$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.042$ & $.007$ & $.008$ & $.008$ & $.954$
& $.954$ & $.000$ \\
$1000$ & $24$ & $.3$ & $.6$ & $.000$ & $-.003$ & $-.024$ & $.008$ & $.008$ &
$.007$ & $.950$ & $.932$ & $.053$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $-.016$ & $.007$ & $.007$ & $.007$ & $.954$
& $.941$ & $.364$ \\
$1000$ & $24$ & $1$ & $.6$ & $.000$ & $-.002$ & $-.011$ & $.007$ & $.007$ & $%
.007$ & $.946$ & $.937$ & $.568$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $.001$ & $.007$ & $.007$ & $.007$ & $.950$
& $.948$ & $.935$ \\
$1000$ & $24$ & $2$ & $.6$ & $.000$ & $-.001$ & $.009$ & $.006$ & $.007$ & $%
.006$ & $.949$ & $.947$ & $.690$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.026$ & $.006$ & $.006$ & $.006$ & $.950$
& $.950$ & $.014$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $.3$ & $1$ & $-.092$ & --- & $-.859$ & $.134$ & --- & $.085$
& $.831$ & $.931$ & $.000$ \\
&  &  & $-.2$ & $-.120$ & --- & $-.379$ & $.202$ & --- & $.225$ & $.960$ & $%
.935$ & $.223$ \\
$1000$ & $2$ & $1$ & $1$ & $-.091$ & --- & $-.859$ & $.134$ & --- & $.046$ &
$.833$ & $.928$ & $.000$ \\
&  &  & $-.2$ & $-.057$ & --- & $-.378$ & $.085$ & --- & $.072$ & $.890$ & $%
.930$ & $.000$ \\
$1000$ & $2$ & $2$ & $1$ & $-.093$ & --- & $-.859$ & $.136$ & --- & $.041$ &
$.825$ & $.932$ & $.000$ \\
&  &  & $-.2$ & $-.051$ & --- & $-.376$ & $.075$ & --- & $.041$ & $.844$ & $%
.933$ & $.000$ \\
$1000$ & $4$ & $.3$ & $1$ & $-.029$ & $-.028$ & $-.359$ & $.060$ & $.090$ & $%
.019$ & $.876$ & $.935$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.003$ & $.011$ & $.029$ & $.029$ & $.030$ & $.969$
& $.948$ & $.791$ \\
$1000$ & $4$ & $1$ & $1$ & $.008$ & $-.003$ & $-.211$ & $.051$ & $.032$ & $%
.017$ & $.960$ & $.946$ & $.000$ \\
&  &  & $-.2$ & $.002$ & $.000$ & $.013$ & $.026$ & $.025$ & $.027$ & $.956$
& $.947$ & $.782$ \\
$1000$ & $4$ & $2$ & $1$ & $.000$ & $-.001$ & $-.022$ & $.018$ & $.016$ & $%
.012$ & $.948$ & $.948$ & $.589$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.061$ & $.019$ & $.019$ & $.022$ & $.953$
& $.954$ & $.096$ \\
$1000$ & $6$ & $.3$ & $1$ & $-.005$ & $-.013$ & $-.214$ & $.039$ & $.040$ & $%
.015$ & $.916$ & $.934$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.002$ & $-.024$ & $.021$ & $.019$ & $.019$ & $.967
$ & $.950$ & $.583$ \\
$1000$ & $6$ & $1$ & $1$ & $.001$ & $-.003$ & $-.103$ & $.022$ & $.020$ & $%
.013$ & $.957$ & $.951$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.031$ & $.017$ & $.017$ & $.018$ & $.948$
& $.945$ & $.439$ \\
$1000$ & $6$ & $2$ & $1$ & $.000$ & $-.001$ & $.017$ & $.012$ & $.012$ & $%
.010$ & $.946$ & $.949$ & $.677$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.088$ & $.013$ & $.013$ & $.015$ & $.950$
& $.948$ & $.000$ \\
$1000$ & $8$ & $.3$ & $1$ & $.004$ & $-.008$ & $-.135$ & $.029$ & $.024$ & $%
.013$ & $.954$ & $.935$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.002$ & $-.029$ & $.018$ & $.015$ & $.014$ & $.965
$ & $.948$ & $.358$ \\
$1000$ & $8$ & $1$ & $1$ & $.000$ & $-.002$ & $-.063$ & $.015$ & $.015$ & $%
.012$ & $.954$ & $.948$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.013$ & $.013$ & $.014$ & $.947$
& $.950$ & $.409$ \\
$1000$ & $8$ & $2$ & $1$ & $.000$ & $-.001$ & $.015$ & $.010$ & $.011$ & $%
.009$ & $.953$ & $.948$ & $.654$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.076$ & $.011$ & $.011$ & $.011$ & $.947$
& $.950$ & $.000$ \\
$1000$ & $16$ & $.3$ & $1$ & $.000$ & $-.003$ & $-.036$ & $.009$ & $.010$ & $%
.009$ & $.947$ & $.944$ & $.010$ \\
&  &  & $-.2$ & $.000$ & $-.001$ & $-.014$ & $.009$ & $.009$ & $.009$ & $.947
$ & $.945$ & $.599$ \\
$1000$ & $16$ & $1$ & $1$ & $.000$ & $-.002$ & $-.019$ & $.009$ & $.009$ & $%
.008$ & $.948$ & $.946$ & $.316$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.008$ & $.008$ & $.008$ & $.947$
& $.946$ & $.752$ \\
$1000$ & $16$ & $2$ & $1$ & $.000$ & $-.001$ & $.005$ & $.007$ & $.008$ & $%
.007$ & $.949$ & $.949$ & $.889$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.038$ & $.007$ & $.007$ & $.008$ & $.953$
& $.951$ & $.001$ \\
$1000$ & $24$ & $.3$ & $1$ & $.000$ & $-.002$ & $-.015$ & $.007$ & $.007$ & $%
.007$ & $.952$ & $.944$ & $.342$ \\
&  &  & $-.2$ & $.000$ & $-.001$ & $-.007$ & $.007$ & $.007$ & $.007$ & $.950
$ & $.951$ & $.811$ \\
$1000$ & $24$ & $1$ & $1$ & $.000$ & $-.002$ & $-.009$ & $.007$ & $.007$ & $%
.007$ & $.946$ & $.939$ & $.690$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.005$ & $.007$ & $.007$ & $.007$ & $.949$
& $.949$ & $.874$ \\
$1000$ & $24$ & $2$ & $1$ & $.000$ & $-.001$ & $.002$ & $.006$ & $.006$ & $%
.006$ & $.947$ & $.947$ & $.934$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.006$ & $.006$ & $.006$ & $.951$
& $.951$ & $.044$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $.3$ & $.6$ & $-.074$ & --- & $-.843$ & $.104$ & --- & $.060$
& $.831$ & $.931$ & $.000$ \\
&  &  & $.2$ & $-.158$ & --- & $-.731$ & $.168$ & --- & $.159$ & $.931$ & $%
.931$ & $.000$ \\
$2500$ & $2$ & $1$ & $.6$ & $-.075$ & --- & $-.843$ & $.103$ & --- & $.033$
& $.836$ & $.931$ & $.000$ \\
&  &  & $.2$ & $-.075$ & --- & $-.729$ & $.094$ & --- & $.055$ & $.855$ & $%
.932$ & $.000$ \\
$2500$ & $2$ & $2$ & $.6$ & $-.076$ & --- & $-.844$ & $.105$ & --- & $.030$
& $.827$ & $.935$ & $.000$ \\
&  &  & $.2$ & $-.070$ & --- & $-.731$ & $.093$ & --- & $.036$ & $.836$ & $%
.935$ & $.000$ \\
$2500$ & $4$ & $.3$ & $.6$ & $-.022$ & $-.014$ & $-.325$ & $.048$ & $.067$ &
$.013$ & $.878$ & $.946$ & $.000$ \\
&  &  & $.2$ & $-.009$ & $-.006$ & $-.116$ & $.028$ & $.033$ & $.019$ & $.916
$ & $.944$ & $.000$ \\
$2500$ & $4$ & $1$ & $.6$ & $.007$ & $-.002$ & $-.202$ & $.044$ & $.024$ & $%
.012$ & $.953$ & $.947$ & $.000$ \\
&  &  & $.2$ & $.004$ & $-.001$ & $-.049$ & $.026$ & $.018$ & $.017$ & $.968$
& $.951$ & $.077$ \\
$2500$ & $4$ & $2$ & $.6$ & $.000$ & $.000$ & $-.005$ & $.015$ & $.013$ & $%
.009$ & $.949$ & $.949$ & $.930$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.074$ & $.014$ & $.013$ & $.013$ & $.950$
& $.950$ & $.000$ \\
$2500$ & $6$ & $.3$ & $.6$ & $-.003$ & $-.008$ & $-.205$ & $.032$ & $.031$ &
$.010$ & $.919$ & $.939$ & $.000$ \\
&  &  & $.2$ & $-.001$ & $-.004$ & $-.083$ & $.021$ & $.018$ & $.012$ & $.941
$ & $.942$ & $.000$ \\
$2500$ & $6$ & $1$ & $.6$ & $.001$ & $-.002$ & $-.109$ & $.020$ & $.015$ & $%
.009$ & $.963$ & $.950$ & $.000$ \\
&  &  & $.2$ & $.001$ & $-.001$ & $-.011$ & $.014$ & $.012$ & $.011$ & $.958$
& $.951$ & $.737$ \\
$2500$ & $6$ & $2$ & $.6$ & $.000$ & $.000$ & $.026$ & $.009$ & $.009$ & $%
.007$ & $.949$ & $.953$ & $.062$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.086$ & $.009$ & $.009$ & $.009$ & $.947$
& $.947$ & $.000$ \\
$2500$ & $8$ & $.3$ & $.6$ & $.003$ & $-.005$ & $-.143$ & $.025$ & $.019$ & $%
.009$ & $.941$ & $.948$ & $.000$ \\
&  &  & $.2$ & $.002$ & $-.003$ & $-.064$ & $.017$ & $.013$ & $.009$ & $.957$
& $.944$ & $.000$ \\
$2500$ & $8$ & $1$ & $.6$ & $.000$ & $-.002$ & $-.070$ & $.012$ & $.011$ & $%
.008$ & $.954$ & $.946$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $-.002$ & $.010$ & $.009$ & $.009$ & $.949$
& $.949$ & $.908$ \\
$2500$ & $8$ & $2$ & $.6$ & $.000$ & $.000$ & $.027$ & $.007$ & $.007$ & $%
.006$ & $.949$ & $.950$ & $.013$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.076$ & $.007$ & $.007$ & $.008$ & $.950$
& $.948$ & $.000$ \\
$2500$ & $16$ & $.3$ & $.6$ & $.000$ & $-.002$ & $-.050$ & $.007$ & $.007$ &
$.006$ & $.951$ & $.940$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $-.029$ & $.007$ & $.006$ & $.006$ & $.953$
& $.947$ & $.000$ \\
$2500$ & $16$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.023$ & $.006$ & $.006$ & $%
.005$ & $.951$ & $.945$ & $.007$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.001$ & $.006$ & $.006$ & $.005$ & $.950$
& $.948$ & $.924$ \\
$2500$ & $16$ & $2$ & $.6$ & $.000$ & $-.001$ & $.015$ & $.005$ & $.005$ & $%
.005$ & $.951$ & $.952$ & $.102$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.042$ & $.005$ & $.005$ & $.005$ & $.944$
& $.944$ & $.000$ \\
$2500$ & $24$ & $.3$ & $.6$ & $.000$ & $-.001$ & $-.024$ & $.005$ & $.005$ &
$.004$ & $.948$ & $.941$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $-.016$ & $.005$ & $.005$ & $.004$ & $.952$
& $.945$ & $.048$ \\
$2500$ & $24$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.011$ & $.004$ & $.005$ & $%
.004$ & $.950$ & $.946$ & $.206$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.001$ & $.004$ & $.004$ & $.004$ & $.949$
& $.947$ & $.933$ \\
$2500$ & $24$ & $2$ & $.6$ & $.000$ & $-.001$ & $.009$ & $.004$ & $.004$ & $%
.004$ & $.949$ & $.948$ & $.366$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.026$ & $.004$ & $.004$ & $.004$ & $.949$
& $.949$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $.3$ & $1$ & $-.074$ & --- & $-.859$ & $.104$ & --- & $.053$
& $.831$ & $.931$ & $.000$ \\
&  &  & $-.2$ & $-.093$ & --- & $-.377$ & $.134$ & --- & $.141$ & $.956$ & $%
.931$ & $.043$ \\
$2500$ & $2$ & $1$ & $1$ & $-.075$ & --- & $-.858$ & $.104$ & --- & $.029$ &
$.836$ & $.931$ & $.000$ \\
&  &  & $-.2$ & $-.045$ & --- & $-.376$ & $.062$ & --- & $.045$ & $.878$ & $%
.931$ & $.000$ \\
$2500$ & $2$ & $2$ & $1$ & $-.076$ & --- & $-.859$ & $.105$ & --- & $.026$ &
$.827$ & $.935$ & $.000$ \\
&  &  & $-.2$ & $-.041$ & --- & $-.377$ & $.056$ & --- & $.026$ & $.845$ & $%
.935$ & $.000$ \\
$2500$ & $4$ & $.3$ & $1$ & $-.019$ & $-.011$ & $-.358$ & $.047$ & $.058$ & $%
.012$ & $.885$ & $.945$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.001$ & $.010$ & $.019$ & $.018$ & $.019$ & $.968$
& $.950$ & $.747$ \\
$2500$ & $4$ & $1$ & $1$ & $.003$ & $-.001$ & $-.211$ & $.030$ & $.020$ & $%
.011$ & $.959$ & $.949$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $.014$ & $.016$ & $.016$ & $.017$ & $.956$
& $.951$ & $.702$ \\
$2500$ & $4$ & $2$ & $1$ & $.000$ & $.000$ & $-.022$ & $.011$ & $.010$ & $%
.007$ & $.949$ & $.949$ & $.192$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.060$ & $.012$ & $.012$ & $.014$ & $.946$
& $.949$ & $.002$ \\
$2500$ & $6$ & $.3$ & $1$ & $.001$ & $-.005$ & $-.213$ & $.031$ & $.025$ & $%
.010$ & $.931$ & $.943$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.001$ & $-.024$ & $.014$ & $.012$ & $.012$ & $.969
$ & $.947$ & $.309$ \\
$2500$ & $6$ & $1$ & $1$ & $.000$ & $-.001$ & $-.103$ & $.014$ & $.012$ & $%
.008$ & $.952$ & $.950$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.031$ & $.010$ & $.010$ & $.011$ & $.951$
& $.950$ & $.118$ \\
$2500$ & $6$ & $2$ & $1$ & $.000$ & $.000$ & $.017$ & $.008$ & $.008$ & $.006
$ & $.948$ & $.950$ & $.290$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.088$ & $.008$ & $.008$ & $.009$ & $.947$
& $.945$ & $.000$ \\
$2500$ & $8$ & $.3$ & $1$ & $.004$ & $-.003$ & $-.135$ & $.021$ & $.015$ & $%
.008$ & $.955$ & $.950$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $-.001$ & $-.029$ & $.012$ & $.010$ & $.009$ & $.969
$ & $.947$ & $.061$ \\
$2500$ & $8$ & $1$ & $1$ & $.000$ & $-.001$ & $-.063$ & $.010$ & $.010$ & $%
.007$ & $.952$ & $.945$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.008$ & $.008$ & $.009$ & $.951$
& $.949$ & $.093$ \\
$2500$ & $8$ & $2$ & $1$ & $.000$ & $.000$ & $.015$ & $.007$ & $.007$ & $.006
$ & $.949$ & $.951$ & $.277$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.076$ & $.007$ & $.007$ & $.007$ & $.949$
& $.949$ & $.000$ \\
$2500$ & $16$ & $.3$ & $1$ & $.000$ & $-.001$ & $-.036$ & $.006$ & $.006$ & $%
.005$ & $.951$ & $.946$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.014$ & $.006$ & $.006$ & $.005$ & $.948$
& $.949$ & $.242$ \\
$2500$ & $16$ & $1$ & $1$ & $.000$ & $-.001$ & $-.019$ & $.005$ & $.006$ & $%
.005$ & $.949$ & $.948$ & $.034$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.005$ & $.005$ & $.005$ & $.950$
& $.951$ & $.497$ \\
$2500$ & $16$ & $2$ & $1$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $%
.004$ & $.951$ & $.950$ & $.800$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.038$ & $.005$ & $.005$ & $.005$ & $.948$
& $.946$ & $.000$ \\
$2500$ & $24$ & $.3$ & $1$ & $.000$ & $-.001$ & $-.016$ & $.004$ & $.005$ & $%
.004$ & $.949$ & $.945$ & $.038$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.007$ & $.004$ & $.004$ & $.004$ & $.949$
& $.948$ & $.641$ \\
$2500$ & $24$ & $1$ & $1$ & $.000$ & $-.001$ & $-.009$ & $.004$ & $.004$ & $%
.004$ & $.951$ & $.947$ & $.390$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.005$ & $.004$ & $.004$ & $.004$ & $.949$
& $.951$ & $.781$ \\
$2500$ & $24$ & $2$ & $1$ & $.000$ & $.000$ & $.002$ & $.004$ & $.004$ & $%
.004$ & $.951$ & $.949$ & $.898$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.004$ & $.004$ & $.004$ & $.951$
& $.950$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $.3$ & $.6$ & $-.064$ & --- & $-.843$ & $.087$ & --- & $.042$
& $.831$ & $.931$ & $.000$ \\
&  &  & $.2$ & $-.124$ & --- & $-.729$ & $.125$ & --- & $.113$ & $.926$ & $%
.933$ & $.000$ \\
$5000$ & $2$ & $1$ & $.6$ & $-.063$ & --- & $-.843$ & $.086$ & --- & $.023$
& $.839$ & $.930$ & $.000$ \\
&  &  & $.2$ & $-.063$ & --- & $-.729$ & $.077$ & --- & $.038$ & $.853$ & $%
.929$ & $.000$ \\
$5000$ & $2$ & $2$ & $.6$ & $-.061$ & --- & $-.843$ & $.086$ & --- & $.021$
& $.841$ & $.928$ & $.000$ \\
&  &  & $.2$ & $-.056$ & --- & $-.730$ & $.076$ & --- & $.025$ & $.844$ & $%
.928$ & $.000$ \\
$5000$ & $4$ & $.3$ & $.6$ & $-.016$ & $-.007$ & $-.325$ & $.040$ & $.048$ &
$.009$ & $.880$ & $.944$ & $.000$ \\
&  &  & $.2$ & $-.007$ & $-.003$ & $-.116$ & $.022$ & $.023$ & $.013$ & $.916
$ & $.946$ & $.000$ \\
$5000$ & $4$ & $1$ & $.6$ & $.004$ & $-.001$ & $-.202$ & $.030$ & $.017$ & $%
.009$ & $.959$ & $.953$ & $.000$ \\
&  &  & $.2$ & $.002$ & $.000$ & $-.048$ & $.018$ & $.013$ & $.012$ & $.966$
& $.948$ & $.004$ \\
$5000$ & $4$ & $2$ & $.6$ & $.000$ & $.000$ & $-.005$ & $.010$ & $.009$ & $%
.007$ & $.948$ & $.949$ & $.890$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.074$ & $.010$ & $.009$ & $.009$ & $.950$
& $.948$ & $.000$ \\
$5000$ & $6$ & $.3$ & $.6$ & $-.001$ & $-.004$ & $-.205$ & $.027$ & $.022$ &
$.007$ & $.922$ & $.942$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.002$ & $-.082$ & $.017$ & $.013$ & $.008$ & $.940$
& $.948$ & $.000$ \\
$5000$ & $6$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.109$ & $.014$ & $.011$ & $%
.007$ & $.952$ & $.947$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.011$ & $.010$ & $.008$ & $.008$ & $.952$
& $.949$ & $.595$ \\
$5000$ & $6$ & $2$ & $.6$ & $.000$ & $.000$ & $.026$ & $.007$ & $.006$ & $%
.005$ & $.950$ & $.951$ & $.002$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.086$ & $.006$ & $.006$ & $.007$ & $.952$
& $.952$ & $.000$ \\
$5000$ & $8$ & $.3$ & $.6$ & $.003$ & $-.003$ & $-.143$ & $.021$ & $.013$ & $%
.006$ & $.949$ & $.946$ & $.000$ \\
&  &  & $.2$ & $.002$ & $-.001$ & $-.064$ & $.014$ & $.009$ & $.006$ & $.960$
& $.945$ & $.000$ \\
$5000$ & $8$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.070$ & $.009$ & $.008$ & $%
.006$ & $.954$ & $.949$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.002$ & $.007$ & $.006$ & $.006$ & $.948$
& $.950$ & $.892$ \\
$5000$ & $8$ & $2$ & $.6$ & $.000$ & $.000$ & $.027$ & $.005$ & $.005$ & $%
.004$ & $.950$ & $.951$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.076$ & $.005$ & $.005$ & $.005$ & $.947$
& $.947$ & $.000$ \\
$5000$ & $16$ & $.3$ & $.6$ & $.000$ & $-.001$ & $-.050$ & $.005$ & $.005$ &
$.004$ & $.950$ & $.947$ & $.000$ \\
&  &  & $.2$ & $.000$ & $-.001$ & $-.029$ & $.005$ & $.005$ & $.004$ & $.950$
& $.949$ & $.000$ \\
$5000$ & $16$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.023$ & $.004$ & $.004$ & $%
.004$ & $.950$ & $.949$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.002$ & $.004$ & $.004$ & $.004$ & $.952$
& $.951$ & $.909$ \\
$5000$ & $16$ & $2$ & $.6$ & $.000$ & $.000$ & $.015$ & $.004$ & $.004$ & $%
.003$ & $.950$ & $.948$ & $.005$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.042$ & $.003$ & $.003$ & $.004$ & $.948$
& $.949$ & $.000$ \\
$5000$ & $24$ & $.3$ & $.6$ & $.000$ & $-.001$ & $-.024$ & $.003$ & $.004$ &
$.003$ & $.955$ & $.950$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.016$ & $.003$ & $.003$ & $.003$ & $.950$
& $.948$ & $.001$ \\
$5000$ & $24$ & $1$ & $.6$ & $.000$ & $.000$ & $-.011$ & $.003$ & $.003$ & $%
.003$ & $.950$ & $.945$ & $.025$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.001$ & $.003$ & $.003$ & $.003$ & $.949$
& $.950$ & $.928$ \\
$5000$ & $24$ & $2$ & $.6$ & $.000$ & $.000$ & $.009$ & $.003$ & $.003$ & $%
.003$ & $.952$ & $.951$ & $.094$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.026$ & $.003$ & $.003$ & $.003$ & $.948$
& $.949$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $.3$ & $1$ & $-.064$ & --- & $-.859$ & $.087$ & --- & $.038$
& $.831$ & $.931$ & $.000$ \\
&  &  & $-.2$ & $-.074$ & --- & $-.377$ & $.098$ & --- & $.100$ & $.951$ & $%
.934$ & $.002$ \\
$5000$ & $2$ & $1$ & $1$ & $-.063$ & --- & $-.859$ & $.086$ & --- & $.020$ &
$.839$ & $.930$ & $.000$ \\
&  &  & $-.2$ & $-.037$ & --- & $-.377$ & $.049$ & --- & $.032$ & $.871$ & $%
.929$ & $.000$ \\
$5000$ & $2$ & $2$ & $1$ & $-.061$ & --- & $-.859$ & $.086$ & --- & $.019$ &
$.841$ & $.928$ & $.000$ \\
&  &  & $-.2$ & $-.033$ & --- & $-.377$ & $.046$ & --- & $.019$ & $.850$ & $%
.927$ & $.000$ \\
$5000$ & $4$ & $.3$ & $1$ & $-.013$ & $-.005$ & $-.358$ & $.040$ & $.041$ & $%
.008$ & $.889$ & $.948$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $.010$ & $.014$ & $.013$ & $.013$ & $.970$
& $.949$ & $.697$ \\
$5000$ & $4$ & $1$ & $1$ & $.001$ & $-.001$ & $-.211$ & $.020$ & $.014$ & $%
.007$ & $.960$ & $.953$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.014$ & $.011$ & $.011$ & $.012$ & $.950$
& $.951$ & $.596$ \\
$5000$ & $4$ & $2$ & $1$ & $.000$ & $.000$ & $-.022$ & $.008$ & $.007$ & $%
.005$ & $.949$ & $.951$ & $.020$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.060$ & $.008$ & $.008$ & $.010$ & $.949$
& $.948$ & $.000$ \\
$5000$ & $6$ & $.3$ & $1$ & $.003$ & $-.003$ & $-.213$ & $.026$ & $.018$ & $%
.007$ & $.935$ & $.945$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $-.024$ & $.010$ & $.009$ & $.009$ & $.969$
& $.950$ & $.090$ \\
$5000$ & $6$ & $1$ & $1$ & $.000$ & $-.001$ & $-.103$ & $.010$ & $.009$ & $%
.006$ & $.946$ & $.948$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.031$ & $.007$ & $.008$ & $.008$ & $.946$
& $.946$ & $.010$ \\
$5000$ & $6$ & $2$ & $1$ & $.000$ & $.000$ & $.017$ & $.005$ & $.005$ & $.004
$ & $.950$ & $.951$ & $.040$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.088$ & $.006$ & $.006$ & $.006$ & $.951$
& $.951$ & $.000$ \\
$5000$ & $8$ & $.3$ & $1$ & $.002$ & $-.002$ & $-.135$ & $.016$ & $.011$ & $%
.006$ & $.962$ & $.946$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $-.029$ & $.008$ & $.007$ & $.006$ & $.968$
& $.946$ & $.002$ \\
$5000$ & $8$ & $1$ & $1$ & $.000$ & $-.001$ & $-.063$ & $.007$ & $.007$ & $%
.005$ & $.952$ & $.949$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.006$ & $.006$ & $.006$ & $.951$
& $.950$ & $.005$ \\
$5000$ & $8$ & $2$ & $1$ & $.000$ & $.000$ & $.015$ & $.005$ & $.005$ & $.004
$ & $.952$ & $.949$ & $.045$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.076$ & $.005$ & $.005$ & $.005$ & $.950$
& $.948$ & $.000$ \\
$5000$ & $16$ & $.3$ & $1$ & $.000$ & $-.001$ & $-.036$ & $.004$ & $.005$ & $%
.004$ & $.952$ & $.949$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.014$ & $.004$ & $.004$ & $.004$ & $.951$
& $.952$ & $.040$ \\
$5000$ & $16$ & $1$ & $1$ & $.000$ & $.000$ & $-.019$ & $.004$ & $.004$ & $%
.004$ & $.953$ & $.949$ & $.001$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.004$ & $.004$ & $.004$ & $.954$
& $.953$ & $.223$ \\
$5000$ & $16$ & $2$ & $1$ & $.000$ & $.000$ & $.005$ & $.003$ & $.003$ & $%
.003$ & $.949$ & $.949$ & $.631$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.038$ & $.003$ & $.003$ & $.003$ & $.950$
& $.949$ & $.000$ \\
$5000$ & $24$ & $.3$ & $1$ & $.000$ & $.000$ & $-.016$ & $.003$ & $.003$ & $%
.003$ & $.949$ & $.951$ & $.001$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.007$ & $.003$ & $.003$ & $.003$ & $.949$
& $.952$ & $.377$ \\
$5000$ & $24$ & $1$ & $1$ & $.000$ & $.000$ & $-.009$ & $.003$ & $.003$ & $%
.003$ & $.948$ & $.949$ & $.120$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.005$ & $.003$ & $.003$ & $.003$ & $.948$
& $.949$ & $.618$ \\
$5000$ & $24$ & $2$ & $1$ & $.000$ & $.000$ & $.002$ & $.003$ & $.003$ & $%
.003$ & $.950$ & $.950$ & $.845$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.003$ & $.003$ & $.003$ & $.950$
& $.949$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $.3$ & $.6$ & $-.054$ & --- & $-.843$ & $.071$ & --- & $.030$
& $.836$ & $.928$ & $.000$ \\
&  &  & $.2$ & $-.098$ & --- & $-.730$ & $.092$ & --- & $.079$ & $.914$ & $%
.928$ & $.000$ \\
$10000$ & $2$ & $1$ & $.6$ & $-.054$ & --- & $-.843$ & $.072$ & --- & $.016$
& $.839$ & $.932$ & $.000$ \\
&  &  & $.2$ & $-.053$ & --- & $-.730$ & $.063$ & --- & $.027$ & $.849$ & $%
.932$ & $.000$ \\
$10000$ & $2$ & $2$ & $.6$ & $-.054$ & --- & $-.843$ & $.071$ & --- & $.015$
& $.835$ & $.923$ & $.000$ \\
&  &  & $.2$ & $-.049$ & --- & $-.729$ & $.063$ & --- & $.018$ & $.840$ & $%
.924$ & $.000$ \\
$10000$ & $4$ & $.3$ & $.6$ & $-.011$ & $-.003$ & $-.324$ & $.033$ & $.034$
& $.006$ & $.889$ & $.946$ & $.000$ \\
&  &  & $.2$ & $-.005$ & $-.001$ & $-.116$ & $.018$ & $.017$ & $.009$ & $.915
$ & $.947$ & $.000$ \\
$10000$ & $4$ & $1$ & $.6$ & $.002$ & $.000$ & $-.202$ & $.020$ & $.012$ & $%
.006$ & $.962$ & $.950$ & $.000$ \\
&  &  & $.2$ & $.001$ & $.000$ & $-.048$ & $.013$ & $.009$ & $.008$ & $.962$
& $.949$ & $.000$ \\
$10000$ & $4$ & $2$ & $.6$ & $.000$ & $.000$ & $-.005$ & $.007$ & $.006$ & $%
.005$ & $.951$ & $.952$ & $.819$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.074$ & $.007$ & $.006$ & $.007$ & $.950$
& $.951$ & $.000$ \\
$10000$ & $6$ & $.3$ & $.6$ & $.001$ & $-.002$ & $-.205$ & $.023$ & $.015$ &
$.005$ & $.929$ & $.948$ & $.000$ \\
&  &  & $.2$ & $.001$ & $-.001$ & $-.083$ & $.013$ & $.009$ & $.006$ & $.944$
& $.952$ & $.000$ \\
$10000$ & $6$ & $1$ & $.6$ & $.000$ & $-.001$ & $-.109$ & $.009$ & $.008$ & $%
.005$ & $.954$ & $.948$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.011$ & $.007$ & $.006$ & $.006$ & $.953$
& $.951$ & $.370$ \\
$10000$ & $6$ & $2$ & $.6$ & $.000$ & $.000$ & $.026$ & $.005$ & $.005$ & $%
.004$ & $.950$ & $.950$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.086$ & $.005$ & $.004$ & $.005$ & $.954$
& $.954$ & $.000$ \\
$10000$ & $8$ & $.3$ & $.6$ & $.003$ & $-.001$ & $-.143$ & $.016$ & $.009$ &
$.004$ & $.951$ & $.947$ & $.000$ \\
&  &  & $.2$ & $.001$ & $.000$ & $-.064$ & $.010$ & $.006$ & $.005$ & $.958$
& $.948$ & $.000$ \\
$10000$ & $8$ & $1$ & $.6$ & $.000$ & $.000$ & $-.070$ & $.006$ & $.005$ & $%
.004$ & $.953$ & $.951$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.002$ & $.005$ & $.005$ & $.004$ & $.949$
& $.951$ & $.878$ \\
$10000$ & $8$ & $2$ & $.6$ & $.000$ & $.000$ & $.027$ & $.004$ & $.004$ & $%
.003$ & $.949$ & $.949$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.076$ & $.004$ & $.004$ & $.004$ & $.950$
& $.952$ & $.000$ \\
$10000$ & $16$ & $.3$ & $.6$ & $.000$ & $.000$ & $-.050$ & $.004$ & $.004$ &
$.003$ & $.948$ & $.950$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.029$ & $.003$ & $.003$ & $.003$ & $.951$
& $.947$ & $.000$ \\
$10000$ & $16$ & $1$ & $.6$ & $.000$ & $.000$ & $-.023$ & $.003$ & $.003$ & $%
.003$ & $.948$ & $.946$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.002$ & $.003$ & $.003$ & $.003$ & $.950$
& $.951$ & $.886$ \\
$10000$ & $16$ & $2$ & $.6$ & $.000$ & $.000$ & $.015$ & $.002$ & $.003$ & $%
.002$ & $.948$ & $.948$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.042$ & $.002$ & $.002$ & $.002$ & $.953$
& $.952$ & $.000$ \\
$10000$ & $24$ & $.3$ & $.6$ & $.000$ & $.000$ & $-.024$ & $.002$ & $.003$ &
$.002$ & $.951$ & $.946$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $-.016$ & $.002$ & $.002$ & $.002$ & $.951$
& $.950$ & $.000$ \\
$10000$ & $24$ & $1$ & $.6$ & $.000$ & $.000$ & $-.011$ & $.002$ & $.002$ & $%
.002$ & $.951$ & $.949$ & $.000$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.001$ & $.002$ & $.002$ & $.002$ & $.947$
& $.949$ & $.919$ \\
$10000$ & $24$ & $2$ & $.6$ & $.000$ & $.000$ & $.009$ & $.002$ & $.002$ & $%
.002$ & $.948$ & $.948$ & $.004$ \\
&  &  & $.2$ & $.000$ & $.000$ & $.026$ & $.002$ & $.002$ & $.002$ & $.951$
& $.950$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}









\clearpage
\addtocounter{table}{-1}
\begin{table}[h]   \small
\caption{Simulation results for the second-order autoregression (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{cccc|rrr|rrr|rrr}
\begin{tabular}{ccccrrrrrrrrr}
\hline\hline
&  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
\cmidrule(l{6pt}r{6pt}){5-7}
\cmidrule(l{6pt}r{6pt}){8-10}
\cmidrule(l{6pt}r{6pt}){11-13}
{$N$} & {$T$} & {$\psi$} & {$\rho _{0}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $.3$ & $1$ & $-.054$ & --- & $-.859$ & $.071$ & --- & $.026$
& $.836$ & $.928$ & $.000$ \\
&  &  & $-.2$ & $-.059$ & --- & $-.376$ & $.072$ & --- & $.070$ & $.941$ & $%
.929$ & $.000$ \\
$10000$ & $2$ & $1$ & $1$ & $-.054$ & --- & $-.859$ & $.072$ & --- & $.014$
& $.839$ & $.932$ & $.000$ \\
&  &  & $-.2$ & $-.031$ & --- & $-.377$ & $.040$ & --- & $.022$ & $.863$ & $%
.932$ & $.000$ \\
$10000$ & $2$ & $2$ & $1$ & $-.054$ & --- & $-.859$ & $.071$ & --- & $.013$
& $.835$ & $.923$ & $.000$ \\
