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\begin{document}

\title{{\sc Supplementary Material on} \textquotedblleft {\sc a general
class of non-nested test statistics for models defined through moment
restrictions}\textquotedblright }
\author{Paulo M.D.C. Parente \\
%EndAName
Instituto Universit\'{a}rio de Lisboa (ISCTE-IUL)\\
Business Research Unit (BRU-IUL)}
\date{This version: March 2017}
\maketitle

In this supplement we provide the proofs of the theorems presented in
Sections 4 and 5 of the paper, investigate the behavior of the random
variable $W_{i,j}^{\ast }(\delta _{h})$ $i=1,2,$ $j=3,4$ (defined in
subsection 5.1 of the paper) as some elements of $\delta _{h}$ approach
infinity, and present the results of the Monte Carlo study for the tests
based on the exponential tilting estimator. This supplement is organized as
follows. In Section SM1 we prove the relevant theorems of section 4. Section
SM2 provides the proofs of the theorems of section 5 and analyzes the limit
of $W_{i,j}^{\ast }(\delta _{h})$ $i=1,2,$ $j=3,4$ as some elements of $%
\delta _{h}$ approach infinity. Finally, SM3 presents the additional results
obtained in the Monte Carlo study for the tests based on the exponential
tilting estimator.

In what follows CR, CS, L, and T denote the $c_{r}$, Cauchy-Schwarz,
Lyapunov and triangle inequalities respectively. Furthermore, `with
probability approaching one' is abbreviated as `wpa$1$'. Unless stated
otherwise `LLN' corresponds to the Khinchin law of large numbers, `UWL'
denotes a uniform weak law of large numbers, as Lemma 2.4 of Newey and
McFadden (1994) or a uniform weak law of large numbers at the true parameter
as Lemma 4.3 of Newey and McFadden (1994) and `CLT' refers to the Lindeberg-L%
\'{e}vy central limit theorem. NS refers to Newey and Smith (2004).

\renewcommand{\thesection}{}

\setcounter{page}{1}\renewcommand{\thepage}{[SM.\arabic{page}]}%
\renewcommand{\thesection}{SM1}

\section{Proofs of results of section 4}

\setcounter{equation}{0}

\subsection{Proofs of the results of subsection 4.1}

The following Lemma generalizes Lemma A.1 of Ramalho and Smith (2004). Let $%
g_{i}(\beta )=g(z_{i},\beta )$ and $\hat{g}(\beta )=\sum
\nolimits_{i=1}^{n}g_{i}(\beta )/n$.

\begin{lemma}
\label{Lemma_a1}\noindent Let Assumptions 2.1, 2.2 and 4.1 (a) hold. Then $n%
\hat{p}_{i}^{v_{c}}=1+o_{p}(1)$ and 
\[
n^{1/2}\left( \hat{p}_{i}^{v_{c}}-\frac{1}{n}\right) =\kappa _{v_{c}}\frac{1%
}{n}\hat{g}_{i}^{\prime }n^{1/2}\hat{\lambda}(1+o_{p}(1))+O_{p}(n^{-3/2}), 
\]%
uniformly $(i=1,...,n)$ where $\hat{g}_{i}\equiv g(z_{i},\hat{\beta})$ and $%
\kappa _{v_{c}}=v_{c,1}(0)/v_{c}(0).$
\end{lemma}

{\bf Proof: }Let $b_{i}\equiv \sup_{\beta \in B}\left \vert g_{i}(\beta
)\right \vert $. From the proof of Lemma A1 and Theorem 3.1 in NS we have $%
\max_{1\leq i\leq n}b_{i}$ $=O_{p}(n^{-1/\alpha })$ and $\hat{\lambda}%
=O_{p}(n^{-1/2})$. Thus, $\sup_{\beta \in B,1\leq i\leq n}%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
\hat{\lambda}^{\prime }g_{i}(\beta )%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
=O_{p}(n^{-(1/2-1/\alpha )})$. A first order Taylor expansion of $v_{c}(\hat{%
\lambda}^{\prime }\hat{g}_{i})$ around zero yields $v_{c}(\hat{\lambda}%
^{\prime }\hat{g}_{i})=v_{c}(0)+v_{c,1}(\dot{\lambda}^{\prime }\hat{g}_{i})%
\hat{\lambda}^{\prime }\hat{g}_{i}$, where $\dot{\lambda}$ is on a line
joining $\hat{\lambda}$ and zero. Now $\max_{1\leq i\leq n}%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
v_{c,1}(\hat{\lambda}^{\prime }g_{i}(\hat{\beta}))-v_{c,1}(0)%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
=o_{p}(1)$ as $\sup_{\beta \in B,1\leq i\leq n}%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
\hat{\lambda}^{\prime }g_{i}(\beta )%
%TCIMACRO{\U{a6}}%
%BeginExpansion
{\vert}%
%EndExpansion
=o_{p}(1)$ and so $v_{c,1}(\hat{\lambda}^{\prime }g_{i}(\hat{\beta}))\hat{%
\lambda}^{\prime }\hat{g}_{i}=$ $v_{c,1}(0)\hat{\lambda}^{\prime }\hat{g}%
_{i}(1+o_{p}(1))$. Therefore $v_{c}(\hat{\lambda}^{\prime }\hat{g}%
_{i})=v_{c}(0)+v_{c,1}(0)\hat{\lambda}^{\prime }\hat{g}_{i}(1+o_{p}(1))$
uniformly ($i=1,...,n$). Similarly, 
\[
\left[ \sum_{j=1}^{n}v_{c}(\hat{\lambda}^{\prime }\hat{g}_{j})\right]
^{-1}=(v_{c}(0)n)^{-1}(1+O_{p}(n^{-1})) 
\]%
as $\sum_{j=1}^{n}\hat{g}_{j}/n=O_{p}(n^{-1/2})$ and $\hat{\lambda}%
=O_{p}(n^{-1/2})$ by Theorem 3.1 of NS. Hence $\hat{p}%
_{i}^{v_{c}}=[v_{c}(0)+v_{c,1}(0)\hat{\lambda}^{\prime }\hat{g}%
_{i}(1+o_{p}(1))](v_{c}(0)n)^{-1}(1+O_{p}(n^{-1}))$. It follows by Lemma A.1
of NS that $n\hat{p}_{i}^{v_{c}}=1+(v_{c,1}(0)/v_{c}(0))o_{p}(1)$ and that 
\[
n^{1/2}(\hat{p}_{i}^{v_{c}}-1/n)=(v_{c,1}(0)/v_{c}(0))n^{-1}\hat{g}%
_{i}^{\prime }n^{1/2}\hat{\lambda}(1+o_{p}(1))+O_{p}(n^{-3/2}). 
\]%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

\noindent {\bf Proof of Theorem 4.1:} By the mean value theorem $v_{a}\left( 
\hat{p}_{i}n\right) =v_{a}(1)+v_{a,1}\left( \hat{\sigma}_{i}\right) \left( 
\hat{p}_{i}n-1\right) $, where $\hat{\sigma}_{i}=\alpha _{i}+\left( 1-\alpha
_{i}\right) \hat{p}_{i}n$ and $\alpha _{i}\in \left( 0,1\right) $ and
consequently 
\[
{\cal S}_{v}=\sum\limits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}_{i}\right) 
\sqrt{n}\left( \hat{p}_{i}-\frac{1}{n}\right) n\hat{p}_{i}\left[ v_{b}\left( 
\frac{\hat{q}_{i}^{v_{c}}}{\hat{p}_{i}^{v_{c}}}\right) -\sum\limits_{\ell
=1}^{n}v\left( \frac{\hat{q}_{\ell }^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}%
\right) \hat{p}_{\ell }\right] . 
\]%
By Lemma \ref{Lemma_a1} ${\cal S}_{v}=\sum_{j=1}^{4}R_{j,n},$ where $%
R_{j,n},j=1,...,4$ are defined below.

Let us consider first 
\[
R_{1,n}\equiv \frac{1}{n}\sum\limits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) n\hat{p}_{i}v_{b}\left( \frac{v_{c}\left( \hat{\eta}^{\prime
}h_{i}\left( \hat{\gamma}\right) \right) }{n\hat{p}_{i}^{v_{c}}\hat{v}_{c,h}}%
\right) \hat{g}_{i}^{\prime }\sqrt{n}\hat{\lambda}(1+o_{p}(1)), 
\]%
where $\hat{v}_{c,h}\equiv \dsum\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}%
^{\prime }h_{i}\left( \hat{\gamma}\right) \right) /n$. Now by Lemma \ref%
{Lemma_a1}, $v_{a,1}\left( \hat{\sigma}_{i}\right) =v_{a,1}\left( 1\right)
+o_{p}(1)$. Additionally $n\hat{p}_{i}=1+o_{p}\left( 1\right) $ and $n\hat{p}%
_{i}^{v_{c}}=1+o_{p}\left( 1\right) $ uniformly in $i=1,...,n$ by Lemma \ref%
{Lemma_a1}. Also $\hat{v}_{c,h}={\rm E}_{\text{{\sc p}}_{0}}\left[
v_{c}\left( \eta ^{\ast \prime }h\left( z,\gamma ^{\ast }\right) \right) %
\right] +o_{p}(1)$ by a UWL. It follows using the fact that $\sqrt{n}\hat{%
\lambda}=O_{p}(1)$, and a UWL that 
\begin{equation}
R_{1,n}=A_{v}\sqrt{n}\hat{\lambda}(1+o_{p}(1))+o_{p}\left( 1\right) .
\label{R1n}
\end{equation}%
Hence by Theorem 3.2 of NS we have $R_{1,n}\overset{d}{\rightarrow }{\cal N}%
\left( 0,\sigma _{0}^{2}\right) $.

Let us consider now 
\[
R_{2,n}\equiv \frac{1}{n}\sum \limits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) n\hat{p}_{i}v_{b}\left( \frac{v_{c}\left( \hat{\eta}^{\prime
}h_{i}\left( \hat{\gamma}\right) \right) }{n\hat{p}_{i}^{v_{c}}\hat{v}_{c,h}}%
\right) O_{p}(n^{-1/2}). 
\]%
Using the same arguments as above $n^{-1}\sum
\nolimits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}_{i}\right) n\hat{p}%
_{i}v_{b}(v_{c}\left( \hat{\eta}^{\prime }h_{i}\left( \hat{\gamma}\right)
\right) /[n\hat{p}_{i}^{v_{c}}\hat{v}_{c,h}])=O_{p}(1)$. It follows that $%
R_{2,n}=O_{p}(n^{-1/2})$. Now define 
\[
R_{3,n}\equiv \frac{1}{n}\sum \limits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) n\hat{p}_{i}\hat{g}_{i}^{\prime }\sqrt{n}\hat{\lambda}%
(1+o_{p}(1))\sum \limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q}_{\ell
}^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }. 
\]%
Note that $n^{-1}\sum \nolimits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) n\hat{p}_{i}\hat{g}_{i}^{\prime }=o_{p}(1)$ by a UWL, $\sqrt{n}%
\hat{\lambda}(1+o_{p}(1))=O_{p}(1)$ by Theorem 3.2 of NS and that 
\[
\sum \limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q}_{\ell }^{v_{c}}}{\hat{p}%
_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }=\frac{1}{n}\sum \limits_{\ell
=1}^{n}v_{b}\left( \frac{v_{c}\left( \hat{\eta}^{\prime }h_{\ell }\left( 
\hat{\gamma}\right) \right) }{n\hat{p}_{\ell }^{v_{c}}\hat{v}_{c,h}}\right) n%
\hat{p}_{\ell } 
\]%
converges in probability to ${\rm E}_{\text{{\sc p}}_{0}}[v_{b}(v_{c}\left(
\eta ^{\ast \prime }h\left( z,\gamma ^{\ast }\right) \right) /{\rm E}_{\text{%
{\sc p}}_{0}}\left[ v_{c}\left( \eta ^{\ast \prime }h\left( z,\gamma ^{\ast
}\right) \right) \right] )]+o_{p}\left( 1\right) $ by a UWL. Hence $%
R_{3,n}=o_{p}(1)$. Finally, consider 
\[
R_{4,n}\equiv \frac{1}{n}\sum \limits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) O_{p}(n^{-1/2})\sum \limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q%
}_{\ell }^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }. 
\]%
Since by a UWL $n^{-1}\sum \nolimits_{i=1}^{n}v_{a,1}\left( \hat{\sigma}%
_{i}\right) =O_{p}(n^{-1/2})$ and $\sum \nolimits_{\ell =1}^{n}v_{b}\left( 
\hat{q}_{\ell }^{v_{c}}/\hat{p}_{\ell }^{v_{c}}\right) \hat{p}_{\ell
}=O_{p}(1)$ we have $R_{4,n}=o_{p}(1)$. Hence ${\cal S}_{v}\overset{d}{%
\rightarrow }{\cal N}\left( 0,\sigma _{0}^{2}\right) .$

The fact that ${\cal \tilde{S}}_{v}={\cal S}_{v}+o_{p}(1)$ follows from the
arguments above and the fact that $n\hat{p}_{i}^{v_{c}}=1+o_{p}\left(
1\right) $ by Lemma \ref{Lemma_a1}.

Concerning the Lagrange multiplier statistic ${\cal LM}_{v}=\hat{A}_{v}\sqrt{%
n}\hat{\lambda}$, note that ${\cal S}_{v}=R_{1.n}+o_{p}(1)$ and using $%
\left( \ref{R1n}\right) $ one obtains $R_{1.n}-{\cal LM}_{v}=[A_{0,v}-\hat{A}%
_{v}]\sqrt{n}\hat{\lambda}+A_{v}\sqrt{n}\hat{\lambda}o_{p}(1)+o_{p}(1)$.
Because $A_{0,v}-\hat{A}_{v}=o_{p}(1)$ and $\sqrt{n}\hat{\lambda}=O_{p}(1)$
it follows that $R_{1.n}-{\cal LM}_{v}=o_{p}(1)$.\ 

Finally we consider the statistic ${\cal J}_{v}=-\hat{A}_{v}\hat{\Omega}%
_{g}^{-1}\sqrt{n}\hat{g}(\hat{\beta})$. Note that NS proved in the proof of
Theorem 3.2 (p. 240) that $\hat{g}(\hat{\beta})=-\Omega _{g}\hat{\lambda}%
+o_{p}(n^{1/2})$. Therefore ${\cal J}_{v}-{\cal LM}_{v}=\left[ \hat{A}_{v}%
\hat{\Omega}_{g}^{-1}\Omega _{g}-\hat{A}_{v}\right] \sqrt{n}\hat{\lambda}%
+o_{p}(1),$ as $\hat{A}_{v}\hat{\Omega}_{g}^{-1}=O_{p}(1)$ and $\sqrt{n}\hat{%
\lambda}=O_{p}(1)$. Since $\hat{\Omega}_{g}^{-1}=\Omega _{g}^{-1}+o_{p}(1)$
and $\hat{A}_{v}=A_{0,v}+o_{p}(1)$, it follows that ${\cal J}_{v}-{\cal LM}%
_{v}=o_{p}(1)$.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\bigskip

\noindent {\bf Proof of Theorem 4.2:} We prove here only consistency of $%
\hat{\sigma}_{3}^{2}$ for $\sigma _{0}^{2}$ as the proof for the other
estimators $\hat{\sigma}_{j}^{2},j=1,2,4$ is simpler. First note that $\hat{P%
}\overset{p}{\rightarrow }P_{0,g}$ by a UWL and the Slutsky theorems under
Assumptions 2.1 and 2.3. Now note that 
\[
\hat{A}_{v,3}=\frac{1}{n}\dsum\nolimits_{i=1}^{n}v_{b}\left( \hat{q}%
_{i}^{v_{c}}/\hat{p}_{i}^{v_{c}}\right) \hat{g}_{i}^{\prime }n\hat{p}_{i}=%
\frac{1}{n}\dsum\nolimits_{i=1}^{n}v_{b}\left( \frac{v_{c}(\hat{\eta}%
^{\prime }h_{i}(\hat{\gamma}))}{\frac{1}{n}\sum\nolimits_{i=1}^{n}v_{c}(\hat{%
\eta}^{\prime }h_{i}(\hat{\gamma}))}\frac{1}{n\hat{p}_{i}}\right) \hat{g}%
_{i}^{\prime }n\hat{p}_{i} 
\]%
By a UWL $\sum\nolimits_{i=1}^{n}v_{c}(\hat{\eta}^{\prime }h_{i}(\hat{\gamma}%
))/n\overset{p}{\rightarrow }{\rm E}_{\text{{\sc p}}_{0}}(v_{c}(\eta ^{\ast
\prime }h(z,\gamma ^{\ast })))$. Also by Lemma \ref{Lemma_a1}, $n\hat{p}%
_{i}=1+o_{p}(1)$. Hence, by a UWL and continuity of $v_{b}\left( \cdot
\right) $ it follows that $\hat{A}_{v,3}={\rm E}_{\text{{\sc p}}%
_{0}}(v_{b}(v_{c}(\eta ^{\ast \prime }h(z,\gamma ^{\ast }))\left/ {\rm E}_{%
\text{{\sc p}}_{0}}[v_{c}(\eta ^{\ast \prime }h(z,\gamma ^{\ast }))]\right.
)g(z,\beta _{0}))+o_{p}(1)$. Hence consistency of $\hat{\sigma}_{3}^{2}$ for 
$\sigma _{0}^{2}$ follows from the Slutsky theorem.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\subsection{Proofs of the results of subsection 4.2}

\noindent {\bf Proof of Theorem 4.2:} Let us first consider ${\cal S}_{v}$.
Note that 
\begin{eqnarray*}
{\cal S}_{v}/\sqrt{n} &=&\frac{1}{n}\sum\limits_{i=1}^{n}v_{a}\left( \hat{p}%
_{i}n\right) n\hat{p}_{i}v_{b}\left( \frac{n\hat{q}_{i}^{v_{c}}}{n\hat{p}%
_{i}^{v_{c}}}\right) - \\
&&\frac{1}{n}\sum\limits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}%
_{i}\frac{1}{n}\sum\limits_{\ell =1}^{n}n\hat{p}_{\ell }v_{b}\left( \frac{n%
\hat{q}_{\ell }^{v_{c}}}{n\hat{p}_{\ell }^{v_{c}}}\right) .
\end{eqnarray*}%
Now notice that $n\hat{q}_{i}^{v_{c}}=v_{c}(\hat{\eta}^{\prime }h_{i}(\hat{%
\gamma}))\left/ \left[ \sum\nolimits_{i=1}^{n}v_{c}(\hat{\eta}^{\prime
}h_{i}(\hat{\gamma}))/n\right] \right. $, $n\hat{p}_{i}^{v_{c}}=v_{c}(\hat{%
\lambda}^{\prime }g_{i}(\hat{\beta}))\left/ \left[ \sum%
\nolimits_{i=1}^{n}v_{c}(\hat{\lambda}^{\prime }g_{i}(\hat{\beta}))/n\right]
\right. $ and $n\hat{p}_{i}=\rho _{1}(\hat{\lambda}^{\prime }g_{i}(\hat{\beta%
}))\left/ \left[ \sum\nolimits_{i=1}^{n}\rho _{1}(\hat{\lambda}^{\prime
}g_{i}(\hat{\beta}))/n\right] \right. $. A UWL implies that $%
\sum\nolimits_{i=1}^{n}v_{c}(\hat{\eta}^{\prime }h_{i}(\hat{\gamma})/n={\rm E%
}_{\text{{\sc p}}_{0}}[v_{c}(\eta ^{\ast \prime }h(z,\gamma ^{\ast
}))]+o_{p}(1)$, $\sum\nolimits_{i=1}^{n}v_{c}(\hat{\lambda}^{\prime }g_{i}(%
\hat{\beta}))/n={\rm E}_{\text{{\sc p}}_{0}}[v_{c}(g(z,\beta ^{\ast
}))]+o_{p}(1)$, $\sum\nolimits_{i=1}^{n}\rho _{1}(\hat{\lambda}^{\prime
}g_{i}(\hat{\beta}))/n={\rm E}_{\text{{\sc p}}_{0}}[\rho _{1}(g(z,\beta
^{\ast }))]+o_{p}(1)$. It follows by a UWL that 
\[
\frac{1}{n}\sum\limits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}%
_{i}v_{b}\left( \frac{n\hat{q}_{i}^{v_{c}}}{n\hat{p}_{i}^{v_{c}}}\right) =%
{\rm E}_{\text{{\sc p}}_{0}}\left[ v_{a}(\rho _{1}^{g,z})\rho
_{1}^{g,z}v_{b}\left( \frac{v_{c}^{h,z}}{v_{c}^{g,z}}\right) \right]
+o_{p}(1).
\]

Also 
\[
\frac{1}{n}\sum \limits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}%
_{i}={\rm E}_{\text{{\sc p}}_{0}}\left[ v_{a}(\rho _{1}^{g,z})\rho _{1}^{g,z}%
\right] +o_{p}(1) 
\]%
and 
\[
\frac{1}{n}\sum \nolimits_{\ell =1}^{n}n\hat{p}_{\ell }v_{b}\left( \frac{n%
\hat{q}_{\ell }^{v_{c}}}{n\hat{p}_{\ell }^{v_{c}}}\right) ={\rm E}_{\text{%
{\sc p}}_{0}}[\rho _{1}^{g,z}v_{b}\left( \frac{v_{c}^{h,z}}{v_{c}^{g,z}}%
\right) ]+o_{p}(1). 
\]

Because ${\rm E}_{\text{{\sc p}}_{0}}[v_{a}(\rho _{1}^{g,z})\rho
_{1}^{g,z}v_{b}(v_{c}^{h,z}/v_{c}^{g,z})]\neq {\rm E}_{\text{{\sc p}}%
_{0}}[v_{a}(\rho _{1}^{g,z})\rho _{1}^{g,z}]{\rm E}_{\text{{\sc p}}%
_{0}}[\rho _{1}^{g,z}v_{b}(v_{c}^{h,z}/v_{c}^{g,z})]$, ${\cal S}_{v}\overset{%
p}{\rightarrow }\pm \infty $.

Consider now ${\cal \tilde{S}}_{v}.$ Note that 
\begin{eqnarray*}
{\cal \tilde{S}}_{v}/\sqrt{n} &=&\frac{1}{n}\sum
\limits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}_{i}v_{b}\left( n%
\hat{q}_{i}^{v_{c}}\right) \\
&&-\frac{1}{n}\sum \limits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}%
_{i}\frac{1}{n}\sum \limits_{\ell =1}^{n}v_{b}\left( n\hat{q}_{\ell
}^{v_{c}}\right) n\hat{p}_{\ell }.
\end{eqnarray*}

Using similar arguments to those described above we have $\sum
\nolimits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}_{i}v_{b}\left( n%
\hat{q}_{i}^{v_{c}}\right) /n={\rm E}_{\text{{\sc p}}_{0}}[v_{a}(\rho
_{1}^{g,z})\rho _{1}^{g,z}v_{b}(v_{c}^{h,z})]+o_{p}(1)$, $\sum
\nolimits_{i=1}^{n}v_{a}\left( \hat{p}_{i}n\right) n\hat{p}_{i}/n={\rm E}_{%
\text{{\sc p}}_{0}}[v_{a}(\rho _{1}^{g,z})\rho _{1}^{g,z}]+o_{p}(1)$ and $%
\sum \nolimits_{\ell =1}^{n}n\hat{p}_{\ell }v_{b}\left( n\hat{q}_{\ell
}^{v_{c}}\right) /n={\rm E}_{\text{{\sc p}}_{0}}[\rho
_{1}^{g,z}v_{b}(v_{c}^{h,z})]+o_{p}(1)$. Since ${\rm E}_{\text{{\sc p}}%
_{0}}[v_{a}(\rho _{1}^{g,z})\rho _{1}^{g,z}v_{b}(v_{c}^{h,z})]\neq {\rm E}_{%
\text{{\sc p}}_{0}}[v_{a}(\rho _{1}^{g,z})\rho _{1}^{g,z}]{\rm E}_{\text{%
{\sc p}}_{0}}[\rho _{1}^{g,z}v_{b}(v_{c}^{h,z})]$, ${\cal \tilde{S}}_{v}%
\overset{p}{\rightarrow }\pm \infty .$

Concerning the Lagrange multiplier statistic ${\cal LM}_{v}=\hat{A}_{v}\sqrt{%
n}\hat{\lambda}$, note that ${\cal LM}_{v}/\sqrt{n}=\hat{A}_{v}\hat{\lambda}%
=A_{v}^{\ast }\lambda ^{\ast }+o_{p}(1)$. Since $A_{v}^{\ast }\lambda ^{\ast
}\neq 0$ it follows that ${\cal LM}_{v}\overset{p}{\rightarrow }\pm \infty $%
. Finally we consider the statistic ${\cal J}_{v}/\sqrt{n}=-\hat{A}_{v}\hat{%
\Omega}_{g}^{-1}\hat{g}(\hat{\beta})$. Note that ${\cal J}_{v}/\sqrt{n}%
\overset{p}{\rightarrow }-A_{v}^{\ast }\Omega _{g}^{\ast -1}{\rm E}_{\text{%
{\sc p}}_{0}}[g_{i}(\beta ^{\ast })]$. Given that $A_{v}^{\ast }\Omega
_{g}^{\ast -1}{\rm E}_{\text{{\sc p}}_{0}}[g_{i}(\beta ^{\ast })]\neq 0$, $%
{\cal J}_{v}\overset{p}{\rightarrow }\pm \infty $.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\subsection{Proofs of the results of subsection 4.3}

Let $\left\{ z_{in}\right\} _{i=1}^{n}$ be a triangular array which we
assume to be row wise independent and identically distributed (iid). Let $%
g_{in}(\beta )=g(z_{i,n},\beta )$, $\hat{g}(\beta
)=\sum\nolimits_{i=1}^{n}g_{i,n}(\beta )/n$, $\hat{g}_{in}\equiv g(z_{i,n},%
\hat{\beta})$ and $h_{in}\left( \gamma \right) =h\left( z_{i,n},\gamma
\right) $.

\begin{lemma}
\label{UWL_LOCalt}Under Assumption 4.6 the following result holds $%
\sup_{\beta \in {\cal B}}\left \Vert \hat{g}(\beta )-{\rm E}_{\text{{\sc p}}%
_{0n}}[g_{in}(\beta )]\right \Vert =o_{p}(1)$, and $\left \{ {\rm E}_{\text{%
{\sc p}}_{0n}}[g_{in}(\beta )]\right \} _{n=1}^{\infty }$ is uniformly
equicontinuous in $\beta \in {\cal B}$.
\end{lemma}

\noindent {\bf Proof:} We use the UWL corresponding to Theorem 4 of Andrews
(1992) together with the Weak Law of Large Numbers for Triangular Arrays
(Davidson, 1994, 19.9 Corollary p.301). Note that by the UWL $\sup_{\beta
\in {\cal B}}\left \vert \hat{g}(\beta )-{\rm E}_{\text{{\sc p}}%
_{n}}[g_{i,n}(\beta )]\right \vert =o_{p}(1)$, where the UWL applies because
the four sufficient conditions are satisfied. In particular the total
boundedness condition (BD) holds by Assumption 4.6 (c). Assumption 4.6 (e)
implies both the pointwise convergence condition (P-WLLN) (by the LLN) and
the domination condition (DM). The termwise stochastic equicontinuity (TSE)
condition is satisfied because 
\begin{equation}
{\rm E}_{\text{{\sc p}}_{0n}}[\sup_{\beta ,\beta ^{\prime }\in {\cal B}%
:\left \Vert \beta -\beta ^{\prime }\right \Vert \leq d}\left \Vert
g_{in}(\beta )-g_{in}(\beta ^{\prime })\right \Vert ]\leq {\rm E}_{\text{%
{\sc p}}_{0n}}[\sup_{\beta \in {\cal B}}\left \Vert \frac{\partial
g_{i,n}(\beta )}{\partial \beta }\right \Vert ]d\leq Cd,  \label{eqcont_g}
\end{equation}%
where the first inequality holds by a mean-value expansion (which relies on
Assumption 4.6 (d)) and the second holds by Assumption 4.6 (e). \ In
addition to guaranteeing TSE, equation (\ref{eqcont_g}) also shows the
uniform equicontinuity of $\left \{ {\rm E}_{\text{{\sc p}}%
_{0n}}[g(z_{in},\beta )]\right \} _{n=1}^{\infty }$.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{Lemma_convmat}Under Assumptions 4.6, 4.7 and 4.8 the following
results hold:

\begin{enumerate}
\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\hat{g}_{in}\hat{g}_{in}^{\prime
}-\Omega _{0,g}=o_{p}(1)$ if $\hat{\beta}\overset{p}{\rightarrow }\beta
_{0}; $

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\left \Vert \hat{g}%
_{in}\right
\Vert ^{2}-{\rm E}_{\text{{\sc p}}_{0n}}[\left \Vert
g_{in}(\beta _{0})\right \Vert ^{2}]=o_{p}(1)$ if $\hat{\beta}\overset{p}{%
\rightarrow }\beta _{0};$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\frac{\partial g_{in}(\hat{\beta})%
}{\partial \beta ^{\prime }}-D_{0,g}=o_{p}(1)$ if $\hat{\beta}\overset{p}{%
\rightarrow }\beta _{0};$