&  &  & $-.2$ & $-.029$ & --- & $-.377$ & $.037$ & --- & $.013$ & $.845$ & $%
.924$ & $.000$ \\
$10000$ & $4$ & $.3$ & $1$ & $-.008$ & $-.002$ & $-.358$ & $.033$ & $.030$ &
$.006$ & $.896$ & $.947$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.010$ & $.009$ & $.009$ & $.968$
& $.949$ & $.581$ \\
$10000$ & $4$ & $1$ & $1$ & $.001$ & $.000$ & $-.211$ & $.014$ & $.010$ & $%
.005$ & $.953$ & $.952$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.014$ & $.008$ & $.008$ & $.009$ & $.951$
& $.951$ & $.420$ \\
$10000$ & $4$ & $2$ & $1$ & $.000$ & $.000$ & $-.022$ & $.006$ & $.005$ & $%
.004$ & $.952$ & $.953$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.060$ & $.006$ & $.006$ & $.007$ & $.950$
& $.951$ & $.000$ \\
$10000$ & $6$ & $.3$ & $1$ & $.003$ & $-.001$ & $-.213$ & $.022$ & $.013$ & $%
.005$ & $.941$ & $.949$ & $.000$ \\
&  &  & $-.2$ & $.001$ & $.000$ & $-.024$ & $.008$ & $.006$ & $.006$ & $.968$
& $.951$ & $.006$ \\
$10000$ & $6$ & $1$ & $1$ & $.000$ & $.000$ & $-.103$ & $.007$ & $.006$ & $%
.004$ & $.951$ & $.951$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.031$ & $.005$ & $.005$ & $.006$ & $.952$
& $.954$ & $.000$ \\
$10000$ & $6$ & $2$ & $1$ & $.000$ & $.000$ & $.017$ & $.004$ & $.004$ & $%
.003$ & $.950$ & $.949$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.087$ & $.004$ & $.004$ & $.005$ & $.952$
& $.953$ & $.000$ \\
$10000$ & $8$ & $.3$ & $1$ & $.001$ & $-.001$ & $-.135$ & $.011$ & $.007$ & $%
.004$ & $.963$ & $.948$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.029$ & $.006$ & $.005$ & $.005$ & $.962$
& $.948$ & $.000$ \\
$10000$ & $8$ & $1$ & $1$ & $.000$ & $.000$ & $-.063$ & $.005$ & $.005$ & $%
.004$ & $.953$ & $.951$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.026$ & $.004$ & $.004$ & $.004$ & $.953$
& $.951$ & $.000$ \\
$10000$ & $8$ & $2$ & $1$ & $.000$ & $.000$ & $.015$ & $.003$ & $.003$ & $%
.003$ & $.949$ & $.949$ & $.001$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.076$ & $.003$ & $.003$ & $.004$ & $.952$
& $.953$ & $.000$ \\
$10000$ & $16$ & $.3$ & $1$ & $.000$ & $.000$ & $-.036$ & $.003$ & $.003$ & $%
.003$ & $.947$ & $.952$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.014$ & $.003$ & $.003$ & $.003$ & $.948$
& $.946$ & $.001$ \\
$10000$ & $16$ & $1$ & $1$ & $.000$ & $.000$ & $-.019$ & $.003$ & $.003$ & $%
.003$ & $.948$ & $.946$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.010$ & $.003$ & $.003$ & $.003$ & $.948$
& $.950$ & $.033$ \\
$10000$ & $16$ & $2$ & $1$ & $.000$ & $.000$ & $.005$ & $.002$ & $.002$ & $%
.002$ & $.948$ & $.948$ & $.370$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.038$ & $.002$ & $.002$ & $.002$ & $.951$
& $.951$ & $.000$ \\
$10000$ & $24$ & $.3$ & $1$ & $.000$ & $.000$ & $-.016$ & $.002$ & $.002$ & $%
.002$ & $.950$ & $.947$ & $.000$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $-.007$ & $.002$ & $.002$ & $.002$ & $.951$
& $.948$ & $.112$ \\
$10000$ & $24$ & $1$ & $1$ & $.000$ & $.000$ & $-.009$ & $.002$ & $.002$ & $%
.002$ & $.950$ & $.947$ & $.007$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.005$ & $.002$ & $.002$ & $.002$ & $.948$
& $.950$ & $.373$ \\
$10000$ & $24$ & $2$ & $1$ & $.000$ & $.000$ & $.002$ & $.002$ & $.002$ & $%
.002$ & $.950$ & $.950$ & $.744$ \\
&  &  & $-.2$ & $.000$ & $.000$ & $.022$ & $.002$ & $.002$ & $.002$ & $.950$
& $.950$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\rho_{01}y_{it-1}+\rho_{02}y_{it-2} +\alpha _{i}+\varepsilon_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, and $\psi$ the degree of outlyingness of the initial observations $(y_{i0},y_{i,-1})$. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\rho }_{\mathrm{al}}$), Arellano-Bond ($\widehat{\rho }_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\rho }_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}


\clearpage
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.5$ & $.5$ & $-.089$ & --- & $-.661$ & $.267$ & --- & $%
.153$ & $.845$ & $.884$ & $.001$ \\
&  &  &  & $.5$ & $-.025$ & --- & $-.168$ & $.249$ & --- & $.191$ & $.959$ &
$.940$ & $.660$ \\
$100$ & $2$ & $1$ & $.5$ & $.5$ & $.017$ & --- & $-.416$ & $.270$ & --- & $%
.146$ & $.897$ & $.915$ & $.056$ \\
&  &  &  & $.5$ & $-.002$ & --- & $.005$ & $.251$ & --- & $.205$ & $.969$ & $%
.956$ & $.834$ \\
$100$ & $2$ & $2$ & $.5$ & $.5$ & $.026$ & --- & $.078$ & $.181$ & --- & $%
.118$ & $.946$ & $.937$ & $.904$ \\
&  &  &  & $.5$ & $-.010$ & --- & $-.022$ & $.257$ & --- & $.258$ & $.945$ &
$.942$ & $.840$ \\
$100$ & $4$ & $0$ & $.5$ & $.5$ & $.013$ & $-.094$ & $-.248$ & $.142$ & $.118
$ & $.068$ & $.932$ & $.845$ & $.028$ \\
&  &  &  & $.5$ & $.000$ & $-.013$ & $-.032$ & $.127$ & $.124$ & $.121$ & $%
.958$ & $.942$ & $.885$ \\
$100$ & $4$ & $1$ & $.5$ & $.5$ & $.011$ & $-.070$ & $-.127$ & $.119$ & $.104
$ & $.067$ & $.943$ & $.878$ & $.394$ \\
&  &  &  & $.5$ & $-.004$ & $.006$ & $.013$ & $.127$ & $.124$ & $.121$ & $%
.951$ & $.945$ & $.906$ \\
$100$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $-.025$ & $.076$ & $.063$ & $.062$
& $.056$ & $.943$ & $.919$ & $.646$ \\
&  &  &  & $.5$ & $-.004$ & $.006$ & $-.032$ & $.127$ & $.126$ & $.128$ & $%
.947$ & $.946$ & $.909$ \\
$100$ & $6$ & $0$ & $.5$ & $.5$ & $.003$ & $-.065$ & $-.116$ & $.079$ & $.069
$ & $.050$ & $.956$ & $.834$ & $.282$ \\
&  &  &  & $.5$ & $.001$ & $.000$ & $-.001$ & $.091$ & $.091$ & $.091$ & $%
.947$ & $.947$ & $.918$ \\
$100$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.054$ & $-.053$ & $.061$ & $.063
$ & $.048$ & $.952$ & $.858$ & $.726$ \\
&  &  &  & $.5$ & $.001$ & $.011$ & $.010$ & $.091$ & $.091$ & $.090$ & $.948
$ & $.945$ & $.923$ \\
$100$ & $6$ & $2$ & $.5$ & $.5$ & $-.001$ & $-.026$ & $.055$ & $.042$ & $.044
$ & $.040$ & $.945$ & $.905$ & $.639$ \\
&  &  &  & $.5$ & $.001$ & $.010$ & $-.021$ & $.092$ & $.092$ & $.091$ & $%
.948$ & $.945$ & $.928$ \\
$100$ & $8$ & $0$ & $.5$ & $.5$ & $-.001$ & $-.052$ & $-.063$ & $.051$ & $%
.051$ & $.041$ & $.947$ & $.820$ & $.584$ \\
&  &  &  & $.5$ & $.001$ & $.003$ & $.004$ & $.075$ & $.075$ & $.075$ & $.944
$ & $.944$ & $.927$ \\
$100$ & $8$ & $1$ & $.5$ & $.5$ & $-.001$ & $-.047$ & $-.026$ & $.045$ & $%
.049$ & $.039$ & $.941$ & $.833$ & $.842$ \\
&  &  &  & $.5$ & $.001$ & $.011$ & $.006$ & $.076$ & $.076$ & $.075$ & $.944
$ & $.942$ & $.932$ \\
$100$ & $8$ & $2$ & $.5$ & $.5$ & $-.001$ & $-.026$ & $.041$ & $.034$ & $.037
$ & $.034$ & $.940$ & $.883$ & $.685$ \\
&  &  &  & $.5$ & $.000$ & $.010$ & $-.015$ & $.076$ & $.076$ & $.075$ & $%
.943$ & $.942$ & $.931$ \\
$100$ & $16$ & $0$ & $.5$ & $.5$ & $-.001$ & $-.037$ & $-.011$ & $.026$ & $%
.027$ & $.025$ & $.944$ & $.728$ & $.903$ \\
&  &  &  & $.5$ & $.001$ & $.007$ & $.002$ & $.048$ & $.048$ & $.047$ & $.947
$ & $.947$ & $.945$ \\
$100$ & $16$ & $1$ & $.5$ & $.5$ & $-.001$ & $-.036$ & $-.003$ & $.025$ & $%
.026$ & $.024$ & $.945$ & $.738$ & $.926$ \\
&  &  &  & $.5$ & $.001$ & $.010$ & $.001$ & $.048$ & $.048$ & $.047$ & $.947
$ & $.945$ & $.946$ \\
$100$ & $16$ & $2$ & $.5$ & $.5$ & $-.001$ & $-.027$ & $.017$ & $.022$ & $%
.023$ & $.022$ & $.945$ & $.787$ & $.840$ \\
&  &  &  & $.5$ & $.001$ & $.010$ & $-.006$ & $.048$ & $.048$ & $.047$ & $%
.947$ & $.944$ & $.944$ \\
$100$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.032$ & $-.003$ & $.020$ & $%
.020$ & $.019$ & $.944$ & $.646$ & $.929$ \\
&  &  &  & $.5$ & $.000$ & $.008$ & $.001$ & $.037$ & $.038$ & $.037$ & $.950
$ & $.945$ & $.952$ \\
$100$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.031$ & $.001$ & $.019$ & $.020
$ & $.019$ & $.946$ & $.655$ & $.929$ \\
&  &  &  & $.5$ & $.000$ & $.009$ & $.000$ & $.037$ & $.038$ & $.037$ & $.950
$ & $.941$ & $.952$ \\
$100$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.026$ & $.010$ & $.018$ & $.018
$ & $.018$ & $.944$ & $.702$ & $.887$ \\
&  &  &  & $.5$ & $.000$ & $.009$ & $-.003$ & $.038$ & $.038$ & $.037$ & $%
.951$ & $.941$ & $.953$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}




\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.5$ & $.9$ & $-.141$ & --- & $-.546$ & $.267$ & --- & $%
.153$ & $.817$ & $.819$ & $.018$ \\
&  &  &  & $.1$ & $-.009$ & --- & $-.030$ & $.233$ & --- & $.192$ & $.973$ &
$.976$ & $.826$ \\
$100$ & $2$ & $1$ & $.5$ & $.9$ & $-.106$ & --- & $-.488$ & $.268$ & --- & $%
.152$ & $.836$ & $.824$ & $.042$ \\
&  &  &  & $.1$ & $.003$ & --- & $.022$ & $.236$ & --- & $.197$ & $.974$ & $%
.976$ & $.833$ \\
$100$ & $2$ & $2$ & $.5$ & $.9$ & $-.019$ & --- & $-.318$ & $.270$ & --- & $%
.148$ & $.880$ & $.882$ & $.257$ \\
&  &  &  & $.1$ & $.001$ & --- & $.046$ & $.249$ & --- & $.213$ & $.970$ & $%
.957$ & $.825$ \\
$100$ & $4$ & $0$ & $.5$ & $.9$ & $-.081$ & $-.544$ & $-.292$ & $.126$ & $%
.238$ & $.072$ & $.846$ & $.349$ & $.007$ \\
&  &  &  & $.1$ & $-.007$ & $-.026$ & $-.017$ & $.122$ & $.115$ & $.113$ & $%
.972$ & $.943$ & $.901$ \\
$100$ & $4$ & $1$ & $.5$ & $.9$ & $-.047$ & $-.469$ & $-.236$ & $.126$ & $%
.235$ & $.072$ & $.878$ & $.432$ & $.045$ \\
&  &  &  & $.1$ & $.001$ & $.037$ & $.017$ & $.124$ & $.115$ & $.115$ & $.974
$ & $.937$ & $.903$ \\
$100$ & $4$ & $2$ & $.5$ & $.9$ & $.008$ & $-.156$ & $-.099$ & $.126$ & $.140
$ & $.067$ & $.924$ & $.762$ & $.568$ \\
&  &  &  & $.1$ & $-.006$ & $.032$ & $.018$ & $.129$ & $.123$ & $.121$ & $%
.967$ & $.938$ & $.906$ \\
$100$ & $6$ & $0$ & $.5$ & $.9$ & $-.051$ & $-.367$ & $-.201$ & $.086$ & $%
.133$ & $.050$ & $.861$ & $.191$ & $.006$ \\
&  &  &  & $.1$ & $-.001$ & $-.012$ & $-.007$ & $.090$ & $.089$ & $.086$ & $%
.971$ & $.943$ & $.918$ \\
$100$ & $6$ & $1$ & $.5$ & $.9$ & $-.022$ & $-.304$ & $-.149$ & $.085$ & $%
.127$ & $.049$ & $.901$ & $.293$ & $.070$ \\
&  &  &  & $.1$ & $.003$ & $.033$ & $.016$ & $.091$ & $.088$ & $.087$ & $.968
$ & $.932$ & $.918$ \\
$100$ & $6$ & $2$ & $.5$ & $.9$ & $.009$ & $-.109$ & $-.037$ & $.082$ & $.074
$ & $.044$ & $.942$ & $.680$ & $.802$ \\
&  &  &  & $.1$ & $-.002$ & $.028$ & $.009$ & $.094$ & $.090$ & $.090$ & $%
.960$ & $.934$ & $.926$ \\
$100$ & $8$ & $0$ & $.5$ & $.9$ & $-.033$ & $-.267$ & $-.153$ & $.066$ & $%
.089$ & $.039$ & $.879$ & $.120$ & $.009$ \\
&  &  &  & $.1$ & $-.001$ & $-.007$ & $-.004$ & $.075$ & $.075$ & $.073$ & $%
.968$ & $.941$ & $.921$ \\
$100$ & $8$ & $1$ & $.5$ & $.9$ & $-.009$ & $-.221$ & $-.105$ & $.066$ & $%
.085$ & $.038$ & $.914$ & $.207$ & $.118$ \\
&  &  &  & $.1$ & $.001$ & $.025$ & $.012$ & $.075$ & $.074$ & $.073$ & $.966
$ & $.930$ & $.925$ \\
$100$ & $8$ & $2$ & $.5$ & $.9$ & $.008$ & $-.084$ & $-.012$ & $.060$ & $.050
$ & $.033$ & $.949$ & $.600$ & $.888$ \\
&  &  &  & $.1$ & $-.002$ & $.022$ & $.003$ & $.077$ & $.075$ & $.075$ & $%
.951$ & $.933$ & $.932$ \\
$100$ & $16$ & $0$ & $.5$ & $.9$ & $-.003$ & $-.127$ & $-.071$ & $.038$ & $%
.034$ & $.022$ & $.926$ & $.025$ & $.046$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.047$ & $.048$ & $.047$ & $.966
$ & $.949$ & $.938$ \\
$100$ & $16$ & $1$ & $.5$ & $.9$ & $.002$ & $-.110$ & $-.042$ & $.037$ & $%
.033$ & $.021$ & $.943$ & $.054$ & $.345$ \\
&  &  &  & $.1$ & $.000$ & $.013$ & $.005$ & $.048$ & $.048$ & $.047$ & $.957
$ & $.941$ & $.941$ \\
$100$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.053$ & $.008$ & $.023$ & $.022
$ & $.017$ & $.955$ & $.307$ & $.869$ \\
&  &  &  & $.1$ & $.000$ & $.013$ & $-.002$ & $.048$ & $.048$ & $.047$ & $%
.947$ & $.941$ & $.943$ \\
$100$ & $24$ & $0$ & $.5$ & $.9$ & $.002$ & $-.085$ & $-.041$ & $.027$ & $%
.021$ & $.016$ & $.951$ & $.009$ & $.172$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.001$ & $.037$ & $.038$ & $.037$ & $.955
$ & $.950$ & $.946$ \\
$100$ & $24$ & $1$ & $.5$ & $.9$ & $.001$ & $-.076$ & $-.023$ & $.022$ & $%
.020$ & $.015$ & $.955$ & $.018$ & $.545$ \\
&  &  &  & $.1$ & $.000$ & $.009$ & $.003$ & $.037$ & $.038$ & $.037$ & $.950
$ & $.943$ & $.947$ \\
$100$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.043$ & $.007$ & $.014$ & $.015
$ & $.012$ & $.948$ & $.136$ & $.841$ \\
&  &  &  & $.1$ & $.000$ & $.009$ & $-.001$ & $.037$ & $.038$ & $.037$ & $%
.950$ & $.941$ & $.951$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.5$ & $.99$ & $-.143$ & --- & $-.504$ & $.267$ & --- & $%
.153$ & $.816$ & $.812$ & $.041$ \\
&  &  &  & $.01$ & $-.003$ & --- & $-.005$ & $.232$ & --- & $.195$ & $.975$
& $.979$ & $.833$ \\
$100$ & $2$ & $1$ & $.5$ & $.99$ & $-.140$ & --- & $-.498$ & $.267$ & --- & $%
.153$ & $.817$ & $.811$ & $.043$ \\
&  &  &  & $.01$ & $.001$ & --- & $.010$ & $.232$ & --- & $.195$ & $.975$ & $%
.980$ & $.832$ \\
$100$ & $2$ & $2$ & $.5$ & $.99$ & $-.127$ & --- & $-.477$ & $.268$ & --- & $%
.153$ & $.825$ & $.818$ & $.057$ \\
&  &  &  & $.01$ & $.005$ & --- & $.025$ & $.234$ & --- & $.197$ & $.974$ & $%
.976$ & $.831$ \\
$100$ & $4$ & $0$ & $.5$ & $.99$ & $-.087$ & $-.673$ & $-.259$ & $.122$ & $%
.258$ & $.072$ & $.837$ & $.242$ & $.023$ \\
&  &  &  & $.01$ & $-.004$ & $-.006$ & $-.005$ & $.121$ & $.115$ & $.113$ & $%
.973$ & $.952$ & $.909$ \\
$100$ & $4$ & $1$ & $.5$ & $.99$ & $-.083$ & $-.658$ & $-.252$ & $.122$ & $%
.259$ & $.072$ & $.844$ & $.256$ & $.028$ \\
&  &  &  & $.01$ & $-.001$ & $.021$ & $.006$ & $.121$ & $.115$ & $.113$ & $%
.974$ & $.945$ & $.908$ \\
$100$ & $4$ & $2$ & $.5$ & $.99$ & $-.070$ & $-.620$ & $-.232$ & $.122$ & $%
.259$ & $.072$ & $.857$ & $.283$ & $.050$ \\
&  &  &  & $.01$ & $.002$ & $.045$ & $.015$ & $.122$ & $.115$ & $.114$ & $%
.974$ & $.930$ & $.904$ \\
$100$ & $6$ & $0$ & $.5$ & $.99$ & $-.062$ & $-.493$ & $-.175$ & $.081$ & $%
.152$ & $.049$ & $.839$ & $.060$ & $.019$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.089$ & $.089$ & $.085$ & $%
.970$ & $.945$ & $.922$ \\
$100$ & $6$ & $1$ & $.5$ & $.99$ & $-.057$ & $-.477$ & $-.168$ & $.081$ & $%
.152$ & $.049$ & $.847$ & $.070$ & $.026$ \\
&  &  &  & $.01$ & $.003$ & $.022$ & $.008$ & $.089$ & $.089$ & $.085$ & $%
.969$ & $.941$ & $.922$ \\
$100$ & $6$ & $2$ & $.5$ & $.99$ & $-.045$ & $-.435$ & $-.148$ & $.080$ & $%
.149$ & $.048$ & $.864$ & $.101$ & $.062$ \\
&  &  &  & $.01$ & $.004$ & $.042$ & $.015$ & $.090$ & $.088$ & $.086$ & $%
.970$ & $.923$ & $.921$ \\
$100$ & $8$ & $0$ & $.5$ & $.99$ & $-.047$ & $-.380$ & $-.133$ & $.061$ & $%
.104$ & $.037$ & $.843$ & $.014$ & $.019$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.074$ & $.076$ & $.072$ & $%
.969$ & $.941$ & $.926$ \\
$100$ & $8$ & $1$ & $.5$ & $.99$ & $-.042$ & $-.366$ & $-.126$ & $.061$ & $%
.104$ & $.037$ & $.853$ & $.019$ & $.030$ \\
&  &  &  & $.01$ & $.002$ & $.018$ & $.006$ & $.074$ & $.075$ & $.072$ & $%
.971$ & $.935$ & $.927$ \\
$100$ & $8$ & $2$ & $.5$ & $.99$ & $-.030$ & $-.326$ & $-.106$ & $.060$ & $%
.101$ & $.037$ & $.878$ & $.037$ & $.088$ \\
&  &  &  & $.01$ & $.003$ & $.033$ & $.011$ & $.074$ & $.074$ & $.072$ & $%
.971$ & $.923$ & $.927$ \\
$100$ & $16$ & $0$ & $.5$ & $.99$ & $-.024$ & $-.202$ & $-.068$ & $.031$ & $%
.041$ & $.020$ & $.851$ & $.000$ & $.019$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.047$ & $.049$ & $.046$ & $%
.974$ & $.950$ & $.941$ \\
$100$ & $16$ & $1$ & $.5$ & $.99$ & $-.019$ & $-.192$ & $-.061$ & $.031$ & $%
.041$ & $.020$ & $.869$ & $.000$ & $.043$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.003$ & $.047$ & $.048$ & $.047$ & $%
.974$ & $.945$ & $.941$ \\
$100$ & $16$ & $2$ & $.5$ & $.99$ & $-.009$ & $-.159$ & $-.044$ & $.030$ & $%
.038$ & $.019$ & $.902$ & $.001$ & $.212$ \\
&  &  &  & $.01$ & $.001$ & $.017$ & $.005$ & $.047$ & $.048$ & $.047$ & $%
.972$ & $.937$ & $.943$ \\
$100$ & $24$ & $0$ & $.5$ & $.99$ & $-.016$ & $-.138$ & $-.047$ & $.021$ & $%
.023$ & $.013$ & $.858$ & $.000$ & $.017$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.037$ & $.038$ & $.037$ & $%
.972$ & $.951$ & $.948$ \\
$100$ & $24$ & $1$ & $.5$ & $.99$ & $-.012$ & $-.129$ & $-.040$ & $.021$ & $%
.023$ & $.013$ & $.880$ & $.000$ & $.054$ \\
&  &  &  & $.01$ & $.001$ & $.007$ & $.002$ & $.037$ & $.038$ & $.037$ & $%
.972$ & $.946$ & $.949$ \\
$100$ & $24$ & $2$ & $.5$ & $.99$ & $-.003$ & $-.103$ & $-.024$ & $.020$ & $%
.021$ & $.012$ & $.919$ & $.000$ & $.349$ \\
&  &  &  & $.01$ & $.001$ & $.011$ & $.003$ & $.037$ & $.038$ & $.037$ & $%
.970$ & $.939$ & $.949$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.99$ & $.5$ & $.006$ & --- & $.392$ & $.088$ & --- & $%
.091$ & $.948$ & $.946$ & $.747$ \\
&  &  &  & $.5$ & $.000$ & --- & $.094$ & $.288$ & --- & $.371$ & $.940$ & $%
.939$ & $.841$ \\
$100$ & $2$ & $1$ & $.99$ & $.5$ & $-.082$ & --- & $-.648$ & $.268$ & --- & $%
.152$ & $.848$ & $.884$ & $.001$ \\
&  &  &  & $.5$ & $-.002$ & --- & $.003$ & $.277$ & --- & $.217$ & $.973$ & $%
.964$ & $.830$ \\
$100$ & $2$ & $2$ & $.99$ & $.5$ & $.003$ & --- & $.444$ & $.078$ & --- & $%
.086$ & $.942$ & $.942$ & $.521$ \\
&  &  &  & $.5$ & $-.003$ & --- & $-.117$ & $.287$ & --- & $.396$ & $.938$ &
$.938$ & $.798$ \\
$100$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $-.010$ & $.201$ & $.042$ & $.040
$ & $.044$ & $.941$ & $.935$ & $.005$ \\
&  &  &  & $.5$ & $-.004$ & $-.005$ & $.007$ & $.127$ & $.127$ & $.141$ & $%
.941$ & $.942$ & $.913$ \\
$100$ & $4$ & $1$ & $.99$ & $.5$ & $.015$ & $-.084$ & $-.222$ & $.139$ & $%
.112$ & $.068$ & $.932$ & $.857$ & $.059$ \\
&  &  &  & $.5$ & $-.007$ & $.016$ & $.047$ & $.132$ & $.128$ & $.124$ & $%
.958$ & $.944$ & $.876$ \\
$100$ & $4$ & $2$ & $.99$ & $.5$ & $-.001$ & $-.009$ & $.227$ & $.038$ & $%
.037$ & $.041$ & $.941$ & $.937$ & $.000$ \\
&  &  &  & $.5$ & $-.004$ & $.000$ & $-.124$ & $.128$ & $.128$ & $.147$ & $%
.944$ & $.944$ & $.809$ \\
$100$ & $6$ & $0$ & $.99$ & $.5$ & $-.001$ & $-.013$ & $.126$ & $.031$ & $%
.031$ & $.033$ & $.943$ & $.925$ & $.026$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.013$ & $.084$ & $.084$ & $.086$ & $%
.941$ & $.941$ & $.934$ \\
$100$ & $6$ & $1$ & $.99$ & $.5$ & $.001$ & $-.059$ & $-.095$ & $.072$ & $%
.065$ & $.049$ & $.956$ & $.846$ & $.419$ \\
&  &  &  & $.5$ & $.000$ & $.023$ & $.036$ & $.088$ & $.087$ & $.085$ & $.945
$ & $.936$ & $.895$ \\
$100$ & $6$ & $2$ & $.99$ & $.5$ & $-.001$ & $-.011$ & $.145$ & $.028$ & $%
.028$ & $.030$ & $.945$ & $.928$ & $.002$ \\
&  &  &  & $.5$ & $.000$ & $.007$ & $-.095$ & $.086$ & $.086$ & $.090$ & $%
.943$ & $.942$ & $.777$ \\
$100$ & $8$ & $0$ & $.99$ & $.5$ & $-.001$ & $-.015$ & $.088$ & $.027$ & $%
.027$ & $.027$ & $.942$ & $.910$ & $.085$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.021$ & $.063$ & $.064$ & $.062$ & $%
.940$ & $.940$ & $.929$ \\
$100$ & $8$ & $1$ & $.99$ & $.5$ & $-.001$ & $-.048$ & $-.046$ & $.048$ & $%
.048$ & $.040$ & $.944$ & $.829$ & $.722$ \\
&  &  &  & $.5$ & $.000$ & $.023$ & $.022$ & $.067$ & $.067$ & $.065$ & $.942
$ & $.931$ & $.918$ \\
$100$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $-.012$ & $.103$ & $.024$ & $.025
$ & $.026$ & $.942$ & $.917$ & $.016$ \\
&  &  &  & $.5$ & $.000$ & $.008$ & $-.075$ & $.065$ & $.066$ & $.065$ & $%
.943$ & $.942$ & $.763$ \\
$100$ & $16$ & $0$ & $.99$ & $.5$ & $-.001$ & $-.019$ & $.035$ & $.019$ & $%
.019$ & $.019$ & $.944$ & $.834$ & $.485$ \\
&  &  &  & $.5$ & $.000$ & $.010$ & $-.019$ & $.033$ & $.033$ & $.031$ & $%
.945$ & $.936$ & $.913$ \\
$100$ & $16$ & $1$ & $.99$ & $.5$ & $-.002$ & $-.034$ & $-.003$ & $.025$ & $%
.026$ & $.024$ & $.944$ & $.743$ & $.924$ \\
&  &  &  & $.5$ & $.001$ & $.023$ & $.002$ & $.035$ & $.036$ & $.034$ & $.943
$ & $.900$ & $.943$ \\
$100$ & $16$ & $2$ & $.99$ & $.5$ & $-.001$ & $-.016$ & $.042$ & $.017$ & $%
.018$ & $.018$ & $.945$ & $.849$ & $.299$ \\
&  &  &  & $.5$ & $.000$ & $.013$ & $-.035$ & $.034$ & $.035$ & $.033$ & $%
.948$ & $.927$ & $.815$ \\
$100$ & $24$ & $0$ & $.99$ & $.5$ & $-.001$ & $-.020$ & $.020$ & $.015$ & $%
.015$ & $.015$ & $.950$ & $.760$ & $.700$ \\
&  &  &  & $.5$ & $.000$ & $.013$ & $-.013$ & $.023$ & $.024$ & $.023$ & $%
.944$ & $.913$ & $.908$ \\
$100$ & $24$ & $1$ & $.99$ & $.5$ & $-.001$ & $-.029$ & $.003$ & $.018$ & $%
.019$ & $.018$ & $.948$ & $.671$ & $.935$ \\
&  &  &  & $.5$ & $.001$ & $.023$ & $-.002$ & $.025$ & $.026$ & $.025$ & $%
.945$ & $.855$ & $.946$ \\
$100$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $-.018$ & $.024$ & $.015$ & $%
.015$ & $.015$ & $.946$ & $.777$ & $.574$ \\
&  &  &  & $.5$ & $.000$ & $.016$ & $-.021$ & $.025$ & $.025$ & $.024$ & $%
.942$ & $.898$ & $.854$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.99$ & $.9$ & $-.045$ & --- & $-.382$ & $.266$ & --- & $%
.151$ & $.870$ & $.915$ & $.145$ \\
&  &  &  & $.1$ & $-.005$ & --- & $-.022$ & $.284$ & --- & $.240$ & $.970$ &
$.955$ & $.828$ \\
$100$ & $2$ & $1$ & $.99$ & $.9$ & $-.138$ & --- & $-.541$ & $.268$ & --- & $%
.153$ & $.819$ & $.814$ & $.020$ \\
&  &  &  & $.1$ & $-.002$ & --- & $.001$ & $.270$ & --- & $.222$ & $.974$ & $%
.978$ & $.829$ \\
$100$ & $2$ & $2$ & $.99$ & $.9$ & $-.004$ & --- & $-.278$ & $.270$ & --- & $%
.147$ & $.888$ & $.925$ & $.346$ \\
&  &  &  & $.1$ & $-.002$ & --- & $.015$ & $.289$ & --- & $.250$ & $.970$ & $%
.950$ & $.831$ \\
$100$ & $4$ & $0$ & $.99$ & $.9$ & $-.008$ & $-.119$ & $-.155$ & $.126$ & $%
.122$ & $.070$ & $.910$ & $.808$ & $.268$ \\
&  &  &  & $.1$ & $-.005$ & $-.009$ & $-.011$ & $.127$ & $.123$ & $.121$ & $%
.965$ & $.943$ & $.903$ \\
$100$ & $4$ & $1$ & $.99$ & $.9$ & $-.077$ & $-.539$ & $-.285$ & $.126$ & $%
.238$ & $.072$ & $.852$ & $.352$ & $.009$ \\
&  &  &  & $.1$ & $.000$ & $.028$ & $.013$ & $.124$ & $.122$ & $.117$ & $.972
$ & $.942$ & $.897$ \\
$100$ & $4$ & $2$ & $.99$ & $.9$ & $.012$ & $-.074$ & $-.078$ & $.125$ & $%
.097$ & $.066$ & $.928$ & $.857$ & $.685$ \\
&  &  &  & $.1$ & $-.006$ & $.008$ & $.008$ & $.130$ & $.125$ & $.124$ & $%
.963$ & $.944$ & $.907$ \\
$100$ & $6$ & $0$ & $.99$ & $.9$ & $.002$ & $-.101$ & $-.086$ & $.086$ & $%
.072$ & $.047$ & $.928$ & $.700$ & $.421$ \\
&  &  &  & $.1$ & $.000$ & $-.003$ & $-.003$ & $.085$ & $.083$ & $.083$ & $%
.959$ & $.940$ & $.919$ \\
$100$ & $6$ & $1$ & $.99$ & $.9$ & $-.046$ & $-.354$ & $-.193$ & $.085$ & $%
.131$ & $.050$ & $.865$ & $.201$ & $.010$ \\
&  &  &  & $.1$ & $.005$ & $.039$ & $.021$ & $.084$ & $.086$ & $.082$ & $.968
$ & $.920$ & $.900$ \\
$100$ & $6$ & $2$ & $.99$ & $.9$ & $.010$ & $-.063$ & $-.024$ & $.081$ & $%
.056$ & $.043$ & $.942$ & $.786$ & $.868$ \\
&  &  &  & $.1$ & $-.002$ & $.016$ & $.006$ & $.088$ & $.084$ & $.084$ & $%
.953$ & $.936$ & $.925$ \\
$100$ & $8$ & $0$ & $.99$ & $.9$ & $.004$ & $-.087$ & $-.055$ & $.066$ & $%
.050$ & $.036$ & $.933$ & $.581$ & $.544$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.001$ & $.064$ & $.064$ & $.063$ & $%
.956$ & $.940$ & $.925$ \\
$100$ & $8$ & $1$ & $.99$ & $.9$ & $-.027$ & $-.251$ & $-.143$ & $.066$ & $%
.087$ & $.039$ & $.888$ & $.142$ & $.018$ \\
&  &  &  & $.1$ & $.004$ & $.038$ & $.022$ & $.064$ & $.067$ & $.063$ & $.967
$ & $.908$ & $.897$ \\
$100$ & $8$ & $2$ & $.99$ & $.9$ & $.007$ & $-.054$ & $-.003$ & $.057$ & $%
.040$ & $.032$ & $.951$ & $.718$ & $.908$ \\
&  &  &  & $.1$ & $-.002$ & $.017$ & $.000$ & $.066$ & $.064$ & $.063$ & $%
.949$ & $.934$ & $.933$ \\
$100$ & $16$ & $0$ & $.99$ & $.9$ & $.001$ & $-.060$ & $-.016$ & $.030$ & $%
.023$ & $.019$ & $.953$ & $.256$ & $.799$ \\
&  &  &  & $.1$ & $.000$ & $.004$ & $.001$ & $.031$ & $.033$ & $.031$ & $.947
$ & $.942$ & $.941$ \\
$100$ & $16$ & $1$ & $.99$ & $.9$ & $-.001$ & $-.115$ & $-.060$ & $.038$ & $%
.033$ & $.021$ & $.933$ & $.041$ & $.110$ \\
&  &  &  & $.1$ & $.000$ & $.033$ & $.017$ & $.033$ & $.035$ & $.033$ & $.962
$ & $.842$ & $.890$ \\
$100$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.040$ & $.013$ & $.022$ & $%
.018$ & $.017$ & $.950$ & $.420$ & $.806$ \\
&  &  &  & $.1$ & $.000$ & $.020$ & $-.007$ & $.033$ & $.034$ & $.032$ & $%
.944$ & $.899$ & $.941$ \\
$100$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.049$ & $-.007$ & $.017$ & $%
.015$ & $.014$ & $.951$ & $.101$ & $.870$ \\
&  &  &  & $.1$ & $.000$ & $.008$ & $.001$ & $.021$ & $.023$ & $.021$ & $.940
$ & $.931$ & $.942$ \\
$100$ & $24$ & $1$ & $.99$ & $.9$ & $.001$ & $-.075$ & $-.031$ & $.025$ & $%
.019$ & $.015$ & $.951$ & $.017$ & $.364$ \\
&  &  &  & $.1$ & $.000$ & $.030$ & $.012$ & $.023$ & $.025$ & $.022$ & $.951
$ & $.764$ & $.900$ \\
$100$ & $24$ & $2$ & $.99$ & $.9$ & $-.001$ & $-.034$ & $.012$ & $.014$ & $%
.013$ & $.012$ & $.948$ & $.217$ & $.740$ \\
&  &  &  & $.1$ & $.000$ & $.022$ & $-.008$ & $.023$ & $.023$ & $.022$ & $%
.941$ & $.841$ & $.935$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$100$ & $2$ & $0$ & $.99$ & $.99$ & $-.142$ & --- & $-.503$ & $.267$ & --- &
$.153$ & $.815$ & $.812$ & $.042$ \\
&  &  &  & $.01$ & $-.003$ & --- & $-.005$ & $.269$ & --- & $.226$ & $.973$
& $.980$ & $.831$ \\
$100$ & $2$ & $1$ & $.99$ & $.99$ & $-.142$ & --- & $-.502$ & $.267$ & --- &
$.153$ & $.814$ & $.816$ & $.042$ \\
&  &  &  & $.01$ & $-.002$ & --- & $.000$ & $.269$ & --- & $.226$ & $.973$ &
$.981$ & $.832$ \\
$100$ & $2$ & $2$ & $.99$ & $.99$ & $-.131$ & --- & $-.485$ & $.268$ & --- &
$.153$ & $.821$ & $.829$ & $.052$ \\
&  &  &  & $.01$ & $.000$ & --- & $.005$ & $.271$ & --- & $.227$ & $.974$ & $%
.976$ & $.830$ \\
$100$ & $4$ & $0$ & $.99$ & $.99$ & $-.086$ & $-.660$ & $-.257$ & $.122$ & $%
.260$ & $.072$ & $.838$ & $.252$ & $.023$ \\
&  &  &  & $.01$ & $-.005$ & $-.007$ & $-.006$ & $.123$ & $.125$ & $.116$ & $%
.972$ & $.946$ & $.905$ \\
$100$ & $4$ & $1$ & $.99$ & $.99$ & $-.086$ & $-.662$ & $-.257$ & $.122$ & $%
.260$ & $.072$ & $.838$ & $.249$ & $.024$ \\
&  &  &  & $.01$ & $-.003$ & $.006$ & $-.001$ & $.123$ & $.125$ & $.116$ & $%
.972$ & $.946$ & $.904$ \\
$100$ & $4$ & $2$ & $.99$ & $.99$ & $-.075$ & $-.481$ & $-.240$ & $.122$ & $%
.244$ & $.072$ & $.850$ & $.405$ & $.040$ \\
&  &  &  & $.01$ & $-.002$ & $.012$ & $.004$ & $.124$ & $.121$ & $.117$ & $%
.971$ & $.945$ & $.905$ \\
$100$ & $6$ & $0$ & $.99$ & $.99$ & $-.061$ & $-.482$ & $-.174$ & $.081$ & $%
.154$ & $.049$ & $.838$ & $.071$ & $.019$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.083$ & $.091$ & $.080$ & $%
.969$ & $.945$ & $.918$ \\
$100$ & $6$ & $1$ & $.99$ & $.99$ & $-.061$ & $-.481$ & $-.174$ & $.081$ & $%
.152$ & $.049$ & $.842$ & $.066$ & $.021$ \\
&  &  &  & $.01$ & $.002$ & $.013$ & $.005$ & $.083$ & $.090$ & $.080$ & $%
.970$ & $.942$ & $.917$ \\
$100$ & $6$ & $2$ & $.99$ & $.99$ & $-.050$ & $-.351$ & $-.157$ & $.080$ & $%
.137$ & $.049$ & $.858$ & $.177$ & $.043$ \\
&  &  &  & $.01$ & $.003$ & $.020$ & $.009$ & $.083$ & $.085$ & $.080$ & $%
.970$ & $.939$ & $.917$ \\
$100$ & $8$ & $0$ & $.99$ & $.99$ & $-.047$ & $-.371$ & $-.132$ & $.061$ & $%
.105$ & $.037$ & $.844$ & $.017$ & $.021$ \\
&  &  &  & $.01$ & $-.001$ & $-.002$ & $-.001$ & $.062$ & $.070$ & $.061$ & $%
.972$ & $.945$ & $.924$ \\
$100$ & $8$ & $1$ & $.99$ & $.99$ & $-.046$ & $-.371$ & $-.131$ & $.061$ & $%
.106$ & $.037$ & $.845$ & $.018$ & $.022$ \\
&  &  &  & $.01$ & $.001$ & $.013$ & $.004$ & $.062$ & $.070$ & $.061$ & $%
.970$ & $.942$ & $.923$ \\
$100$ & $8$ & $2$ & $.99$ & $.99$ & $-.035$ & $-.271$ & $-.115$ & $.061$ & $%
.094$ & $.037$ & $.868$ & $.084$ & $.057$ \\
&  &  &  & $.01$ & $.002$ & $.020$ & $.008$ & $.063$ & $.066$ & $.061$ & $%
.969$ & $.933$ & $.923$ \\
$100$ & $16$ & $0$ & $.99$ & $.99$ & $-.023$ & $-.196$ & $-.067$ & $.031$ & $%
.041$ & $.020$ & $.854$ & $.000$ & $.022$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.031$ & $.038$ & $.032$ & $%
.972$ & $.946$ & $.934$ \\
$100$ & $16$ & $1$ & $.99$ & $.99$ & $-.022$ & $-.196$ & $-.066$ & $.031$ & $%
.041$ & $.020$ & $.855$ & $.000$ & $.025$ \\
&  &  &  & $.01$ & $.001$ & $.014$ & $.004$ & $.031$ & $.038$ & $.032$ & $%
.971$ & $.932$ & $.930$ \\
$100$ & $16$ & $2$ & $.99$ & $.99$ & $-.014$ & $-.145$ & $-.051$ & $.031$ & $%
.036$ & $.019$ & $.888$ & $.002$ & $.113$ \\
&  &  &  & $.01$ & $.002$ & $.021$ & $.007$ & $.032$ & $.035$ & $.031$ & $%
.969$ & $.906$ & $.929$ \\
$100$ & $24$ & $0$ & $.99$ & $.99$ & $-.015$ & $-.134$ & $-.045$ & $.021$ & $%
.023$ & $.013$ & $.861$ & $.000$ & $.021$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.021$ & $.026$ & $.022$ & $%
.970$ & $.943$ & $.933$ \\
$100$ & $24$ & $1$ & $.99$ & $.99$ & $-.014$ & $-.133$ & $-.045$ & $.021$ & $%
.023$ & $.013$ & $.864$ & $.000$ & $.027$ \\
&  &  &  & $.01$ & $.002$ & $.014$ & $.005$ & $.021$ & $.026$ & $.022$ & $%
.969$ & $.911$ & $.929$ \\
$100$ & $24$ & $2$ & $.99$ & $.99$ & $-.007$ & $-.099$ & $-.031$ & $.021$ & $%
.020$ & $.013$ & $.903$ & $.000$ & $.179$ \\
&  &  &  & $.01$ & $.001$ & $.021$ & $.007$ & $.022$ & $.025$ & $.021$ & $%
.969$ & $.857$ & $.928$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.5$ & $.5$ & $-.068$ & --- & $-.662$ & $.200$ & --- & $%
.095$ & $.860$ & $.920$ & $.000$ \\
&  &  &  & $.5$ & $-.018$ & --- & $-.167$ & $.159$ & --- & $.120$ & $.965$ &
$.949$ & $.484$ \\
$250$ & $2$ & $1$ & $.5$ & $.5$ & $.025$ & --- & $-.417$ & $.204$ & --- & $%
.091$ & $.918$ & $.937$ & $.000$ \\
&  &  &  & $.5$ & $-.002$ & --- & $.006$ & $.160$ & --- & $.130$ & $.968$ & $%
.951$ & $.834$ \\
$250$ & $2$ & $2$ & $.5$ & $.5$ & $.009$ & --- & $.076$ & $.104$ & --- & $%
.073$ & $.955$ & $.944$ & $.803$ \\
&  &  &  & $.5$ & $-.004$ & --- & $-.022$ & $.160$ & --- & $.164$ & $.946$ &
$.945$ & $.837$ \\
$250$ & $4$ & $0$ & $.5$ & $.5$ & $.014$ & $-.038$ & $-.246$ & $.104$ & $.080
$ & $.043$ & $.945$ & $.912$ & $.000$ \\
&  &  &  & $.5$ & $.001$ & $-.005$ & $-.030$ & $.081$ & $.079$ & $.077$ & $%
.956$ & $.947$ & $.869$ \\
$250$ & $4$ & $1$ & $.5$ & $.5$ & $.004$ & $-.028$ & $-.125$ & $.072$ & $.067
$ & $.042$ & $.957$ & $.926$ & $.093$ \\
&  &  &  & $.5$ & $-.002$ & $.003$ & $.015$ & $.080$ & $.079$ & $.077$ & $%
.950$ & $.948$ & $.899$ \\
$250$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $-.010$ & $.077$ & $.039$ & $.039$
& $.035$ & $.951$ & $.942$ & $.330$ \\
&  &  &  & $.5$ & $-.001$ & $.003$ & $-.030$ & $.081$ & $.080$ & $.081$ & $%
.947$ & $.948$ & $.891$ \\
$250$ & $6$ & $0$ & $.5$ & $.5$ & $.000$ & $-.028$ & $-.115$ & $.048$ & $.046
$ & $.032$ & $.958$ & $.906$ & $.028$ \\
&  &  &  & $.5$ & $.001$ & $.000$ & $-.001$ & $.058$ & $.058$ & $.058$ & $%
.944$ & $.944$ & $.913$ \\
$250$ & $6$ & $1$ & $.5$ & $.5$ & $-.001$ & $-.024$ & $-.052$ & $.039$ & $%
.042$ & $.031$ & $.950$ & $.911$ & $.489$ \\
&  &  &  & $.5$ & $.001$ & $.005$ & $.010$ & $.058$ & $.058$ & $.057$ & $.946
$ & $.945$ & $.920$ \\
$250$ & $6$ & $2$ & $.5$ & $.5$ & $-.001$ & $-.011$ & $.055$ & $.027$ & $.028
$ & $.026$ & $.948$ & $.930$ & $.334$ \\
&  &  &  & $.5$ & $.001$ & $.005$ & $-.021$ & $.058$ & $.059$ & $.058$ & $%
.947$ & $.945$ & $.913$ \\
$250$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.023$ & $-.062$ & $.032$ & $.033
$ & $.026$ & $.947$ & $.894$ & $.261$ \\
&  &  &  & $.5$ & $.001$ & $.002$ & $.004$ & $.047$ & $.047$ & $.047$ & $.947
$ & $.947$ & $.926$ \\
$250$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $-.020$ & $-.025$ & $.028$ & $.031
$ & $.025$ & $.946$ & $.901$ & $.756$ \\
&  &  &  & $.5$ & $.001$ & $.005$ & $.006$ & $.047$ & $.047$ & $.047$ & $.946
$ & $.945$ & $.931$ \\
$250$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $-.011$ & $.041$ & $.022$ & $.023$
& $.021$ & $.947$ & $.921$ & $.401$ \\
&  &  &  & $.5$ & $.001$ & $.005$ & $-.015$ & $.047$ & $.048$ & $.047$ & $%
.948$ & $.945$ & $.925$ \\
$250$ & $16$ & $0$ & $.5$ & $.5$ & $-.001$ & $-.016$ & $-.011$ & $.017$ & $%
.018$ & $.016$ & $.944$ & $.854$ & $.865$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.001$ & $.030$ & $.030$ & $.030$ & $.947
$ & $.947$ & $.940$ \\
$250$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $-.015$ & $-.002$ & $.016$ & $%
.018$ & $.016$ & $.942$ & $.859$ & $.921$ \\
&  &  &  & $.5$ & $.000$ & $.004$ & $.000$ & $.030$ & $.031$ & $.030$ & $.947
$ & $.945$ & $.942$ \\
$250$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $-.011$ & $.017$ & $.014$ & $.015
$ & $.014$ & $.945$ & $.884$ & $.696$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.007$ & $.031$ & $.031$ & $.030$ & $%
.947$ & $.946$ & $.939$ \\
$250$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.014$ & $-.002$ & $.012$ & $%
.013$ & $.012$ & $.947$ & $.821$ & $.929$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $.001$ & $.024$ & $.024$ & $.024$ & $.950
$ & $.948$ & $.948$ \\
$250$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.013$ & $.001$ & $.012$ & $.013
$ & $.012$ & $.948$ & $.824$ & $.930$ \\
&  &  &  & $.5$ & $.000$ & $.004$ & $.000$ & $.024$ & $.024$ & $.024$ & $.951
$ & $.948$ & $.949$ \\
$250$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.011$ & $.010$ & $.011$ & $.012
$ & $.011$ & $.946$ & $.849$ & $.814$ \\
&  &  &  & $.5$ & $.000$ & $.004$ & $-.003$ & $.024$ & $.024$ & $.024$ & $%
.952$ & $.948$ & $.947$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.5$ & $.9$ & $-.119$ & --- & $-.546$ & $.199$ & --- & $%
.095$ & $.829$ & $.829$ & $.000$ \\
&  &  &  & $.1$ & $-.007$ & --- & $-.029$ & $.148$ & --- & $.121$ & $.976$ &
$.973$ & $.822$ \\
$250$ & $2$ & $1$ & $.5$ & $.9$ & $-.084$ & --- & $-.489$ & $.200$ & --- & $%
.095$ & $.853$ & $.836$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & --- & $.023$ & $.151$ & --- & $.124$ & $.977$ & $%
.975$ & $.824$ \\
$250$ & $2$ & $2$ & $.5$ & $.9$ & $-.003$ & --- & $-.319$ & $.203$ & --- & $%
.092$ & $.901$ & $.918$ & $.032$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.047$ & $.161$ & --- & $.135$ & $.972$ & $%
.956$ & $.803$ \\
$250$ & $4$ & $0$ & $.5$ & $.9$ & $-.064$ & $-.406$ & $-.290$ & $.096$ & $%
.210$ & $.046$ & $.846$ & $.487$ & $.000$ \\
&  &  &  & $.1$ & $-.004$ & $-.018$ & $-.014$ & $.077$ & $.074$ & $.072$ & $%
.972$ & $.941$ & $.894$ \\
$250$ & $4$ & $1$ & $.5$ & $.9$ & $-.031$ & $-.319$ & $-.234$ & $.096$ & $%
.196$ & $.046$ & $.886$ & $.588$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.027$ & $.020$ & $.079$ & $.074$ & $.073$ & $.971
$ & $.935$ & $.891$ \\
$250$ & $4$ & $2$ & $.5$ & $.9$ & $.013$ & $-.071$ & $-.097$ & $.097$ & $.094
$ & $.042$ & $.935$ & $.868$ & $.261$ \\
&  &  &  & $.1$ & $-.004$ & $.015$ & $.021$ & $.083$ & $.080$ & $.077$ & $%
.962$ & $.942$ & $.896$ \\
$250$ & $6$ & $0$ & $.5$ & $.9$ & $-.037$ & $-.253$ & $-.199$ & $.065$ & $%
.109$ & $.032$ & $.865$ & $.354$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.008$ & $-.007$ & $.057$ & $.056$ & $.055$ & $%
.971$ & $.943$ & $.914$ \\
$250$ & $6$ & $1$ & $.5$ & $.9$ & $-.010$ & $-.195$ & $-.147$ & $.067$ & $%
.100$ & $.031$ & $.908$ & $.491$ & $.001$ \\
&  &  &  & $.1$ & $.002$ & $.021$ & $.016$ & $.058$ & $.056$ & $.055$ & $.971
$ & $.930$ & $.905$ \\
$250$ & $6$ & $2$ & $.5$ & $.9$ & $.007$ & $-.051$ & $-.035$ & $.059$ & $.051
$ & $.028$ & $.948$ & $.817$ & $.653$ \\
&  &  &  & $.1$ & $-.001$ & $.013$ & $.009$ & $.060$ & $.058$ & $.057$ & $%
.956$ & $.942$ & $.920$ \\
$250$ & $8$ & $0$ & $.5$ & $.9$ & $-.021$ & $-.176$ & $-.151$ & $.050$ & $%
.070$ & $.025$ & $.890$ & $.284$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.005$ & $-.004$ & $.047$ & $.046$ & $.046$ & $%
.974$ & $.946$ & $.924$ \\
$250$ & $8$ & $1$ & $.5$ & $.9$ & $.000$ & $-.136$ & $-.103$ & $.051$ & $.065
$ & $.024$ & $.928$ & $.421$ & $.003$ \\
&  &  &  & $.1$ & $.000$ & $.016$ & $.012$ & $.047$ & $.046$ & $.046$ & $.968
$ & $.934$ & $.919$ \\
$250$ & $8$ & $2$ & $.5$ & $.9$ & $.004$ & $-.039$ & $-.010$ & $.038$ & $.034
$ & $.021$ & $.958$ & $.779$ & $.866$ \\
&  &  &  & $.1$ & $-.001$ & $.011$ & $.003$ & $.048$ & $.047$ & $.047$ & $%
.951$ & $.941$ & $.933$ \\
$250$ & $16$ & $0$ & $.5$ & $.9$ & $.001$ & $-.074$ & $-.070$ & $.029$ & $%
.025$ & $.014$ & $.941$ & $.161$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.001$ & $.030$ & $.030$ & $.030$ & $%
.964$ & $.946$ & $.934$ \\
$250$ & $16$ & $1$ & $.5$ & $.9$ & $.003$ & $-.064$ & $-.041$ & $.026$ & $%
.024$ & $.013$ & $.952$ & $.229$ & $.054$ \\
&  &  &  & $.1$ & $-.001$ & $.007$ & $.004$ & $.030$ & $.030$ & $.030$ & $%
.952$ & $.941$ & $.934$ \\
$250$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.025$ & $.008$ & $.014$ & $.014
$ & $.011$ & $.950$ & $.598$ & $.810$ \\
&  &  &  & $.1$ & $-.001$ & $.005$ & $-.003$ & $.030$ & $.030$ & $.030$ & $%
.946$ & $.944$ & $.942$ \\
$250$ & $24$ & $0$ & $.5$ & $.9$ & $.002$ & $-.047$ & $-.040$ & $.019$ & $%
.015$ & $.010$ & $.957$ & $.099$ & $.007$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.001$ & $.024$ & $.024$ & $.024$ & $.954
$ & $.950$ & $.942$ \\
$250$ & $24$ & $1$ & $.5$ & $.9$ & $.001$ & $-.043$ & $-.023$ & $.014$ & $%
.014$ & $.009$ & $.961$ & $.130$ & $.213$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $.003$ & $.024$ & $.024$ & $.024$ & $.951
$ & $.946$ & $.944$ \\
$250$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.021$ & $.008$ & $.009$ & $.010
$ & $.008$ & $.946$ & $.420$ & $.741$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $-.002$ & $.024$ & $.024$ & $.024$ & $%
.950$ & $.948$ & $.948$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.5$ & $.99$ & $-.122$ & --- & $-.505$ & $.199$ & --- & $%
.095$ & $.828$ & $.819$ & $.000$ \\
&  &  &  & $.01$ & $-.002$ & --- & $-.004$ & $.148$ & --- & $.123$ & $.977$
& $.985$ & $.834$ \\
$250$ & $2$ & $1$ & $.5$ & $.99$ & $-.118$ & --- & $-.498$ & $.199$ & --- & $%
.095$ & $.832$ & $.822$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.012$ & $.148$ & --- & $.123$ & $.977$ & $%
.982$ & $.833$ \\
$250$ & $2$ & $2$ & $.5$ & $.99$ & $-.105$ & --- & $-.478$ & $.199$ & --- & $%
.095$ & $.841$ & $.826$ & $.001$ \\
&  &  &  & $.01$ & $.004$ & --- & $.026$ & $.150$ & --- & $.125$ & $.977$ & $%
.974$ & $.823$ \\
$250$ & $4$ & $0$ & $.5$ & $.99$ & $-.070$ & $-.673$ & $-.257$ & $.093$ & $%
.261$ & $.046$ & $.834$ & $.248$ & $.000$ \\
&  &  &  & $.01$ & $-.002$ & $-.004$ & $-.003$ & $.077$ & $.074$ & $.072$ & $%
.973$ & $.951$ & $.903$ \\
$250$ & $4$ & $1$ & $.5$ & $.99$ & $-.066$ & $-.642$ & $-.250$ & $.094$ & $%
.261$ & $.046$ & $.841$ & $.270$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.023$ & $.009$ & $.077$ & $.073$ & $.072$ & $%
.974$ & $.941$ & $.902$ \\
$250$ & $4$ & $2$ & $.5$ & $.99$ & $-.053$ & $-.568$ & $-.230$ & $.093$ & $%
.252$ & $.046$ & $.858$ & $.333$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & $.044$ & $.018$ & $.078$ & $.073$ & $.072$ & $%
.974$ & $.914$ & $.894$ \\
$250$ & $6$ & $0$ & $.5$ & $.99$ & $-.048$ & $-.487$ & $-.172$ & $.062$ & $%
.149$ & $.031$ & $.836$ & $.057$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.057$ & $.056$ & $.054$ & $%
.973$ & $.950$ & $.919$ \\
$250$ & $6$ & $1$ & $.5$ & $.99$ & $-.043$ & $-.459$ & $-.166$ & $.062$ & $%
.147$ & $.031$ & $.846$ & $.075$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & $.020$ & $.008$ & $.057$ & $.056$ & $.054$ & $%
.973$ & $.936$ & $.915$ \\
$250$ & $6$ & $2$ & $.5$ & $.99$ & $-.031$ & $-.385$ & $-.146$ & $.062$ & $%
.140$ & $.031$ & $.871$ & $.137$ & $.000$ \\
&  &  &  & $.01$ & $.004$ & $.036$ & $.015$ & $.057$ & $.055$ & $.055$ & $%
.974$ & $.903$ & $.908$ \\
$250$ & $8$ & $0$ & $.5$ & $.99$ & $-.035$ & $-.379$ & $-.131$ & $.046$ & $%
.105$ & $.024$ & $.852$ & $.016$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $.000$ & $.046$ & $.047$ & $.045$ & $%
.976$ & $.949$ & $.930$ \\
$250$ & $8$ & $1$ & $.5$ & $.99$ & $-.031$ & $-.354$ & $-.124$ & $.046$ & $%
.104$ & $.024$ & $.863$ & $.023$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.017$ & $.006$ & $.046$ & $.046$ & $.045$ & $%
.975$ & $.936$ & $.928$ \\
$250$ & $8$ & $2$ & $.5$ & $.99$ & $-.019$ & $-.282$ & $-.104$ & $.046$ & $%
.096$ & $.024$ & $.890$ & $.067$ & $.001$ \\
&  &  &  & $.01$ & $.002$ & $.029$ & $.011$ & $.047$ & $.046$ & $.045$ & $%
.974$ & $.908$ & $.922$ \\
$250$ & $16$ & $0$ & $.5$ & $.99$ & $-.018$ & $-.199$ & $-.068$ & $.023$ & $%
.041$ & $.012$ & $.852$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & $-.001$ & $-.001$ & $.030$ & $.031$ & $.030$ & $%
.975$ & $.947$ & $.938$ \\
$250$ & $16$ & $1$ & $.5$ & $.99$ & $-.014$ & $-.182$ & $-.061$ & $.023$ & $%
.040$ & $.012$ & $.874$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.009$ & $.003$ & $.030$ & $.031$ & $.030$ & $%
.974$ & $.937$ & $.937$ \\
$250$ & $16$ & $2$ & $.5$ & $.99$ & $-.004$ & $-.129$ & $-.043$ & $.023$ & $%
.033$ & $.012$ & $.915$ & $.003$ & $.014$ \\
&  &  &  & $.01$ & $.000$ & $.013$ & $.004$ & $.030$ & $.030$ & $.030$ & $%
.971$ & $.925$ & $.937$ \\
$250$ & $24$ & $0$ & $.5$ & $.99$ & $-.012$ & $-.134$ & $-.046$ & $.016$ & $%
.023$ & $.009$ & $.860$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.023$ & $.024$ & $.023$ & $%
.975$ & $.949$ & $.944$ \\
$250$ & $24$ & $1$ & $.5$ & $.99$ & $-.008$ & $-.121$ & $-.039$ & $.016$ & $%
.023$ & $.008$ & $.887$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.006$ & $.002$ & $.024$ & $.024$ & $.023$ & $%
.974$ & $.940$ & $.944$ \\
$250$ & $24$ & $2$ & $.5$ & $.99$ & $.000$ & $-.081$ & $-.023$ & $.016$ & $%
.018$ & $.008$ & $.929$ & $.000$ & $.067$ \\
&  &  &  & $.01$ & $.000$ & $.009$ & $.003$ & $.024$ & $.024$ & $.023$ & $%
.967$ & $.934$ & $.945$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.99$ & $.5$ & $.002$ & --- & $.388$ & $.055$ & --- & $%
.057$ & $.950$ & $.947$ & $.457$ \\
&  &  &  & $.5$ & $-.001$ & --- & $.095$ & $.179$ & --- & $.232$ & $.945$ & $%
.945$ & $.824$ \\
$250$ & $2$ & $1$ & $.99$ & $.5$ & $-.062$ & --- & $-.649$ & $.200$ & --- & $%
.095$ & $.868$ & $.923$ & $.000$ \\
&  &  &  & $.5$ & $-.001$ & --- & $.004$ & $.175$ & --- & $.136$ & $.976$ & $%
.961$ & $.836$ \\
$250$ & $2$ & $2$ & $.99$ & $.5$ & $.001$ & --- & $.442$ & $.049$ & --- & $%
.053$ & $.950$ & $.948$ & $.252$ \\
&  &  &  & $.5$ & $-.002$ & --- & $-.119$ & $.180$ & --- & $.248$ & $.945$ &
$.945$ & $.764$ \\
$250$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $-.004$ & $.201$ & $.026$ & $.025
$ & $.028$ & $.950$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $-.001$ & $-.001$ & $.012$ & $.081$ & $.081$ & $.090$ & $%
.945$ & $.945$ & $.908$ \\
$250$ & $4$ & $1$ & $.99$ & $.5$ & $.013$ & $-.034$ & $-.220$ & $.100$ & $%
.075$ & $.043$ & $.948$ & $.915$ & $.000$ \\
&  &  &  & $.5$ & $-.004$ & $.007$ & $.050$ & $.085$ & $.083$ & $.079$ & $%
.953$ & $.942$ & $.827$ \\
$250$ & $4$ & $2$ & $.99$ & $.5$ & $-.001$ & $-.004$ & $.228$ & $.023$ & $%
.023$ & $.025$ & $.952$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $-.001$ & $.001$ & $-.121$ & $.082$ & $.082$ & $.094$ & $%
.944$ & $.945$ & $.656$ \\
$250$ & $6$ & $0$ & $.99$ & $.5$ & $.000$ & $-.005$ & $.127$ & $.020$ & $.020
$ & $.021$ & $.949$ & $.943$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.013$ & $.052$ & $.052$ & $.054$ & $%
.948$ & $.948$ & $.931$ \\
$250$ & $6$ & $1$ & $.99$ & $.5$ & $-.001$ & $-.025$ & $-.094$ & $.044$ & $%
.043$ & $.032$ & $.953$ & $.905$ & $.101$ \\
&  &  &  & $.5$ & $.001$ & $.010$ & $.036$ & $.055$ & $.055$ & $.053$ & $.950
$ & $.947$ & $.859$ \\
$250$ & $6$ & $2$ & $.99$ & $.5$ & $-.001$ & $-.004$ & $.145$ & $.018$ & $%
.018$ & $.019$ & $.949$ & $.939$ & $.000$ \\
&  &  &  & $.5$ & $.001$ & $.003$ & $-.095$ & $.053$ & $.053$ & $.056$ & $%
.950$ & $.950$ & $.563$ \\
$250$ & $8$ & $0$ & $.99$ & $.5$ & $.000$ & $-.006$ & $.089$ & $.017$ & $.017
$ & $.017$ & $.946$ & $.936$ & $.001$ \\
&  &  &  & $.5$ & $.001$ & $.002$ & $-.020$ & $.040$ & $.040$ & $.039$ & $%
.943$ & $.941$ & $.909$ \\
$250$ & $8$ & $1$ & $.99$ & $.5$ & $-.001$ & $-.021$ & $-.045$ & $.030$ & $%
.031$ & $.025$ & $.947$ & $.899$ & $.483$ \\
&  &  &  & $.5$ & $.001$ & $.011$ & $.022$ & $.042$ & $.043$ & $.041$ & $.945
$ & $.938$ & $.891$ \\
$250$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $-.005$ & $.103$ & $.015$ & $.015
$ & $.016$ & $.946$ & $.936$ & $.000$ \\
&  &  &  & $.5$ & $.001$ & $.004$ & $-.074$ & $.041$ & $.041$ & $.041$ & $%
.942$ & $.942$ & $.533$ \\
$250$ & $16$ & $0$ & $.99$ & $.5$ & $.000$ & $-.008$ & $.035$ & $.012$ & $%
.012$ & $.012$ & $.949$ & $.904$ & $.129$ \\
&  &  &  & $.5$ & $.000$ & $.004$ & $-.019$ & $.021$ & $.021$ & $.020$ & $%
.948$ & $.944$ & $.844$ \\
$250$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $-.014$ & $-.002$ & $.016$ & $%
.017$ & $.015$ & $.945$ & $.862$ & $.924$ \\
&  &  &  & $.5$ & $.000$ & $.009$ & $.001$ & $.023$ & $.023$ & $.022$ & $.947
$ & $.928$ & $.943$ \\
$250$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $-.006$ & $.042$ & $.011$ & $%
.011$ & $.011$ & $.950$ & $.913$ & $.027$ \\
&  &  &  & $.5$ & $.000$ & $.005$ & $-.036$ & $.022$ & $.022$ & $.021$ & $%
.949$ & $.942$ & $.606$ \\
$250$ & $24$ & $0$ & $.99$ & $.5$ & $.000$ & $-.008$ & $.021$ & $.010$ & $%
.010$ & $.010$ & $.946$ & $.871$ & $.402$ \\
&  &  &  & $.5$ & $.000$ & $.006$ & $-.013$ & $.015$ & $.015$ & $.014$ & $%
.950$ & $.932$ & $.843$ \\
$250$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $-.012$ & $.003$ & $.012$ & $%
.012$ & $.012$ & $.949$ & $.826$ & $.922$ \\
&  &  &  & $.5$ & $.000$ & $.010$ & $-.003$ & $.016$ & $.016$ & $.016$ & $%
.949$ & $.909$ & $.945$ \\
$250$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $-.007$ & $.025$ & $.009$ & $%
.009$ & $.009$ & $.947$ & $.882$ & $.213$ \\
&  &  &  & $.5$ & $.000$ & $.007$ & $-.022$ & $.015$ & $.016$ & $.015$ & $%
.950$ & $.930$ & $.701$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.99$ & $.9$ & $-.028$ & --- & $-.384$ & $.202$ & --- & $%
.094$ & $.885$ & $.935$ & $.005$ \\
&  &  &  & $.1$ & $-.003$ & --- & $-.021$ & $.178$ & --- & $.150$ & $.973$ &
$.955$ & $.826$ \\
$250$ & $2$ & $1$ & $.99$ & $.9$ & $-.116$ & --- & $-.541$ & $.199$ & --- & $%
.095$ & $.833$ & $.825$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.002$ & $.170$ & --- & $.139$ & $.977$ & $%
.983$ & $.833$ \\
$250$ & $2$ & $2$ & $.99$ & $.9$ & $.010$ & --- & $-.279$ & $.204$ & --- & $%
.091$ & $.909$ & $.944$ & $.070$ \\
&  &  &  & $.1$ & $-.003$ & --- & $.016$ & $.182$ & --- & $.157$ & $.972$ & $%
.949$ & $.834$ \\
$250$ & $4$ & $0$ & $.99$ & $.9$ & $.003$ & $-.051$ & $-.153$ & $.098$ & $%
.082$ & $.044$ & $.926$ & $.892$ & $.029$ \\
&  &  &  & $.1$ & $-.001$ & $-.003$ & $-.008$ & $.082$ & $.080$ & $.077$ & $%
.965$ & $.946$ & $.898$ \\
$250$ & $4$ & $1$ & $.99$ & $.9$ & $-.059$ & $-.395$ & $-.283$ & $.096$ & $%
.208$ & $.046$ & $.852$ & $.496$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & $.023$ & $.016$ & $.080$ & $.077$ & $.075$ & $.973
$ & $.938$ & $.887$ \\
$250$ & $4$ & $2$ & $.99$ & $.9$ & $.014$ & $-.031$ & $-.076$ & $.095$ & $%
.063$ & $.042$ & $.940$ & $.912$ & $.443$ \\
&  &  &  & $.1$ & $-.003$ & $.004$ & $.011$ & $.083$ & $.081$ & $.079$ & $%
.958$ & $.945$ & $.899$ \\
$250$ & $6$ & $0$ & $.99$ & $.9$ & $.007$ & $-.045$ & $-.083$ & $.066$ & $%
.048$ & $.030$ & $.939$ & $.835$ & $.117$ \\
&  &  &  & $.1$ & $.001$ & $-.001$ & $-.002$ & $.053$ & $.052$ & $.051$ & $%
.960$ & $.948$ & $.921$ \\
$250$ & $6$ & $1$ & $.99$ & $.9$ & $-.032$ & $-.235$ & $-.190$ & $.066$ & $%
.105$ & $.032$ & $.872$ & $.388$ & $.000$ \\
&  &  &  & $.1$ & $.004$ & $.026$ & $.021$ & $.052$ & $.052$ & $.051$ & $.973
$ & $.925$ & $.888$ \\
$250$ & $6$ & $2$ & $.99$ & $.9$ & $.006$ & $-.027$ & $-.022$ & $.056$ & $%
.037$ & $.027$ & $.949$ & $.879$ & $.808$ \\
&  &  &  & $.1$ & $-.001$ & $.007$ & $.006$ & $.054$ & $.052$ & $.052$ & $%
.954$ & $.947$ & $.926$ \\
$250$ & $8$ & $0$ & $.99$ & $.9$ & $.007$ & $-.039$ & $-.052$ & $.049$ & $%
.033$ & $.023$ & $.950$ & $.783$ & $.240$ \\
&  &  &  & $.1$ & $.001$ & $.000$ & $.000$ & $.040$ & $.040$ & $.040$ & $.951
$ & $.942$ & $.921$ \\
$250$ & $8$ & $1$ & $.99$ & $.9$ & $-.016$ & $-.159$ & $-.141$ & $.051$ & $%
.067$ & $.025$ & $.898$ & $.330$ & $.000$ \\
&  &  &  & $.1$ & $.003$ & $.025$ & $.022$ & $.041$ & $.041$ & $.040$ & $.970
$ & $.902$ & $.866$ \\
$250$ & $8$ & $2$ & $.99$ & $.9$ & $.003$ & $-.023$ & $-.001$ & $.035$ & $%
.026$ & $.020$ & $.961$ & $.846$ & $.909$ \\
&  &  &  & $.1$ & $.000$ & $.008$ & $.001$ & $.042$ & $.041$ & $.040$ & $.946
$ & $.936$ & $.928$ \\
$250$ & $16$ & $0$ & $.99$ & $.9$ & $.000$ & $-.028$ & $-.015$ & $.018$ & $%
.015$ & $.012$ & $.959$ & $.565$ & $.671$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.001$ & $.020$ & $.020$ & $.020$ & $.946
$ & $.946$ & $.937$ \\
$250$ & $16$ & $1$ & $.99$ & $.9$ & $.003$ & $-.064$ & $-.059$ & $.028$ & $%
.024$ & $.013$ & $.944$ & $.217$ & $.003$ \\
&  &  &  & $.1$ & $-.001$ & $.019$ & $.017$ & $.021$ & $.022$ & $.021$ & $%
.964$ & $.861$ & $.834$ \\
$250$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.017$ & $.014$ & $.013$ & $%
.012$ & $.010$ & $.951$ & $.698$ & $.648$ \\
&  &  &  & $.1$ & $.000$ & $.009$ & $-.007$ & $.021$ & $.021$ & $.020$ & $%
.948$ & $.929$ & $.931$ \\
$250$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.023$ & $-.006$ & $.011$ & $%
.010$ & $.009$ & $.951$ & $.378$ & $.831$ \\
&  &  &  & $.1$ & $.000$ & $.004$ & $.001$ & $.013$ & $.014$ & $.013$ & $.950
$ & $.943$ & $.947$ \\
$250$ & $24$ & $1$ & $.99$ & $.9$ & $.001$ & $-.041$ & $-.030$ & $.016$ & $%
.014$ & $.009$ & $.960$ & $.148$ & $.062$ \\
&  &  &  & $.1$ & $.000$ & $.016$ & $.012$ & $.014$ & $.015$ & $.014$ & $.956
$ & $.814$ & $.847$ \\
$250$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $-.015$ & $.013$ & $.008$ & $%
.008$ & $.007$ & $.946$ & $.539$ & $.500$ \\
&  &  &  & $.1$ & $.000$ & $.010$ & $-.008$ & $.014$ & $.014$ & $.014$ & $%
.953$ & $.900$ & $.914$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$250$ & $2$ & $0$ & $.99$ & $.99$ & $-.120$ & --- & $-.503$ & $.199$ & --- &
$.095$ & $.830$ & $.830$ & $.000$ \\
&  &  &  & $.01$ & $-.002$ & --- & $-.004$ & $.170$ & --- & $.141$ & $.977$
& $.982$ & $.831$ \\
$250$ & $2$ & $1$ & $.99$ & $.99$ & $-.120$ & --- & $-.503$ & $.199$ & --- &
$.095$ & $.830$ & $.824$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & --- & $.001$ & $.170$ & --- & $.141$ & $.977$ &
$.982$ & $.834$ \\
$250$ & $2$ & $2$ & $.99$ & $.99$ & $-.109$ & --- & $-.485$ & $.199$ & --- &
$.095$ & $.837$ & $.859$ & $.001$ \\
&  &  &  & $.01$ & $.000$ & --- & $.006$ & $.171$ & --- & $.143$ & $.977$ & $%
.977$ & $.834$ \\
$250$ & $4$ & $0$ & $.99$ & $.99$ & $-.069$ & $-.622$ & $-.255$ & $.093$ & $%
.263$ & $.046$ & $.836$ & $.286$ & $.000$ \\
&  &  &  & $.01$ & $-.002$ & $-.004$ & $-.003$ & $.079$ & $.079$ & $.074$ & $%
.972$ & $.952$ & $.899$ \\
$250$ & $4$ & $1$ & $.99$ & $.99$ & $-.069$ & $-.628$ & $-.255$ & $.094$ & $%
.263$ & $.046$ & $.837$ & $.281$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.008$ & $.003$ & $.079$ & $.079$ & $.074$ & $%
.973$ & $.949$ & $.897$ \\
$250$ & $4$ & $2$ & $.99$ & $.99$ & $-.058$ & $-.320$ & $-.238$ & $.093$ & $%
.196$ & $.046$ & $.853$ & $.566$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.007$ & $.079$ & $.076$ & $.074$ & $%
.973$ & $.946$ & $.896$ \\
$250$ & $6$ & $0$ & $.99$ & $.99$ & $-.047$ & $-.449$ & $-.171$ & $.062$ & $%
.152$ & $.031$ & $.839$ & $.088$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.051$ & $.055$ & $.050$ & $%
.975$ & $.952$ & $.919$ \\
$250$ & $6$ & $1$ & $.99$ & $.99$ & $-.047$ & $-.454$ & $-.170$ & $.062$ & $%
.150$ & $.031$ & $.840$ & $.082$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.012$ & $.005$ & $.051$ & $.056$ & $.050$ & $%
.975$ & $.946$ & $.917$ \\
$250$ & $6$ & $2$ & $.99$ & $.99$ & $-.036$ & $-.239$ & $-.154$ & $.062$ & $%
.111$ & $.031$ & $.862$ & $.347$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.014$ & $.009$ & $.052$ & $.051$ & $.050$ & $%
.975$ & $.944$ & $.917$ \\
$250$ & $8$ & $0$ & $.99$ & $.99$ & $-.034$ & $-.350$ & $-.129$ & $.046$ & $%