\item $\frac{1}{n}\sum\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime
}h_{in}\left( \hat{\gamma}\right) \right) -{\rm E}_{\text{{\sc p}}%
_{0n}}[v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast }\right)
)]=o_{p}(1)$ if $\hat{\beta}\overset{p}{\rightarrow }\beta _{0}$, $\hat{%
\gamma}\overset{p}{\rightarrow }\gamma ^{\ast }$ and $\hat{\eta}\overset{p}{%
\rightarrow }\eta ^{\ast };$

\item $\frac{1}{n}\sum\nolimits_{i=1}^{n}v_{b}\left( \frac{v_{c}\left( \hat{%
\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) }{%
\sum\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime }h_{in}\left( \hat{%
\gamma}\right) \right) /n}\right) -{\rm E}_{\text{{\sc p}}_{0n}}[v_{b}(\frac{%
v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast }\right) )}{{\rm E}_{%
\text{{\sc p}}_{0n}}[v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast
}\right) )]})]=o_{p}(1)$ if $\hat{\beta}\overset{p}{\rightarrow }\beta _{0}$%
, $\hat{\gamma}\overset{p}{\rightarrow }\gamma ^{\ast }$ and $\hat{\eta}%
\overset{p}{\rightarrow }\eta ^{\ast };$

\item $\frac{1}{n}\sum\nolimits_{i=1}^{n}v_{b}\left( \frac{v_{c}\left( \hat{%
\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) }{%
\sum\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime }h_{in}\left( \hat{%
\gamma}\right) \right) /n}\right) )\hat{g}_{in}-A_{0,v}=o_{p}(1),$ if $\hat{%
\beta}\overset{p}{\rightarrow }\beta _{0}$, $\hat{\gamma}\overset{p}{%
\rightarrow }\gamma ^{\ast }$ and $\hat{\eta}\overset{p}{\rightarrow }\eta
^{\ast };$
\end{enumerate}
\end{lemma}

\noindent {\bf Proof: }We prove results (1), (5) and (6) as the proofs of
the remaining results are similar. Proof of 1: Using a proof similar to that
Lemma \ref{UWL_LOCalt} we have $\sup_{\beta \in {\cal B}}\left\Vert
\sum\nolimits_{i=1}^{n}g_{i,n}(\beta )g_{i,n}(\beta )^{\prime }/n-{\rm E}_{%
\text{{\sc p}}_{0n}}[g_{i,n}(\beta )g_{i,n}(\beta )^{\prime }]\right\Vert
=o_{p}(1)$ which relies on Assumptions 4.6 (c), (d) and (e) and CS. Now we
use the fact that ${\rm E}_{\text{{\sc p}}_{0n}}[g_{i,n}(\beta
_{0})g_{i,n}(\beta _{0})^{\prime }]\rightarrow \Omega _{0,g}$ by Assumption
4.6 (f).

Concerning (5) and (6), write $\hat{a}_{n}=\sum\nolimits_{i=1}^{n}v_{c}%
\left( \hat{\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) /n$ and $%
a_{n}={\rm E}_{\text{{\sc p}}_{0n}}[v_{c}(\eta ^{\ast \prime }h_{in}\left(
\gamma ^{\ast }\right) )]$, we know by Assumption 4.8 (c) that $a_{n}\in 
{\cal A}$, $n\geq 1$. Let $\psi =\left( \beta ^{\prime },\gamma ^{\prime
},\eta ^{\prime },a^{\prime }\right) ^{\prime }$. Using a proof similar to
that of Lemma \ref{UWL_LOCalt} we have $\sup_{\psi \in {\cal B}\times {\cal G%
}\times {\cal H}\times {\cal A}}||\sum\nolimits_{i=1}^{n}v_{b}\left(
v_{c}\left( \eta ^{\prime }h_{in}\left( \gamma \right) \right) \left/
a\right. \right) )g\left( z,\beta \right) /n-{\rm E}_{\text{{\sc p}}%
_{0n}}[v_{b}\left( v_{c}\left( \eta ^{\prime }h_{in}\left( \gamma \right)
\right) \left/ a\right. \right) )g_{in}\left( \beta \right) ]||=o_{p}(1)$
using Assumptions 4.6 (c), (d) , (e), 4.7 (a), 4.8 (a), (b) and (c) and CS
and consequently by result (4) we have $\sum\nolimits_{i=1}^{n}v_{b}\left(
v_{c}\left( \hat{\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) /%
\hat{a}_{n}\right) \hat{g}_{in}/n-{\rm E}_{\text{{\sc p}}_{0n}}[v_{b}(v_{c}(%
\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast }\right) )/a_{n})g_{in}\left(
\beta _{0}\right) ]=o_{p}(1)$ which proves (5). The conclusion (6) follows
from the fact that ${\rm E}_{\text{{\sc p}}_{0n}}[v_{b}(v_{c}(\eta ^{\ast
\prime }h_{in}\left( \gamma ^{\ast }\right) )/a_{n})g_{in}\left( \beta
_{0}\right) ]\rightarrow A_{0,v}$ by Assumption 4.8 (d).%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{Lemma_CLTlocalt}If Assumption 4.6 is satisfied, then $\sqrt{n}\hat{g}%
\left( \beta _{0,n}\right) \overset{D}{\rightarrow }{\cal N}(\delta
_{g},\Omega _{0,g})$.
\end{lemma}

\noindent {\bf Proof:} First we use the Cram\'{e}r Wold device to show that 
\[
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}[g_{in}(\beta _{0,n})-{\rm E}_{\text{{\sc p}}%
_{0n}}(g_{in}(\beta _{0,n}))]\overset{D}{\rightarrow }{\cal N}(0,\Omega
_{0,g}). 
\]%
That is, we show that for a fixed $\lambda \neq 0$ 
\begin{equation}
\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\lambda ^{\prime }[g_{in}(\beta
_{0,n})-{\rm E}_{\text{{\sc p}}_{0n}}(g_{in}(\beta _{0,n}))]}{\sqrt{\lambda
^{\prime }B_{n}\lambda }}\overset{D}{\rightarrow }{\cal N}(0,1),
\label{CLTloc}
\end{equation}%
where $B_{n}={\rm E}_{\text{{\sc p}}_{0n}}([g_{in}(\beta _{0,n})-{\rm E}_{%
\text{{\sc p}}_{0n}}(g_{in}(\beta _{0,n}))][g_{in}(\beta _{0,n})-{\rm E}_{%
\text{{\sc p}}_{0n}}(g_{in}(\beta _{0,n}))]^{\prime })\rightarrow \Omega
_{0,g}$. Note first that 
\[
B_{n}={\rm E}_{\text{{\sc p}}_{0n}}\left[ g_{in}(\beta _{0,n})g_{in}(\beta
_{0,n})^{\prime }\right] -n{\rm E}_{\text{{\sc p}}_{0n}}(g_{in}(\beta
_{0,n})){\rm E}_{\text{{\sc p}}_{0n}}(g_{in}(\beta _{0,n}))^{\prime }/n. 
\]%
Now $n{\rm E}_{\text{{\sc p}}_{0n}}(g_{in}(\beta _{0,n})){\rm E}_{\text{{\sc %
p}}_{0n}}(g_{in}(\beta _{0,n}))^{\prime }=\delta _{g}\delta _{g}^{\prime }$.
Additionally, $\left\Vert {\rm E}_{\text{{\sc p}}_{0n}}\left[ g_{in}(\beta
_{0,n})g_{in}(\beta _{0,n})^{\prime }\right] -{\rm E}_{\text{{\sc p}}_{0n}}%
\left[ g_{in}(\beta _{0})g_{in}(\beta _{0})^{\prime }\right] \right\Vert
\rightarrow 0$ as $\beta _{0,n}\rightarrow \beta _{0}$ because%
\[
{\rm E}_{\text{{\sc p}}_{0n}}[\sup_{\beta _{a},\beta _{b}\in {\cal B}%
:\left\Vert \beta _{a}-\beta _{b}\right\Vert \leq d}\left\Vert g_{in}(\beta
_{a})g_{in}(\beta _{a})^{\prime }-g_{in}(\beta _{b})g_{in}(\beta
_{b})^{\prime }\right\Vert ]\leq Cd, 
\]%
by a mean-value expansion (which holds by Assumption 4.6 (d)), Assumption
4.6 (e) and CS. It follows from Assumption 4.6 (f) that $B_{n}\rightarrow
\Omega _{0,g}$. Also note that $B_{n}$ is positive definite for $n$ large
enough because $\Omega _{0,g}$ is positive definite. Now for $a=2+\delta $
we have by CR, L, CS and Assumption 4.6 (e) 
\begin{eqnarray*}
{\rm E}_{\text{{\sc p}}_{0n}}\left[ \left\vert \lambda ^{\prime
}[g_{in}(\beta _{0,n})-{\rm E}_{\text{{\sc p}}_{0n}}(g_{in}(\beta
_{0,n}))]\right\vert ^{a}\right] &\leq &2^{a-1}\left[ {\rm E}_{\text{{\sc p}}%
_{0n}}\left\Vert \lambda ^{\prime }g_{in}(\beta _{0,n})\right\Vert
^{a}+\left\vert {\rm E}_{\text{{\sc p}}_{0n}}(\lambda ^{\prime }g_{in}(\beta
_{0,n}))\right\vert ^{a}\right] \\
&\leq &2^{a}\left[ \left\Vert \lambda \right\Vert ^{a}{\rm E}_{\text{{\sc p}}%
_{0n}}[\left\Vert g_{in}(\beta _{0,n})\right\Vert ^{a}\right]
<2^{a}\left\Vert \lambda \right\Vert ^{a}C.
\end{eqnarray*}%
Therefore 
\[
\frac{1}{n^{a/2}}\sum_{i=1}^{n}{\rm E}_{\text{{\sc p}}_{0n}}\left[
\left\vert \lambda ^{\prime }[g_{in}(\beta _{0,n})-{\rm E}_{\text{{\sc p}}%
_{0n}}(g_{in}(\beta _{0,n}))]\right\vert ^{a}\right] \leq \frac{%
2^{a}\left\Vert \lambda \right\Vert ^{a}C}{n^{a/2-1}}\rightarrow 0. 
\]%
Hence by the Lyapunov CLT (Serfling, 1980, p.31-32, Corollary) it follows
that $\left( \ref{CLTloc}\right) $ holds. Now note that 
\[
\sqrt{n}\hat{g}\left( \beta _{0,n}\right) =\sqrt{n}\left[ \hat{g}\left(
\beta _{0,n}\right) -{\rm E}_{\text{{\sc p}}_{0n}}(\hat{g}\left( \beta
_{0,n}\right) )\right] +\sqrt{n}{\rm E}_{\text{{\sc p}}_{0n}}(\hat{g}\left(
\beta _{0,n}\right) ) 
\]%
and the first term converges to ${\cal N}(0,\Omega _{0,g})$ while the second
is equal to $\delta _{g}$ which proves the result. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
.

Lemmata \ref{LA1} to \ref{LA3} correspond to versions of Lemmata A1 to A3 of
NS for iid triangular arrays and the proofs are similar to those Lemmata
given in NS (with $\beta _{0}$ replaced by $\beta _{0,n}$ in those proofs)
and therefore are omitted.

\begin{lemma}
\label{LA1}If Assumption 4.6 is satisfied, then for any $1/\alpha <\zeta
<1/2 $ and $\Lambda _{n}=\left\{ \lambda :\left\Vert \lambda \right\Vert
\leq n^{-\zeta }\right\} $ and with wpa$1$ $\Lambda _{n}\subseteq \hat{%
\Lambda}_{n}(\beta )$ for all $\beta \in {\cal B}$.
\end{lemma}

\begin{lemma}
\label{LA2}If Assumption 4.6 is satisfied, $\bar{\beta}\in {\cal B}$, $\bar{%
\beta}-\beta _{0,n}\overset{p}{\rightarrow }0$ and $\hat{g}(\bar{\beta}%
)=O_{p}(n^{-1/2})$, then $\bar{\lambda}=\arg \max_{\lambda \in \hat{\Lambda}%
_{n}(\bar{\beta})}\hat{P}_{n}^{g}(\bar{\beta},\lambda )$ exists wpa$1$, and $%
\bar{\lambda}=O_{p}(n^{-1/2})$, $\sup_{\lambda \in \hat{\Lambda}_{n}(\bar{%
\beta})}\hat{P}_{n}^{g}(\bar{\beta},\lambda )\leq O_{p}(n^{-1})$.
\end{lemma}

\begin{lemma}
\label{LA3}If Assumption 4.6 is satisfied, then $\left \Vert \hat{g}(\hat{%
\beta})\right \Vert =O_{p}(n^{-1/2})$.
\end{lemma}

The proof of the following Lemma follows the same steps of the proof of
Theorem 3.1 of NS, but since there are some small differences we present it
below.

\begin{lemma}
\label{LA_GELcons}If Assumption 4.6 is satisfied, then $\hat{\beta}\overset{p%
}{\rightarrow }\beta _{0}$, $\hat{\beta}-\beta _{0,n}\overset{p}{\rightarrow 
}0$, $g(\hat{\beta})=O_{p}(n^{-1/2})$, $\hat{\lambda}=\arg \max_{\lambda \in 
\hat{\Lambda}_{n}(\hat{\beta})}\sum\nolimits_{i=1}^{n}\rho (\lambda ^{\prime
}g_{in}(\hat{\beta}))/n$ exists wpa$1$, and $\hat{\lambda}=O_{p}(n^{-1/2})$.
\end{lemma}

{}{\bf Proof:} Let $g_{n}(\beta )={\rm E}_{\text{{\sc p}}_{0n}}[g(z_{in},%
\beta )]$. By Lemma \ref{LA3} $\hat{g}(\hat{\beta})=o_{p}(1)$, and by Lemma %
\ref{UWL_LOCalt} $\sup_{\beta \in {\cal B}}\left\Vert \hat{g}(\beta
)-g_{n}(\beta )]\right\Vert \overset{p}{\rightarrow }0$ and $\left\{
g_{n}(\beta )\right\} _{n=1}^{\infty }$ is uniformly equicontinuous.
Additionally, since $\lim_{n\rightarrow \infty }g_{n}(\beta )=g(\beta )$ for
each $\beta \in {\cal B}$, we have $\sup_{\beta \in {\cal B}}\left\Vert
g_{n}(\beta )-g(\beta )\right\Vert \rightarrow 0$ (see Rudin, 1976, Exercise
16, p.168). Hence by T $g(\hat{\beta})\overset{p}{\rightarrow }0$. Since $%
g(\beta )=0$ has a unique zero at $\beta _{0}$, $g(\beta )$ must be bounded
away from zero outside any neighborhood of $\beta _{0}$. Therefore, $\hat{%
\beta}$ must be inside any neighborhood of $\beta _{0}$ wpa$1$, i.e. $\hat{%
\beta}\overset{p}{\rightarrow }\beta _{0}$, giving the first conclusion. The
second conclusion follows from the inequality $\left\Vert \hat{\beta}-\beta
_{0,n}\right\Vert \leq \left\Vert \hat{\beta}-\beta _{0}\right\Vert
+\left\Vert \beta _{0}-\beta _{0,n}\right\Vert $, the first conclusion and
the fact that $\left\Vert \beta _{0}-\beta _{0,n}\right\Vert \rightarrow 0$
by Assumption 4.6 (a). The third conclusion is due to Lemma \ref{LA3}. Also,
note that by the second and third conclusions the hypotheses of Lemma \ref%
{LA2} are satisfied for $\bar{\beta}=\hat{\beta}$, so that the last
conclusion follows from Lemma \ref{LA2}. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{asymp_norm}If Assumption 4.6 is satisfied, then 
\[
\sqrt{n}\left( 
\begin{array}{c}
\hat{\beta}-\beta _{0,n} \\ 
\hat{\lambda}%
\end{array}%
\right) \overset{d}{\rightarrow }{\cal N}(\left( 
\begin{array}{c}
-H_{0,g}\delta _{g} \\ 
-P_{0,g}\delta _{g}%
\end{array}%
\right) ,\left( 
\begin{array}{cc}
\Sigma _{0,g} & 0 \\ 
0 & P_{0,g}%
\end{array}%
\right) ). 
\]
\end{lemma}

{\bf Proof: }Let $\hat{\theta}=(\hat{\beta}^{\prime },\hat{\lambda}^{\prime
})^{\prime }$ and $\theta _{0,n}=\left( \beta _{0,n}^{\prime },0^{\prime
}\right) ^{\prime }$. Note that since $\beta _{0}\in int\left( {\cal B}%
\right) $ and $\beta _{0,n}\rightarrow \beta _{0}$, then $\beta _{0,n}\in
int\left( {\cal B}\right) $ for $n$ large enough. Using arguments similar to
those of NS in the proof of their Theorem 3.2 (which in our case are based
on a first order Taylor expansion of the first order conditions of the GEL
objective function around $\theta _{0,n}$ and require the fact that $\beta
_{0,n}\rightarrow \beta _{0}$, Lemma \ref{LA_GELcons} and the Lemma \ref%
{Lemma_convmat}) we have 
\begin{equation}
\sqrt{n}(\hat{\theta}-\theta _{0,n})=-(H_{0,g}^{\prime },-P_{0,g})\sqrt{n}%
\hat{g}(\beta _{0,n})+o_{p}(1).  \label{lamb_TA}
\end{equation}%
Now apply the CLT given by Lemma \ref{Lemma_CLTlocalt}. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

The following Lemma generalizes the results of Lemma \ref{Lemma_a1} for
triangular arrays and its proof is similar and therefore omitted, though
available upon request.

\begin{lemma}
\label{Lemma_a2}Let Assumptions 4.6, 4.7 and 4.8 hold. Then $n\hat{p}%
_{i}^{v_{c}}=1+o_{p}(1)$ and 
\[
n^{1/2}\left( \hat{p}_{i}^{v_{c}}-\frac{1}{n}\right) =\kappa _{v_{c}}\frac{1%
}{n}\hat{g}_{in}^{\prime }n^{1/2}\hat{\lambda}(1+o_{p}(1))+O_{p}(n^{-3/2}), 
\]%
uniformly $(i=1,...,n)$ where $\kappa _{v_{c}}=v_{c,1}(0)/v_{c}(0)$.
\end{lemma}

\noindent {\bf Proof of Theorem 4.3: }Using the similar arguments to those
used in the proof of Theorem 4.1, which require \ref{Lemma_convmat} rather
than Lemma 4.3 of Newey and McFadden (1994) and Lemma \ref{Lemma_a2} we have 
${\cal S}_{v}=A_{0,v}\sqrt{n}\hat{\lambda}(1+o_{p}(1))+o_{p}\left( 1\right) $%
. Now, given\ that by Lemma \ref{asymp_norm} $\sqrt{n}\hat{\lambda}\overset{d%
}{\rightarrow }{\cal N}\left( -P_{0,g}\delta _{g},P_{0,g}\right) $, it
follows that ${\cal N}\left( -A_{0,v}P_{0,g}\delta _{g},\sigma
_{0}^{2}\right) $.

The demonstration of the asymptotic equivalence of the statistics ${\cal S}%
_{v}$, ${\cal \tilde{S}}_{v}$, ${\cal LM}_{v}$ and ${\cal J}_{v}$ is similar
to the proof of asymptotic equivalence of these statistics given in the
proof of Theorem 4.2. 
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion

\renewcommand{\thesection}{SM2}

\section{Proofs of the results of section 5 and discussion}

In this section we provide the proofs of the theorems presented in section 5
of the paper and investigate the behavior of the random variable $%
W_{i,j}^{\ast }(\delta _{h})$ $i=1,2$, $j=3,4$ (defined in subsection 5.1 of
the paper) as some elements of $\delta _{h}$ approach infinity.

\subsection{Proofs of the results of section 5}

We start by compiling a number of Lemmata without presenting their proofs
either because the proofs are very similar to those given in NS or to those
provided in the previous sections. Let $g_{in}(\beta )=g(z_{i,n},\beta )$, $%
\hat{g}(\beta )=\sum\nolimits_{i=1}^{n}g_{i,n}(\beta )/n$, $\hat{g}%
_{in}=g_{in}(\hat{\beta})$ and $h_{in}\left( \gamma \right) =h\left(
z_{i,n},\gamma \right) $, $\hat{h}_{in}\equiv h(z_{i,n},\hat{\gamma})$, $%
\hat{h}(\gamma )=\sum\nolimits_{i=1}^{n}h_{in}(\gamma )/n,$\ $s_{in}(\varphi
)=s(z_{i,n},\varphi )$ and $\hat{s}(\varphi
)=\sum\nolimits_{i=1}^{n}s_{in}(\varphi )/n$.

The proofs of Lemmata \ref{UWL_LOC1} to \ref{Lemma_CLTthloc} are similar to
the proofs of Lemmata \ref{UWL_LOCalt} to \ref{Lemma_CLTlocalt} above.

\begin{lemma}
\label{UWL_LOC1}Suppose Assumption 5.1 holds. Under a sequence $\left \{ 
\text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\cal P}$ the following
results hold:

\begin{enumerate}
\item $\sup_{\beta \in {\cal B}}\left \Vert \hat{g}(\beta )-{\rm E}_{\text{%
{\sc p}}_{n}}[g_{in}(\beta )]\right \Vert =o_{p}(1)$, and $\left \{ {\rm E}_{%
\text{{\sc p}}_{n}}[g_{in}(\beta )]\right \} _{n=1}^{\infty }$ is uniformly
equicontinuous in $\beta \in {\cal B}$.

\item $\sup_{\gamma \in {\cal G}}\left \Vert \hat{h}(\gamma )-{\rm E}_{\text{%
{\sc p}}_{n}}[h_{in}(\gamma )]\right \Vert =o_{p}(1)$, and $\left \{ {\rm E}%
_{\text{{\sc p}}_{n}}[h_{in}(\gamma )]\right \} _{n=1}^{\infty }$ is
uniformly equicontinuous in $\gamma \in {\cal G}$.
\end{enumerate}
\end{lemma}

\begin{lemma}
\label{Lemmata_D_loc}Suppose Assumption 5.1 holds. Under a sequence $\left\{ 
\text{{\sc p}}_{n}\right\} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $ we have:

\begin{enumerate}
\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}s_{in}(\hat{\varphi})s_{in}(\hat{%
\varphi})^{\prime }-\Omega =o_{p}(1)$ if $\hat{\varphi}\overset{p}{%
\rightarrow }\varphi ^{\ast };$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\left \Vert s_{in}(\hat{\varphi}%
)\right \Vert ^{2}-{\rm E}_{\text{{\sc p}}_{n}}[\left \Vert s_{in}(\varphi
^{\ast })\right \Vert ^{2}]=o_{p}(1)$ if $\hat{\varphi}\overset{p}{%
\rightarrow }\varphi ^{\ast };$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\frac{\partial g_{in}(\hat{\beta})%
}{\partial \beta ^{\prime }}-D_{g}=o_{p}(1)$ if $\hat{\beta}\overset{p}{%
\rightarrow }\beta _{0};$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}\frac{\partial h_{in}(\hat{\gamma})%
}{\partial \gamma ^{\prime }}-D_{h}=o_{p}(1)$ if $\hat{\gamma}\overset{p}{%
\rightarrow }\gamma ^{\ast };$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime
}h_{in}\left( \hat{\gamma}\right) \right) -{\rm E}_{\text{{\sc p}}%
_{n}}[v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast }\right)
)]=o_{p}(1)$ if $\hat{\beta}\overset{p}{\rightarrow }\beta _{0}$, $\hat{%
\gamma}\overset{p}{\rightarrow }\gamma ^{\ast }$ and $\hat{\eta}\overset{p}{%
\rightarrow }\eta ^{\ast };$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}v_{b}\left( \frac{v_{c}\left( \hat{%
\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) }{\sum
\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime }h_{in}\left( \hat{\gamma}%
\right) \right) /n}\right) -{\rm E}_{\text{{\sc p}}_{n}}[v_{b}(\frac{%
v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast }\right) )}{{\rm E}_{%
\text{{\sc p}}_{n}}[v_{c}(\eta ^{\ast \prime }h_{in}\left( \gamma ^{\ast
}\right) )]})]=o_{p}(1)$ if $\hat{\beta}\overset{p}{\rightarrow }\beta _{0}$%
, $\hat{\gamma}\overset{p}{\rightarrow }\gamma ^{\ast }$ and $\hat{\eta}%
\overset{p}{\rightarrow }\eta ^{\ast };$

\item $\frac{1}{n}\sum \nolimits_{i=1}^{n}v_{b}\left( \frac{v_{c}\left( \hat{%
\eta}^{\prime }h_{in}\left( \hat{\gamma}\right) \right) }{\sum
\nolimits_{i=1}^{n}v_{c}\left( \hat{\eta}^{\prime }h_{in}\left( \hat{\gamma}%
\right) \right) /n}\right) )\hat{g}_{in}-A_{v}=o_{p}(1),$ if $\hat{\beta}%
\overset{p}{\rightarrow }\beta _{0}$, $\hat{\gamma}\overset{p}{\rightarrow }%
\gamma ^{\ast }$ and $\hat{\eta}\overset{p}{\rightarrow }\eta ^{\ast }.$
\end{enumerate}
\end{lemma}

\begin{lemma}
\label{Lemma_CLTthloc}Suppose Assumption 5.1 holds. Under a sequence $%
\left\{ \text{{\sc p}}_{n}\right\} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $ with $\left\Vert \delta _{h}\right\Vert _{\infty
}<\infty $ and satisfying $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta
_{0,\text{{\sc p}}_{n}}))\rightarrow \delta _{g}$ with $\left\Vert \delta
_{g}\right\Vert _{\infty }<\infty $, we have $\sqrt{n}\hat{s}\left( \varphi
_{n}^{\ast }\right) \overset{D}{\rightarrow }{\cal N}(\delta ,\Omega )$,
where $\varphi _{n}^{\ast }=(\beta _{0,\text{{\sc p}}_{n}}^{\prime },\gamma
_{\text{{\sc p}}_{n}}^{\ast \prime })^{\prime }$ and $\delta =\left( \delta
_{g}^{\prime },\delta _{h}^{\prime }\right) ^{\prime }$.
\end{lemma}

The proofs of Lemmata \ref{LA1_loc} to \ref{g_hat_loc} are similar to the
proofs of Lemma A1 to A3 of NS. The proofs of these Lemmata of NS required a
LLN, a UWL and a CLT. In our framework these are replaced by the LLN for
triangular arrays in Davidson (1994, 19.9 Corollary p.301), Lemmata \ref%
{Lemmata_D_loc} and \ref{Lemma_CLTthloc} respectively. The proof of part (2)
of Lemma \ref{Lambda_lemma_loc} is similar to that of Lemma A2 of NS, but
uses the assumptions that $H_{n}\subset {\cal H}$, $n\geq 1$ and ${\cal H}$
is a convex set.

\begin{lemma}
\label{LA1_loc}Suppose Assumption 5.1 holds. Under a sequence $\left \{ 
\text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\cal P}$ for any $%
1/(2+\delta )<\zeta <1/2$, the following results hold:

\begin{enumerate}
\item $\sup_{\beta \in {\cal B},\lambda \in \Lambda _{n},1\leq i\leq
n}\left
\vert \lambda ^{\prime }g_{in}\left( \beta \right) \right \vert 
\overset{p}{\rightarrow }0$, where $\Lambda _{n}=\left \{ \lambda
:\left
\Vert \lambda \right \Vert \leq n^{-\zeta }\right \} $ and wpa$1$ $%
\Lambda _{n}\subseteq \hat{\Lambda}_{n}(\beta )$ for all $\beta \in {\cal B}$%
.

\item $\sup_{\gamma \in {\cal G},\eta \in H_{n},1\leq i\leq n}\left\vert
\eta ^{\prime }h_{in}\left( \gamma \right) \right\vert \overset{p}{%
\rightarrow }0$, where $H_{n}=\left\{ \eta :\left\Vert \eta \right\Vert \leq
n^{-\zeta }\right\} $.
\end{enumerate}
\end{lemma}

\begin{lemma}
\label{Lambda_lemma_loc}Suppose Assumption 5.1 holds. Under a sequence $%
\left \{ \text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left(
\beta _{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $:

\begin{enumerate}
\item if $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta _{0,\text{{\sc p}%
}_{n}}))\rightarrow \delta _{g},$ with $\left\Vert \delta _{g}\right\Vert
_{\infty }<+\infty $, $\bar{\beta}\in {\cal B}$, $\bar{\beta}-\beta _{0,%
\text{{\sc p}}_{n}}\overset{p}{\rightarrow }0$ and $\hat{g}(\bar{\beta}%
)=O_{p}(n^{-1/2})$, then $\bar{\lambda}=\arg \max_{\lambda \in \hat{\Lambda}%
_{n}(\bar{\beta})}\hat{P}_{g}(\bar{\beta},\lambda )$ exists wpa$1$, $\bar{%
\lambda}=O_{p}(n^{-1/2})$ and $\sup_{\lambda \in \hat{\Lambda}_{n}(\bar{\beta%
})}\hat{P}_{g}(\bar{\beta},\lambda )\leq \rho _{0}+O_{p}(1/n)$.