.104$ & $.024$ & $.853$ & $.023$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.039$ & $.044$ & $.039$ & $%
.973$ & $.948$ & $.919$ \\
$250$ & $8$ & $1$ & $.99$ & $.99$ & $-.034$ & $-.351$ & $-.128$ & $.046$ & $%
.105$ & $.024$ & $.853$ & $.025$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.013$ & $.005$ & $.040$ & $.044$ & $.039$ & $%
.973$ & $.940$ & $.918$ \\
$250$ & $8$ & $2$ & $.99$ & $.99$ & $-.024$ & $-.185$ & $-.112$ & $.046$ & $%
.073$ & $.024$ & $.878$ & $.199$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.014$ & $.009$ & $.040$ & $.040$ & $.039$ & $%
.972$ & $.931$ & $.914$ \\
$250$ & $16$ & $0$ & $.99$ & $.99$ & $-.017$ & $-.183$ & $-.066$ & $.023$ & $%
.040$ & $.012$ & $.856$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.020$ & $.023$ & $.020$ & $%
.973$ & $.949$ & $.932$ \\
$250$ & $16$ & $1$ & $.99$ & $.99$ & $-.017$ & $-.182$ & $-.065$ & $.023$ & $%
.040$ & $.012$ & $.859$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.013$ & $.004$ & $.020$ & $.023$ & $.020$ & $%
.972$ & $.916$ & $.928$ \\
$250$ & $16$ & $2$ & $.99$ & $.99$ & $-.008$ & $-.099$ & $-.050$ & $.023$ & $%
.027$ & $.012$ & $.901$ & $.013$ & $.003$ \\
&  &  &  & $.01$ & $.001$ & $.014$ & $.007$ & $.020$ & $.021$ & $.020$ & $%
.971$ & $.897$ & $.918$ \\
$250$ & $24$ & $0$ & $.99$ & $.99$ & $-.011$ & $-.124$ & $-.045$ & $.016$ & $%
.023$ & $.009$ & $.868$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.013$ & $.016$ & $.013$ & $%
.976$ & $.950$ & $.937$ \\
$250$ & $24$ & $1$ & $.99$ & $.99$ & $-.010$ & $-.122$ & $-.044$ & $.016$ & $%
.022$ & $.009$ & $.870$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.013$ & $.005$ & $.013$ & $.016$ & $.013$ & $%
.975$ & $.873$ & $.920$ \\
$250$ & $24$ & $2$ & $.99$ & $.99$ & $-.003$ & $-.069$ & $-.030$ & $.016$ & $%
.015$ & $.008$ & $.915$ & $.000$ & $.011$ \\
&  &  &  & $.01$ & $.001$ & $.015$ & $.007$ & $.014$ & $.014$ & $.013$ & $%
.974$ & $.831$ & $.911$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.5$ & $.5$ & $-.052$ & --- & $-.662$ & $.166$ & --- & $%
.067$ & $.867$ & $.934$ & $.000$ \\
&  &  &  & $.5$ & $-.012$ & --- & $-.165$ & $.115$ & --- & $.085$ & $.965$ &
$.954$ & $.284$ \\
$500$ & $2$ & $1$ & $.5$ & $.5$ & $.031$ & --- & $-.417$ & $.170$ & --- & $%
.064$ & $.929$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.008$ & $.112$ & --- & $.091$ & $.967$ & $%
.954$ & $.831$ \\
$500$ & $2$ & $2$ & $.5$ & $.5$ & $.005$ & --- & $.075$ & $.070$ & --- & $%
.052$ & $.955$ & $.947$ & $.648$ \\
&  &  &  & $.5$ & $-.001$ & --- & $-.020$ & $.112$ & --- & $.114$ & $.952$ &
$.951$ & $.837$ \\
$500$ & $4$ & $0$ & $.5$ & $.5$ & $.011$ & $-.019$ & $-.245$ & $.080$ & $.058
$ & $.030$ & $.950$ & $.928$ & $.000$ \\
&  &  &  & $.5$ & $.002$ & $-.002$ & $-.029$ & $.057$ & $.056$ & $.054$ & $%
.955$ & $.949$ & $.847$ \\
$500$ & $4$ & $1$ & $.5$ & $.5$ & $.003$ & $-.014$ & $-.124$ & $.050$ & $.049
$ & $.030$ & $.958$ & $.936$ & $.006$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.016$ & $.056$ & $.056$ & $.054$ & $.950
$ & $.949$ & $.896$ \\
$500$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $-.005$ & $.078$ & $.028$ & $.028$
& $.025$ & $.945$ & $.945$ & $.085$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.030$ & $.057$ & $.057$ & $.057$ & $%
.949$ & $.949$ & $.872$ \\
$500$ & $6$ & $0$ & $.5$ & $.5$ & $.001$ & $-.014$ & $-.114$ & $.033$ & $.033
$ & $.022$ & $.953$ & $.927$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.002$ & $.040$ & $.040$ & $.041$ & $%
.947$ & $.946$ & $.919$ \\
$500$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.012$ & $-.051$ & $.027$ & $.030
$ & $.021$ & $.953$ & $.935$ & $.240$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.010$ & $.041$ & $.041$ & $.040$ & $.947
$ & $.945$ & $.918$ \\
$500$ & $6$ & $2$ & $.5$ & $.5$ & $.000$ & $-.005$ & $.056$ & $.019$ & $.020$
& $.018$ & $.951$ & $.940$ & $.081$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.022$ & $.041$ & $.041$ & $.041$ & $%
.946$ & $.946$ & $.895$ \\
$500$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.011$ & $-.061$ & $.022$ & $.024
$ & $.018$ & $.951$ & $.926$ & $.054$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.003$ & $.033$ & $.033$ & $.033$ & $.947
$ & $.947$ & $.929$ \\
$500$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $-.010$ & $-.025$ & $.020$ & $.022
$ & $.017$ & $.950$ & $.927$ & $.619$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.005$ & $.033$ & $.033$ & $.033$ & $.948
$ & $.947$ & $.932$ \\
$500$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $-.005$ & $.042$ & $.015$ & $.016$
& $.015$ & $.949$ & $.936$ & $.138$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.016$ & $.033$ & $.033$ & $.033$ & $%
.947$ & $.947$ & $.908$ \\
$500$ & $16$ & $0$ & $.5$ & $.5$ & $.000$ & $-.008$ & $-.010$ & $.012$ & $%
.013$ & $.011$ & $.949$ & $.905$ & $.818$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.002$ & $.021$ & $.021$ & $.021$ & $.950
$ & $.950$ & $.943$ \\
$500$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $-.008$ & $-.002$ & $.011$ & $%
.012$ & $.011$ & $.951$ & $.909$ & $.928$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.001$ & $.021$ & $.022$ & $.021$ & $.950
$ & $.949$ & $.944$ \\
$500$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $-.006$ & $.017$ & $.010$ & $.011
$ & $.010$ & $.950$ & $.920$ & $.503$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.006$ & $.022$ & $.022$ & $.021$ & $%
.950$ & $.949$ & $.935$ \\
$500$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.007$ & $-.002$ & $.009$ & $%
.009$ & $.009$ & $.946$ & $.883$ & $.922$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.000$ & $.017$ & $.017$ & $.017$ & $.948
$ & $.947$ & $.945$ \\
$500$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.007$ & $.001$ & $.009$ & $.009
$ & $.008$ & $.948$ & $.883$ & $.930$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.000$ & $.017$ & $.017$ & $.017$ & $.949
$ & $.947$ & $.945$ \\
$500$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.006$ & $.010$ & $.008$ & $.008
$ & $.008$ & $.948$ & $.897$ & $.704$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.004$ & $.017$ & $.017$ & $.017$ & $%
.949$ & $.948$ & $.941$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.5$ & $.9$ & $-.103$ & --- & $-.546$ & $.164$ & --- & $%
.067$ & $.829$ & $.834$ & $.000$ \\
&  &  &  & $.1$ & $-.005$ & --- & $-.027$ & $.105$ & --- & $.085$ & $.977$ &
$.966$ & $.813$ \\
$500$ & $2$ & $1$ & $.5$ & $.9$ & $-.066$ & --- & $-.489$ & $.165$ & --- & $%
.067$ & $.859$ & $.842$ & $.000$ \\
&  &  &  & $.1$ & $.004$ & --- & $.025$ & $.107$ & --- & $.087$ & $.978$ & $%
.968$ & $.817$ \\
$500$ & $2$ & $2$ & $.5$ & $.9$ & $.009$ & --- & $-.319$ & $.169$ & --- & $%
.065$ & $.912$ & $.936$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.049$ & $.113$ & --- & $.095$ & $.973$ & $%
.959$ & $.780$ \\
$500$ & $4$ & $0$ & $.5$ & $.9$ & $-.051$ & $-.283$ & $-.289$ & $.079$ & $%
.175$ & $.032$ & $.850$ & $.611$ & $.000$ \\
&  &  &  & $.1$ & $-.002$ & $-.012$ & $-.013$ & $.055$ & $.053$ & $.051$ & $%
.974$ & $.942$ & $.890$ \\
$500$ & $4$ & $1$ & $.5$ & $.9$ & $-.019$ & $-.205$ & $-.232$ & $.080$ & $%
.158$ & $.032$ & $.891$ & $.711$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & $.018$ & $.021$ & $.056$ & $.054$ & $.051$ & $.975
$ & $.940$ & $.884$ \\
$500$ & $4$ & $2$ & $.5$ & $.9$ & $.013$ & $-.036$ & $-.096$ & $.078$ & $.069
$ & $.030$ & $.942$ & $.908$ & $.060$ \\
&  &  &  & $.1$ & $-.003$ & $.008$ & $.021$ & $.059$ & $.057$ & $.054$ & $%
.964$ & $.947$ & $.887$ \\
$500$ & $6$ & $0$ & $.5$ & $.9$ & $-.028$ & $-.164$ & $-.198$ & $.053$ & $%
.086$ & $.023$ & $.875$ & $.547$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.005$ & $-.007$ & $.040$ & $.039$ & $.039$ & $%
.975$ & $.944$ & $.914$ \\
$500$ & $6$ & $1$ & $.5$ & $.9$ & $-.003$ & $-.118$ & $-.146$ & $.054$ & $%
.076$ & $.022$ & $.923$ & $.669$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.013$ & $.016$ & $.041$ & $.040$ & $.039$ & $.969
$ & $.938$ & $.899$ \\
$500$ & $6$ & $2$ & $.5$ & $.9$ & $.005$ & $-.026$ & $-.034$ & $.042$ & $.037
$ & $.020$ & $.958$ & $.888$ & $.475$ \\
&  &  &  & $.1$ & $-.001$ & $.007$ & $.009$ & $.042$ & $.041$ & $.040$ & $%
.952$ & $.943$ & $.921$ \\
$500$ & $8$ & $0$ & $.5$ & $.9$ & $-.015$ & $-.109$ & $-.150$ & $.041$ & $%
.055$ & $.017$ & $.897$ & $.499$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.003$ & $-.005$ & $.033$ & $.032$ & $.032$ & $%
.972$ & $.947$ & $.923$ \\
$500$ & $8$ & $1$ & $.5$ & $.9$ & $.004$ & $-.082$ & $-.103$ & $.042$ & $.049
$ & $.017$ & $.934$ & $.609$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.009$ & $.012$ & $.033$ & $.033$ & $.032$ & $%
.966$ & $.942$ & $.913$ \\
$500$ & $8$ & $2$ & $.5$ & $.9$ & $.002$ & $-.021$ & $-.010$ & $.026$ & $.024
$ & $.015$ & $.962$ & $.860$ & $.833$ \\
&  &  &  & $.1$ & $-.001$ & $.005$ & $.002$ & $.034$ & $.033$ & $.033$ & $%
.948$ & $.946$ & $.932$ \\
$500$ & $16$ & $0$ & $.5$ & $.9$ & $.002$ & $-.043$ & $-.070$ & $.024$ & $%
.019$ & $.010$ & $.946$ & $.391$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.021$ & $.021$ & $.021$ & $.962
$ & $.949$ & $.937$ \\
$500$ & $16$ & $1$ & $.5$ & $.9$ & $.001$ & $-.037$ & $-.041$ & $.018$ & $%
.018$ & $.009$ & $.959$ & $.448$ & $.002$ \\
&  &  &  & $.1$ & $.000$ & $.004$ & $.005$ & $.021$ & $.021$ & $.021$ & $.950
$ & $.947$ & $.933$ \\
$500$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.013$ & $.008$ & $.010$ & $.011
$ & $.008$ & $.948$ & $.760$ & $.720$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $-.002$ & $.021$ & $.021$ & $.021$ & $%
.950$ & $.948$ & $.941$ \\
$500$ & $24$ & $0$ & $.5$ & $.9$ & $.001$ & $-.027$ & $-.040$ & $.013$ & $%
.011$ & $.007$ & $.962$ & $.300$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.017$ & $.017$ & $.017$ & $.949
$ & $.948$ & $.940$ \\
$500$ & $24$ & $1$ & $.5$ & $.9$ & $.000$ & $-.025$ & $-.022$ & $.009$ & $%
.011$ & $.007$ & $.956$ & $.337$ & $.031$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.002$ & $.017$ & $.017$ & $.017$ & $.949
$ & $.945$ & $.941$ \\
$500$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.011$ & $.008$ & $.006$ & $.007
$ & $.005$ & $.949$ & $.643$ & $.584$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $-.002$ & $.017$ & $.017$ & $.017$ & $%
.949$ & $.947$ & $.943$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.5$ & $.99$ & $-.105$ & --- & $-.505$ & $.164$ & --- & $%
.067$ & $.828$ & $.814$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.002$ & $.105$ & --- & $.087$ & $.978$ &
$.984$ & $.837$ \\
$500$ & $2$ & $1$ & $.5$ & $.99$ & $-.100$ & --- & $-.498$ & $.164$ & --- & $%
.067$ & $.833$ & $.815$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & --- & $.013$ & $.105$ & --- & $.087$ & $.977$ & $%
.979$ & $.829$ \\
$500$ & $2$ & $2$ & $.5$ & $.99$ & $-.087$ & --- & $-.478$ & $.164$ & --- & $%
.067$ & $.842$ & $.823$ & $.000$ \\
&  &  &  & $.01$ & $.006$ & --- & $.028$ & $.106$ & --- & $.088$ & $.977$ & $%
.963$ & $.813$ \\
$500$ & $4$ & $0$ & $0.5$ & $.99$ & $-.058$ & $-.670$ & $-.255$ & $.077$ & $%
.259$ & $.032$ & $.837$ & $.251$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & $-.004$ & $-.002$ & $.054$ & $.052$ & $.050$ & $%
.976$ & $.955$ & $.907$ \\
$500$ & $4$ & $1$ & $.5$ & $.99$ & $-.053$ & $-.617$ & $-.248$ & $.077$ & $%
.259$ & $.032$ & $.841$ & $.290$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.022$ & $.009$ & $.055$ & $.051$ & $.050$ & $%
.977$ & $.937$ & $.904$ \\
$500$ & $4$ & $2$ & $.5$ & $.99$ & $-.041$ & $-.500$ & $-.228$ & $.077$ & $%
.243$ & $.032$ & $.861$ & $.389$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & $.039$ & $.019$ & $.055$ & $.051$ & $.051$ & $%
.975$ & $.896$ & $.888$ \\
$500$ & $6$ & $0$ & $0.5$ & $.99$ & $-.039$ & $-.481$ & $-.172$ & $.050$ & $%
.150$ & $.022$ & $.844$ & $.065$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.040$ & $.040$ & $.038$ & $%
.974$ & $.949$ & $.923$ \\
$500$ & $6$ & $1$ & $.5$ & $.99$ & $-.035$ & $-.433$ & $-.165$ & $.050$ & $%
.149$ & $.022$ & $.854$ & $.101$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.020$ & $.008$ & $.040$ & $.039$ & $.038$ & $%
.973$ & $.924$ & $.917$ \\
$500$ & $6$ & $2$ & $.5$ & $.99$ & $-.022$ & $-.320$ & $-.145$ & $.050$ & $%
.131$ & $.022$ & $.882$ & $.217$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.031$ & $.014$ & $.040$ & $.039$ & $.038$ & $%
.973$ & $.886$ & $.902$ \\
$500$ & $8$ & $0$ & $0.5$ & $.99$ & $-.030$ & $-.372$ & $-.130$ & $.038$ & $%
.104$ & $.017$ & $.850$ & $.017$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.032$ & $.033$ & $.032$ & $%
.974$ & $.952$ & $.930$ \\
$500$ & $8$ & $1$ & $.5$ & $.99$ & $-.025$ & $-.330$ & $-.123$ & $.037$ & $%
.101$ & $.017$ & $.863$ & $.032$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.016$ & $.006$ & $.033$ & $.032$ & $.032$ & $%
.975$ & $.930$ & $.926$ \\
$500$ & $8$ & $2$ & $.5$ & $.99$ & $-.013$ & $-.228$ & $-.104$ & $.038$ & $%
.083$ & $.016$ & $.893$ & $.121$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.023$ & $.011$ & $.033$ & $.032$ & $.032$ & $%
.974$ & $.898$ & $.919$ \\
$500$ & $16$ & $0$ & $0.5$ & $.99$ & $-.015$ & $-.193$ & $-.067$ & $.019$ & $%
.040$ & $.009$ & $.847$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.021$ & $.022$ & $.021$ & $%
.976$ & $.951$ & $.940$ \\
$500$ & $16$ & $1$ & $.5$ & $.99$ & $-.011$ & $-.166$ & $-.060$ & $.019$ & $%
.038$ & $.009$ & $.871$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.009$ & $.003$ & $.021$ & $.021$ & $.021$ & $%
.975$ & $.933$ & $.938$ \\
$500$ & $16$ & $2$ & $.5$ & $.99$ & $-.002$ & $-.098$ & $-.043$ & $.020$ & $%
.028$ & $.008$ & $.917$ & $.014$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.011$ & $.005$ & $.021$ & $.021$ & $.021$ & $%
.971$ & $.922$ & $.937$ \\
$500$ & $24$ & $0$ & $0.5$ & $.99$ & $-.010$ & $-.128$ & $-.046$ & $.013$ & $%
.023$ & $.006$ & $.862$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.017$ & $.017$ & $.017$ & $%
.976$ & $.950$ & $.942$ \\
$500$ & $24$ & $1$ & $.5$ & $.99$ & $-.005$ & $-.110$ & $-.039$ & $.013$ & $%
.022$ & $.006$ & $.897$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.002$ & $.017$ & $.017$ & $.017$ & $%
.975$ & $.939$ & $.942$ \\
$500$ & $24$ & $2$ & $.5$ & $.99$ & $.001$ & $-.060$ & $-.023$ & $.013$ & $%
.014$ & $.006$ & $.938$ & $.002$ & $.003$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.002$ & $.017$ & $.017$ & $.017$ & $%
.965$ & $.934$ & $.942$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.99$ & $.5$ & $.001$ & --- & $.387$ & $.039$ & --- & $%
.041$ & $.947$ & $.949$ & $.266$ \\
&  &  &  & $.5$ & $.001$ & --- & $.098$ & $.128$ & --- & $.165$ & $.949$ & $%
.948$ & $.794$ \\
$500$ & $2$ & $1$ & $.99$ & $.5$ & $-.044$ & --- & $-.649$ & $.166$ & --- & $%
.067$ & $.873$ & $.938$ & $.000$ \\
&  &  &  & $.5$ & $.002$ & --- & $.007$ & $.125$ & --- & $.097$ & $.977$ & $%
.958$ & $.831$ \\
$500$ & $2$ & $2$ & $.99$ & $.5$ & $.001$ & --- & $.442$ & $.034$ & --- & $%
.038$ & $.949$ & $.946$ & $.102$ \\
&  &  &  & $.5$ & $.001$ & --- & $-.117$ & $.127$ & --- & $.175$ & $.948$ & $%
.948$ & $.714$ \\
$500$ & $4$ & $0$ & $0.99$ & $.5$ & $.000$ & $-.002$ & $.201$ & $.019$ & $%
.018$ & $.020$ & $.950$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.013$ & $.057$ & $.057$ & $.063$ & $.951
$ & $.951$ & $.905$ \\
$500$ & $4$ & $1$ & $.99$ & $.5$ & $.009$ & $-.017$ & $-.219$ & $.073$ & $%
.054$ & $.031$ & $.952$ & $.931$ & $.000$ \\
&  &  &  & $.5$ & $-.002$ & $.004$ & $.051$ & $.060$ & $.058$ & $.055$ & $%
.953$ & $.949$ & $.763$ \\
$500$ & $4$ & $2$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.228$ & $.017$ & $.016
$ & $.018$ & $.947$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.120$ & $.058$ & $.057$ & $.067$ & $%
.947$ & $.949$ & $.454$ \\
$500$ & $6$ & $0$ & $0.99$ & $.5$ & $.000$ & $-.002$ & $.127$ & $.014$ & $%
.014$ & $.015$ & $.946$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.013$ & $.037$ & $.037$ & $.038$ & $%
.948$ & $.947$ & $.920$ \\
$500$ & $6$ & $1$ & $.99$ & $.5$ & $.000$ & $-.012$ & $-.093$ & $.031$ & $%
.031$ & $.022$ & $.951$ & $.933$ & $.007$ \\
&  &  &  & $.5$ & $.000$ & $.005$ & $.036$ & $.039$ & $.039$ & $.038$ & $.948
$ & $.945$ & $.792$ \\
$500$ & $6$ & $2$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.146$ & $.013$ & $.013
$ & $.013$ & $.948$ & $.944$ & $.000$ \\
&  &  &  & $.5$ & $.001$ & $.002$ & $-.095$ & $.038$ & $.038$ & $.040$ & $%
.948$ & $.948$ & $.292$ \\
$500$ & $8$ & $0$ & $0.99$ & $.5$ & $.000$ & $-.003$ & $.089$ & $.012$ & $%
.012$ & $.012$ & $.948$ & $.940$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.021$ & $.028$ & $.028$ & $.027$ & $%
.949$ & $.949$ & $.872$ \\
$500$ & $8$ & $1$ & $.99$ & $.5$ & $.000$ & $-.010$ & $-.044$ & $.021$ & $%
.022$ & $.018$ & $.949$ & $.928$ & $.233$ \\
&  &  &  & $.5$ & $.000$ & $.005$ & $.021$ & $.030$ & $.030$ & $.029$ & $.950
$ & $.947$ & $.861$ \\
$500$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.103$ & $.011$ & $.011
$ & $.011$ & $.949$ & $.942$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.075$ & $.029$ & $.029$ & $.029$ & $%
.947$ & $.947$ & $.241$ \\
$500$ & $16$ & $0$ & $0.99$ & $.5$ & $.000$ & $-.004$ & $.036$ & $.008$ & $%
.009$ & $.009$ & $.945$ & $.927$ & $.009$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.019$ & $.015$ & $.015$ & $.014$ & $%
.948$ & $.945$ & $.738$ \\
$500$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $-.007$ & $-.001$ & $.011$ & $%
.012$ & $.011$ & $.947$ & $.909$ & $.924$ \\
&  &  &  & $.5$ & $.000$ & $.005$ & $.001$ & $.016$ & $.016$ & $.015$ & $.946
$ & $.938$ & $.941$ \\
$500$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $-.003$ & $.042$ & $.008$ & $%
.008$ & $.008$ & $.951$ & $.929$ & $.001$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.035$ & $.016$ & $.016$ & $.015$ & $%
.946$ & $.942$ & $.338$ \\
$500$ & $24$ & $0$ & $0.99$ & $.5$ & $.000$ & $-.004$ & $.021$ & $.007$ & $%
.007$ & $.007$ & $.947$ & $.908$ & $.131$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.014$ & $.010$ & $.011$ & $.010$ & $%
.947$ & $.940$ & $.729$ \\
$500$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $-.006$ & $.004$ & $.008$ & $%
.009$ & $.008$ & $.949$ & $.888$ & $.907$ \\
&  &  &  & $.5$ & $.000$ & $.005$ & $-.003$ & $.011$ & $.012$ & $.011$ & $%
.949$ & $.930$ & $.939$ \\
$500$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $-.004$ & $.025$ & $.007$ & $%
.007$ & $.007$ & $.948$ & $.909$ & $.034$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.022$ & $.011$ & $.011$ & $.011$ & $%
.949$ & $.940$ & $.475$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.99$ & $.9$ & $-.016$ & --- & $-.383$ & $.168$ & --- & $%
.066$ & $.891$ & $.944$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & --- & $-.018$ & $.127$ & --- & $.106$ & $.975$ & $%
.954$ & $.826$ \\
$500$ & $2$ & $1$ & $.99$ & $.9$ & $-.099$ & --- & $-.541$ & $.164$ & --- & $%
.067$ & $.833$ & $.828$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & --- & $.005$ & $.122$ & --- & $.099$ & $.978$ & $%
.981$ & $.835$ \\
$500$ & $2$ & $2$ & $.99$ & $.9$ & $.020$ & --- & $-.279$ & $.170$ & --- & $%
.064$ & $.920$ & $.945$ & $.004$ \\
&  &  &  & $.1$ & $.000$ & --- & $.019$ & $.130$ & --- & $.111$ & $.969$ & $%
.952$ & $.830$ \\
$500$ & $4$ & $0$ & $0.99$ & $.9$ & $.008$ & $-.027$ & $-.152$ & $.081$ & $%
.060$ & $.031$ & $.932$ & $.917$ & $.001$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.006$ & $.057$ & $.056$ & $.054$ & $%
.968$ & $.951$ & $.906$ \\
$500$ & $4$ & $1$ & $.99$ & $.9$ & $-.046$ & $-.271$ & $-.281$ & $.079$ & $%
.172$ & $.032$ & $.854$ & $.625$ & $.000$ \\
&  &  &  & $.1$ & $.003$ & $.017$ & $.017$ & $.056$ & $.054$ & $.052$ & $.974
$ & $.943$ & $.882$ \\
$500$ & $4$ & $2$ & $.99$ & $.9$ & $.012$ & $-.015$ & $-.074$ & $.075$ & $%
.045$ & $.030$ & $.947$ & $.932$ & $.200$ \\
&  &  &  & $.1$ & $-.002$ & $.002$ & $.012$ & $.059$ & $.057$ & $.055$ & $%
.957$ & $.949$ & $.901$ \\
$500$ & $6$ & $0$ & $0.99$ & $.9$ & $.009$ & $-.023$ & $-.083$ & $.054$ & $%
.034$ & $.021$ & $.947$ & $.895$ & $.011$ \\
&  &  &  & $.1$ & $.001$ & $.000$ & $-.002$ & $.037$ & $.037$ & $.036$ & $%
.958$ & $.947$ & $.922$ \\
$500$ & $6$ & $1$ & $.99$ & $.9$ & $-.023$ & $-.149$ & $-.189$ & $.053$ & $%
.083$ & $.023$ & $.885$ & $.581$ & $.000$ \\
&  &  &  & $.1$ & $.003$ & $.017$ & $.021$ & $.037$ & $.037$ & $.036$ & $.971
$ & $.926$ & $.861$ \\
$500$ & $6$ & $2$ & $.99$ & $.9$ & $.003$ & $-.014$ & $-.021$ & $.038$ & $%
.026$ & $.019$ & $.960$ & $.913$ & $.736$ \\
&  &  &  & $.1$ & $.000$ & $.004$ & $.006$ & $.038$ & $.038$ & $.037$ & $.948
$ & $.946$ & $.921$ \\
$500$ & $8$ & $0$ & $0.99$ & $.9$ & $.005$ & $-.020$ & $-.052$ & $.037$ & $%
.024$ & $.016$ & $.954$ & $.862$ & $.051$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.001$ & $.028$ & $.028$ & $.028$ & $%
.952$ & $.949$ & $.927$ \\
$500$ & $8$ & $1$ & $.99$ & $.9$ & $-.010$ & $-.096$ & $-.140$ & $.041$ & $%
.051$ & $.017$ & $.907$ & $.547$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.015$ & $.021$ & $.029$ & $.029$ & $.028$ & $.973
$ & $.920$ & $.829$ \\
$500$ & $8$ & $2$ & $.99$ & $.9$ & $.001$ & $-.012$ & $-.001$ & $.024$ & $%
.018$ & $.014$ & $.960$ & $.900$ & $.912$ \\
&  &  &  & $.1$ & $-.001$ & $.004$ & $.000$ & $.029$ & $.028$ & $.028$ & $%
.948$ & $.944$ & $.932$ \\
$500$ & $16$ & $0$ & $0.99$ & $.9$ & $.000$ & $-.014$ & $-.014$ & $.013$ & $%
.011$ & $.009$ & $.956$ & $.748$ & $.485$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.001$ & $.014$ & $.014$ & $.014$ & $.947
$ & $.946$ & $.936$ \\
$500$ & $16$ & $1$ & $.99$ & $.9$ & $.002$ & $-.037$ & $-.059$ & $.022$ & $%
.017$ & $.009$ & $.951$ & $.455$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.011$ & $.017$ & $.015$ & $.015$ & $.015$ & $%
.960$ & $.892$ & $.738$ \\
$500$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.009$ & $.014$ & $.009$ & $%
.009$ & $.007$ & $.947$ & $.816$ & $.431$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $-.007$ & $.015$ & $.015$ & $.014$ & $%
.946$ & $.932$ & $.909$ \\
$500$ & $24$ & $0$ & $0.99$ & $.9$ & $.000$ & $-.012$ & $-.006$ & $.008$ & $%
.007$ & $.006$ & $.949$ & $.606$ & $.752$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.001$ & $.010$ & $.010$ & $.010$ & $.946
$ & $.940$ & $.940$ \\
$500$ & $24$ & $1$ & $.99$ & $.9$ & $.000$ & $-.023$ & $-.030$ & $.011$ & $%
.010$ & $.007$ & $.961$ & $.374$ & $.002$ \\
&  &  &  & $.1$ & $.000$ & $.009$ & $.012$ & $.010$ & $.010$ & $.010$ & $.946
$ & $.861$ & $.749$ \\
$500$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $-.008$ & $.013$ & $.006$ & $%
.006$ & $.005$ & $.949$ & $.724$ & $.246$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $-.008$ & $.010$ & $.010$ & $.010$ & $%
.948$ & $.917$ & $.860$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$500$ & $2$ & $0$ & $.99$ & $.99$ & $-.104$ & --- & $-.503$ & $.164$ & --- &
$.067$ & $.829$ & $.827$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $-.002$ & $.121$ & --- & $.100$ & $.978$ &
$.983$ & $.833$ \\
$500$ & $2$ & $1$ & $.99$ & $.99$ & $-.103$ & --- & $-.503$ & $.164$ & --- &
$.067$ & $.830$ & $.827$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & --- & $.004$ & $.121$ & --- & $.100$ & $.978$ & $%
.981$ & $.835$ \\
$500$ & $2$ & $2$ & $.99$ & $.99$ & $-.091$ & --- & $-.485$ & $.164$ & --- &
$.067$ & $.840$ & $.888$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & --- & $.009$ & $.122$ & --- & $.101$ & $.979$ & $%
.971$ & $.833$ \\
$500$ & $4$ & $0$ & $0.99$ & $.99$ & $-.057$ & $-.566$ & $-.254$ & $.077$ & $%
.259$ & $.032$ & $.840$ & $.331$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.003$ & $-.002$ & $.055$ & $.055$ & $.052$ & $%
.976$ & $.957$ & $.906$ \\
$500$ & $4$ & $1$ & $.99$ & $.99$ & $-.056$ & $-.575$ & $-.253$ & $.077$ & $%
.259$ & $.032$ & $.837$ & $.325$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.008$ & $.004$ & $.055$ & $.055$ & $.052$ & $%
.975$ & $.957$ & $.905$ \\
$500$ & $4$ & $2$ & $.99$ & $.99$ & $-.045$ & $-.199$ & $-.236$ & $.077$ & $%
.152$ & $.032$ & $.853$ & $.706$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.007$ & $.008$ & $.056$ & $.054$ & $.052$ & $%
.974$ & $.951$ & $.904$ \\
$500$ & $6$ & $0$ & $0.99$ & $.99$ & $-.038$ & $-.407$ & $-.170$ & $.050$ & $%
.148$ & $.022$ & $.845$ & $.126$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.037$ & $.038$ & $.035$ & $%
.974$ & $.953$ & $.918$ \\
$500$ & $6$ & $1$ & $.99$ & $.99$ & $-.038$ & $-.411$ & $-.170$ & $.050$ & $%
.150$ & $.022$ & $.847$ & $.116$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.011$ & $.005$ & $.037$ & $.039$ & $.035$ & $%
.973$ & $.944$ & $.915$ \\
$500$ & $6$ & $2$ & $.99$ & $.99$ & $-.027$ & $-.152$ & $-.153$ & $.050$ & $%
.084$ & $.022$ & $.870$ & $.535$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.009$ & $.009$ & $.037$ & $.036$ & $.035$ & $%
.971$ & $.942$ & $.910$ \\
$500$ & $8$ & $0$ & $0.99$ & $.99$ & $-.029$ & $-.315$ & $-.129$ & $.038$ & $%
.101$ & $.017$ & $.852$ & $.042$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.028$ & $.030$ & $.027$ & $%
.975$ & $.952$ & $.924$ \\
$500$ & $8$ & $1$ & $.99$ & $.99$ & $-.028$ & $-.317$ & $-.128$ & $.038$ & $%
.101$ & $.017$ & $.854$ & $.040$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.011$ & $.004$ & $.028$ & $.030$ & $.027$ & $%
.975$ & $.937$ & $.922$ \\
$500$ & $8$ & $2$ & $.99$ & $.99$ & $-.018$ & $-.119$ & $-.112$ & $.038$ & $%
.056$ & $.017$ & $.883$ & $.394$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.009$ & $.008$ & $.028$ & $.028$ & $.027$ & $%