\item if $\left\Vert \delta _{h}\right\Vert _{\infty }<+\infty $, $\bar{%
\gamma}\in {\cal G}$, $\bar{\gamma}-\gamma _{\text{{\sc p}}_{n}}^{\ast }%
\overset{p}{\rightarrow }0$ and $\hat{h}(\bar{\gamma})=O_{p}(n^{-1/2})$,
then $\bar{\eta}=\arg \max_{\eta \in {\cal H}}\hat{P}_{h}(\bar{\gamma},\eta
) $ exists wpa$1$, $\bar{\eta}=O_{p}(n^{-1/2})$ and $\sup_{\eta \in {\cal H}}%
\hat{P}_{h}(\bar{\gamma},\eta )\leq \rho _{0}+O_{p}(1/n)$.
\end{enumerate}
\end{lemma}

\begin{lemma}
\label{g_hat_loc}Suppose Assumption 5.1 holds. Under a sequence $\left \{ 
\text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $:

\begin{enumerate}
\item if $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta _{0,\text{{\sc p}%
}_{n}}))\rightarrow \delta _{g}$, with $\left \Vert \delta _{g}\right \Vert
_{\infty }<+\infty $, then $\left \Vert \hat{g}(\hat{\beta})\right \Vert
=O_{p}(n^{-1/2})$.

\item if $\left \Vert \delta _{h}\right \Vert _{\infty }<+\infty $, then $%
\left \Vert \hat{h}(\hat{\gamma})\right \Vert =O_{p}(n^{-1/2})$.
\end{enumerate}
\end{lemma}

The proof of the following Lemma is similar to that of Lemma \ref{LA_GELcons}%
.

\begin{lemma}
\label{Th_cons_locg}Suppose Assumption 5.1 holds. Under a sequence $\left \{ 
\text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $:

\begin{enumerate}
\item if $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta _{0,\text{{\sc p}%
}_{n}}))\rightarrow \delta _{g}$ with $\left\Vert \delta _{g}\right\Vert
_{\infty }<+\infty $, then $\hat{\beta}\overset{p}{\rightarrow }\beta _{0}$, 
$\hat{\beta}-\beta _{0,\text{{\sc p}}_{n}}\overset{p}{\rightarrow }0$, $\hat{%
\lambda}=\arg \max_{\lambda \in \hat{\Lambda}_{n}(\hat{\beta}%
)}\sum\nolimits_{i=1}^{n}\rho (\lambda ^{\prime }g_{in}(\hat{\beta}))/n$
exists wpa$1$, and $\hat{\lambda}=O_{p}(n^{-1/2})$.

\item if $\left\Vert \delta _{h}\right\Vert _{\infty }<+\infty $, then $\hat{%
\gamma}\overset{p}{\rightarrow }\gamma ^{\ast }$, $\hat{\gamma}-\gamma _{%
\text{{\sc p}}_{n}}^{\ast }\overset{p}{\rightarrow }0$, $\hat{\eta}=\arg
\max_{\eta \in {\cal H}}\sum\nolimits_{i=1}^{n}\rho (\eta ^{\prime }h_{in}(%
\hat{\gamma}))/n$ exists wpa$1$, and $\hat{\eta}=O_{p}(n^{-1/2})$.
\end{enumerate}
\end{lemma}

The proof of the following Lemma is similar to that of Lemma \ref{asymp_norm}%
.

\begin{lemma}
\label{asymp_norm_loc}Suppose Assumption 5.1 holds. Under a sequence $%
\left
\{ \text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left(
\beta _{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $:

\begin{enumerate}
\item if $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta _{0,\text{{\sc p}%
}_{n}}))\rightarrow \delta _{g}$ with $\left\Vert \delta _{g}\right\Vert
_{\infty }<+\infty $, then 
\[
\sqrt{n}\left( 
\begin{array}{c}
\hat{\beta}-\beta _{0,\text{{\sc p}}_{n}} \\ 
\hat{\lambda}%
\end{array}%
\right) \overset{d}{\rightarrow }{\cal N}(\left( 
\begin{array}{c}
-H_{0,g}\delta _{g} \\ 
-P_{0,g}\delta _{g}%
\end{array}%
\right) ,\left( 
\begin{array}{cc}
\Sigma _{0,g} & 0 \\ 
0 & P_{0,g}%
\end{array}%
\right) ). 
\]

\item if $\left\Vert \delta _{h}\right\Vert _{\infty }<+\infty $, then 
\[
\sqrt{n}\left( 
\begin{array}{c}
\hat{\gamma}-\gamma _{\text{{\sc p}}_{n}}^{\ast } \\ 
\hat{\eta}%
\end{array}%
\right) \overset{d}{\rightarrow }{\cal N}(\left( 
\begin{array}{c}
-H_{0,h}\delta _{h} \\ 
-P_{0,h}\delta _{h}%
\end{array}%
\right) ,\left( 
\begin{array}{cc}
\Sigma _{0,h} & 0 \\ 
0 & P_{0,h}%
\end{array}%
\right) ).
\]
\end{enumerate}
\end{lemma}

The proof of the following Lemma is similar to that of Lemma \ref{Lemma_a1}.

\begin{lemma}
\label{Lemma_pn_loc}Suppose Assumption 5.1 holds. Under a sequence $\left \{ 
\text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $:

\begin{enumerate}
\item if $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta _{0,\text{{\sc p}%
}_{n}}))\rightarrow \delta _{g}$ with $\left \Vert \delta _{g}\right \Vert
_{\infty }<+\infty $, we have $n\hat{p}_{i}^{v_{c}}=1+o_{p}(1)$ and 
\[
n^{1/2}\left( \hat{p}_{i}^{v_{c}}-\frac{1}{n}\right) =\kappa _{v_{c}}\frac{1%
}{n}\hat{g}_{in}^{\prime }n^{1/2}\hat{\lambda}(1+o_{p}(1))+O_{p}(n^{-3/2}), 
\]%
uniformly $(i=1,...,n)$ where $\kappa _{v_{c}}=v_{c,1}(0)/v_{c}(0).$

\item if $\left \Vert \delta _{h}\right \Vert _{\infty }<+\infty $, we have $%
n\hat{q}_{i}^{v_{c}}=1+o_{p}(1)$ and 
\[
n^{1/2}\left( \hat{q}_{i}^{v_{c}}-\frac{1}{n}\right) =\kappa _{v_{c}}\frac{1%
}{n}\hat{h}_{in}^{\prime }n^{1/2}\hat{\eta}(1+o_{p}(1))+O_{p}(n^{-3/2}), 
\]%
uniformly $(i=1,...,n)$ where $\kappa _{v_{c}}=v_{c,1}(0)/v_{c}(0)$.
\end{enumerate}
\end{lemma}

The proofs of the following two Lemmata are similar to those of Theorem 4.3
and Theorem 4.2.

\begin{lemma}
\label{Lemma_normloc_deltag}Suppose Assumption 5.1 holds. Under a sequence $%
\left \{ \text{{\sc p}}_{n}\right \} _{n=1}^{\infty }\in {\rm Seq}\left(
\beta _{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $ that satisfies $\sqrt{n}{\rm E}_{\text{{\sc p}}%
_{n}}(g_{in}(\beta _{0,\text{{\sc p}}_{n}}))\rightarrow \delta _{g}$ with $%
\left \Vert \delta _{g}\right \Vert _{\infty }<+\infty $ and Assumption 5.2
and if $\left \Vert \delta _{h}\right \Vert =+\infty $, then ${\cal S}_{v}$
converges in distribution to ${\cal N}\left( -A_{v}P_{g}\delta _{g},\sigma
^{2}\right) $. Furthermore, ${\cal \tilde{S}}_{v},$ ${\cal LM}_{v}$ and $%
{\cal J}_{v}$ are asymptotically equivalent to ${\cal S}_{v}$.
\end{lemma}

\begin{lemma}
\label{Lemma_sigma_deltag}Suppose Assumption 5.1 holds. Under a sequence $%
\left\{ \text{{\sc p}}_{n}\right\} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $ satisfying $\sqrt{n}{\rm E}_{\text{{\sc p}}%
_{n}}(g_{in}(\beta _{0,\text{{\sc p}}_{n}}))\rightarrow \delta _{g}$ and
Assumption 5.2 and if $\left\Vert \delta _{h}\right\Vert =+\infty $, $\hat{%
\sigma}_{j}^{2}\overset{p}{\rightarrow }\sigma ^{2}$, $j=1,...4$.
\end{lemma}

\begin{lemma}
\label{Lemma_obs_eq}Suppose Assumptions 5.1 holds. Under a sequence $\left\{ 
\text{{\sc p}}_{n}\right\} _{n=1}^{\infty }\in {\rm Seq}\left( \beta
_{0},\gamma ^{\ast },\eta ^{\ast },\delta _{h},\Omega
,D_{g},D_{h},A_{v}\right) $ with $\left\Vert \delta _{h}\right\Vert _{\infty
}<\infty $ and satisfying $\sqrt{n}{\rm E}_{\text{{\sc p}}_{n}}(g_{in}(\beta
_{0,\text{{\sc p}}_{n}}))\rightarrow \delta _{g}$ with $\left\Vert \delta
_{g}\right\Vert _{\infty }<+\infty $ and let $\delta =(\delta _{g}^{\prime
},\delta _{h}^{\prime })^{\prime }$, then

\begin{enumerate}
\item $\sqrt{n}{\cal S}_{v}\overset{d}{\rightarrow }{\cal T}\left( \delta
,\Omega ,Q_{1}\right) $ and $\sqrt{n}{\cal J}_{v,2}$, $\sqrt{n}{\cal J}%
_{v,3} $, $\sqrt{n}{\cal LM}_{v,2}$ and $\sqrt{n}{\cal LM}_{v,3}$ are
asymptotically equivalent to $\sqrt{n}{\cal S}_{v}$;

\item $\sqrt{n}{\cal \tilde{S}}_{v}\overset{d}{\rightarrow }{\cal T}\left(
\delta ,\Omega ,Q_{2}\right) $ and $\sqrt{n}{\cal J}_{v,1}$, $\sqrt{n}{\cal J%
}_{v,4}$, $\sqrt{n}{\cal LM}_{v,1}$ and $\sqrt{n}{\cal LM}_{v,4}$ are
asymptotically equivalent to $\sqrt{n}{\cal \tilde{S}}_{v}$;

\item $n\hat{\sigma}_{1}^{2}\overset{d}{\rightarrow }{\cal T}\left( \delta
,\Omega ,Q_{3}\right) $ and $n\hat{\sigma}_{4}^{2}$ is asymptotically
equivalent to $n\hat{\sigma}_{1}^{2}$;

\item $n\hat{\sigma}_{2}^{2}\overset{d}{\rightarrow }{\cal T}\left( \delta
,\Omega ,Q_{4}\right) $ and $n\hat{\sigma}_{3}^{2}$ is asymptotically
equivalent to $n\hat{\sigma}_{2}^{2}$;

\item $n\hat{\sigma}_{j}^{2}$ is non-negative with probability approaching
one for $j=1,2,3,4$.
\end{enumerate}
\end{lemma}

\noindent {\bf Proofs:}

{\bf Proof of 1:} {$\sqrt{n}{\cal S}_{v}$ is considered first. By a second
order Taylor expansion, 
\begin{equation}
v_{a}\left( \hat{p}_{i}n\right) =v_{a}(1)+\left( \hat{p}_{i}n-1\right)
+v_{a,2}\left( \hat{\sigma}_{i}\right) \left( \hat{p}_{i}n-1\right) ^{2}/2,
\label{phi_taylor}
\end{equation}%
where $\hat{\sigma}_{i}=\alpha _{i}+\left( 1-\alpha _{i}\right) \hat{p}_{i}n$
and $\alpha _{i}\in \left( 0,1\right) $. Hence, 
\begin{align*}
\sqrt{n}{\cal S}_{v}& =\sqrt{n}\sum \limits_{i=1}^{n}\sqrt{n}\left( \hat{p}%
_{i}-\frac{1}{n}\right) n\hat{p}_{i}\left[ v_{b}\left( \frac{\hat{q}%
_{i}^{v_{c}}}{\hat{p}_{i}^{v_{c}}}\right) -\sum \limits_{\ell
=1}^{n}v_{b}\left( \frac{\hat{q}_{\ell }^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}%
\right) \hat{p}_{\ell }\right] \\
& +n\sum \limits_{i=1}^{n}\frac{v_{a,2}\left( \hat{\sigma}_{i}\right) }{2}%
\left( \hat{p}_{i}n-1\right) ^{2}\hat{p}_{i}\left[ v_{b}\left( \frac{\hat{q}%
_{i}^{v_{c}}}{\hat{p}_{i}^{v_{c}}}\right) -\sum \limits_{\ell
=1}^{n}v_{b}\left( \frac{\hat{q}_{\ell }^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}%
\right) \hat{p}_{\ell }\right] =B_{1,n}+B_{2,n}.
\end{align*}%
}

{Using Lemma \ref{Lemma_pn_loc} $n^{1/2}(\hat{p}_{i}-1/n)=n^{-1}\hat{g}%
_{in}^{\prime }\sqrt{n}\hat{\lambda}\left( 1+o_{p}\left( 1\right) \right)
+O_{p}(n^{-3/2})$ and $\sum\nolimits_{i=1}^{n}\hat{p}_{i}\hat{g}%
_{in}^{\prime }=0$ we have $B_{1,n}=\sqrt{n}\hat{A}_{v,3}\sqrt{n}\hat{\lambda%
}\left( 1+o_{p}\left( 1\right) \right) ]+B_{1r,n}$, where $B_{1r,n}$ is
defined below.}

{Note that by a first-order Taylor expansion }%
\[
{v_{b}\left( \frac{n\hat{q}_{i}^{v_{c}}}{n\hat{p}_{i}^{v_{c}}}\right) }%
=v_{b}\left( 1\right) +v_{b,1}\left( \frac{\hat{\sigma}_{1,i}}{\hat{\sigma}%
_{2,i}}\right) \frac{\left( n\hat{q}_{i}^{v_{c}}-1\right) }{\hat{\sigma}%
_{2,i}}{-}\frac{{\hat{\sigma}_{1,i}}}{{\hat{\sigma}_{2,i}^{2}}}{%
v_{b,1}\left( \frac{\hat{\sigma}_{1,i}}{\hat{\sigma}_{2,i}}\right) \left( n%
\hat{p}_{i}^{v_{c}}-1\right) ,} 
\]%
{\ where $\hat{\sigma}_{1,i}=\alpha _{1,i}+\left( 1-\alpha _{1,i}\right) n%
\hat{q}_{i}^{v_{c}}$ and $\alpha _{1,i}\in \left( 0,1\right) $ and $\hat{%
\sigma}_{2,i}=\alpha _{2,i}+\left( 1-\alpha _{2,i}\right) n\hat{p}%
_{i}^{v_{c}}$ and $\alpha _{2,2}\in \left( 0,1\right) $. Thus, }%
\begin{align*}
\sqrt{n}\hat{A}_{v,3}{\ }& {=\sum \nolimits_{i=1}^{n}n\hat{p}_{i}\hat{g}%
_{in}^{\prime }v_{b,1}\left( \frac{\hat{\sigma}_{1,i}}{\hat{\sigma}_{2,i}}%
\right) \sqrt{n}}\left( {\hat{q}_{i}^{v_{c}}-}\frac{1}{n}\right) \frac{1}{%
\hat{\sigma}_{2,i}} \\
& -{\sum \nolimits_{i=1}^{n}n\hat{p}_{i}\hat{g}_{in}^{\prime }\frac{{\hat{%
\sigma}_{1,i}}}{{\hat{\sigma}_{2,i}^{2}}}{v_{b,1}\left( \frac{\hat{\sigma}%
_{1,i}}{\hat{\sigma}_{2,i}}\right) }\sqrt{n}\left( \hat{p}_{i}^{v_{c}}-\frac{%
1}{n}\right) } \\
{}& {=W_{1,n}-W_{2,n}.}
\end{align*}%
\ 

Now by Lemmata \ref{Lemma_pn_loc} $\sqrt{n}(\hat{p}_{i}^{v_{c}}-1/n)=n^{-1}%
\hat{g}_{in}^{\prime }\sqrt{n}\hat{\lambda}\left( 1+o_{p}\left( 1\right)
\right) +O_{p}(n^{-3/2})$ and $\sqrt{n}(\hat{q}_{i}^{v_{c}}-1/n)=n^{-1}\hat{h%
}_{in}^{\prime }\sqrt{n}\hat{\eta}\left( 1+o_{p}\left( 1\right) \right)
+O_{p}(n^{-3/2})$ as $\kappa _{v_{c}}=1$. Additionally, note that similarly
to (\ref{lamb_TA}) we have\ $\sqrt{n}\hat{\lambda}=-P_{g}\sqrt{n}\hat{g}%
(\beta _{0,\text{{\sc p}}_{n}})+o_{p}(1)$, and $\sqrt{n}\hat{\eta}=-P_{h}%
\sqrt{n}\hat{h}(\gamma _{\text{{\sc p}}_{n}}^{\ast })+o_{p}(1)$. Combining
these results with the fact that $v_{b,1}=1$ and Lemma \ref{Lemmata_D_loc},\
we obtain 
\begin{align}
{W_{1,n}}& =\sqrt{n}\hat{\eta}^{\prime }\frac{1}{n}\sum\limits_{i=1}^{n}\hat{%
h}_{in}\hat{g}_{in}^{\prime }v_{b,1}{{\left( \frac{\hat{\sigma}_{1,i}}{\hat{%
\sigma}_{2,i}}\right) }}\frac{1}{\hat{\sigma}_{2,i}}+O_{p}(n^{-2})  \nonumber
\\
{\ }& {=-\sqrt{n}\hat{s}({\varphi }_{n}^{\ast })^{\prime }S_{h}^{\prime
}P_{h}\Omega _{hg}+O_{p}(n^{-2}),}  \label{w1n}
\end{align}%
\ \ and 
\begin{eqnarray*}
W_{2,n} &=&\sum\limits_{i=1}^{n}n\hat{p}_{i}\hat{g}_{in}^{\prime }\frac{\hat{%
\sigma}_{1,i}}{\hat{\sigma}_{2,i}^{2}}v_{b,1}\left( \frac{\hat{\sigma}_{1,i}%
}{\hat{\sigma}_{2,i}}\right) \left[ \frac{1}{n}\hat{g}_{in}^{\prime }\sqrt{n}%
\hat{\lambda}\left( 1+o_{p}\left( 1\right) \right) +O_{p}(n^{-3/2})\right] \\
&=&-{\sqrt{n}\hat{s}(\varphi }_{n}^{\ast }{)^{\prime }S_{g}P_{g}\Omega
_{g}+O_{p}(n^{-2}),}
\end{eqnarray*}%
where $\varphi _{n}^{\ast }=(\beta _{0,\text{{\sc p}}_{n}}^{\prime },\gamma
_{\text{{\sc p}}_{n}}^{\ast \prime })^{\prime }$.

Hence%
\begin{eqnarray}
\sqrt{n}\hat{A}_{3,v} &=&\frac{1}{\sqrt{n}}\sum\limits_{i=1}^{n}n\hat{p}_{i}%
\hat{g}_{in}^{\prime }v_{b}\left( \frac{\hat{q}_{i}^{v_{c}}}{\hat{p}%
_{i}^{v_{c}}}\right) =-\sqrt{n}{\hat{s}({\varphi }_{n}^{\ast })}^{\prime
}S_{h}^{\prime }P_{h}\Omega _{hg}  \label{aj1} \\
&&+\sqrt{n}{\hat{s}({\varphi }_{n}^{\ast })}^{\prime }S_{g}^{\prime
}P_{g}\Omega _{g}+O_{p}(n^{-2}).  \nonumber
\end{eqnarray}

Now note that $B_{1r,n}\equiv n^{-1}\sum \nolimits_{i=1}^{n}n\hat{p}%
_{i}[v_{b}\left( \hat{q}_{i}^{v_{c}}/\hat{p}_{i}^{v_{c}}\right) -n^{-1}\sum
\nolimits_{i=1}^{n}v_{b}\left( \hat{q}_{i}^{v_{c}}/\hat{p}%
_{i}^{v_{c}}\right) n\hat{p}_{i}]O_{p}(1)=o_{p}(1)$.

Additionally, by Lemma \ref{Lemma_pn_loc} we have%
\begin{eqnarray*}
B_{2,n} &\equiv &n\sum \limits_{i=1}^{n}\frac{v_{a,2}\left( \hat{\sigma}%
_{i}\right) }{2}\left[ \sqrt{n}\left( \hat{p}_{i}-\frac{1}{n}\right) \right]
^{2}n\hat{p}_{i}\left[ v_{b}\left( \frac{\hat{q}_{i}^{v_{c}}}{\hat{p}%
_{i}^{v_{c}}}\right) -\sum \limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q}%
_{\ell }^{v_{c}}}{\hat{p}_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }\right] \\
&=&n\sum \limits_{i=1}^{n}\frac{v_{a,2}\left( \hat{\sigma}_{i}\right) }{2}%
\left[ \frac{1}{n}\hat{g}_{in}^{\prime }\sqrt{n}\hat{\lambda}\left(
1+o_{p}\left( 1\right) \right) +O_{p}(n^{-3/2})\right] ^{2}n\hat{p}_{i}\left[
v_{b}\left( \frac{\hat{q}_{i}^{v_{c}}}{\hat{p}_{i}^{v_{c}}}\right) -\sum
\limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q}_{\ell }^{v_{c}}}{\hat{p}%
_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }\right] .
\end{eqnarray*}

By T and the fact that $(a+b)^{2}\leq 2a^{2}+2b^{2}$ we have 
\[
\left \vert B_{2,n}\right \vert \leq \frac{2}{n}\sum
\limits_{i=1}^{n}[\left
\Vert \hat{g}_{in}\right \Vert ^{2}\left \Vert 
\sqrt{n}\hat{\lambda}\right
\Vert ^{2}\left( 1+o_{p}\left( 1\right) \right)
+O_{p}(n^{-3})]\left \vert \frac{v_{a,2}\left( \hat{\sigma}_{i}\right) }{2}n%
\hat{p}_{i}\left[ v_{b}\left( \frac{\hat{q}_{i}^{v_{c}}}{\hat{p}_{i}^{v_{c}}}%
\right) -\sum \limits_{\ell =1}^{n}v_{b}\left( \frac{\hat{q}_{\ell }^{v_{c}}%
}{\hat{p}_{\ell }^{v_{c}}}\right) \hat{p}_{\ell }\right] \right \vert . 
\]

Since $\sum \nolimits_{i=1}^{n}v_{b}\left( \hat{q}_{i}^{v_{c}}/\hat{p}%
_{i}^{v_{c}}\right) \hat{p}_{i}=v_{b}\left( 1\right) +o_{p}(1)$ by Lemmata %
\ref{Lemma_pn_loc} and $\left \Vert \sqrt{n}\hat{\lambda}\right \Vert
^{2}=O_{p}(1)$ by Lemma \ref{asymp_norm_loc} it follows by Lemma \ref%
{Lemmata_D_loc} that $\left \vert B_{2,n}\right \vert =o_{p}(1)$.

Therefore, 
\begin{align*}
{\sqrt{n}{\cal S}_{v}}& {=-\sqrt{n}\hat{s}({\varphi }_{n}^{\ast })^{\prime
}S_{h}^{\prime }P_{h}\Omega _{hg}\sqrt{n}\hat{\lambda}+\sqrt{n}\hat{s}({%
\varphi }_{n}^{\ast })^{\prime }S_{g}^{\prime }P_{g}\Omega _{g}\sqrt{n}\hat{%
\lambda}+O_{p}(n^{-2})} \\
{}& {=\sqrt{n}\hat{s}({\varphi }_{n}^{\ast })^{\prime }S_{h}^{\prime
}P_{h}\Omega _{hg}P_{g}S_{g}\sqrt{n}\hat{s}({\varphi }_{n}^{\ast })-\sqrt{n}%
\hat{s}({\varphi }_{n}^{\ast })^{\prime }S_{g}^{\prime }P_{g}S_{g}\sqrt{n}%
\hat{s}({\varphi }_{n}^{\ast })+O_{p}(n^{-2})}
\end{align*}%
\ because $\sqrt{n}\hat{\lambda}=-P_{g}\sqrt{n}\hat{g}(\beta _{0,\text{{\sc p%
}}_{n}})+o_{p}(1)$ and the fact $P_{g}\Omega _{g}P_{g}=P_{g}$. Hence, since $%
\sqrt{n}\hat{s}({{\varphi }_{n}^{\ast }})\overset{d}{\rightarrow }{\cal N}%
(\delta ,\Omega )$, $\delta =(\delta _{g}^{\prime },\delta _{h}^{\prime
})^{\prime }$ by Lemma \ref{asymp_norm_loc} it follows that $\sqrt{n}{\cal S}%
_{v}\overset{d}{\rightarrow }{\cal T}\left( \delta ,\Omega ,Q_{1}\right) $.

Consider now $\sqrt{n}{\cal J}_{v,2}=-\sqrt{n}\hat{A}_{v,2}\hat{\Omega}%
_{g}^{-1}\sqrt{n}\hat{g}(\hat{\beta})$, note that 
\begin{eqnarray*}
\sqrt{n}{\cal J}_{v,2} &=&n^{-1/2}\dsum \limits_{i=1}^{n}(n\hat{p}%
_{i}-1)v_{b}(n\hat{q}_{i}/(n\hat{p}_{i}))\hat{g}_{in}^{\prime }\hat{\Omega}%
_{g}^{-1}\sqrt{n}\hat{g}(\hat{\beta}) \\
&&-\sqrt{n}\hat{A}_{v,3}\hat{\Omega}_{g}^{-1}\sqrt{n}\hat{g}(\hat{\beta}).{\ 
}
\end{eqnarray*}

We prove that the first term converges in probability to zero. Note that by
Lemma \ref{Lemma_pn_loc} 
\begin{eqnarray*}
n^{-1/2}\dsum\limits_{i=1}^{n}(n\hat{p}_{i}-1)v_{b}\left( \frac{n\hat{q}%
_{i}^{v_{c}}}{n\hat{p}_{i}}\right) \hat{g}_{in}^{\prime } &=&\frac{1}{n}%
\dsum\limits_{i=1}^{n}v_{b}\left( \frac{n\hat{q}_{i}^{v_{c}}}{n\hat{p}_{i}}%
\right) \hat{g}_{in}\hat{g}_{in}^{\prime }\sqrt{n}\hat{\lambda}\left(
1+o_{p}\left( 1\right) \right) \\
&&+\dsum\limits_{i=1}^{n}O_{p}(n^{-3/2})v\left( n\hat{q}_{i}\right) \hat{g}%
_{in}^{\prime }.
\end{eqnarray*}%
Both terms of the rhs converge to zero in probability by Lemma \ref%
{Lemma_pn_loc}, Lemma \ref{Lemmata_D_loc}, the facts that\ $\sqrt{n}\hat{%
\lambda}=-P_{g}\sqrt{n}\hat{g}_{n}(\beta _{0,\text{{\sc p}}_{n}})+o_{p}(1),$ 
$\Omega P=0$ and $\hat{g}(\hat{\beta})\overset{p}{\rightarrow }0$. Hence 
\begin{equation}
n^{-1/2}\dsum\limits_{i=1}^{n}(n\hat{p}_{i}-1)v_{b}\left( \frac{n\hat{q}%
_{i}^{v_{c}}}{n\hat{p}_{i}}\right) \hat{g}_{in}^{\prime }=o_{p}(1).
\label{aj2}
\end{equation}

Combining $\left( \ref{aj1}\right) $ and $\left( \ref{aj2}\right) $, one
obtains 
\begin{equation}
\sqrt{n}\hat{A}_{v,2}=\sqrt{n}\hat{A}_{v,3}+o_{p}(1).  \label{eq_J1_3}
\end{equation}%
Consequently as 
\begin{align}
-\sqrt{n}\Omega _{g}^{-1}\hat{g}(\hat{\beta})& =\sqrt{n}\hat{\lambda}%
+o_{p}(1)  \nonumber \\
& =-P_{g}\sqrt{n}\hat{g}(\beta _{0,\text{{\sc p}}_{n}})+o_{p}(1),
\label{g_beta_hat}
\end{align}%
we have $\sqrt{n}{\cal J}_{2,v}=\sqrt{n}{\cal J}_{3,v}+o_{p}(1)=\sqrt{n}%
{\cal S}_{v}+o_{p}(1)$.

Concerning the Lagrange multiplier test statistics note that\ for $\sqrt{n}%
{\cal LM}_{j,v}\ j=2,3$\ we have\ $\sqrt{n}{\cal LM}_{v,j}-\sqrt{n}{\cal J}%
_{v,j}=n^{1/2}\hat{A}_{v,j}\left[ n^{1/2}\hat{\lambda}+\hat{\Omega}%
_{g}^{-1}n^{1/2}\hat{g}(\hat{\beta})\right] =$ $n^{1/2}\hat{A}_{v,j}[-\Omega
_{g}^{-1}+\hat{\Omega}_{g}^{-1}+o_{p}(1)]\sqrt{n}\hat{g}(\hat{\beta}%
)=o_{p}(1)$\ using $\left( \ref{g_beta_hat}\right) $ and the facts that $%
\sqrt{n}\hat{A}_{j,v}=O_{p}(1),$ $\Omega _{g}^{-1}-\hat{\Omega}%
_{g}^{-1}=o_{p}(1)$ by Lemma \ref{Lemmata_D_loc} and $\sqrt{n}\hat{g}(\hat{%
\beta})=O_{p}(1)$.