.975$ & $.938$ & $.913$ \\
$500$ & $16$ & $0$ & $0.99$ & $.99$ & $-.014$ & $-.163$ & $-.066$ & $.019$ &
$.038$ & $.009$ & $.854$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.014$ & $.016$ & $.014$ & $%
.972$ & $.950$ & $.931$ \\
$500$ & $16$ & $1$ & $.99$ & $.99$ & $-.014$ & $-.162$ & $-.065$ & $.019$ & $%
.038$ & $.009$ & $.857$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.012$ & $.005$ & $.014$ & $.016$ & $.014$ & $%
.971$ & $.893$ & $.914$ \\
$500$ & $16$ & $2$ & $.99$ & $.99$ & $-.005$ & $-.065$ & $-.050$ & $.019$ & $%
.021$ & $.008$ & $.900$ & $.082$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.008$ & $.014$ & $.015$ & $.014$ & $%
.970$ & $.897$ & $.898$ \\
$500$ & $24$ & $0$ & $0.99$ & $.99$ & $-.009$ & $-.110$ & $-.044$ & $.013$ &
$.021$ & $.006$ & $.867$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.010$ & $.011$ & $.010$ & $%
.975$ & $.947$ & $.931$ \\
$500$ & $24$ & $1$ & $.99$ & $.99$ & $-.008$ & $-.107$ & $-.043$ & $.013$ & $%
.021$ & $.006$ & $.873$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.011$ & $.005$ & $.010$ & $.011$ & $.010$ & $%
.975$ & $.820$ & $.896$ \\
$500$ & $24$ & $2$ & $.99$ & $.99$ & $-.001$ & $-.045$ & $-.030$ & $.013$ & $%
.012$ & $.006$ & $.925$ & $.013$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.010$ & $.006$ & $.010$ & $.010$ & $.010$ & $%
.973$ & $.832$ & $.875$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.5$ & $.5$ & $-.040$ & --- & $-.662$ & $.137$ & --- & $%
.048$ & $.873$ & $.943$ & $.000$ \\
&  &  &  & $.5$ & $-.010$ & --- & $-.165$ & $.083$ & --- & $.060$ & $.961$ &
$.953$ & $.085$ \\
$1000$ & $2$ & $1$ & $.5$ & $.5$ & $.026$ & --- & $-.417$ & $.138$ & --- & $%
.046$ & $.938$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $-.001$ & --- & $.007$ & $.079$ & --- & $.064$ & $.965$ & $%
.953$ & $.834$ \\
$1000$ & $2$ & $2$ & $.5$ & $.5$ & $.002$ & --- & $.074$ & $.048$ & --- & $%
.037$ & $.954$ & $.952$ & $.404$ \\
&  &  &  & $.5$ & $-.001$ & --- & $-.021$ & $.078$ & --- & $.080$ & $.951$ &
$.952$ & $.827$ \\
$1000$ & $4$ & $0$ & $.5$ & $.5$ & $.006$ & $-.010$ & $-.245$ & $.056$ & $%
.042$ & $.021$ & $.957$ & $.937$ & $.000$ \\
&  &  &  & $.5$ & $.001$ & $-.001$ & $-.028$ & $.040$ & $.039$ & $.038$ & $%
.953$ & $.952$ & $.801$ \\
$1000$ & $4$ & $1$ & $.5$ & $.5$ & $.001$ & $-.007$ & $-.124$ & $.034$ & $%
.035$ & $.021$ & $.958$ & $.944$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.016$ & $.040$ & $.039$ & $.038$ & $.951
$ & $.951$ & $.880$ \\
$1000$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $-.002$ & $.078$ & $.019$ & $.020
$ & $.017$ & $.956$ & $.950$ & $.003$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.030$ & $.040$ & $.040$ & $.040$ & $%
.952$ & $.951$ & $.831$ \\
$1000$ & $6$ & $0$ & $.5$ & $.5$ & $.000$ & $-.007$ & $-.114$ & $.023$ & $%
.023$ & $.016$ & $.953$ & $.941$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.002$ & $.028$ & $.028$ & $.028$ & $%
.955$ & $.956$ & $.922$ \\
$1000$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.006$ & $-.051$ & $.019$ & $%
.021$ & $.015$ & $.954$ & $.943$ & $.045$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.009$ & $.029$ & $.029$ & $.028$ & $.954
$ & $.955$ & $.910$ \\
$1000$ & $6$ & $2$ & $.5$ & $.5$ & $.000$ & $-.003$ & $.056$ & $.013$ & $.014
$ & $.013$ & $.954$ & $.947$ & $.004$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.022$ & $.029$ & $.029$ & $.029$ & $%
.953$ & $.953$ & $.859$ \\
$1000$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.006$ & $-.061$ & $.016$ & $%
.017$ & $.013$ & $.952$ & $.937$ & $.002$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.003$ & $.023$ & $.024$ & $.024$ & $.952
$ & $.952$ & $.928$ \\
$1000$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $-.005$ & $-.025$ & $.014$ & $%
.016$ & $.012$ & $.953$ & $.935$ & $.391$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.005$ & $.024$ & $.024$ & $.023$ & $.951
$ & $.951$ & $.926$ \\
$1000$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $-.003$ & $.042$ & $.011$ & $.012
$ & $.010$ & $.953$ & $.946$ & $.013$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.016$ & $.024$ & $.024$ & $.024$ & $%
.952$ & $.952$ & $.875$ \\
$1000$ & $16$ & $0$ & $.5$ & $.5$ & $.000$ & $-.004$ & $-.010$ & $.008$ & $%
.009$ & $.008$ & $.949$ & $.925$ & $.699$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.002$ & $.015$ & $.015$ & $.015$ & $.945
$ & $.944$ & $.936$ \\
$1000$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $-.004$ & $-.002$ & $.008$ & $%
.009$ & $.008$ & $.950$ & $.930$ & $.921$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.001$ & $.015$ & $.015$ & $.015$ & $.946
$ & $.944$ & $.939$ \\
$1000$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $-.003$ & $.017$ & $.007$ & $%
.008$ & $.007$ & $.950$ & $.935$ & $.229$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.006$ & $.015$ & $.015$ & $.015$ & $%
.947$ & $.944$ & $.926$ \\
$1000$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.004$ & $-.002$ & $.006$ & $%
.007$ & $.006$ & $.953$ & $.917$ & $.917$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.001$ & $.012$ & $.012$ & $.012$ & $.952
$ & $.951$ & $.948$ \\
$1000$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.004$ & $.001$ & $.006$ & $%
.007$ & $.006$ & $.954$ & $.921$ & $.932$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.000$ & $.012$ & $.012$ & $.012$ & $.953
$ & $.951$ & $.950$ \\
$1000$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.003$ & $.010$ & $.005$ & $%
.006$ & $.005$ & $.954$ & $.922$ & $.491$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.004$ & $.012$ & $.012$ & $.012$ & $%
.953$ & $.951$ & $.940$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.5$ & $.9$ & $-.090$ & --- & $-.546$ & $.135$ & --- & $%
.048$ & $.828$ & $.855$ & $.000$ \\
&  &  &  & $.1$ & $-.005$ & --- & $-.027$ & $.074$ & --- & $.060$ & $.976$ &
$.959$ & $.793$ \\
$1000$ & $2$ & $1$ & $.5$ & $.9$ & $-.055$ & --- & $-.489$ & $.135$ & --- & $%
.048$ & $.865$ & $.870$ & $.000$ \\
&  &  &  & $.1$ & $.003$ & --- & $.024$ & $.076$ & --- & $.062$ & $.978$ & $%
.965$ & $.803$ \\
$1000$ & $2$ & $2$ & $.5$ & $.9$ & $.013$ & --- & $-.319$ & $.140$ & --- & $%
.047$ & $.921$ & $.944$ & $.000$ \\
&  &  &  & $.1$ & $-.002$ & --- & $.048$ & $.081$ & --- & $.067$ & $.971$ & $%
.957$ & $.734$ \\
$1000$ & $4$ & $0$ & $.5$ & $.9$ & $-.041$ & $-.174$ & $-.289$ & $.064$ & $%
.142$ & $.023$ & $.856$ & $.736$ & $.000$ \\
&  &  &  & $.1$ & $-.002$ & $-.007$ & $-.013$ & $.039$ & $.038$ & $.036$ & $%
.975$ & $.945$ & $.881$ \\
$1000$ & $4$ & $1$ & $.5$ & $.9$ & $-.011$ & $-.118$ & $-.232$ & $.065$ & $%
.121$ & $.023$ & $.907$ & $.813$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.011$ & $.021$ & $.040$ & $.039$ & $.036$ & $.975
$ & $.945$ & $.853$ \\
$1000$ & $4$ & $2$ & $.5$ & $.9$ & $.010$ & $-.018$ & $-.096$ & $.060$ & $%
.050$ & $.021$ & $.954$ & $.928$ & $.002$ \\
&  &  &  & $.1$ & $-.003$ & $.004$ & $.021$ & $.042$ & $.041$ & $.038$ & $%
.962$ & $.951$ & $.859$ \\
$1000$ & $6$ & $0$ & $.5$ & $.9$ & $-.022$ & $-.096$ & $-.198$ & $.044$ & $%
.068$ & $.016$ & $.872$ & $.703$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.003$ & $-.007$ & $.028$ & $.028$ & $.027$ & $%
.979$ & $.955$ & $.913$ \\
$1000$ & $6$ & $1$ & $.5$ & $.9$ & $.001$ & $-.067$ & $-.146$ & $.046$ & $%
.058$ & $.015$ & $.928$ & $.781$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.007$ & $.016$ & $.029$ & $.028$ & $.027$ & $.972
$ & $.946$ & $.873$ \\
$1000$ & $6$ & $2$ & $.5$ & $.9$ & $.002$ & $-.014$ & $-.034$ & $.029$ & $%
.027$ & $.014$ & $.962$ & $.916$ & $.212$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.009$ & $.029$ & $.029$ & $.028$ & $.955
$ & $.952$ & $.916$ \\
$1000$ & $8$ & $0$ & $.5$ & $.9$ & $-.010$ & $-.063$ & $-.150$ & $.034$ & $%
.041$ & $.012$ & $.895$ & $.687$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.002$ & $-.004$ & $.023$ & $.023$ & $.023$ & $%
.973$ & $.950$ & $.918$ \\
$1000$ & $8$ & $1$ & $.5$ & $.9$ & $.005$ & $-.045$ & $-.103$ & $.035$ & $%
.036$ & $.012$ & $.942$ & $.763$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.005$ & $.012$ & $.024$ & $.024$ & $.023$ & $%
.966$ & $.945$ & $.890$ \\
$1000$ & $8$ & $2$ & $.5$ & $.9$ & $.001$ & $-.010$ & $-.010$ & $.017$ & $%
.017$ & $.010$ & $.956$ & $.909$ & $.764$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.003$ & $.024$ & $.024$ & $.023$ & $.953
$ & $.949$ & $.933$ \\
$1000$ & $16$ & $0$ & $.5$ & $.9$ & $.003$ & $-.024$ & $-.070$ & $.019$ & $%
.014$ & $.007$ & $.951$ & $.609$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.015$ & $.015$ & $.015$ & $.954
$ & $.947$ & $.933$ \\
$1000$ & $16$ & $1$ & $.5$ & $.9$ & $.001$ & $-.020$ & $-.041$ & $.012$ & $%
.013$ & $.006$ & $.965$ & $.658$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.005$ & $.015$ & $.015$ & $.015$ & $.947
$ & $.941$ & $.921$ \\
$1000$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.007$ & $.009$ & $.007$ & $%
.007$ & $.005$ & $.953$ & $.855$ & $.529$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $-.002$ & $.015$ & $.015$ & $.015$ & $%
.947$ & $.943$ & $.939$ \\
$1000$ & $24$ & $0$ & $.5$ & $.9$ & $.000$ & $-.015$ & $-.040$ & $.009$ & $%
.008$ & $.005$ & $.965$ & $.553$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.012$ & $.012$ & $.012$ & $.950
$ & $.950$ & $.942$ \\
$1000$ & $24$ & $1$ & $.5$ & $.9$ & $.000$ & $-.014$ & $-.023$ & $.007$ & $%
.008$ & $.005$ & $.956$ & $.576$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.003$ & $.012$ & $.012$ & $.012$ & $.950
$ & $.950$ & $.939$ \\
$1000$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.006$ & $.008$ & $.004$ & $%
.005$ & $.004$ & $.954$ & $.786$ & $.355$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.002$ & $.012$ & $.012$ & $.012$ & $%
.951$ & $.950$ & $.945$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.5$ & $.99$ & $-.092$ & --- & $-.505$ & $.135$ & --- &
$.048$ & $.825$ & $.824$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & --- & $-.002$ & $.074$ & --- & $.061$ & $.980$
& $.984$ & $.833$ \\
$1000$ & $2$ & $1$ & $.5$ & $.99$ & $-.088$ & --- & $-.498$ & $.135$ & --- &
$.048$ & $.832$ & $.832$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & --- & $.013$ & $.074$ & --- & $.061$ & $.980$ & $%
.975$ & $.822$ \\
$1000$ & $2$ & $2$ & $.5$ & $.99$ & $-.075$ & --- & $-.478$ & $.135$ & --- &
$.048$ & $.843$ & $.838$ & $.000$ \\
&  &  &  & $.01$ & $.004$ & --- & $.027$ & $.075$ & --- & $.062$ & $.977$ & $%
.953$ & $.798$ \\
$1000$ & $4$ & $0$ & $.5$ & $.99$ & $-.048$ & $-.667$ & $-.255$ & $.063$ & $%
.260$ & $.023$ & $.839$ & $.253$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & $-.003$ & $-.002$ & $.039$ & $.037$ & $.036$ & $%
.976$ & $.951$ & $.908$ \\
$1000$ & $4$ & $1$ & $.5$ & $.99$ & $-.044$ & $-.566$ & $-.249$ & $.062$ & $%
.251$ & $.023$ & $.849$ & $.328$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.021$ & $.010$ & $.039$ & $.036$ & $.036$ & $%
.976$ & $.921$ & $.898$ \\
$1000$ & $4$ & $2$ & $.5$ & $.99$ & $-.031$ & $-.395$ & $-.228$ & $.062$ & $%
.214$ & $.023$ & $.874$ & $.488$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.032$ & $.019$ & $.039$ & $.037$ & $.036$ & $%
.976$ & $.873$ & $.864$ \\
$1000$ & $6$ & $0$ & $.5$ & $.99$ & $-.033$ & $-.472$ & $-.172$ & $.042$ & $%
.150$ & $.016$ & $.835$ & $.071$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.028$ & $.028$ & $.027$ & $%
.979$ & $.955$ & $.928$ \\
$1000$ & $6$ & $1$ & $.5$ & $.99$ & $-.029$ & $-.390$ & $-.165$ & $.042$ & $%
.144$ & $.016$ & $.850$ & $.144$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.017$ & $.008$ & $.028$ & $.027$ & $.027$ & $%
.978$ & $.913$ & $.915$ \\
$1000$ & $6$ & $2$ & $.5$ & $.99$ & $-.017$ & $-.240$ & $-.145$ & $.042$ & $%
.110$ & $.015$ & $.881$ & $.348$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.023$ & $.014$ & $.029$ & $.028$ & $.027$ & $%
.977$ & $.874$ & $.884$ \\
$1000$ & $8$ & $0$ & $.5$ & $.99$ & $-.025$ & $-.361$ & $-.130$ & $.031$ & $%
.102$ & $.012$ & $.840$ & $.017$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.023$ & $.024$ & $.023$ & $%
.976$ & $.950$ & $.930$ \\
$1000$ & $8$ & $1$ & $.5$ & $.99$ & $-.020$ & $-.291$ & $-.123$ & $.031$ & $%
.095$ & $.012$ & $.859$ & $.053$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.014$ & $.006$ & $.023$ & $.023$ & $.023$ & $%
.976$ & $.911$ & $.918$ \\
$1000$ & $8$ & $2$ & $.5$ & $.99$ & $-.009$ & $-.163$ & $-.104$ & $.031$ & $%
.066$ & $.012$ & $.897$ & $.252$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.017$ & $.011$ & $.024$ & $.023$ & $.023$ & $%
.975$ & $.889$ & $.899$ \\
$1000$ & $16$ & $0$ & $.5$ & $.99$ & $-.012$ & $-.182$ & $-.067$ & $.016$ & $%
.039$ & $.006$ & $.852$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.015$ & $.015$ & $.015$ & $%
.973$ & $.948$ & $.937$ \\
$1000$ & $16$ & $1$ & $.5$ & $.99$ & $-.008$ & $-.142$ & $-.060$ & $.016$ & $%
.035$ & $.006$ & $.883$ & $.001$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.008$ & $.003$ & $.015$ & $.015$ & $.015$ & $%
.972$ & $.918$ & $.928$ \\
$1000$ & $16$ & $2$ & $.5$ & $.99$ & $.000$ & $-.066$ & $-.042$ & $.016$ & $%
.021$ & $.006$ & $.928$ & $.080$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.007$ & $.005$ & $.015$ & $.015$ & $.015$ & $%
.967$ & $.916$ & $.923$ \\
$1000$ & $24$ & $0$ & $.5$ & $.99$ & $-.008$ & $-.118$ & $-.046$ & $.011$ & $%
.021$ & $.004$ & $.853$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.012$ & $.012$ & $.012$ & $%
.976$ & $.952$ & $.945$ \\
$1000$ & $24$ & $1$ & $.5$ & $.99$ & $-.004$ & $-.093$ & $-.039$ & $.011$ & $%
.019$ & $.004$ & $.893$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.005$ & $.002$ & $.012$ & $.012$ & $.012$ & $%
.974$ & $.934$ & $.942$ \\
$1000$ & $24$ & $2$ & $.5$ & $.99$ & $.001$ & $-.039$ & $-.023$ & $.011$ & $%
.011$ & $.004$ & $.941$ & $.027$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.003$ & $.012$ & $.012$ & $.012$ & $%
.964$ & $.936$ & $.941$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.99$ & $.5$ & $.000$ & --- & $.387$ & $.028$ & --- & $%
.029$ & $.945$ & $.946$ & $.126$ \\
&  &  &  & $.5$ & $.000$ & --- & $.096$ & $.089$ & --- & $.116$ & $.949$ & $%
.949$ & $.734$ \\
$1000$ & $2$ & $1$ & $.99$ & $.5$ & $-.033$ & --- & $-.649$ & $.137$ & --- &
$.048$ & $.881$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.006$ & $.088$ & --- & $.068$ & $.975$ & $%
.953$ & $.832$ \\
$1000$ & $2$ & $2$ & $.99$ & $.5$ & $.000$ & --- & $.441$ & $.024$ & --- & $%
.027$ & $.953$ & $.951$ & $.021$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.118$ & $.089$ & --- & $.123$ & $.949$ & $%
.949$ & $.635$ \\
$1000$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.201$ & $.013$ & $%
.013$ & $.014$ & $.951$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.013$ & $.040$ & $.040$ & $.044$ & $.950
$ & $.950$ & $.899$ \\
$1000$ & $4$ & $1$ & $.99$ & $.5$ & $.004$ & $-.009$ & $-.219$ & $.049$ & $%
.039$ & $.021$ & $.964$ & $.940$ & $.000$ \\
&  &  &  & $.5$ & $-.001$ & $.002$ & $.051$ & $.042$ & $.041$ & $.039$ & $%
.954$ & $.953$ & $.631$ \\
$1000$ & $4$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.228$ & $.012$ & $%
.011$ & $.013$ & $.956$ & $.954$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.121$ & $.040$ & $.040$ & $.046$ & $%
.951$ & $.951$ & $.183$ \\
$1000$ & $6$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.127$ & $.010$ & $%
.010$ & $.010$ & $.949$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.013$ & $.026$ & $.026$ & $.027$ & $%
.949$ & $.949$ & $.907$ \\
$1000$ & $6$ & $1$ & $.99$ & $.5$ & $.000$ & $-.006$ & $-.092$ & $.022$ & $%
.022$ & $.016$ & $.953$ & $.938$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.035$ & $.027$ & $.027$ & $.026$ & $.950
$ & $.951$ & $.671$ \\
$1000$ & $6$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.146$ & $.009$ & $%
.009$ & $.009$ & $.952$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.096$ & $.027$ & $.027$ & $.028$ & $%
.949$ & $.950$ & $.059$ \\
$1000$ & $8$ & $0$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.089$ & $.008$ & $%
.008$ & $.009$ & $.949$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.021$ & $.020$ & $.020$ & $.019$ & $%
.952$ & $.951$ & $.801$ \\
$1000$ & $8$ & $1$ & $.99$ & $.5$ & $.000$ & $-.005$ & $-.044$ & $.015$ & $%
.016$ & $.013$ & $.952$ & $.937$ & $.037$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $.021$ & $.021$ & $.021$ & $.020$ & $.951
$ & $.948$ & $.781$ \\
$1000$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.104$ & $.008$ & $%
.008$ & $.008$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.075$ & $.020$ & $.020$ & $.020$ & $%
.953$ & $.951$ & $.035$ \\
$1000$ & $16$ & $0$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.036$ & $.006$ & $%
.006$ & $.006$ & $.949$ & $.939$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.019$ & $.010$ & $.010$ & $.010$ & $%
.950$ & $.949$ & $.524$ \\
$1000$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $-.004$ & $-.001$ & $.008$ & $%
.009$ & $.008$ & $.948$ & $.925$ & $.920$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $.001$ & $.011$ & $.012$ & $.011$ & $.946
$ & $.940$ & $.943$ \\
$1000$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.042$ & $.005$ & $%
.006$ & $.006$ & $.954$ & $.944$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.036$ & $.011$ & $.011$ & $.011$ & $%
.949$ & $.946$ & $.082$ \\
$1000$ & $24$ & $0$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.021$ & $.005$ & $%
.005$ & $.005$ & $.948$ & $.932$ & $.009$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.014$ & $.007$ & $.007$ & $.007$ & $%
.950$ & $.947$ & $.519$ \\
$1000$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $-.003$ & $.004$ & $.006$ & $%
.006$ & $.006$ & $.952$ & $.920$ & $.887$ \\
&  &  &  & $.5$ & $.000$ & $.003$ & $-.003$ & $.008$ & $.008$ & $.008$ & $%
.951$ & $.940$ & $.933$ \\
$1000$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $-.002$ & $.025$ & $.005$ & $%
.005$ & $.005$ & $.950$ & $.931$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.002$ & $-.022$ & $.008$ & $.008$ & $.008$ & $%
.952$ & $.946$ & $.178$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.99$ & $.9$ & $-.006$ & --- & $-.383$ & $.140$ & --- &
$.047$ & $.899$ & $.946$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & --- & $-.019$ & $.089$ & --- & $.074$ & $.971$ & $%
.951$ & $.821$ \\
$1000$ & $2$ & $1$ & $.99$ & $.9$ & $-.086$ & --- & $-.541$ & $.135$ & --- &
$.048$ & $.833$ & $.844$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $.004$ & $.085$ & --- & $.069$ & $.975$ & $%
.981$ & $.835$ \\
$1000$ & $2$ & $2$ & $.99$ & $.9$ & $.021$ & --- & $-.280$ & $.140$ & --- & $%
.046$ & $.929$ & $.947$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.018$ & $.091$ & --- & $.078$ & $.967$ & $%
.950$ & $.827$ \\
$1000$ & $4$ & $0$ & $.99$ & $.9$ & $.011$ & $-.014$ & $-.152$ & $.068$ & $%
.043$ & $.022$ & $.941$ & $.937$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.007$ & $.040$ & $.040$ & $.038$ & $%
.964$ & $.951$ & $.903$ \\
$1000$ & $4$ & $1$ & $.99$ & $.9$ & $-.036$ & $-.165$ & $-.281$ & $.064$ & $%
.139$ & $.023$ & $.866$ & $.749$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & $.010$ & $.017$ & $.040$ & $.039$ & $.037$ & $.976
$ & $.947$ & $.868$ \\
$1000$ & $4$ & $2$ & $.99$ & $.9$ & $.008$ & $-.008$ & $-.074$ & $.054$ & $%
.032$ & $.021$ & $.957$ & $.945$ & $.029$ \\
&  &  &  & $.1$ & $-.002$ & $.001$ & $.012$ & $.041$ & $.040$ & $.039$ & $%
.956$ & $.951$ & $.895$ \\
$1000$ & $6$ & $0$ & $.99$ & $.9$ & $.006$ & $-.012$ & $-.083$ & $.041$ & $%
.025$ & $.015$ & $.950$ & $.925$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.003$ & $.026$ & $.026$ & $.026$ & $%
.954$ & $.949$ & $.921$ \\
$1000$ & $6$ & $1$ & $.99$ & $.9$ & $-.017$ & $-.085$ & $-.190$ & $.044$ & $%
.065$ & $.016$ & $.886$ & $.730$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & $.009$ & $.021$ & $.026$ & $.026$ & $.025$ & $.975
$ & $.938$ & $.816$ \\
$1000$ & $6$ & $2$ & $.99$ & $.9$ & $.001$ & $-.007$ & $-.021$ & $.026$ & $%
.019$ & $.014$ & $.961$ & $.931$ & $.571$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.005$ & $.027$ & $.026$ & $.026$ & $.952
$ & $.949$ & $.922$ \\
$1000$ & $8$ & $0$ & $.99$ & $.9$ & $.002$ & $-.011$ & $-.052$ & $.026$ & $%
.017$ & $.011$ & $.956$ & $.903$ & $.002$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.001$ & $.020$ & $.020$ & $.020$ & $%
.952$ & $.951$ & $.932$ \\
$1000$ & $8$ & $1$ & $.99$ & $.9$ & $-.005$ & $-.054$ & $-.140$ & $.034$ & $%
.039$ & $.012$ & $.911$ & $.721$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.009$ & $.022$ & $.020$ & $.020$ & $.020$ & $.974
$ & $.932$ & $.732$ \\
$1000$ & $8$ & $2$ & $.99$ & $.9$ & $.001$ & $-.006$ & $-.001$ & $.016$ & $%
.013$ & $.010$ & $.954$ & $.927$ & $.912$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.000$ & $.020$ & $.020$ & $.020$ & $.951
$ & $.951$ & $.937$ \\
$1000$ & $16$ & $0$ & $.99$ & $.9$ & $.000$ & $-.008$ & $-.014$ & $.009$ & $%
.008$ & $.006$ & $.950$ & $.839$ & $.231$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.010$ & $.010$ & $.010$ & $.950
$ & $.950$ & $.938$ \\
$1000$ & $16$ & $1$ & $.99$ & $.9$ & $.002$ & $-.020$ & $-.058$ & $.017$ & $%
.013$ & $.007$ & $.956$ & $.664$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.006$ & $.017$ & $.011$ & $.011$ & $.010$ & $%
.959$ & $.911$ & $.559$ \\
$1000$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.005$ & $.014$ & $.006$ & $%
.006$ & $.005$ & $.953$ & $.887$ & $.163$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $-.007$ & $.010$ & $.010$ & $.010$ & $%
.947$ & $.941$ & $.881$ \\
$1000$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.006$ & $-.006$ & $.005$ & $%
.005$ & $.004$ & $.949$ & $.774$ & $.619$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.001$ & $.007$ & $.007$ & $.007$ & $.951
$ & $.949$ & $.944$ \\
$1000$ & $24$ & $1$ & $.99$ & $.9$ & $.000$ & $-.012$ & $-.030$ & $.007$ & $%
.007$ & $.005$ & $.955$ & $.613$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $.012$ & $.007$ & $.007$ & $.007$ & $.951
$ & $.900$ & $.567$ \\
$1000$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $-.004$ & $.013$ & $.004$ & $%
.004$ & $.004$ & $.951$ & $.835$ & $.043$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $-.008$ & $.007$ & $.007$ & $.007$ & $%
.951$ & $.937$ & $.784$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$1000$ & $2$ & $0$ & $.99$ & $.99$ & $-.091$ & --- & $-.503$ & $.135$ & ---
& $.048$ & $.827$ & $.846$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.002$ & $.085$ & --- & $.070$ & $.975$ &
$.982$ & $.836$ \\
$1000$ & $2$ & $1$ & $.99$ & $.99$ & $-.091$ & --- & $-.502$ & $.135$ & ---
& $.048$ & $.828$ & $.842$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.003$ & $.085$ & --- & $.070$ & $.975$ & $%
.981$ & $.835$ \\
$1000$ & $2$ & $2$ & $.99$ & $.99$ & $-.080$ & --- & $-.485$ & $.135$ & ---
& $.048$ & $.841$ & $.917$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.008$ & $.086$ & --- & $.071$ & $.975$ & $%
.967$ & $.833$ \\
$1000$ & $4$ & $0$ & $0.99$ & $.99$ & $-.047$ & $-.477$ & $-.254$ & $.063$ &
$.245$ & $.023$ & $.839$ & $.417$ & $.000$ \\
&  &  &  & $.01$ & $-.001$ & $-.002$ & $-.002$ & $.039$ & $.038$ & $.036$ & $%
.977$ & $.955$ & $.905$ \\
$1000$ & $4$ & $1$ & $.99$ & $.99$ & $-.046$ & $-.485$ & $-.253$ & $.063$ & $%
.245$ & $.023$ & $.843$ & $.406$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.007$ & $.004$ & $.039$ & $.038$ & $.036$ & $%
.977$ & $.953$ & $.905$ \\
$1000$ & $4$ & $2$ & $.99$ & $.99$ & $-.036$ & $-.111$ & $-.236$ & $.062$ & $%
.114$ & $.023$ & $.866$ & $.817$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.004$ & $.008$ & $.039$ & $.038$ & $.037$ & $%
.976$ & $.952$ & $.898$ \\
$1000$ & $6$ & $0$ & $0.99$ & $.99$ & $-.032$ & $-.340$ & $-.170$ & $.042$ &
$.137$ & $.016$ & $.839$ & $.196$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.026$ & $.026$ & $.025$ & $%
.976$ & $.953$ & $.920$ \\
$1000$ & $6$ & $1$ & $.99$ & $.99$ & $-.032$ & $-.348$ & $-.170$ & $.042$ & $%
.139$ & $.016$ & $.840$ & $.187$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.008$ & $.004$ & $.026$ & $.026$ & $.025$ & $%
.975$ & $.948$ & $.917$ \\
$1000$ & $6$ & $2$ & $.99$ & $.99$ & $-.022$ & $-.087$ & $-.153$ & $.042$ & $%
.063$ & $.015$ & $.869$ & $.704$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.005$ & $.008$ & $.026$ & $.026$ & $.025$ & $%
.976$ & $.949$ & $.906$ \\
$1000$ & $8$ & $0$ & $0.99$ & $.99$ & $-.024$ & $-.263$ & $-.128$ & $.031$ &
$.092$ & $.012$ & $.846$ & $.085$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.019$ & $.020$ & $.019$ & $%
.977$ & $.954$ & $.927$ \\
$1000$ & $8$ & $1$ & $.99$ & $.99$ & $-.023$ & $-.265$ & $-.128$ & $.031$ & $%
.092$ & $.012$ & $.848$ & $.077$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.009$ & $.005$ & $.020$ & $.020$ & $.019$ & $%
.977$ & $.933$ & $.919$ \\
$1000$ & $8$ & $2$ & $.99$ & $.99$ & $-.013$ & $-.068$ & $-.112$ & $.031$ & $%
.040$ & $.012$ & $.883$ & $.614$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.005$ & $.009$ & $.020$ & $.020$ & $.019$ & $%
.976$ & $.943$ & $.900$ \\
$1000$ & $16$ & $0$ & $0.99$ & $.99$ & $-.011$ & $-.135$ & $-.065$ & $.016$
& $.034$ & $.006$ & $.861$ & $.002$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.010$ & $.011$ & $.010$ & $%
.975$ & $.952$ & $.934$ \\
$1000$ & $16$ & $1$ & $.99$ & $.99$ & $-.011$ & $-.133$ & $-.065$ & $.016$ &
$.033$ & $.006$ & $.864$ & $.003$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.005$ & $.010$ & $.011$ & $.010$ & $%