{\bf Proof of 2: }Let us now consider $\sqrt{n}{{\cal \tilde{S}}_{v}}$. By $%
\left( \ref{phi_taylor}\right) $, it follows that\ 
\begin{align*}
\sqrt{n}{{\cal \tilde{S}}_{v}}& =\sqrt{n}\sum\limits_{i=1}^{n}\sqrt{n}\left( 
\hat{p}_{i}-\frac{1}{n}\right) n\hat{p}_{i}\left[ v_{b}\left( n\hat{q}%
_{i}^{v_{c}}\right) -\sum\limits_{\ell =1}^{n}v_{b}\left( n\hat{q}_{\ell
}^{v_{c}}\right) \hat{p}_{\ell }\right]  \\
& +n\sum\limits_{i=1}^{n}\frac{v_{2,a}\left( \hat{\sigma}_{i}\right) }{2}%
\left( \hat{p}_{i}n-1\right) ^{2}\hat{p}_{i}\left[ v_{b}\left( n\hat{q}%
_{i}^{v_{c}}\right) -\sum\limits_{\ell =1}^{n}v_{b}\left( n\hat{q}_{\ell
}^{v_{c}}\right) \hat{p}_{\ell }\right] =C_{1,n}+C_{2,n}.
\end{align*}%
Hence, by Lemma \ref{Lemma_pn_loc}, 
\begin{eqnarray*}
C_{1,n} &=&n^{-1/2}\sum\limits_{i=1}^{n}n\hat{p}_{i}\left[ v_{b}\left( \hat{q%
}_{i}^{v_{c}}n\right) -\sum\limits_{\ell =1}^{n}v_{b}\left( \hat{q}_{\ell
}^{v_{c}}n\right) \hat{p}_{\ell }\right] \hat{g}_{in}^{\prime }\sqrt{n}\hat{%
\lambda}\left( 1+o_{p}\left( 1\right) \right)  \\
&&+C_{1r,n}.
\end{eqnarray*}

Note that $n^{-1/2}\sum \nolimits_{i=1}^{n}n\hat{p}_{i}[v_{b}\left( \hat{q}%
_{i}^{v_{c}}n\right) -\sum \nolimits_{\ell =1}^{n}v_{b}\left( \hat{q}_{\ell
}^{v_{c}}n\right) \hat{p}_{\ell }]\hat{g}_{in}^{\prime }=n^{-1/2}\sum
\nolimits_{i=1}^{n}n\hat{p}_{i}v_{b}\left( \hat{q}_{i}^{v_{c}}n\right) \hat{g%
}_{in}^{\prime }$ because $\sum \nolimits_{i=1}^{n}\hat{p}_{i}\hat{g}%
_{in}^{\prime }=0$. Additionally by a Taylor expansion $v_{b}\left( n\hat{q}%
_{i}^{v_{c}}\right) =v_{b}\left( 1\right) +v_{b,1}\left( \hat{\sigma}%
_{3,i}\right) \left( n\hat{q}_{i}^{v_{c}}-1\right) $, where $\hat{\sigma}%
_{3,i}=\alpha _{3,i}+\left( 1-\alpha _{3,i}\right) n\hat{q}_{i}^{v_{c}}$ and 
$\alpha _{3,i}\in \left( 0,1\right) $. Thus 
\[
\frac{1}{\sqrt{n}}\sum \limits_{i=1}^{n}n\hat{p}_{i}\hat{g}_{in}^{\prime
}v_{b}\left( \hat{q}_{i}^{v_{c}}n\right) =\frac{1}{\sqrt{n}}\sum
\limits_{i=1}^{n}n\hat{p}_{i}\hat{g}_{in}^{\prime }v_{b,1}\left( \hat{\sigma}%
_{3,i}\right) \left( n\hat{q}_{i}^{v_{c}}-1\right) . 
\]

Now note that $\sum\nolimits_{i=1}^{n}n\hat{p}_{i}\hat{g}_{in}^{\prime }%
\sqrt{n}\left( n\hat{q}_{i}^{v_{c}}-1/n\right) =O_{p}(1)$ by Lemma \ref%
{Lemma_pn_loc}, Lemma \ref{Lemmata_D_loc} and the fact that $\sqrt{n}\hat{%
\eta}=O_{p}(1)$. Thus using continuity of $v_{b,1}\left( .\right) $ at $1$,
we have 
\begin{eqnarray}
\sqrt{n}\hat{A}_{4,v} &=&n^{-1/2}\sum\limits_{i=1}^{n}n\hat{p}_{in}\hat{g}%
_{in}^{\prime }v_{b}\left( \hat{q}_{i}^{v_{c}}n\right) ={W_{1,n}+o_{p}(1)}
\label{eq_J1_1} \\
&=&-\sqrt{n}{\hat{s}(\varphi }_{n}^{\ast }{)}^{\prime }S_{h}^{\prime
}P_{h}\Omega _{hg}+{O_{p}(n^{-2})}  \nonumber
\end{eqnarray}%
\ \ by (\ref{w1n}). Also $C_{1r,n}=n^{-1/2}\sum\nolimits_{i=1}^{n}n\hat{p}%
_{i}[v_{b}\left( \hat{q}_{i}^{v_{c}}n\right) -$ $\sum\nolimits_{\ell
=1}^{n}v_{b}\left( \hat{q}_{\ell }^{v_{c}}n\right) \hat{p}_{\ell }]\hat{g}%
_{i}^{\prime }O_{p}(n^{-3/2})=o_{p}(1/n)$ by Lemmata \ref{Lemma_pn_loc} and %
\ref{UWL_LOC1} and $C_{2,n}=o_{p}(1)$ \ using a proof similar to the proof
that $B_{2,n}=o_{p}(1)$. Hence 
\begin{eqnarray*}
\sqrt{n}{{\cal \tilde{S}}_{v}} &=&-\sqrt{n}{\hat{s}({\varphi }_{n}^{\ast })}%
^{\prime }S_{h}^{\prime }P_{h}\Omega _{hg}\sqrt{n}\hat{\lambda}+o_{p}(1) \\
&=&\sqrt{n}{\hat{s}({\varphi }_{n}^{\ast })}^{\prime }S_{h}^{\prime
}P_{h}\Omega _{hg}P_{g}S_{g}\sqrt{n}{\hat{s}({\varphi }_{n}^{\ast })}%
+o_{p}(1).
\end{eqnarray*}%
\ 

Thus, since $\sqrt{n}\hat{s}(\varphi _{n}^{\ast })\overset{d}{\rightarrow }%
{\cal N}(\delta ,\Omega )$ we have $\sqrt{n}{{\cal \tilde{S}}_{v}}\overset{d}%
{\rightarrow }{\cal T}\left( \delta ,\Omega ,Q_{2}\right) $.

Consider now $\sqrt{n}{\cal J}_{v,1}=-n^{-1/2}\dsum\nolimits_{i=1}^{n}v_{b}%
\left( n\hat{q}_{i}^{v_{c}}\right) \hat{g}_{i}^{\prime }\hat{\Omega}%
_{g}^{-1}n^{1/2}\hat{g}(\hat{\beta})$.\ Note that$\sqrt{n}\hat{A}%
_{v,1}=n^{-1/2}\dsum\nolimits_{i=1}^{n}v\left( n\hat{q}_{i}\right) \hat{g}%
_{in}^{\prime }=-n^{-1/2}\dsum\nolimits_{i=1}^{n}(n\hat{p}_{i}-1)v_{b}\left(
n\hat{q}_{i}^{v_{c}}\right) \hat{g}_{in}^{\prime }$ $+\sqrt{n}\hat{A}_{v,4}$%
.\ The second term of the rhs has the asymptotic representation given in (%
\ref{eq_J1_1}). Similarly to the proof of $\left( \ref{aj2}\right) $ the
first term converges in probability to zero and therefore 
\begin{equation}
\sqrt{n}\hat{A}_{v,1}=\sqrt{n}\hat{A}_{v,4}+o_{p}(1).  \label{eq_J1_2}
\end{equation}%
Thus, using equation $\left( \ref{g_beta_hat}\right) $, and the fact that $%
P_{g}\sqrt{n}\hat{g}(\beta _{0,\text{{\sc p}}_{n}})=O_{p}(1)$\ it follows
that $\sqrt{n}{\cal J}_{1,v}=\sqrt{n}{\cal J}_{4,v}+o_{p}(1)=\sqrt{n}{\cal 
\tilde{S}}_{v}+o_{p}(1)$. The proof that $\sqrt{n}{\cal LM}_{v,j}-\sqrt{n}%
{\cal J}_{v,j}=o_{p}(1)$, $j=1,4$ is similar to the case $j=2,3$ and
therefore omitted.

{\bf Proof of 3: }{\bf \ }Let us first consider $n\hat{\sigma}_{1}^{2}=n\hat{%
A}_{1,v}\hat{P}_{g}\hat{A}_{1,v}^{\prime }$. Note that by $\left( \ref%
{eq_J1_2}\right) $ and \ $\left( \ref{eq_J1_1}\right) $, $n\hat{\sigma}%
_{1}^{2}=n\hat{s}(\varphi _{n}^{\ast })^{\prime }K_{h}P_{g}K_{h}^{\prime }%
\hat{s}(\varphi _{n}^{\ast })+o_{p}(1)$. Hence, it follows that \ $n\hat{%
\sigma}_{1}^{2}\overset{d}{\rightarrow }{\cal T}\left( \delta ,\Omega
,Q_{3}\right) $\ because of the fact that $\sqrt{n}\hat{s}(\varphi
_{n}^{\ast })\overset{d}{\rightarrow }{\cal N}(\delta ,\Omega )$. \
Additionally, using $\left( \ref{eq_J1_2}\right) $ and \ $\left( \ref%
{eq_J1_1}\right) $ we have $n\hat{\sigma}_{4}^{2}=n\hat{\sigma}%
_{1}^{2}+o_{p}(1)$.

{\bf Proof of 4: }Now consider $n\hat{\sigma}_{2}^{2}=n\hat{A}_{2,v}\hat{P}%
_{g}\hat{A}_{2,v}^{\prime }$. By $\left( \ref{eq_J1_3}\right) $ and $\left( %
\ref{aj1}\right) $\ we have 
\[
n\hat{\sigma}_{2}^{2}=n{\hat{s}({\varphi }_{n}^{\ast })}^{\prime
}(K_{h}-K_{g})P_{g}(K_{h}-K_{g})^{\prime }{\hat{s}({\varphi }_{n}^{\ast })}%
+o_{p}(1){.} 
\]%
\ Hence $n\hat{\sigma}_{1}^{2}\overset{d}{\rightarrow }{\cal T}\left( \delta
,\Omega ,Q_{4}\right) $ as $\sqrt{n}\hat{s}(\varphi _{n}^{\ast })\overset{d}{%
\rightarrow }{\cal N}(\delta ,\Omega )$.\ Additionally, also by $\left( \ref%
{eq_J1_3}\right) $ and $\left( \ref{aj1}\right) $ we have\ $n\hat{\sigma}%
_{2}^{2}=n\hat{\sigma}_{3}^{2}+o_{p}(1)$.

{\bf Proof of 5: }Because $P_{g}$ is a positive semidefinite matrix, it
follows that $n\hat{\sigma}_{j}^{2}\geq 0$, $j=1,2,3,4$ wpa$1$.{%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
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\endproof%
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}

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\noindent {\bf Proof of Theorem 5.1:} (1) follows from Lemmata \ref%
{Lemma_normloc_deltag} and \ref{Lemma_sigma_deltag} with $\delta _{g}=0$,
while (2) follows from Lemma \ref{Lemma_obs_eq} with $\delta =S_{h}^{\prime
}\delta _{h}$.{%
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\endproof%
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}

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\noindent {\bf Proof of Theorem 5.2: }The proof of this theorem is similar
to the proof of Theorem 4.1 of Shi (2015). First let $\hat{\Omega}_{n}=\hat{%
\Omega}$, $\hat{Q}_{n,i}=\hat{Q}_{i}$ and $\hat{Q}_{n,j}=\hat{Q}_{j}$, $%
cv_{ij,n}^{\ast }=cv^{\ast }(1-\tau ,\hat{\Omega}_{n},\hat{Q}_{n,i},\hat{Q}%
_{n,j}).$ We take a sequence $\{${\sc p}$_{n}\in {\cal P}_{0}\}$ and a
subsequence $\{b_{n}\}$ of $\{n\}$ such that 
\[
\lim \sup_{n\rightarrow \infty }\sup_{\text{{\sc p}}\in P_{0}}\Pr \left(
\left\vert W_{n}^{s}\left( i,j\right) \right\vert >cv_{n}^{\ast }\right) =%
\underset{n\rightarrow \infty }{\lim }\Pr\nolimits_{\text{{\sc p}}%
_{b_{n}}}\left( \left\vert W_{b_{n}}^{s}\left( i,j\right) \right\vert
>cv_{ij,b_{n}}^{\ast }\right) . 
\]%
Such sequences and subsequences always exist. Assumption 5.1 and condition
(c) of Definition 5.1 imply that elements in the arrays $\beta _{o,\text{%
{\sc p}}}$, $\gamma _{\text{{\sc p}}}^{\ast }$, $\eta _{\text{{\sc p}}%
}^{\ast }$, $\Omega _{\text{{\sc p}}}(\varphi ^{\ast })$, $D_{\text{{\sc p}}%
,g}(\beta ^{\ast })$, $D_{\text{{\sc p}},h}(\gamma ^{\ast })$, $A_{\text{%
{\sc p}},v}(\mu ^{\ast })$ are uniformly bounded over {\sc p}$\in {\cal P}$.
Thus, there exists a subsequence $\{a_{n}\}$ of $\{b_{n}\}$ and some $\left(
\beta _{0},\gamma ^{\ast },\delta _{h},\Omega ,D_{g},D_{h},A_{v}\right) $
such that $\left( \beta _{0,\text{{\sc p}}_{a_{n}}},\gamma _{\text{{\sc p}}%
_{a_{n}}}^{\ast },\sqrt{n}{\cal E}_{\text{{\sc p}}_{a_{n}},h},\Omega _{\text{%
{\sc p}}_{a_{n}}}(\varphi ^{\ast }),D_{\text{{\sc p}}_{a_{n}},g}(\beta
^{\ast }),D_{\text{{\sc p}}_{a_{n}},h}(\gamma ^{\ast }),A_{\text{{\sc p}}%
_{a_{n}},v}(\mu ^{\ast })\right) \rightarrow \left( \beta _{0},\gamma ^{\ast
},\delta _{h},\Omega ,D_{g},D_{h},A_{v}\right) $. It suffices to show that $%
\lim\limits_{n\rightarrow \infty }\Pr\nolimits_{\text{{\sc p}}%
_{a_{n}}}\left( \left\vert W_{a_{n}}^{s}\left( i,j\right) \right\vert
>cv_{ij,a_{n}}^{\ast }\right) \leq \tau $.