.974$ & $.866$ & $.901$ \\
$1000$ & $16$ & $2$ & $.99$ & $.99$ & $-.003$ & $-.038$ & $-.050$ & $.016$ &
$.015$ & $.006$ & $.914$ & $.277$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.007$ & $.010$ & $.010$ & $.010$ & $%
.973$ & $.914$ & $.862$ \\
$1000$ & $24$ & $0$ & $0.99$ & $.99$ & $-.007$ & $-.090$ & $-.044$ & $.011$
& $.018$ & $.004$ & $.862$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.007$ & $.007$ & $.007$ & $%
.976$ & $.954$ & $.937$ \\
$1000$ & $24$ & $1$ & $.99$ & $.99$ & $-.007$ & $-.087$ & $-.043$ & $.011$ &
$.018$ & $.004$ & $.871$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.009$ & $.005$ & $.007$ & $.007$ & $.007$ & $%
.976$ & $.773$ & $.874$ \\
$1000$ & $24$ & $2$ & $.99$ & $.99$ & $.000$ & $-.027$ & $-.030$ & $.011$ & $%
.008$ & $.004$ & $.928$ & $.098$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.006$ & $.007$ & $.007$ & $.007$ & $%
.975$ & $.871$ & $.822$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.5$ & $.5$ & $-.024$ & --- & $-.662$ & $.105$ & --- & $%
.030$ & $.889$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $-.006$ & --- & $-.166$ & $.056$ & --- & $.038$ & $.961$ &
$.953$ & $.002$ \\
$2500$ & $2$ & $1$ & $.5$ & $.5$ & $.019$ & --- & $-.417$ & $.100$ & --- & $%
.028$ & $.947$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.007$ & $.050$ & --- & $.041$ & $.956$ & $%
.950$ & $.824$ \\
$2500$ & $2$ & $2$ & $.5$ & $.5$ & $.001$ & --- & $.074$ & $.030$ & --- & $%
.023$ & $.952$ & $.950$ & $.067$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.021$ & $.050$ & --- & $.051$ & $.949$ & $%
.950$ & $.806$ \\
$2500$ & $4$ & $0$ & $.5$ & $.5$ & $.002$ & $-.004$ & $-.245$ & $.033$ & $%
.026$ & $.014$ & $.961$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.028$ & $.025$ & $.025$ & $.024$ & $%
.954$ & $.954$ & $.675$ \\
$2500$ & $4$ & $1$ & $.5$ & $.5$ & $.001$ & $-.003$ & $-.124$ & $.022$ & $%
.022$ & $.014$ & $.950$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.016$ & $.025$ & $.025$ & $.024$ & $.955
$ & $.955$ & $.837$ \\
$2500$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.078$ & $.012$ & $.013
$ & $.011$ & $.949$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.030$ & $.025$ & $.025$ & $.025$ & $%
.954$ & $.953$ & $.715$ \\
$2500$ & $6$ & $0$ & $.5$ & $.5$ & $.000$ & $-.003$ & $-.114$ & $.015$ & $%
.015$ & $.010$ & $.949$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.002$ & $.018$ & $.018$ & $.018$ & $%
.951$ & $.951$ & $.918$ \\
$2500$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.002$ & $-.051$ & $.012$ & $%
.014$ & $.010$ & $.950$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.010$ & $.018$ & $.018$ & $.018$ & $.950
$ & $.952$ & $.883$ \\
$2500$ & $6$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.056$ & $.009$ & $.009
$ & $.008$ & $.950$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.022$ & $.018$ & $.018$ & $.018$ & $%
.951$ & $.952$ & $.741$ \\
$2500$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.002$ & $-.061$ & $.010$ & $%
.011$ & $.008$ & $.948$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.003$ & $.015$ & $.015$ & $.015$ & $.952
$ & $.951$ & $.926$ \\
$2500$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $-.002$ & $-.025$ & $.009$ & $%
.010$ & $.008$ & $.950$ & $.942$ & $.075$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.005$ & $.015$ & $.015$ & $.015$ & $.952
$ & $.951$ & $.921$ \\
$2500$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.041$ & $.007$ & $.007
$ & $.007$ & $.953$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.016$ & $.015$ & $.015$ & $.015$ & $%
.951$ & $.951$ & $.790$ \\
$2500$ & $16$ & $0$ & $.5$ & $.5$ & $.000$ & $-.002$ & $-.010$ & $.005$ & $%
.006$ & $.005$ & $.949$ & $.939$ & $.425$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.002$ & $.009$ & $.009$ & $.009$ & $.954
$ & $.954$ & $.942$ \\
$2500$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $-.002$ & $-.001$ & $.005$ & $%
.006$ & $.005$ & $.951$ & $.942$ & $.915$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.000$ & $.009$ & $.010$ & $.009$ & $.954
$ & $.955$ & $.948$ \\
$2500$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.017$ & $.004$ & $%
.005$ & $.004$ & $.949$ & $.946$ & $.014$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.006$ & $.010$ & $.010$ & $.009$ & $%
.955$ & $.954$ & $.899$ \\
$2500$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.002$ & $.004$ & $%
.004$ & $.004$ & $.948$ & $.933$ & $.889$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.001$ & $.008$ & $.008$ & $.008$ & $.948
$ & $.948$ & $.944$ \\
$2500$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.001$ & $.004$ & $%
.004$ & $.004$ & $.948$ & $.934$ & $.910$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.008$ & $.008$ & $.008$ & $.950
$ & $.948$ & $.945$ \\
$2500$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.010$ & $.003$ & $%
.004$ & $.003$ & $.948$ & $.936$ & $.139$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.004$ & $.008$ & $.008$ & $.008$ & $%
.950$ & $.949$ & $.916$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.5$ & $.9$ & $-.072$ & --- & $-.546$ & $.104$ & --- & $%
.030$ & $.839$ & $.895$ & $.000$ \\
&  &  &  & $.1$ & $-.003$ & --- & $-.027$ & $.048$ & --- & $.038$ & $.976$ &
$.951$ & $.731$ \\
$2500$ & $2$ & $1$ & $.5$ & $.9$ & $-.038$ & --- & $-.489$ & $.105$ & --- & $%
.030$ & $.878$ & $.899$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & --- & $.024$ & $.048$ & --- & $.039$ & $.974$ & $%
.952$ & $.756$ \\
$2500$ & $2$ & $2$ & $.5$ & $.9$ & $.018$ & --- & $-.319$ & $.110$ & --- & $%
.029$ & $.930$ & $.950$ & $.000$ \\
&  &  &  & $.1$ & $-.003$ & --- & $.048$ & $.052$ & --- & $.042$ & $.968$ & $%
.951$ & $.596$ \\
$2500$ & $4$ & $0$ & $.5$ & $.9$ & $-.031$ & $-.080$ & $-.288$ & $.050$ & $%
.100$ & $.015$ & $.861$ & $.855$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.003$ & $-.013$ & $.024$ & $.024$ & $.022$ & $%
.976$ & $.951$ & $.852$ \\
$2500$ & $4$ & $1$ & $.5$ & $.9$ & $-.003$ & $-.051$ & $-.232$ & $.052$ & $%
.082$ & $.015$ & $.913$ & $.888$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.005$ & $.021$ & $.025$ & $.025$ & $.023$ & $.977
$ & $.949$ & $.772$ \\
$2500$ & $4$ & $2$ & $.5$ & $.9$ & $.004$ & $-.007$ & $-.096$ & $.038$ & $%
.032$ & $.014$ & $.960$ & $.942$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.002$ & $.022$ & $.026$ & $.026$ & $.024$ & $%
.959$ & $.953$ & $.786$ \\
$2500$ & $6$ & $0$ & $.5$ & $.9$ & $-.015$ & $-.043$ & $-.198$ & $.034$ & $%
.046$ & $.010$ & $.886$ & $.838$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.007$ & $.018$ & $.018$ & $.017$ & $%
.975$ & $.950$ & $.895$ \\
$2500$ & $6$ & $1$ & $.5$ & $.9$ & $.004$ & $-.029$ & $-.146$ & $.036$ & $%
.038$ & $.010$ & $.937$ & $.882$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.016$ & $.018$ & $.018$ & $.017$ & $.969
$ & $.950$ & $.799$ \\
$2500$ & $6$ & $2$ & $.5$ & $.9$ & $.001$ & $-.005$ & $-.034$ & $.017$ & $%
.017$ & $.009$ & $.958$ & $.937$ & $.013$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.009$ & $.019$ & $.018$ & $.018$ & $.951
$ & $.952$ & $.894$ \\
$2500$ & $8$ & $0$ & $.5$ & $.9$ & $-.005$ & $-.028$ & $-.150$ & $.026$ & $%
.028$ & $.008$ & $.908$ & $.830$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.005$ & $.015$ & $.015$ & $.014$ & $%
.974$ & $.952$ & $.913$ \\
$2500$ & $8$ & $1$ & $.5$ & $.9$ & $.005$ & $-.020$ & $-.103$ & $.027$ & $%
.024$ & $.008$ & $.946$ & $.861$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.002$ & $.012$ & $.015$ & $.015$ & $.014$ & $%
.962$ & $.950$ & $.840$ \\
$2500$ & $8$ & $2$ & $.5$ & $.9$ & $.000$ & $-.005$ & $-.010$ & $.011$ & $%
.011$ & $.007$ & $.950$ & $.929$ & $.561$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.002$ & $.015$ & $.015$ & $.015$ & $.951
$ & $.949$ & $.933$ \\
$2500$ & $16$ & $0$ & $.5$ & $.9$ & $.002$ & $-.010$ & $-.069$ & $.013$ & $%
.009$ & $.004$ & $.957$ & $.808$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.009$ & $.009$ & $.009$ & $.955
$ & $.953$ & $.940$ \\
$2500$ & $16$ & $1$ & $.5$ & $.9$ & $.000$ & $-.009$ & $-.041$ & $.008$ & $%
.008$ & $.004$ & $.957$ & $.834$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.005$ & $.009$ & $.009$ & $.009$ & $.953
$ & $.952$ & $.909$ \\
$2500$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.003$ & $.009$ & $.004$ & $%
.005$ & $.003$ & $.952$ & $.911$ & $.188$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.002$ & $.009$ & $.009$ & $.009$ & $%
.953$ & $.953$ & $.942$ \\
$2500$ & $24$ & $0$ & $.5$ & $.9$ & $.000$ & $-.006$ & $-.040$ & $.005$ & $%
.005$ & $.003$ & $.952$ & $.777$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.008$ & $.008$ & $.008$ & $.946
$ & $.946$ & $.936$ \\
$2500$ & $24$ & $1$ & $.5$ & $.9$ & $.000$ & $-.006$ & $-.022$ & $.004$ & $%
.005$ & $.003$ & $.953$ & $.786$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.003$ & $.008$ & $.008$ & $.008$ & $.948
$ & $.947$ & $.929$ \\
$2500$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.002$ & $.008$ & $.003$ & $%
.003$ & $.002$ & $.950$ & $.884$ & $.058$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.002$ & $.008$ & $.008$ & $.008$ & $%
.948$ & $.946$ & $.938$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.5$ & $.99$ & $-.074$ & --- & $-.505$ & $.104$ & --- &
$.030$ & $.836$ & $.831$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.003$ & $.047$ & --- & $.039$ & $.977$ &
$.984$ & $.829$ \\
$2500$ & $2$ & $1$ & $.5$ & $.99$ & $-.070$ & --- & $-.498$ & $.104$ & --- &
$.030$ & $.841$ & $.841$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & --- & $.013$ & $.047$ & --- & $.039$ & $.976$ & $%
.963$ & $.811$ \\
$2500$ & $2$ & $2$ & $.5$ & $.99$ & $-.057$ & --- & $-.478$ & $.104$ & --- &
$.030$ & $.857$ & $.855$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & --- & $.027$ & $.048$ & --- & $.039$ & $.974$ & $%
.928$ & $.741$ \\
$2500$ & $4$ & $0$ & $.5$ & $.99$ & $-.038$ & $-.634$ & $-.255$ & $.049$ & $%
.254$ & $.015$ & $.842$ & $.273$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.003$ & $-.001$ & $.024$ & $.023$ & $.022$ & $%
.978$ & $.954$ & $.909$ \\
$2500$ & $4$ & $1$ & $.5$ & $.99$ & $-.034$ & $-.445$ & $-.248$ & $.049$ & $%
.227$ & $.015$ & $.851$ & $.439$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.017$ & $.010$ & $.024$ & $.023$ & $.022$ & $%
.977$ & $.896$ & $.880$ \\
$2500$ & $4$ & $2$ & $.5$ & $.99$ & $-.022$ & $-.235$ & $-.228$ & $.049$ & $%
.162$ & $.015$ & $.879$ & $.659$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & $.020$ & $.019$ & $.025$ & $.026$ & $.023$ & $%
.977$ & $.875$ & $.793$ \\
$2500$ & $6$ & $0$ & $.5$ & $.99$ & $-.026$ & $-.441$ & $-.172$ & $.032$ & $%
.147$ & $.010$ & $.844$ & $.088$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.018$ & $.017$ & $.017$ & $%
.974$ & $.954$ & $.925$ \\
$2500$ & $6$ & $1$ & $.5$ & $.99$ & $-.021$ & $-.292$ & $-.165$ & $.032$ & $%
.122$ & $.010$ & $.862$ & $.258$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.014$ & $.008$ & $.018$ & $.017$ & $.017$ & $%
.976$ & $.887$ & $.892$ \\
$2500$ & $6$ & $2$ & $.5$ & $.99$ & $-.010$ & $-.132$ & $-.145$ & $.032$ & $%
.077$ & $.010$ & $.898$ & $.582$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.013$ & $.015$ & $.018$ & $.018$ & $.017$ & $%
.975$ & $.891$ & $.818$ \\
$2500$ & $8$ & $0$ & $.5$ & $.99$ & $-.019$ & $-.334$ & $-.130$ & $.024$ & $%
.098$ & $.008$ & $.847$ & $.028$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.015$ & $.015$ & $.014$ & $%
.976$ & $.952$ & $.933$ \\
$2500$ & $8$ & $1$ & $.5$ & $.99$ & $-.015$ & $-.214$ & $-.123$ & $.024$ & $%
.079$ & $.008$ & $.865$ & $.146$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.006$ & $.015$ & $.014$ & $.014$ & $%
.978$ & $.898$ & $.907$ \\
$2500$ & $8$ & $2$ & $.5$ & $.99$ & $-.004$ & $-.087$ & $-.103$ & $.025$ & $%
.047$ & $.007$ & $.909$ & $.518$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.009$ & $.011$ & $.015$ & $.015$ & $.014$ & $%
.975$ & $.909$ & $.853$ \\
$2500$ & $16$ & $0$ & $.5$ & $.99$ & $-.010$ & $-.155$ & $-.067$ & $.012$ & $%
.035$ & $.004$ & $.853$ & $.001$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $.000$ & $.009$ & $.009$ & $.009$ & $%
.975$ & $.954$ & $.942$ \\
$2500$ & $16$ & $1$ & $.5$ & $.99$ & $-.005$ & $-.099$ & $-.060$ & $.012$ & $%
.027$ & $.004$ & $.889$ & $.014$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.005$ & $.003$ & $.009$ & $.009$ & $.009$ & $%
.976$ & $.916$ & $.926$ \\
$2500$ & $16$ & $2$ & $.5$ & $.99$ & $.001$ & $-.033$ & $-.042$ & $.013$ & $%
.014$ & $.004$ & $.939$ & $.341$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.005$ & $.010$ & $.009$ & $.009$ & $%
.967$ & $.935$ & $.912$ \\
$2500$ & $24$ & $0$ & $.5$ & $.99$ & $-.006$ & $-.095$ & $-.046$ & $.008$ & $%
.018$ & $.003$ & $.857$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.008$ & $.008$ & $.008$ & $%
.973$ & $.947$ & $.940$ \\
$2500$ & $24$ & $1$ & $.5$ & $.99$ & $-.002$ & $-.063$ & $-.039$ & $.008$ & $%
.014$ & $.003$ & $.899$ & $.001$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.003$ & $.002$ & $.008$ & $.008$ & $.008$ & $%
.971$ & $.927$ & $.931$ \\
$2500$ & $24$ & $2$ & $.5$ & $.99$ & $.001$ & $-.019$ & $-.023$ & $.009$ & $%
.007$ & $.002$ & $.948$ & $.219$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.003$ & $.008$ & $.008$ & $.008$ & $%
.957$ & $.940$ & $.930$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.99$ & $.5$ & $.000$ & --- & $.387$ & $.017$ & --- & $%
.018$ & $.949$ & $.948$ & $.022$ \\
&  &  &  & $.5$ & $.000$ & --- & $.097$ & $.057$ & --- & $.073$ & $.954$ & $%
.954$ & $.565$ \\
$2500$ & $2$ & $1$ & $.99$ & $.5$ & $-.018$ & --- & $-.649$ & $.106$ & --- &
$.030$ & $.897$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.006$ & $.056$ & --- & $.043$ & $.976$ & $%
.956$ & $.833$ \\
$2500$ & $2$ & $2$ & $.99$ & $.5$ & $.000$ & --- & $.441$ & $.015$ & --- & $%
.017$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.118$ & $.057$ & --- & $.078$ & $.954$ & $%
.954$ & $.430$ \\
$2500$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.201$ & $.008$ & $.008
$ & $.009$ & $.950$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.013$ & $.025$ & $.025$ & $.028$ & $.952
$ & $.952$ & $.878$ \\
$2500$ & $4$ & $1$ & $.99$ & $.5$ & $.002$ & $-.003$ & $-.219$ & $.029$ & $%
.024$ & $.014$ & $.955$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.051$ & $.026$ & $.026$ & $.025$ & $.951
$ & $.950$ & $.338$ \\
$2500$ & $4$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.228$ & $.007$ & $.007
$ & $.008$ & $.949$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.121$ & $.026$ & $.026$ & $.030$ & $%
.951$ & $.951$ & $.008$ \\
$2500$ & $6$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.127$ & $.006$ & $%
.006$ & $.007$ & $.950$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.013$ & $.017$ & $.017$ & $.017$ & $%
.949$ & $.949$ & $.860$ \\
$2500$ & $6$ & $1$ & $.99$ & $.5$ & $.000$ & $-.003$ & $-.092$ & $.014$ & $%
.014$ & $.010$ & $.950$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.035$ & $.017$ & $.017$ & $.017$ & $.953
$ & $.951$ & $.374$ \\
$2500$ & $6$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.146$ & $.006$ & $.006
$ & $.006$ & $.949$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.096$ & $.017$ & $.017$ & $.018$ & $%
.951$ & $.951$ & $.000$ \\
$2500$ & $8$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.089$ & $.005$ & $%
.005$ & $.006$ & $.948$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.021$ & $.012$ & $.012$ & $.012$ & $%
.950$ & $.949$ & $.582$ \\
$2500$ & $8$ & $1$ & $.99$ & $.5$ & $.000$ & $-.002$ & $-.044$ & $.010$ & $%
.010$ & $.008$ & $.948$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.021$ & $.013$ & $.013$ & $.013$ & $.948
$ & $.946$ & $.586$ \\
$2500$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.103$ & $.005$ & $%
.005$ & $.005$ & $.952$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.075$ & $.013$ & $.013$ & $.013$ & $%
.948$ & $.947$ & $.000$ \\
$2500$ & $16$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.036$ & $.004$ & $%
.004$ & $.004$ & $.948$ & $.943$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.019$ & $.007$ & $.007$ & $.006$ & $%
.948$ & $.947$ & $.155$ \\
$2500$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.001$ & $.005$ & $%
.005$ & $.005$ & $.948$ & $.941$ & $.915$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.001$ & $.007$ & $.007$ & $.007$ & $.950
$ & $.948$ & $.943$ \\
$2500$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.043$ & $.003$ & $%
.004$ & $.004$ & $.953$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.036$ & $.007$ & $.007$ & $.007$ & $%
.948$ & $.949$ & $.000$ \\
$2500$ & $24$ & $0$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.021$ & $.003$ & $%
.003$ & $.003$ & $.950$ & $.944$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.014$ & $.005$ & $.005$ & $.005$ & $%
.946$ & $.944$ & $.142$ \\
$2500$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.004$ & $.004$ & $%
.004$ & $.004$ & $.949$ & $.939$ & $.795$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.003$ & $.005$ & $.005$ & $.005$ & $%
.947$ & $.943$ & $.901$ \\
$2500$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.025$ & $.003$ & $%
.003$ & $.003$ & $.952$ & $.943$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.022$ & $.005$ & $.005$ & $.005$ & $%
.950$ & $.948$ & $.005$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.99$ & $.9$ & $.006$ & --- & $-.383$ & $.110$ & --- & $%
.029$ & $.920$ & $.948$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & --- & $-.019$ & $.057$ & --- & $.047$ & $.973$ & $%
.954$ & $.804$ \\
$2500$ & $2$ & $1$ & $.99$ & $.9$ & $-.069$ & --- & $-.541$ & $.104$ & --- &
$.030$ & $.843$ & $.875$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $.004$ & $.055$ & --- & $.044$ & $.978$ & $%
.976$ & $.837$ \\
$2500$ & $2$ & $2$ & $.99$ & $.9$ & $.021$ & --- & $-.280$ & $.109$ & --- & $%
.029$ & $.938$ & $.949$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.018$ & $.058$ & --- & $.050$ & $.965$ & $%
.954$ & $.808$ \\
$2500$ & $4$ & $0$ & $.99$ & $.9$ & $.009$ & $-.005$ & $-.152$ & $.051$ & $%
.027$ & $.014$ & $.952$ & $.946$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.000$ & $-.006$ & $.025$ & $.025$ & $.024$ & $%
.959$ & $.952$ & $.900$ \\
$2500$ & $4$ & $1$ & $.99$ & $.9$ & $-.027$ & $-.075$ & $-.281$ & $.050$ & $%
.097$ & $.015$ & $.869$ & $.859$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & $.005$ & $.018$ & $.025$ & $.025$ & $.023$ & $.976
$ & $.946$ & $.807$ \\
$2500$ & $4$ & $2$ & $.99$ & $.9$ & $.003$ & $-.003$ & $-.074$ & $.032$ & $%
.020$ & $.013$ & $.961$ & $.945$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.012$ & $.026$ & $.025$ & $.025$ & $.953
$ & $.952$ & $.870$ \\
$2500$ & $6$ & $0$ & $.99$ & $.9$ & $.003$ & $-.005$ & $-.082$ & $.026$ & $%
.016$ & $.009$ & $.959$ & $.938$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.017$ & $.016$ & $.016$ & $%
.950$ & $.950$ & $.919$ \\
$2500$ & $6$ & $1$ & $.99$ & $.9$ & $-.010$ & $-.038$ & $-.189$ & $.034$ & $%
.043$ & $.010$ & $.898$ & $.855$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.004$ & $.021$ & $.017$ & $.017$ & $.016$ & $.973
$ & $.943$ & $.661$ \\
$2500$ & $6$ & $2$ & $.99$ & $.9$ & $.001$ & $-.003$ & $-.020$ & $.016$ & $%
.012$ & $.009$ & $.955$ & $.941$ & $.249$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.005$ & $.017$ & $.017$ & $.016$ & $.952
$ & $.950$ & $.912$ \\
$2500$ & $8$ & $0$ & $.99$ & $.9$ & $.001$ & $-.004$ & $-.051$ & $.015$ & $%
.011$ & $.007$ & $.963$ & $.931$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.001$ & $.012$ & $.012$ & $.012$ & $%
.948$ & $.947$ & $.926$ \\
$2500$ & $8$ & $1$ & $.99$ & $.9$ & $-.001$ & $-.024$ & $-.140$ & $.027$ & $%
.026$ & $.008$ & $.922$ & $.845$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.021$ & $.013$ & $.013$ & $.012$ & $.972
$ & $.940$ & $.522$ \\
$2500$ & $8$ & $2$ & $.99$ & $.9$ & $.000$ & $-.002$ & $-.001$ & $.010$ & $%
.008$ & $.006$ & $.949$ & $.940$ & $.906$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.000$ & $.013$ & $.013$ & $.013$ & $.949
$ & $.948$ & $.933$ \\
$2500$ & $16$ & $0$ & $.99$ & $.9$ & $.000$ & $-.003$ & $-.014$ & $.006$ & $%
.005$ & $.004$ & $.950$ & $.903$ & $.018$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.006$ & $.006$ & $.006$ & $.947
$ & $.947$ & $.933$ \\
$2500$ & $16$ & $1$ & $.99$ & $.9$ & $.001$ & $-.008$ & $-.058$ & $.010$ & $%
.008$ & $.004$ & $.962$ & $.833$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.003$ & $.017$ & $.007$ & $.007$ & $.007$ & $.955
$ & $.936$ & $.206$ \\
$2500$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.002$ & $.014$ & $.004$ & $%
.004$ & $.003$ & $.954$ & $.927$ & $.006$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.007$ & $.007$ & $.007$ & $.006$ & $%
.950$ & $.946$ & $.778$ \\
$2500$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.003$ & $-.006$ & $.003$ & $%
.003$ & $.003$ & $.950$ & $.880$ & $.334$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.004$ & $.004$ & $.004$ & $.948
$ & $.948$ & $.935$ \\
$2500$ & $24$ & $1$ & $.99$ & $.9$ & $.000$ & $-.005$ & $-.030$ & $.005$ & $%
.005$ & $.003$ & $.951$ & $.805$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.012$ & $.005$ & $.005$ & $.004$ & $.948
$ & $.925$ & $.224$ \\
$2500$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $-.002$ & $.013$ & $.003$ & $%
.003$ & $.002$ & $.951$ & $.907$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.008$ & $.005$ & $.005$ & $.004$ & $%
.947$ & $.943$ & $.530$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$2500$ & $2$ & $0$ & $.99$ & $.99$ & $-.073$ & --- & $-.503$ & $.104$ & ---
& $.030$ & $.838$ & $.864$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.003$ & $.055$ & --- & $.045$ & $.977$ &
$.977$ & $.836$ \\
$2500$ & $2$ & $1$ & $.99$ & $.99$ & $-.073$ & --- & $-.502$ & $.104$ & ---
& $.030$ & $.839$ & $.859$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.003$ & $.055$ & --- & $.045$ & $.979$ & $%
.978$ & $.837$ \\
$2500$ & $2$ & $2$ & $.99$ & $.99$ & $-.062$ & --- & $-.485$ & $.104$ & ---
& $.030$ & $.851$ & $.938$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.008$ & $.055$ & --- & $.045$ & $.978$ & $%
.961$ & $.830$ \\
$2500$ & $4$ & $0$ & $.99$ & $.99$ & $-.037$ & $-.310$ & $-.253$ & $.049$ & $%
.198$ & $.015$ & $.846$ & $.590$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.025$ & $.024$ & $.023$ & $%
.976$ & $.954$ & $.907$ \\
$2500$ & $4$ & $1$ & $.99$ & $.99$ & $-.037$ & $-.323$ & $-.253$ & $.049$ & $%
.199$ & $.015$ & $.843$ & $.567$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.005$ & $.004$ & $.025$ & $.024$ & $.023$ & $%
.977$ & $.951$ & $.900$ \\
$2500$ & $4$ & $2$ & $.99$ & $.99$ & $-.026$ & $-.048$ & $-.236$ & $.049$ & $%
.076$ & $.015$ & $.869$ & $.894$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.002$ & $.009$ & $.025$ & $.025$ & $.023$ & $%
.977$ & $.952$ & $.884$ \\
$2500$ & $6$ & $0$ & $.99$ & $.99$ & $-.025$ & $-.225$ & $-.170$ & $.032$ & $%
.109$ & $.010$ & $.851$ & $.375$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.016$ & $.016$ & $.016$ & $%
.974$ & $.952$ & $.920$ \\
$2500$ & $6$ & $1$ & $.99$ & $.99$ & $-.024$ & $-.231$ & $-.169$ & $.032$ & $%
.108$ & $.010$ & $.853$ & $.366$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.006$ & $.004$ & $.016$ & $.016$ & $.016$ & $%
.974$ & $.938$ & $.907$ \\
$2500$ & $6$ & $2$ & $.99$ & $.99$ & $-.014$ & $-.038$ & $-.153$ & $.032$ & $%
.042$ & $.010$ & $.884$ & $.841$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.002$ & $.009$ & $.016$ & $.016$ & $.016$ & $%
.975$ & $.949$ & $.874$ \\
$2500$ & $8$ & $0$ & $.99$ & $.99$ & $-.018$ & $-.174$ & $-.128$ & $.024$ & $%
.070$ & $.008$ & $.854$ & $.228$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.012$ & $.012$ & $.012$ & $%
.973$ & $.948$ & $.921$ \\
$2500$ & $8$ & $1$ & $.99$ & $.99$ & $-.018$ & $-.177$ & $-.127$ & $.024$ & $%
.070$ & $.008$ & $.855$ & $.218$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.004$ & $.012$ & $.012$ & $.012$ & $%
.975$ & $.925$ & $.904$ \\
$2500$ & $8$ & $2$ & $.99$ & $.99$ & $-.008$ & $-.031$ & $-.111$ & $.024$ & $%
.027$ & $.007$ & $.894$ & $.787$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.008$ & $.013$ & $.012$ & $.012$ & $%
.974$ & $.945$ & $.860$ \\
$2500$ & $16$ & $0$ & $.99$ & $.99$ & $-.008$ & $-.088$ & $-.065$ & $.012$ &
$.025$ & $.004$ & $.863$ & $.025$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.006$ & $.006$ & $.006$ & $%
.972$ & $.949$ & $.931$ \\
$2500$ & $16$ & $1$ & $.99$ & $.99$ & $-.008$ & $-.086$ & $-.064$ & $.012$ &
$.024$ & $.004$ & $.868$ & $.027$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.006$ & $.005$ & $.006$ & $.007$ & $.006$ & $%
.973$ & $.849$ & $.854$ \\
$2500$ & $16$ & $2$ & $.99$ & $.99$ & $-.001$ & $-.017$ & $-.050$ & $.013$ &
$.010$ & $.004$ & $.923$ & $.599$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.003$ & $.007$ & $.007$ & $.006$ & $.006$ & $%
.972$ & $.932$ & $.745$ \\
$2500$ & $24$ & $0$ & $.99$ & $.99$ & $-.005$ & $-.057$ & $-.044$ & $.008$ &
$.013$ & $.003$ & $.871$ & $.003$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.004$ & $.004$ & $.004$ & $%
.973$ & $.948$ & $.929$ \\
$2500$ & $24$ & $1$ & $.99$ & $.99$ & $-.005$ & $-.055$ & $-.043$ & $.008$ &
$.013$ & $.003$ & $.877$ & $.004$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.006$ & $.005$ & $.004$ & $.005$ & $.004$ & $%
.972$ & $.754$ & $.771$ \\
$2500$ & $24$ & $2$ & $.99$ & $.99$ & $.001$ & $-.012$ & $-.030$ & $.009$ & $%
.006$ & $.003$ & $.934$ & $.416$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.003$ & $.006$ & $.005$ & $.004$ & $.004$ & $%
.968$ & $.907$ & $.644$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.5$ & $.5$ & $-.015$ & --- & $-.662$ & $.089$ & --- & $%
.021$ & $.891$ & $.944$ & $.000$ \\
&  &  &  & $.5$ & $-.004$ & --- & $-.165$ & $.041$ & --- & $.027$ & $.960$ &
$.952$ & $.000$ \\
$5000$ & $2$ & $1$ & $.5$ & $.5$ & $.012$ & --- & $-.417$ & $.074$ & --- & $%
.020$ & $.953$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.007$ & $.035$ & --- & $.029$ & $.955$ & $%
.953$ & $.823$ \\
$5000$ & $2$ & $2$ & $.5$ & $.5$ & $.000$ & --- & $.074$ & $.021$ & --- & $%
.016$ & $.952$ & $.951$ & $.002$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.021$ & $.035$ & --- & $.036$ & $.951$ & $%
.950$ & $.771$ \\
$5000$ & $4$ & $0$ & $.5$ & $.5$ & $.001$ & $-.002$ & $-.245$ & $.023$ & $%
.019$ & $.010$ & $.951$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.028$ & $.018$ & $.018$ & $.017$ & $%
.949$ & $.950$ & $.496$ \\
$5000$ & $4$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.124$ & $.015$ & $%
.016$ & $.010$ & $.947$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.016$ & $.018$ & $.018$ & $.017$ & $.951
$ & $.949$ & $.760$ \\
$5000$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.078$ & $.009$ & $.009$
& $.008$ & $.948$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.030$ & $.018$ & $.018$ & $.018$ & $%
.951$ & $.949$ & $.539$ \\
$5000$ & $6$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.114$ & $.011$ & $%
.011$ & $.007$ & $.951$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.002$ & $.013$ & $.013$ & $.013$ & $%
.951$ & $.951$ & $.917$ \\
$5000$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.051$ & $.009$ & $%
.010$ & $.007$ & $.951$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.009$ & $.013$ & $.013$ & $.013$ & $.951
$ & $.950$ & $.847$ \\
$5000$ & $6$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.056$ & $.006$ & $.006
$ & $.006$ & $.950$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.022$ & $.013$ & $.013$ & $.013$ & $%
.950$ & $.949$ & $.561$ \\
$5000$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.061$ & $.007$ & $%
.007$ & $.006$ & $.954$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.003$ & $.010$ & $.010$ & $.010$ & $.952
$ & $.951$ & $.919$ \\
$5000$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.025$ & $.006$ & $%
.007$ & $.005$ & $.953$ & $.951$ & $.003$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.005$ & $.010$ & $.010$ & $.010$ & $.952
$ & $.952$ & $.903$ \\
$5000$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.041$ & $.005$ & $.005
$ & $.005$ & $.951$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.016$ & $.011$ & $.011$ & $.010$ & $%
.952$ & $.951$ & $.645$ \\
$5000$ & $16$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.010$ & $.004$ & $%
.004$ & $.004$ & $.951$ & $.947$ & $.156$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.002$ & $.007$ & $.007$ & $.007$ & $.951
$ & $.950$ & $.934$ \\
$5000$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.001$ & $.004$ & $%
.004$ & $.003$ & $.953$ & $.950$ & $.902$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.007$ & $.007$ & $.007$ & $.951
$ & $.951$ & $.944$ \\
$5000$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.017$ & $.003$ & $%
.003$ & $.003$ & $.953$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.006$ & $.007$ & $.007$ & $.007$ & $%
.951$ & $.950$ & $.841$ \\
$5000$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.002$ & $.003$ & $%
.003$ & $.003$ & $.951$ & $.945$ & $.850$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.952
$ & $.951$ & $.947$ \\
$5000$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.001$ & $.003$ & $%
.003$ & $.003$ & $.951$ & $.942$ & $.889$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.950
$ & $.949$ & $.945$ \\
$5000$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $-.001$ & $.010$ & $.002$ & $%
.003$ & $.002$ & $.948$ & $.942$ & $.011$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.004$ & $.005$ & $.005$ & $.005$ & $%
.952$ & $.952$ & $.892$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.5$ & $.9$ & $-.062$ & --- & $-.546$ & $.087$ & --- & $%
.021$ & $.836$ & $.911$ & $.000$ \\
&  &  &  & $.1$ & $-.003$ & --- & $-.027$ & $.034$ & --- & $.027$ & $.976$ &
$.951$ & $.638$ \\
$5000$ & $2$ & $1$ & $.5$ & $.9$ & $-.028$ & --- & $-.489$ & $.088$ & --- & $%
.021$ & $.881$ & $.919$ & $.000$ \\
&  &  &  & $.1$ & $.002$ & --- & $.024$ & $.034$ & --- & $.028$ & $.976$ & $%
.953$ & $.681$ \\
$5000$ & $2$ & $2$ & $.5$ & $.9$ & $.017$ & --- & $-.319$ & $.091$ & --- & $%
.021$ & $.939$ & $.949$ & $.000$ \\
&  &  &  & $.1$ & $-.002$ & --- & $.048$ & $.038$ & --- & $.030$ & $.968$ & $%
.950$ & $.415$ \\
$5000$ & $4$ & $0$ & $.5$ & $.9$ & $-.026$ & $-.041$ & $-.289$ & $.042$ & $%
.075$ & $.010$ & $.854$ & $.898$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.002$ & $-.013$ & $.018$ & $.018$ & $.016$ & $%
.976$ & $.950$ & $.802$ \\
$5000$ & $4$ & $1$ & $.5$ & $.9$ & $.000$ & $-.026$ & $-.232$ & $.044$ & $%
.060$ & $.010$ & $.917$ & $.916$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.021$ & $.018$ & $.018$ & $.016$ & $.972
$ & $.946$ & $.646$ \\
$5000$ & $4$ & $2$ & $.5$ & $.9$ & $.002$ & $-.004$ & $-.096$ & $.025$ & $%
.023$ & $.010$ & $.959$ & $.947$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.022$ & $.018$ & $.018$ & $.017$ & $.952
$ & $.949$ & $.662$ \\
$5000$ & $6$ & $0$ & $.5$ & $.9$ & $-.010$ & $-.023$ & $-.198$ & $.028$ & $%
.034$ & $.007$ & $.890$ & $.894$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $-.001$ & $-.007$ & $.013$ & $.013$ & $.012$ & $%
.972$ & $.949$ & $.866$ \\
$5000$ & $6$ & $1$ & $.5$ & $.9$ & $.005$ & $-.015$ & $-.146$ & $.030$ & $%
.028$ & $.007$ & $.947$ & $.913$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $.002$ & $.016$ & $.013$ & $.013$ & $.012$ & $%
.965$ & $.948$ & $.686$ \\
$5000$ & $6$ & $2$ & $.5$ & $.9$ & $.000$ & $-.003$ & $-.034$ & $.012$ & $%
.012$ & $.006$ & $.952$ & $.943$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.009$ & $.013$ & $.013$ & $.013$ & $.952
$ & $.950$ & $.861$ \\
$5000$ & $8$ & $0$ & $.5$ & $.9$ & $-.002$ & $-.014$ & $-.150$ & $.022$ & $%
.020$ & $.006$ & $.914$ & $.892$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.005$ & $.010$ & $.010$ & $.010$ & $%
.972$ & $.951$ & $.902$ \\
$5000$ & $8$ & $1$ & $.5$ & $.9$ & $.003$ & $-.010$ & $-.103$ & $.020$ & $%
.017$ & $.005$ & $.953$ & $.909$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.012$ & $.011$ & $.011$ & $.010$ & $.957
$ & $.951$ & $.749$ \\
$5000$ & $8$ & $2$ & $.5$ & $.9$ & $.000$ & $-.002$ & $-.010$ & $.008$ & $%
.008$ & $.005$ & $.949$ & $.942$ & $.313$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.003$ & $.011$ & $.011$ & $.010$ & $.951
$ & $.951$ & $.929$ \\
$5000$ & $16$ & $0$ & $.5$ & $.9$ & $.001$ & $-.005$ & $-.069$ & $.009$ & $%
.006$ & $.003$ & $.958$ & $.874$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.007$ & $.007$ & $.007$ & $.953
$ & $.953$ & $.937$ \\
$5000$ & $16$ & $1$ & $.5$ & $.9$ & $.000$ & $-.004$ & $-.041$ & $.005$ & $%
.006$ & $.003$ & $.952$ & $.888$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.005$ & $.007$ & $.007$ & $.007$ & $.952
$ & $.951$ & $.874$ \\
$5000$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.001$ & $.009$ & $.003$ & $%
.003$ & $.002$ & $.950$ & $.928$ & $.028$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.007$ & $.007$ & $.007$ & $%
.952$ & $.951$ & $.934$ \\
$5000$ & $24$ & $0$ & $.5$ & $.9$ & $.000$ & $-.003$ & $-.040$ & $.004$ & $%
.004$ & $.002$ & $.953$ & $.861$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.951
$ & $.951$ & $.943$ \\
$5000$ & $24$ & $1$ & $.5$ & $.9$ & $.000$ & $-.003$ & $-.022$ & $.003$ & $%
.004$ & $.002$ & $.951$ & $.865$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.002$ & $.005$ & $.005$ & $.005$ & $.951
$ & $.952$ & $.919$ \\
$5000$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.001$ & $.008$ & $.002$ & $%
.002$ & $.002$ & $.949$ & $.918$ & $.002$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.005$ & $.005$ & $.005$ & $%
.951$ & $.952$ & $.932$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.5$ & $.99$ & $-.064$ & --- & $-.505$ & $.087$ & --- &
$.021$ & $.832$ & $.821$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.002$ & $.034$ & --- & $.027$ & $.978$ &
$.984$ & $.833$ \\
$5000$ & $2$ & $1$ & $.5$ & $.99$ & $-.060$ & --- & $-.498$ & $.087$ & --- &
$.021$ & $.839$ & $.841$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & --- & $.013$ & $.034$ & --- & $.027$ & $.978$ & $%
.949$ & $.789$ \\
$5000$ & $2$ & $2$ & $.5$ & $.99$ & $-.047$ & --- & $-.478$ & $.087$ & --- &
$.021$ & $.854$ & $.873$ & $.000$ \\
&  &  &  & $.01$ & $.003$ & --- & $.027$ & $.034$ & --- & $.028$ & $.976$ & $%
.917$ & $.649$ \\
$5000$ & $4$ & $0$ & $.5$ & $.99$ & $-.033$ & $-.596$ & $-.255$ & $.041$ & $%
.242$ & $.010$ & $.826$ & $.296$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.003$ & $-.001$ & $.017$ & $.016$ & $.016$ & $%
.976$ & $.954$ & $.906$ \\
$5000$ & $4$ & $1$ & $.5$ & $.99$ & $-.029$ & $-.326$ & $-.248$ & $.041$ & $%
.193$ & $.010$ & $.842$ & $.563$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.013$ & $.010$ & $.017$ & $.017$ & $.016$ & $%
.977$ & $.883$ & $.841$ \\
$5000$ & $4$ & $2$ & $.5$ & $.99$ & $-.017$ & $-.139$ & $-.228$ & $.041$ & $%
.127$ & $.010$ & $.878$ & $.778$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.012$ & $.019$ & $.018$ & $.020$ & $.016$ & $%
.975$ & $.894$ & $.685$ \\
$5000$ & $6$ & $0$ & $.5$ & $.99$ & $-.021$ & $-.405$ & $-.172$ & $.027$ & $%
.137$ & $.007$ & $.844$ & $.113$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.002$ & $-.001$ & $.013$ & $.012$ & $.012$ & $%
.974$ & $.950$ & $.924$ \\
$5000$ & $6$ & $1$ & $.5$ & $.99$ & $-.017$ & $-.208$ & $-.165$ & $.027$ & $%
.099$ & $.007$ & $.862$ & $.397$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.010$ & $.008$ & $.013$ & $.012$ & $.012$ & $%
.975$ & $.888$ & $.864$ \\
$5000$ & $6$ & $2$ & $.5$ & $.99$ & $-.006$ & $-.076$ & $-.145$ & $.027$ & $%
.058$ & $.007$ & $.903$ & $.736$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.007$ & $.014$ & $.013$ & $.013$ & $.012$ & $%
.973$ & $.913$ & $.720$ \\
$5000$ & $8$ & $0$ & $.5$ & $.99$ & $-.016$ & $-.295$ & $-.130$ & $.020$ & $%
.090$ & $.005$ & $.848$ & $.044$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.010$ & $.010$ & $.010$ & $%
.974$ & $.953$ & $.931$ \\
$5000$ & $8$ & $1$ & $.5$ & $.99$ & $-.012$ & $-.148$ & $-.123$ & $.020$ & $%
.062$ & $.005$ & $.871$ & $.297$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.007$ & $.006$ & $.010$ & $.010$ & $.010$ & $%
.975$ & $.899$ & $.883$ \\
$5000$ & $8$ & $2$ & $.5$ & $.99$ & $-.002$ & $-.049$ & $-.103$ & $.021$ & $%
.034$ & $.005$ & $.914$ & $.697$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.005$ & $.011$ & $.011$ & $.011$ & $.010$ & $%
.973$ & $.925$ & $.775$ \\
$5000$ & $16$ & $0$ & $.5$ & $.99$ & $-.008$ & $-.125$ & $-.067$ & $.010$ & $%
.030$ & $.003$ & $.851$ & $.003$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.007$ & $.007$ & $.007$ & $%
.976$ & $.952$ & $.942$ \\
$5000$ & $16$ & $1$ & $.5$ & $.99$ & $-.004$ & $-.065$ & $-.060$ & $.010$ & $%
.020$ & $.003$ & $.890$ & $.081$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.003$ & $.007$ & $.007$ & $.007$ & $%
.975$ & $.919$ & $.908$ \\
$5000$ & $16$ & $2$ & $.5$ & $.99$ & $.002$ & $-.018$ & $-.042$ & $.011$ & $%
.010$ & $.003$ & $.942$ & $.578$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.005$ & $.007$ & $.007$ & $.007$ & $%
.965$ & $.940$ & $.876$ \\
$5000$ & $24$ & $0$ & $.5$ & $.99$ & $-.005$ & $-.071$ & $-.046$ & $.007$ & $%
.015$ & $.002$ & $.857$ & $.000$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $%
.976$ & $.953$ & $.946$ \\
$5000$ & $24$ & $1$ & $.5$ & $.99$ & $-.001$ & $-.040$ & $-.039$ & $.007$ & $%
.011$ & $.002$ & $.907$ & $.021$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.002$ & $.005$ & $.005$ & $.005$ & $%
.974$ & $.935$ & $.927$ \\
$5000$ & $24$ & $2$ & $.5$ & $.99$ & $.001$ & $-.010$ & $-.023$ & $.006$ & $%
.005$ & $.002$ & $.951$ & $.473$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.002$ & $.005$ & $.005$ & $.005$ & $%
.957$ & $.950$ & $.919$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.99$ & $.5$ & $.000$ & --- & $.386$ & $.012$ & --- & $%
.013$ & $.949$ & $.951$ & $.002$ \\
&  &  &  & $.5$ & $.000$ & --- & $.097$ & $.040$ & --- & $.052$ & $.951$ & $%
.952$ & $.339$ \\
$5000$ & $2$ & $1$ & $.99$ & $.5$ & $-.010$ & --- & $-.649$ & $.090$ & --- &
$.021$ & $.903$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.006$ & $.040$ & --- & $.031$ & $.976$ & $%
.954$ & $.823$ \\
$5000$ & $2$ & $2$ & $.99$ & $.5$ & $.000$ & --- & $.441$ & $.011$ & --- & $%
.012$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.117$ & $.040$ & --- & $.056$ & $.953$ & $%
.952$ & $.225$ \\
$5000$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.201$ & $.006$ & $.006
$ & $.006$ & $.949$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.013$ & $.018$ & $.018$ & $.020$ & $.948
$ & $.949$ & $.844$ \\
$5000$ & $4$ & $1$ & $.99$ & $.5$ & $.000$ & $-.002$ & $-.219$ & $.021$ & $%
.018$ & $.010$ & $.949$ & $.942$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.051$ & $.019$ & $.018$ & $.017$ & $.947
$ & $.949$ & $.102$ \\
$5000$ & $4$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.228$ & $.005$ & $.005
$ & $.006$ & $.949$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.121$ & $.018$ & $.018$ & $.021$ & $%
.948$ & $.949$ & $.000$ \\
$5000$ & $6$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.127$ & $.004$ & $.004
$ & $.005$ & $.950$ & $.953$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.013$ & $.012$ & $.012$ & $.012$ & $%
.950$ & $.950$ & $.780$ \\
$5000$ & $6$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.092$ & $.010$ & $%
.010$ & $.007$ & $.952$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.035$ & $.012$ & $.012$ & $.012$ & $.950
$ & $.950$ & $.124$ \\
$5000$ & $6$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.146$ & $.004$ & $.004
$ & $.004$ & $.949$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.095$ & $.012$ & $.012$ & $.013$ & $%
.950$ & $.951$ & $.000$ \\
$5000$ & $8$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.089$ & $.004$ & $.004
$ & $.004$ & $.952$ & $.953$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.021$ & $.009$ & $.009$ & $.009$ & $%
.952$ & $.952$ & $.320$ \\
$5000$ & $8$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.044$ & $.007$ & $%
.007$ & $.006$ & $.957$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.021$ & $.009$ & $.009$ & $.009$ & $.954
$ & $.952$ & $.308$ \\
$5000$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.103$ & $.003$ & $.003
$ & $.004$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.075$ & $.009$ & $.009$ & $.009$ & $%
.953$ & $.954$ & $.000$ \\
$5000$ & $16$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.036$ & $.003$ & $%
.003$ & $.003$ & $.949$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.019$ & $.005$ & $.005$ & $.004$ & $%
.949$ & $.951$ & $.011$ \\
$5000$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.001$ & $.004$ & $%
.004$ & $.003$ & $.948$ & $.944$ & $.907$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $.001$ & $.005$ & $.005$ & $.005$ & $.950
$ & $.950$ & $.942$ \\
$5000$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.042$ & $.002$ & $%
.003$ & $.003$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.036$ & $.005$ & $.005$ & $.005$ & $%
.950$ & $.950$ & $.000$ \\
$5000$ & $24$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.021$ & $.002$ & $%
.002$ & $.002$ & $.949$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.014$ & $.003$ & $.003$ & $.003$ & $%
.946$ & $.945$ & $.010$ \\
$5000$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $.004$ & $.003$ & $%
.003$ & $.003$ & $.948$ & $.945$ & $.652$ \\
&  &  &  & $.5$ & $.000$ & $.001$ & $-.003$ & $.004$ & $.004$ & $.004$ & $%
.948$ & $.946$ & $.861$ \\
$5000$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.025$ & $.002$ & $%
.002$ & $.002$ & $.949$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.022$ & $.003$ & $.004$ & $.003$ & $%
.949$ & $.948$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.99$ & $.9$ & $.010$ & --- & $-.383$ & $.093$ & --- & $%
.021$ & $.923$ & $.950$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $-.019$ & $.041$ & --- & $.034$ & $.971$ & $%
.951$ & $.766$ \\
$5000$ & $2$ & $1$ & $.99$ & $.9$ & $-.058$ & --- & $-.541$ & $.087$ & --- &
$.021$ & $.841$ & $.896$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $.004$ & $.039$ & --- & $.031$ & $.979$ & $%
.974$ & $.827$ \\
$5000$ & $2$ & $2$ & $.99$ & $.9$ & $.017$ & --- & $-.280$ & $.087$ & --- & $%
.021$ & $.946$ & $.950$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.018$ & $.041$ & --- & $.035$ & $.962$ & $%
.953$ & $.779$ \\
$5000$ & $4$ & $0$ & $.99$ & $.9$ & $.006$ & $-.003$ & $-.152$ & $.038$ & $%
.019$ & $.010$ & $.950$ & $.948$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.006$ & $.018$ & $.018$ & $.017$ & $%
.953$ & $.948$ & $.882$ \\
$5000$ & $4$ & $1$ & $.99$ & $.9$ & $-.022$ & $-.038$ & $-.281$ & $.042$ & $%
.073$ & $.010$ & $.866$ & $.898$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.003$ & $.017$ & $.018$ & $.018$ & $.016$ & $.974
$ & $.946$ & $.719$ \\
$5000$ & $4$ & $2$ & $.99$ & $.9$ & $.001$ & $-.002$ & $-.074$ & $.022$ & $%
.015$ & $.009$ & $.957$ & $.947$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.012$ & $.018$ & $.018$ & $.017$ & $.949
$ & $.949$ & $.830$ \\
$5000$ & $6$ & $0$ & $.99$ & $.9$ & $.001$ & $-.002$ & $-.082$ & $.017$ & $%
.011$ & $.007$ & $.961$ & $.946$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.012$ & $.012$ & $.012$ & $%
.949$ & $.949$ & $.913$ \\
$5000$ & $6$ & $1$ & $.99$ & $.9$ & $-.006$ & $-.020$ & $-.189$ & $.029$ & $%
.032$ & $.007$ & $.904$ & $.901$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.002$ & $.021$ & $.012$ & $.012$ & $.011$ & $.972
$ & $.947$ & $.453$ \\
$5000$ & $6$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.020$ & $.011$ & $%
.008$ & $.006$ & $.951$ & $.945$ & $.049$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.005$ & $.012$ & $.012$ & $.012$ & $.950
$ & $.951$ & $.896$ \\
$5000$ & $8$ & $0$ & $.99$ & $.9$ & $.001$ & $-.002$ & $-.051$ & $.011$ & $%
.008$ & $.005$ & $.956$ & $.942$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.001$ & $.009$ & $.009$ & $.009$ & $%
.953$ & $.953$ & $.927$ \\
$5000$ & $8$ & $1$ & $.99$ & $.9$ & $.001$ & $-.012$ & $-.140$ & $.023$ & $%
.018$ & $.006$ & $.927$ & $.900$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.002$ & $.022$ & $.009$ & $.009$ & $.009$ & $.975
$ & $.947$ & $.240$ \\
$5000$ & $8$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.001$ & $.007$ & $%
.006$ & $.005$ & $.949$ & $.944$ & $.908$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.009$ & $.009$ & $.009$ & $.951
$ & $.952$ & $.936$ \\
$5000$ & $16$ & $0$ & $.99$ & $.9$ & $.000$ & $-.002$ & $-.014$ & $.004$ & $%
.003$ & $.003$ & $.952$ & $.932$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.004$ & $.004$ & $.004$ & $.952
$ & $.953$ & $.937$ \\
$5000$ & $16$ & $1$ & $.99$ & $.9$ & $.000$ & $-.004$ & $-.058$ & $.007$ & $%
.006$ & $.003$ & $.958$ & $.885$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.017$ & $.005$ & $.005$ & $.005$ & $.952
$ & $.942$ & $.030$ \\
$5000$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $.014$ & $.003$ & $%
.003$ & $.002$ & $.946$ & $.937$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.007$ & $.005$ & $.005$ & $.004$ & $%
.953$ & $.950$ & $.619$ \\
$5000$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.006$ & $.002$ & $%
.002$ & $.002$ & $.954$ & $.914$ & $.095$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.003$ & $.003$ & $.003$ & $.950
$ & $.949$ & $.931$ \\
$5000$ & $24$ & $1$ & $.99$ & $.9$ & $.000$ & $-.003$ & $-.030$ & $.003$ & $%
.003$ & $.002$ & $.948$ & $.876$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.012$ & $.003$ & $.003$ & $.003$ & $.951
$ & $.941$ & $.033$ \\
$5000$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $.013$ & $.002$ & $%
.002$ & $.002$ & $.952$ & $.933$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $-.008$ & $.003$ & $.003$ & $.003$ & $%
.948$ & $.947$ & $.239$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$5000$ & $2$ & $0$ & $.99$ & $.99$ & $-.063$ & --- & $-.503$ & $.087$ & ---
& $.021$ & $.834$ & $.890$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.002$ & $.039$ & --- & $.032$ & $.978$ &
$.973$ & $.827$ \\
$5000$ & $2$ & $1$ & $.99$ & $.99$ & $-.063$ & --- & $-.503$ & $.087$ & ---
& $.021$ & $.834$ & $.891$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.003$ & $.039$ & --- & $.032$ & $.978$ & $%
.975$ & $.828$ \\
$5000$ & $2$ & $2$ & $.99$ & $.99$ & $-.052$ & --- & $-.485$ & $.087$ & ---
& $.021$ & $.848$ & $.948$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.008$ & $.039$ & --- & $.032$ & $.979$ & $%
.956$ & $.818$ \\
$5000$ & $4$ & $0$ & $.99$ & $.99$ & $-.032$ & $-.189$ & $-.253$ & $.041$ & $%
.148$ & $.010$ & $.830$ & $.723$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.018$ & $.017$ & $.016$ & $%
.975$ & $.954$ & $.902$ \\
$5000$ & $4$ & $1$ & $.99$ & $.99$ & $-.032$ & $-.202$ & $-.253$ & $.041$ & $%
.155$ & $.010$ & $.832$ & $.705$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.003$ & $.004$ & $.018$ & $.017$ & $.016$ & $%
.975$ & $.951$ & $.897$ \\
$5000$ & $4$ & $2$ & $.99$ & $.99$ & $-.021$ & $-.024$ & $-.236$ & $.041$ & $%
.056$ & $.010$ & $.867$ & $.917$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.001$ & $.009$ & $.018$ & $.018$ & $.016$ & $%
.974$ & $.949$ & $.859$ \\
$5000$ & $6$ & $0$ & $.99$ & $.99$ & $-.020$ & $-.142$ & $-.170$ & $.027$ & $%
.081$ & $.007$ & $.849$ & $.560$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.012$ & $.011$ & $.011$ & $%
.976$ & $.952$ & $.920$ \\
$5000$ & $6$ & $1$ & $.99$ & $.99$ & $-.020$ & $-.148$ & $-.169$ & $.027$ & $%
.082$ & $.007$ & $.851$ & $.542$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.004$ & $.004$ & $.012$ & $.012$ & $.011$ & $%
.976$ & $.940$ & $.894$ \\
$5000$ & $6$ & $2$ & $.99$ & $.99$ & $-.010$ & $-.019$ & $-.153$ & $.027$ & $%
.030$ & $.007$ & $.889$ & $.897$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.001$ & $.009$ & $.012$ & $.012$ & $.011$ & $%
.975$ & $.948$ & $.832$ \\
$5000$ & $8$ & $0$ & $.99$ & $.99$ & $-.015$ & $-.109$ & $-.128$ & $.020$ & $%
.052$ & $.005$ & $.854$ & $.428$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.009$ & $.009$ & $.008$ & $%
.977$ & $.954$ & $.924$ \\
$5000$ & $8$ & $1$ & $.99$ & $.99$ & $-.015$ & $-.113$ & $-.127$ & $.020$ & $%
.053$ & $.005$ & $.856$ & $.415$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.004$ & $.009$ & $.009$ & $.008$ & $%
.975$ & $.931$ & $.888$ \\
$5000$ & $8$ & $2$ & $.99$ & $.99$ & $-.006$ & $-.016$ & $-.111$ & $.020$ & $%
.020$ & $.005$ & $.898$ & $.867$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.008$ & $.009$ & $.009$ & $.008$ & $%
.975$ & $.952$ & $.792$ \\
$5000$ & $16$ & $0$ & $.99$ & $.99$ & $-.007$ & $-.055$ & $-.065$ & $.010$ &
$.018$ & $.003$ & $.863$ & $.126$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.004$ & $.004$ & $.004$ & $%
.977$ & $.952$ & $.934$ \\
$5000$ & $16$ & $1$ & $.99$ & $.99$ & $-.006$ & $-.054$ & $-.064$ & $.010$ &
$.018$ & $.003$ & $.868$ & $.130$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.005$ & $.004$ & $.005$ & $.004$ & $%
.976$ & $.869$ & $.786$ \\
$5000$ & $16$ & $2$ & $.99$ & $.99$ & $.000$ & $-.009$ & $-.050$ & $.011$ & $%
.007$ & $.003$ & $.928$ & $.756$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.007$ & $.005$ & $.004$ & $.004$ & $%
.974$ & $.942$ & $.574$ \\
$5000$ & $24$ & $0$ & $.99$ & $.99$ & $-.004$ & $-.036$ & $-.044$ & $.007$ &
$.010$ & $.002$ & $.873$ & $.039$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.003$ & $.003$ & $.003$ & $%
.971$ & $.946$ & $.930$ \\
$5000$ & $24$ & $1$ & $.99$ & $.99$ & $-.004$ & $-.034$ & $-.043$ & $.007$ &
$.010$ & $.002$ & $.881$ & $.047$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.004$ & $.005$ & $.003$ & $.003$ & $.003$ & $%
.970$ & $.805$ & $.633$ \\
$5000$ & $24$ & $2$ & $.99$ & $.99$ & $.001$ & $-.006$ & $-.030$ & $.007$ & $%
.004$ & $.002$ & $.940$ & $.651$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.006$ & $.003$ & $.003$ & $.003$ & $%
.966$ & $.925$ & $.410$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.5$ & $.5$ & $-.007$ & --- & $-.662$ & $.074$ & --- &
$.015$ & $.905$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $-.002$ & --- & $-.166$ & $.030$ & --- & $.019$ & $.960$ &
$.953$ & $.000$ \\
$10000$ & $2$ & $1$ & $.5$ & $.5$ & $.006$ & --- & $-.417$ & $.050$ & --- & $%
.014$ & $.958$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.007$ & $.024$ & --- & $.020$ & $.954$ & $%
.953$ & $.817$ \\
$10000$ & $2$ & $2$ & $.5$ & $.5$ & $.001$ & --- & $.074$ & $.015$ & --- & $%
.012$ & $.946$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.021$ & $.025$ & --- & $.025$ & $.953$ & $%
.952$ & $.705$ \\
$10000$ & $4$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.245$ & $.016$ & $%
.013$ & $.007$ & $.951$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.028$ & $.013$ & $.013$ & $.012$ & $%
.949$ & $.949$ & $.243$ \\
$10000$ & $4$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.124$ & $.011$ & $%
.011$ & $.007$ & $.951$ & $.953$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.016$ & $.013$ & $.013$ & $.012$ & $.949
$ & $.949$ & $.629$ \\
$10000$ & $4$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.078$ & $.006$ & $.006
$ & $.006$ & $.952$ & $.953$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.030$ & $.013$ & $.013$ & $.013$ & $%
.948$ & $.949$ & $.283$ \\
$10000$ & $6$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.114$ & $.007$ & $%
.007$ & $.005$ & $.949$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.002$ & $.009$ & $.009$ & $.009$ & $%
.951$ & $.951$ & $.915$ \\
$10000$ & $6$ & $1$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.051$ & $.006$ & $%
.007$ & $.005$ & $.953$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.010$ & $.009$ & $.009$ & $.009$ & $.951
$ & $.951$ & $.760$ \\
$10000$ & $6$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.056$ & $.004$ & $.005
$ & $.004$ & $.951$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.022$ & $.009$ & $.009$ & $.009$ & $%
.952$ & $.952$ & $.299$ \\
$10000$ & $8$ & $0$ & $.5$ & $.5$ & $.000$ & $-.001$ & $-.061$ & $.005$ & $%
.005$ & $.004$ & $.947$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.003$ & $.007$ & $.007$ & $.008$ & $.946
$ & $.945$ & $.897$ \\
$10000$ & $8$ & $1$ & $.5$ & $.5$ & $.000$ & $.000$ & $-.025$ & $.004$ & $%
.005$ & $.004$ & $.950$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.005$ & $.008$ & $.008$ & $.007$ & $.946
$ & $.946$ & $.860$ \\
$10000$ & $8$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.042$ & $.003$ & $.004
$ & $.003$ & $.947$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.016$ & $.008$ & $.008$ & $.007$ & $%
.946$ & $.946$ & $.403$ \\
$10000$ & $16$ & $0$ & $.5$ & $.5$ & $.000$ & $.000$ & $-.010$ & $.003$ & $%
.003$ & $.003$ & $.951$ & $.948$ & $.013$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.002$ & $.005$ & $.005$ & $.005$ & $.950
$ & $.949$ & $.926$ \\
$10000$ & $16$ & $1$ & $.5$ & $.5$ & $.000$ & $.000$ & $-.001$ & $.002$ & $%
.003$ & $.002$ & $.949$ & $.947$ & $.872$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.950
$ & $.949$ & $.943$ \\
$10000$ & $16$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.017$ & $.002$ & $%
.002$ & $.002$ & $.948$ & $.945$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.006$ & $.005$ & $.005$ & $.005$ & $%
.950$ & $.949$ & $.726$ \\
$10000$ & $24$ & $0$ & $.5$ & $.5$ & $.000$ & $.000$ & $-.002$ & $.002$ & $%
.002$ & $.002$ & $.948$ & $.947$ & $.765$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.001$ & $.004$ & $.004$ & $.004$ & $.947
$ & $.947$ & $.942$ \\
$10000$ & $24$ & $1$ & $.5$ & $.5$ & $.000$ & $.000$ & $.001$ & $.002$ & $%
.002$ & $.002$ & $.954$ & $.946$ & $.844$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.000$ & $.004$ & $.004$ & $.004$ & $.954
$ & $.956$ & $.942$ \\
$10000$ & $24$ & $2$ & $.5$ & $.5$ & $.000$ & $.000$ & $.010$ & $.002$ & $%
.002$ & $.002$ & $.951$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.004$ & $.004$ & $.004$ & $.004$ & $%
.951$ & $.951$ & $.836$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% confiydence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.5$ & $.9$ & $-.051$ & --- & $-.546$ & $.072$ & --- &
$.015$ & $.843$ & $.933$ & $.000$ \\
&  &  &  & $.1$ & $-.003$ & --- & $-.027$ & $.024$ & --- & $.019$ & $.976$ &
$.957$ & $.473$ \\
$10000$ & $2$ & $1$ & $.5$ & $.9$ & $-.018$ & --- & $-.489$ & $.073$ & --- &
$.015$ & $.892$ & $.935$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $.024$ & $.024$ & --- & $.019$ & $.977$ & $%
.957$ & $.557$ \\
$10000$ & $2$ & $2$ & $.5$ & $.9$ & $.015$ & --- & $-.319$ & $.073$ & --- & $%
.015$ & $.947$ & $.947$ & $.000$ \\
&  &  &  & $.1$ & $-.002$ & --- & $.048$ & $.027$ & --- & $.021$ & $.966$ & $%
.952$ & $.191$ \\
$10000$ & $4$ & $0$ & $.5$ & $.9$ & $-.020$ & $-.022$ & $-.288$ & $.035$ & $%
.054$ & $.007$ & $.864$ & $.924$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & $-.001$ & $-.013$ & $.012$ & $.013$ & $.011$ & $%
.976$ & $.948$ & $.699$ \\
$10000$ & $4$ & $1$ & $.5$ & $.9$ & $.003$ & $-.014$ & $-.232$ & $.037$ & $%
.043$ & $.007$ & $.930$ & $.934$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.021$ & $.013$ & $.013$ & $.011$ & $.970
$ & $.948$ & $.426$ \\
$10000$ & $4$ & $2$ & $.5$ & $.9$ & $.001$ & $-.002$ & $-.096$ & $.017$ & $%
.016$ & $.007$ & $.955$ & $.951$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.022$ & $.013$ & $.013$ & $.012$ & $.949
$ & $.950$ & $.457$ \\
$10000$ & $6$ & $0$ & $.5$ & $.9$ & $-.007$ & $-.012$ & $-.198$ & $.023$ & $%
.025$ & $.005$ & $.901$ & $.918$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.007$ & $.009$ & $.009$ & $.009$ & $%
.973$ & $.951$ & $.816$ \\
$10000$ & $6$ & $1$ & $.5$ & $.9$ & $.004$ & $-.008$ & $-.145$ & $.024$ & $%
.020$ & $.005$ & $.952$ & $.925$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.016$ & $.009$ & $.009$ & $.009$ & $.962
$ & $.949$ & $.469$ \\
$10000$ & $6$ & $2$ & $.5$ & $.9$ & $.000$ & $-.002$ & $-.034$ & $.008$ & $%
.009$ & $.004$ & $.952$ & $.945$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.009$ & $.009$ & $.009$ & $.009$ & $.953
$ & $.952$ & $.789$ \\
$10000$ & $8$ & $0$ & $.5$ & $.9$ & $.000$ & $-.007$ & $-.150$ & $.019$ & $%
.014$ & $.004$ & $.919$ & $.919$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.004$ & $.007$ & $.007$ & $.007$ & $%
.968$ & $.946$ & $.867$ \\
$10000$ & $8$ & $1$ & $.5$ & $.9$ & $.002$ & $-.005$ & $-.103$ & $.014$ & $%
.012$ & $.004$ & $.958$ & $.932$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.012$ & $.008$ & $.008$ & $.007$ & $.948
$ & $.945$ & $.569$ \\
$10000$ & $8$ & $2$ & $.5$ & $.9$ & $.000$ & $-.001$ & $-.010$ & $.005$ & $%
.006$ & $.003$ & $.949$ & $.947$ & $.089$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.003$ & $.008$ & $.008$ & $.007$ & $.946
$ & $.947$ & $.912$ \\
$10000$ & $16$ & $0$ & $.5$ & $.9$ & $.000$ & $-.003$ & $-.069$ & $.006$ & $%
.005$ & $.002$ & $.963$ & $.914$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.950
$ & $.949$ & $.934$ \\
$10000$ & $16$ & $1$ & $.5$ & $.9$ & $.000$ & $-.002$ & $-.041$ & $.004$ & $%
.004$ & $.002$ & $.951$ & $.920$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $.005$ & $.951
$ & $.951$ & $.817$ \\
$10000$ & $16$ & $2$ & $.5$ & $.9$ & $.000$ & $-.001$ & $.009$ & $.002$ & $%
.002$ & $.002$ & $.951$ & $.939$ & $.001$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.005$ & $.005$ & $.005$ & $%
.950$ & $.951$ & $.920$ \\
$10000$ & $24$ & $0$ & $.5$ & $.9$ & $.000$ & $-.002$ & $-.040$ & $.003$ & $%
.003$ & $.002$ & $.951$ & $.902$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.004$ & $.004$ & $.004$ & $.949
$ & $.949$ & $.936$ \\
$10000$ & $24$ & $1$ & $.5$ & $.9$ & $.000$ & $-.001$ & $-.022$ & $.002$ & $%
.002$ & $.001$ & $.953$ & $.910$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.003$ & $.004$ & $.004$ & $.004$ & $.948
$ & $.948$ & $.889$ \\
$10000$ & $24$ & $2$ & $.5$ & $.9$ & $.000$ & $-.001$ & $.008$ & $.001$ & $%
.002$ & $.001$ & $.950$ & $.935$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.004$ & $.004$ & $.004$ & $%
.949$ & $.949$ & $.918$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.5$ & $.99$ & $-.053$ & --- & $-.505$ & $.072$ & ---
& $.015$ & $.839$ & $.831$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.003$ & $.024$ & --- & $.019$ & $.978$ &
$.983$ & $.833$ \\
$10000$ & $2$ & $1$ & $.5$ & $.99$ & $-.049$ & --- & $-.498$ & $.072$ & ---
& $.015$ & $.847$ & $.871$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.013$ & $.024$ & --- & $.019$ & $.977$ & $%
.940$ & $.753$ \\
$10000$ & $2$ & $2$ & $.5$ & $.99$ & $-.036$ & --- & $-.478$ & $.072$ & ---
& $.015$ & $.865$ & $.902$ & $.000$ \\
&  &  &  & $.01$ & $.002$ & --- & $.027$ & $.024$ & --- & $.019$ & $.977$ & $%
.926$ & $.501$ \\
$10000$ & $4$ & $0$ & $.5$ & $.99$ & $-.027$ & $-.532$ & $-.255$ & $.034$ & $%
.235$ & $.007$ & $.839$ & $.349$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.003$ & $-.001$ & $.012$ & $.011$ & $.011$ & $%
.975$ & $.949$ & $.908$ \\
$10000$ & $4$ & $1$ & $.5$ & $.99$ & $-.023$ & $-.207$ & $-.248$ & $.034$ & $%
.152$ & $.007$ & $.855$ & $.699$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.008$ & $.010$ & $.012$ & $.013$ & $.011$ & $%
.974$ & $.897$ & $.785$ \\
$10000$ & $4$ & $2$ & $.5$ & $.99$ & $-.011$ & $-.075$ & $-.228$ & $.034$ & $%
.095$ & $.007$ & $.888$ & $.861$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.006$ & $.019$ & $.013$ & $.014$ & $.011$ & $%
.973$ & $.922$ & $.487$ \\
$10000$ & $6$ & $0$ & $.5$ & $.99$ & $-.018$ & $-.342$ & $-.171$ & $.022$ & $%
.126$ & $.005$ & $.850$ & $.176$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.009$ & $.009$ & $.009$ & $%
.974$ & $.946$ & $.922$ \\
$10000$ & $6$ & $1$ & $.5$ & $.99$ & $-.013$ & $-.130$ & $-.165$ & $.022$ & $%
.075$ & $.005$ & $.871$ & $.585$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.006$ & $.008$ & $.009$ & $.009$ & $.009$ & $%
.975$ & $.895$ & $.797$ \\
$10000$ & $6$ & $2$ & $.5$ & $.99$ & $-.003$ & $-.041$ & $-.145$ & $.023$ & $%
.043$ & $.005$ & $.914$ & $.832$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.004$ & $.015$ & $.009$ & $.010$ & $.009$ & $%
.974$ & $.926$ & $.534$ \\
$10000$ & $8$ & $0$ & $.5$ & $.99$ & $-.014$ & $-.238$ & $-.130$ & $.017$ & $%
.079$ & $.004$ & $.845$ & $.098$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.007$ & $.007$ & $.007$ & $%
.973$ & $.944$ & $.928$ \\
$10000$ & $8$ & $1$ & $.5$ & $.99$ & $-.009$ & $-.090$ & $-.123$ & $.017$ & $%
.047$ & $.004$ & $.874$ & $.506$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.005$ & $.006$ & $.007$ & $.007$ & $.007$ & $%
.974$ & $.902$ & $.821$ \\
$10000$ & $8$ & $2$ & $.5$ & $.99$ & $.000$ & $-.026$ & $-.103$ & $.018$ & $%
.025$ & $.004$ & $.921$ & $.822$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.003$ & $.011$ & $.008$ & $.008$ & $.007$ & $%
.971$ & $.929$ & $.610$ \\
$10000$ & $16$ & $0$ & $.5$ & $.99$ & $-.007$ & $-.089$ & $-.067$ & $.008$ &
$.024$ & $.002$ & $.851$ & $.019$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.005$ & $.005$ & $.005$ & $%
.976$ & $.949$ & $.939$ \\
$10000$ & $16$ & $1$ & $.5$ & $.99$ & $-.003$ & $-.039$ & $-.060$ & $.008$ &
$.015$ & $.002$ & $.900$ & $.273$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.003$ & $.005$ & $.005$ & $.005$ & $%
.976$ & $.932$ & $.881$ \\
$10000$ & $16$ & $2$ & $.5$ & $.99$ & $.001$ & $-.009$ & $-.042$ & $.009$ & $%
.007$ & $.002$ & $.953$ & $.756$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.005$ & $.005$ & $.005$ & $.005$ & $%
.959$ & $.946$ & $.816$ \\
$10000$ & $24$ & $0$ & $.5$ & $.99$ & $-.004$ & $-.048$ & $-.046$ & $.006$ &
$.011$ & $.001$ & $.857$ & $.008$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.004$ & $.004$ & $.004$ & $%
.976$ & $.949$ & $.940$ \\
$10000$ & $24$ & $1$ & $.5$ & $.99$ & $-.001$ & $-.024$ & $-.039$ & $.006$ &
$.008$ & $.001$ & $.918$ & $.141$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.002$ & $.004$ & $.004$ & $.004$ & $%
.970$ & $.936$ & $.906$ \\
$10000$ & $24$ & $2$ & $.5$ & $.99$ & $.000$ & $-.005$ & $-.023$ & $.004$ & $%
.004$ & $.001$ & $.960$ & $.695$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.003$ & $.004$ & $.004$ & $.004$ & $%
.950$ & $.946$ & $.888$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.99$ & $.5$ & $.000$ & --- & $.386$ & $.009$ & --- & $%
.009$ & $.948$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.097$ & $.028$ & --- & $.036$ & $.952$ & $%
.952$ & $.113$ \\
$10000$ & $2$ & $1$ & $.99$ & $.5$ & $-.001$ & --- & $-.649$ & $.075$ & ---
& $.015$ & $.914$ & $.948$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $.006$ & $.028$ & --- & $.021$ & $.973$ & $%
.953$ & $.826$ \\
$10000$ & $2$ & $2$ & $.99$ & $.5$ & $.000$ & --- & $.441$ & $.008$ & --- & $%
.009$ & $.948$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & --- & $-.118$ & $.028$ & --- & $.038$ & $.953$ & $%
.953$ & $.046$ \\
$10000$ & $4$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.201$ & $.004$ & $%
.004$ & $.004$ & $.952$ & $.952$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.013$ & $.013$ & $.013$ & $.014$ & $.950
$ & $.949$ & $.783$ \\
$10000$ & $4$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.219$ & $.015$ & $%
.012$ & $.007$ & $.949$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.051$ & $.013$ & $.013$ & $.012$ & $.949
$ & $.948$ & $.007$ \\
$10000$ & $4$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.228$ & $.004$ & $%
.004$ & $.004$ & $.951$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.121$ & $.013$ & $.013$ & $.015$ & $%
.947$ & $.948$ & $.000$ \\
$10000$ & $6$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.127$ & $.003$ & $%
.003$ & $.003$ & $.946$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.013$ & $.008$ & $.008$ & $.009$ & $%
.948$ & $.948$ & $.623$ \\
$10000$ & $6$ & $1$ & $.99$ & $.5$ & $.000$ & $-.001$ & $-.092$ & $.007$ & $%
.007$ & $.005$ & $.947$ & $.946$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.035$ & $.009$ & $.009$ & $.008$ & $.949
$ & $.949$ & $.008$ \\
$10000$ & $6$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.146$ & $.003$ & $%
.003$ & $.003$ & $.950$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.095$ & $.009$ & $.009$ & $.009$ & $%
.949$ & $.948$ & $.000$ \\
$10000$ & $8$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.089$ & $.003$ & $%
.003$ & $.003$ & $.953$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.021$ & $.006$ & $.006$ & $.006$ & $%
.951$ & $.951$ & $.066$ \\
$10000$ & $8$ & $1$ & $.99$ & $.5$ & $.000$ & $.000$ & $-.044$ & $.005$ & $%
.005$ & $.004$ & $.949$ & $.950$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.021$ & $.007$ & $.007$ & $.006$ & $.948
$ & $.948$ & $.075$ \\
$10000$ & $8$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.103$ & $.002$ & $%
.002$ & $.003$ & $.949$ & $.949$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.075$ & $.006$ & $.006$ & $.006$ & $%
.949$ & $.947$ & $.000$ \\
$10000$ & $16$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.036$ & $.002$ & $%
.002$ & $.002$ & $.953$ & $.951$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.019$ & $.003$ & $.003$ & $.003$ & $%
.950$ & $.950$ & $.000$ \\
$10000$ & $16$ & $1$ & $.99$ & $.5$ & $.000$ & $.000$ & $-.001$ & $.002$ & $%
.003$ & $.002$ & $.953$ & $.950$ & $.887$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $.001$ & $.004$ & $.004$ & $.003$ & $.951
$ & $.951$ & $.939$ \\
$10000$ & $16$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.042$ & $.002$ & $%
.002$ & $.002$ & $.950$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.036$ & $.003$ & $.003$ & $.003$ & $%
.951$ & $.951$ & $.000$ \\
$10000$ & $24$ & $0$ & $.99$ & $.5$ & $.000$ & $.000$ & $.021$ & $.002$ & $%
.002$ & $.002$ & $.948$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.014$ & $.002$ & $.002$ & $.002$ & $%
.951$ & $.950$ & $.000$ \\
$10000$ & $24$ & $1$ & $.99$ & $.5$ & $.000$ & $.000$ & $.004$ & $.002$ & $%
.002$ & $.002$ & $.947$ & $.945$ & $.416$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.003$ & $.003$ & $.003$ & $.002$ & $%
.951$ & $.949$ & $.773$ \\
$10000$ & $24$ & $2$ & $.99$ & $.5$ & $.000$ & $.000$ & $.025$ & $.001$ & $%
.002$ & $.002$ & $.951$ & $.947$ & $.000$ \\
&  &  &  & $.5$ & $.000$ & $.000$ & $-.022$ & $.002$ & $.002$ & $.002$ & $%
.946$ & $.947$ & $.000$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.99$ & $.9$ & $.013$ & --- & $-.383$ & $.077$ & --- &
$.015$ & $.933$ & $.949$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & --- & $-.019$ & $.028$ & --- & $.023$ & $.967$ & $%
.953$ & $.701$ \\
$10000$ & $2$ & $1$ & $.99$ & $.9$ & $-.048$ & --- & $-.541$ & $.072$ & ---
& $.015$ & $.849$ & $.919$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & --- & $.004$ & $.027$ & --- & $.022$ & $.977$ & $%
.969$ & $.838$ \\
$10000$ & $2$ & $2$ & $.99$ & $.9$ & $.012$ & --- & $-.280$ & $.065$ & --- &
$.014$ & $.953$ & $.947$ & $.000$ \\
&  &  &  & $.1$ & $-.001$ & --- & $.018$ & $.028$ & --- & $.024$ & $.957$ & $%
.953$ & $.735$ \\
$10000$ & $4$ & $0$ & $.99$ & $.9$ & $.003$ & $-.001$ & $-.152$ & $.027$ & $%
.014$ & $.007$ & $.961$ & $.949$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.006$ & $.013$ & $.013$ & $.012$ & $%
.950$ & $.949$ & $.859$ \\
$10000$ & $4$ & $1$ & $.99$ & $.9$ & $-.016$ & $-.020$ & $-.281$ & $.035$ & $%
.052$ & $.007$ & $.877$ & $.924$ & $.000$ \\
&  &  &  & $.1$ & $.001$ & $.001$ & $.017$ & $.013$ & $.013$ & $.012$ & $.972
$ & $.947$ & $.567$ \\
$10000$ & $4$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.074$ & $.015$ & $%
.010$ & $.007$ & $.953$ & $.951$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.012$ & $.013$ & $.013$ & $.012$ & $.949
$ & $.948$ & $.759$ \\
$10000$ & $6$ & $0$ & $.99$ & $.9$ & $.001$ & $-.001$ & $-.082$ & $.012$ & $%
.008$ & $.005$ & $.959$ & $.950$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.002$ & $.008$ & $.008$ & $.008$ & $%
.947$ & $.948$ & $.908$ \\
$10000$ & $6$ & $1$ & $.99$ & $.9$ & $-.003$ & $-.010$ & $-.189$ & $.024$ & $%
.023$ & $.005$ & $.915$ & $.924$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.021$ & $.009$ & $.009$ & $.008$ & $.974
$ & $.946$ & $.196$ \\
$10000$ & $6$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.020$ & $.008$ & $%
.006$ & $.004$ & $.952$ & $.948$ & $.001$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.005$ & $.009$ & $.008$ & $.008$ & $.949
$ & $.948$ & $.871$ \\
$10000$ & $8$ & $0$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.051$ & $.007$ & $%
.005$ & $.004$ & $.953$ & $.946$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.001$ & $.006$ & $.006$ & $.006$ & $%
.949$ & $.950$ & $.925$ \\
$10000$ & $8$ & $1$ & $.99$ & $.9$ & $.002$ & $-.006$ & $-.140$ & $.019$ & $%
.013$ & $.004$ & $.933$ & $.927$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.022$ & $.007$ & $.007$ & $.006$ & $.971
$ & $.946$ & $.046$ \\
$10000$ & $8$ & $2$ & $.99$ & $.9$ & $.000$ & $-.001$ & $.000$ & $.005$ & $%
.004$ & $.003$ & $.949$ & $.947$ & $.906$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.000$ & $.006$ & $.006$ & $.006$ & $.949
$ & $.948$ & $.935$ \\
$10000$ & $16$ & $0$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.014$ & $.003$ & $%
.002$ & $.002$ & $.949$ & $.936$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.003$ & $.003$ & $.003$ & $.952
$ & $.952$ & $.931$ \\
$10000$ & $16$ & $1$ & $.99$ & $.9$ & $.000$ & $-.002$ & $-.058$ & $.005$ & $%
.004$ & $.002$ & $.957$ & $.922$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.017$ & $.003$ & $.003$ & $.003$ & $.952
$ & $.948$ & $.001$ \\
$10000$ & $16$ & $2$ & $.99$ & $.9$ & $.000$ & $.000$ & $.014$ & $.002$ & $%
.002$ & $.002$ & $.951$ & $.944$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.007$ & $.003$ & $.003$ & $.003$ & $%
.951$ & $.951$ & $.347$ \\
$10000$ & $24$ & $0$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.006$ & $.002$ & $%
.002$ & $.001$ & $.953$ & $.936$ & $.005$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $.001$ & $.002$ & $.002$ & $.002$ & $.955
$ & $.955$ & $.926$ \\
$10000$ & $24$ & $1$ & $.99$ & $.9$ & $.000$ & $-.001$ & $-.029$ & $.002$ & $%
.002$ & $.001$ & $.950$ & $.912$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.001$ & $.012$ & $.002$ & $.002$ & $.002$ & $.952
$ & $.946$ & $.001$ \\
$10000$ & $24$ & $2$ & $.99$ & $.9$ & $.000$ & $.000$ & $.013$ & $.001$ & $%
.001$ & $.001$ & $.949$ & $.937$ & $.000$ \\
&  &  &  & $.1$ & $.000$ & $.000$ & $-.008$ & $.002$ & $.002$ & $.002$ & $%
.953$ & $.952$ & $.038$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}





\clearpage
\addtocounter{table}{-1}
\begin{table}[h]    \small
\caption{Simulation results for the f\mbox{}irst-order autoregression with a covariate (cont'd)}
%\vspace{-.25cm}
\begin{center}
%\begin{tabular}{ccccc|rrr|rrr|rrr}
\begin{tabular}{cccccrrrrrrrrr}
\hline\hline
 & &  &  & & &  bias  & &  & std &   &  & $\mathrm{ci}_{.95}$ &    \T \\
 \cmidrule(l{6pt}r{6pt}){6-8}
\cmidrule(l{6pt}r{6pt}){9-11}
\cmidrule(l{6pt}r{6pt}){12-14}
{$N$} & {$T$} & {$\psi$} & {$\gamma$} & {$\theta_0$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$} &
{$\widehat{\rho }_{\mathrm{al}}$} & {$\widehat{\rho }_{\mathrm{ab}}$} & {$\widehat{\rho }_{\mathrm{hk}}$}
\T\B  \\ \hline \T
$10000$ & $2$ & $0$ & $.99$ & $.99$ & $-.052$ & --- & $-.503$ & $.072$ & ---
& $.015$ & $.841$ & $.911$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $-.003$ & $.027$ & --- & $.022$ & $.978$ &
$.970$ & $.841$ \\
$10000$ & $2$ & $1$ & $.99$ & $.99$ & $-.052$ & --- & $-.503$ & $.072$ & ---
& $.015$ & $.842$ & $.912$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & --- & $.003$ & $.027$ & --- & $.022$ & $.977$ & $%
.972$ & $.840$ \\
$10000$ & $2$ & $2$ & $.99$ & $.99$ & $-.041$ & --- & $-.485$ & $.072$ & ---
& $.015$ & $.858$ & $.947$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & --- & $.008$ & $.027$ & --- & $.022$ & $.977$ & $%
.955$ & $.819$ \\
$10000$ & $4$ & $0$ & $.99$ & $.99$ & $-.026$ & $-.106$ & $-.253$ & $.034$ &
$.112$ & $.007$ & $.843$ & $.823$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $-.001$ & $-.001$ & $.012$ & $.012$ & $.012$ & $%
.973$ & $.951$ & $.904$ \\
$10000$ & $4$ & $1$ & $.99$ & $.99$ & $-.026$ & $-.114$ & $-.253$ & $.034$ &
$.115$ & $.007$ & $.845$ & $.811$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.004$ & $.012$ & $.012$ & $.012$ & $%
.973$ & $.947$ & $.887$ \\
$10000$ & $4$ & $2$ & $.99$ & $.99$ & $-.016$ & $-.013$ & $-.236$ & $.034$ &
$.040$ & $.007$ & $.875$ & $.935$ & $.000$ \\
&  &  &  & $.01$ & $.001$ & $.000$ & $.009$ & $.013$ & $.013$ & $.012$ & $%
.972$ & $.947$ & $.821$ \\
$10000$ & $6$ & $0$ & $.99$ & $.99$ & $-.017$ & $-.080$ & $-.170$ & $.022$ &
$.060$ & $.005$ & $.853$ & $.722$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $-.001$ & $.008$ & $.008$ & $.008$ & $%
.973$ & $.948$ & $.916$ \\
$10000$ & $6$ & $1$ & $.99$ & $.99$ & $-.016$ & $-.085$ & $-.169$ & $.022$ &
$.061$ & $.005$ & $.856$ & $.715$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.004$ & $.008$ & $.008$ & $.008$ & $%
.973$ & $.943$ & $.875$ \\
$10000$ & $6$ & $2$ & $.99$ & $.99$ & $-.007$ & $-.010$ & $-.153$ & $.023$ &
$.022$ & $.005$ & $.900$ & $.922$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.009$ & $.008$ & $.008$ & $.008$ & $%
.973$ & $.946$ & $.745$ \\
$10000$ & $8$ & $0$ & $.99$ & $.99$ & $-.012$ & $-.062$ & $-.128$ & $.017$ &
$.039$ & $.004$ & $.853$ & $.636$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $-.001$ & $.006$ & $.006$ & $.006$ & $%
.975$ & $.949$ & $.922$ \\
$10000$ & $8$ & $1$ & $.99$ & $.99$ & $-.012$ & $-.065$ & $-.127$ & $.017$ &
$.039$ & $.004$ & $.854$ & $.624$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.005$ & $.006$ & $.006$ & $.006$ & $%
.975$ & $.938$ & $.842$ \\
$10000$ & $8$ & $2$ & $.99$ & $.99$ & $-.003$ & $-.008$ & $-.111$ & $.017$ &
$.014$ & $.004$ & $.909$ & $.907$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.008$ & $.006$ & $.006$ & $.006$ & $%
.974$ & $.948$ & $.651$ \\
$10000$ & $16$ & $0$ & $.99$ & $.99$ & $-.006$ & $-.032$ & $-.065$ & $.008$
& $.013$ & $.002$ & $.863$ & $.352$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.003$ & $.003$ & $.003$ & $%
.976$ & $.954$ & $.936$ \\
$10000$ & $16$ & $1$ & $.99$ & $.99$ & $-.005$ & $-.031$ & $-.064$ & $.008$
& $.013$ & $.002$ & $.874$ & $.363$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.005$ & $.003$ & $.003$ & $.003$ & $%
.975$ & $.898$ & $.630$ \\
$10000$ & $16$ & $2$ & $.99$ & $.99$ & $.001$ & $-.004$ & $-.050$ & $.009$ &
$.005$ & $.002$ & $.938$ & $.861$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.007$ & $.003$ & $.003$ & $.003$ & $%
.972$ & $.948$ & $.303$ \\
$10000$ & $24$ & $0$ & $.99$ & $.99$ & $-.003$ & $-.020$ & $-.044$ & $.006$
& $.007$ & $.001$ & $.880$ & $.200$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.000$ & $.000$ & $.002$ & $.002$ & $.002$ & $%
.979$ & $.954$ & $.939$ \\
$10000$ & $24$ & $1$ & $.99$ & $.99$ & $-.003$ & $-.019$ & $-.043$ & $.006$
& $.007$ & $.001$ & $.889$ & $.222$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.002$ & $.005$ & $.002$ & $.002$ & $.002$ & $%
.973$ & $.851$ & $.384$ \\
$10000$ & $24$ & $2$ & $.99$ & $.99$ & $.001$ & $-.003$ & $-.030$ & $.006$ &
$.003$ & $.001$ & $.952$ & $.794$ & $.000$ \\
&  &  &  & $.01$ & $.000$ & $.001$ & $.006$ & $.002$ & $.002$ & $.002$ & $%
.968$ & $.941$ & $.130$%

\\ \hline
\end{tabular}
\end{center}
\emph{Notes: }Data generated as $y_{it}=\theta_{01}y_{it-1}+\theta_{02}x_{it} +\alpha _{i}+\varepsilon_{it}$,
$x_{it}=.5\alpha _{i}+\gamma x_{it-1}+u_{it}$ $(i=1,...,N;t=1,...T)$ with $\alpha _{i}\sim\mathcal{N}(0,1)$, $\varepsilon_{it}\sim\mathcal{N}(0,1)$, $u_{it}\sim\mathcal{N}(0,.25)$, $\psi$ the degree of outlyingness of the initial observations $y_{i0}$, and $x_{i0}$ drawn from the stationary distribution. Entries: bias, standard deviation (std), and coverage rate of 95\% conf\mbox{}idence interval ($\mathrm{ci}_{.95}$) of adjusted likelihood ($\widehat{\theta}_{\mathrm{al}}$), Arellano-Bond ($\widehat{\theta}_{\mathrm{ab}}$), and Hahn-Kuersteiner ($\widehat{\theta}_{\mathrm{hk}}$) estimators; `---' indicates non-existence of the moment; $10,000$ Monte Carlo replications.
\end{table}




















\end{document} 