Note now that 
\[
W_{a_{n}}^{s}\left( i,j\right) =\frac{T_{a_{n}}\left( i\right) -tr(\hat{Q}%
_{a_{n},i}\hat{\Omega}_{a_{n}})/\sqrt{a_{n}}}{\sqrt{V_{a_{n}}(j)+c_{ij}\cdot
tr(\hat{Q}_{a_{n},j}\hat{\Omega}_{a_{n}})/a_{n}}},i=1,2,j=3,4. 
\]%
By Theorem 5.1 (1) if $\left\Vert \delta _{h}\right\Vert _{\infty }=+\infty $%
, we have $T_{a_{n}}\left( i\right) \overset{d}{\rightarrow }{\cal N}%
(0,\sigma ^{2})$ and $V_{a_{n}}(j)\overset{p}{\rightarrow }\sigma ^{2}$.
Additionally, $\hat{Q}_{a_{n},i}\overset{p}{\rightarrow }Q_{i}$, $\hat{Q}%
_{a_{n},j}\overset{p}{\rightarrow }Q_{j}$, $\hat{\Omega}_{a_{n}}\overset{p}{%
\rightarrow }\Omega $ by Lemma \ref{Lemmata_D_loc}\ and the Slutsky
theorems. Therefore $W_{a_{n}}^{s}\left( i,j\right) \overset{d}{\rightarrow }%
{\cal N}(0,1)$ and 
\begin{eqnarray*}
\lim\limits_{n\rightarrow \infty }\Pr\nolimits_{\text{{\sc P}}%
_{a_{n}}}\left( \left\vert W_{a_{n}}^{s}\left( i,j\right) \right\vert
>cv_{ij,a_{n}}^{\ast }\right) &\leq &\lim\limits_{n\rightarrow \infty
}\Pr\nolimits_{\text{{\sc P}}_{a_{n}}}\left( \left\vert W_{a_{n}}^{s}\left(
i,j\right) \right\vert >z_{1-\tau /2}\right) \\
&=&\tau
\end{eqnarray*}%
because $cv_{ij,a_{n}}^{\ast }\geq z_{1-\tau /2}$.

Suppose now that $\left\Vert \delta _{h}\right\Vert _{\infty }<+\infty $ in
this case 
\[
W_{a_{n}}^{s}\left( i,j\right) \overset{d}{\rightarrow }{\cal W}^{s}\left(
S_{h}^{\prime }\delta _{h},i,j\right) =\frac{{\cal T}\left( S_{h}^{\prime
}\delta _{h},\Omega ,Q_{i}\right) -tr(Q_{i}\Omega )}{\sqrt{{\cal T}\left(
S_{h}^{\prime }\delta _{h},\Omega ,Q_{j}\right) +c_{ij}\cdot tr(Q_{j}\Omega )%
}}
\]%
by Lemma \ref{Lemmata_D_loc} and Theorem 5.1 (2).

Note that $cv_{ij,a_{n}}^{\ast }\geq cv(1-\tau ,\hat{\Omega}_{a_{n}},\hat{Q}%
_{a_{n},i},\hat{Q}_{a_{n},j}).$ Hence 
\[
\Pr \nolimits_{\text{{\sc p}}_{a_{n}}}\left( \left \vert W_{a_{n}}^{s}\left(
i,j\right) \right \vert >cv_{ij,a_{n}}^{\ast }\right) \leq \Pr \nolimits_{%
\text{{\sc p}}_{a_{n}}}\left( \left \vert W_{a_{n}}^{s}\left( i,j\right)
\right \vert >cv(1-\tau ,\hat{\Omega}_{a_{n}},\hat{Q}_{a_{n},i},\hat{Q}%
_{a_{n},j})\right) . 
\]

Note that by inspection ${\cal W}^{s}\left( S_{h}^{\prime }\delta
_{h},i,j\right) $ is continuous in $(\delta _{h},\Omega ,Q_{i},Q_{j})$; the
continuity of the Cholesky decomposition follows from Lemma 2.1.6 p. 295 of
Schatzman (2002). Additionally, ${\cal W}^{s}\left( S_{h}^{\prime }\delta
_{h},i,j\right) $ is a continuous random variable and consequently $cv\left(
1-\tau ,\delta _{h},\Omega ,Q_{i},Q_{j}\right) $ is continuous in $(\delta
_{h},\Omega ,Q_{i},Q_{j})$. Since $\left[ 0,c_{h}\right] $ is a compact set
it follows by the maximum theorem that $cv(1-\tau ,\Omega ,Q_{i},Q_{j})$ is
continuous in $\Omega ,Q_{i},Q_{j}$. Now by Lemma \ref{Lemmata_D_loc} $\hat{%
\Omega}_{a_{n}}\overset{p}{\rightarrow }\Omega $, $\hat{Q}_{a_{n},i}\overset{%
p}{\rightarrow }Q_{i}$ and $\hat{Q}_{a_{n},j}\overset{p}{\rightarrow }Q_{j}$
and consequently $cv(1-\tau ,\hat{\Omega}_{a_{n}},\hat{Q}_{a_{n},i},\hat{Q}%
_{a_{n},j})\overset{p}{\rightarrow }cv_{ij}(1-\tau ,\left[ 0,c_{h}\right] )$%
. Therefore 
\[
\lim_{n\rightarrow \infty }\Pr \nolimits_{\text{{\sc P}}_{a_{n}}}\left(
\left \vert W_{a_{n}}^{s}\left( i,j\right) \right \vert >cv(1-\tau ,\hat{%
\Omega}_{a_{n}},\hat{Q}_{a_{n},i},\hat{Q}_{a_{n},j})\right) =\Pr \left(
\left \vert {\cal W}^{s}\left( S_{h}^{\prime }\delta _{h},i,j\right)
\right
\vert >cv_{ij}(1-\tau ,\left[ 0,c_{h}\right] )\right) . 
\]

Now by Assumption 5.2 $cv_{ij}(1-\tau ,\left[ 0,c_{h}\right]
)=cv_{ij}(1-\tau ,[0,+\infty ))$, consequently for any $\delta _{h}\in
\lbrack 0,+\infty )$ 
\begin{eqnarray*}
\Pr \left( \left \vert {\cal W}^{s}\left( S_{h}^{\prime }\delta
_{h},i,j\right) \right \vert >cv(1-\tau ,\left[ 0,c_{h}\right] )\right)
&\leq &\Pr \left( \left \vert {\cal W}^{s}\left( S_{h}^{\prime }\delta
_{h},i,j\right) \right \vert >cv\left( 1-\tau ,S_{h}^{\prime }\delta
_{h},\Omega ,Q_{i},Q_{j},c_{ij}\right) \right) \\
&=&\tau .
\end{eqnarray*}%
{%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
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}%
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\noindent {\bf Proof of Theorem 5.3: }Using the same arguments as in the
proof of Theorem 5.1 $cv(1-\tau ,\Omega ,Q_{i},Q_{j})$ is continuous in $%
\Omega ,Q_{i},Q_{j}$ and therefore $cv(1-\tau ,\hat{\Omega},\hat{Q}_{i},\hat{%
Q}_{j})\overset{p}{\rightarrow }cv(1-\tau ,\Omega ,Q_{i},Q_{j})$ because $%
\hat{\Omega}\overset{p}{\rightarrow }\Omega $, $\hat{Q}_{i}\overset{p}{%
\rightarrow }Q_{i}$ and $\hat{Q}_{,j}\overset{p}{\rightarrow }Q_{j}$ for $%
\left \Vert \delta _{h}\right \Vert _{\infty }\leq +\infty $ by Lemma \ref%
{Lemmata_D_loc}. consequently $cv^{\ast }(1-\tau ,\hat{\Omega},\hat{Q}_{i},%
\hat{Q}_{j})\overset{p}{\rightarrow }cv_{ij}^{\ast }$.

Now let us consider first the case{\bf \ }$\left\Vert \delta _{h}\right\Vert
_{\infty }<+\infty $. In this case $W_{n}^{s}\left( i,j\right) \overset{d}{%
\rightarrow }{\cal W}(\delta ,\Omega ,Q_{i},Q_{j},c_{ij})$ by Lemmata \ref%
{Lemmata_D_loc} and \ref{Lemma_obs_eq}, It follows that $\underset{%
n\rightarrow \infty }{\lim }\Pr\nolimits_{\text{{\sc p}}_{n}}\left(
\left\vert W_{n}^{s}\left( i,j\right) \right\vert >cv^{\ast }(1-\tau ,\hat{%
\Omega},\hat{Q}_{i},\hat{Q}_{j})\right) =1-F_{\left\vert {\cal W}%
_{ij}\right\vert }(cv_{ij}^{\ast })$.

Consider now the case{\bf \ }$\left \Vert \delta _{h}\right \Vert _{\infty
}=+\infty .$ In this case $W_{n}^{s}\left( i,j\right) \overset{d}{%
\rightarrow }{\cal N}\left( -A_{v}P_{g}\delta _{g}/\sigma ,1\right) $ by
Lemmata \ref{Lemmata_D_loc} and \ref{Lemma_obs_eq}. It follows that $%
\underset{n\rightarrow \infty }{\lim }\Pr \nolimits_{\text{{\sc p}}%
_{n}}\left( \left \vert W_{n}^{s}\left( i,j\right) \right \vert >cv^{\ast
}(1-\tau ,\hat{\Omega},\hat{Q}_{i},\hat{Q}_{j})\right) =\Pr \left[
\left
\vert x\right \vert >cv_{ij}^{\ast }\right] $, where $x\sim {\cal N}%
\left( -A_{v}P_{g}\delta _{g}/\sigma ,1\right) $. Now note that $\Pr \left[
\left
\vert x\right \vert >cv_{ij}^{\ast }\right] =\Phi (-cv_{ij}^{\ast
}-A_{v}P_{g}\delta _{g}/\sigma )+\Phi (-cv_{ij}^{\ast }+A_{v}P_{g}\delta
_{g}/\sigma )$. {%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
}

\subsection{The limit behavior of $W_{i,j}^{\ast }(\protect\delta _{h})$}

In this subsection we investigate the behavior of the random variable $%
W_{i,j}^{\ast }(\delta _{h})=\left. {\cal T}\left( S_{h}^{\prime }\delta
_{h},\Omega ,Q_{i}\right) \right/ {\cal T}\left( S_{h}^{\prime }\delta
_{h},\Omega ,Q_{j}\right) ^{1/2}$, $i=1,2$, $j=3,4$, as some of the elements
of $\delta _{h}$ approach infinity. These random variables are defined in
section 5.1 of the paper, page 17. We start by presenting two useful Lemmata
that allow us to analyze this limit. The proofs of the Lemmata are presented
at the end of this subsection.

\begin{lemma}
\label{Lemma_rep} The random variables $W_{i,j}^{\ast }(\delta _{h})=\left. 
{\cal T}\left( S_{h}^{\prime }\delta _{h},\Omega ,Q_{i}\right) \right/ {\cal %
T}\left( S_{h}^{\prime }\delta _{h},\Omega ,Q_{j}\right) ^{1/2}$, $i=1,2$, $%
j=3,4$ have the following representation: 
\begin{equation}
W_{i,j}^{\ast }(\delta _{h})=\frac{s_{0}\left( \delta _{h}\right)
x_{0}+z^{\prime }C_{\Omega }^{\prime }Q_{i}C_{\Omega }z}{\left[
s_{0}^{2}\left( \delta _{h}\right) +2s_{j}\left( \delta _{h}\right)
x_{j}+z^{\prime }C_{\Omega }^{\prime }Q_{j}C_{\Omega }z\right] ^{1/2}},
\label{W_delta}
\end{equation}%
where $z\sim {\cal N}(0,I_{m})$, $x_{0}\sim {\cal N}(0,1)$, $s_{0}^{2}\left(
\delta _{h}\right) =\delta _{h}^{\prime }L_{0}\delta _{h}\geq 0$, $%
L_{0}\equiv P_{h}\Omega _{hg}P_{g}\Omega _{hg}^{\prime }P_{h}$, $%
s_{3}^{2}\left( \delta _{h}\right) =\delta _{h}^{\prime }L_{1}\delta
_{h}\geq 0$, where $L_{1}\equiv L_{0}\Omega _{h}L_{0}$, $s_{4}^{2}\left(
\delta _{h}\right) =s_{0}^{2}\left( \delta _{h}\right) -s_{3}^{2}\left(
\delta _{h}\right) \geq 0$ and $x_{j}\sim {\cal N}(0,1)$.
\end{lemma}

We can see from Lemma \ref{Lemma_rep} that $W_{i,j}^{\ast }(\delta _{h})$
only depends on $\delta _{h}$ via the quadratic forms $\delta _{h}^{\prime
}L_{0}\delta _{h}$ and $\delta _{h}^{\prime }L_{1}\delta _{h}$.
Additionally, it is apparent from equation $\left( \ref{W_delta}\right) $
that if $s_{0}^{2}\left( \delta _{h}\right) =\delta _{h}^{\prime
}L_{0}\delta _{h}\rightarrow \infty $, then $W_{i,j}^{\ast }(\delta
_{h})\rightarrow {\cal N}(0,1)$ (note that $0\leq s_{j}\left( \delta
_{h}\right) /s_{0}^{2}\left( \delta _{h}\right) \leq s_{0}^{-1}\left( \delta
_{h}\right) $, $j=3,4$). However, given that the matrix $L_{0}$ is positive
semidefinite (because $P_{g}$ is positive semidefinite), the quadratic form $%
\delta _{h}^{\prime }L_{0}\delta _{h}$ does not necessarily diverge as any
of the elements of $\delta _{h}$ approach infinity. In fact the following
lemma shows that this limit is path dependent. We use the following
notation. Let $N(A)$ denote the null space of a matrix $A$ and let $%
\left\Vert \cdot \right\Vert $ be the Euclidean norm of $\cdot $. Denote $U$
a matrix of eigenvectors of the matrix $L_{0}$ chosen such that they are
orthogonal to each other. Let also $U_{a}$ be the submatrix of $U$ that
contains the eigenvectors corresponding to the positive eigenvalues of $L_{0}
$ and $U_{b}$ be the submatrix of $U$ that contains the eigenvectors
corresponding to the zero eigenvectors of $L_{0}$.

\begin{lemma}
\label{linalgebra_aux} If $rank(L_{0})=r>0$ and $\delta _{h}\left(
t_{a},t_{b}\right) =U_{a}t_{a}+U_{b}t_{b}$ where $t_{a}\in {\Bbb R}^{r}$ and 
$t_{b}\in {\Bbb R}^{m_{h}-r}$, then we have $\left\Vert \delta _{h}\left(
t_{a},t_{b}\right) \right\Vert =\left\Vert t_{a}\right\Vert +\left\Vert
t_{b}\right\Vert $, and the following results hold for $i=1,2$, $j=3,4$:

\begin{enumerate}
\item $W_{i,j}^{\ast }(\delta _{h}\left( t_{a},t_{b}\right) )=W_{i,j}^{\ast
}(U_{a}t_{a});$

\item $\lim_{\left\Vert t_{b}\right\Vert \rightarrow \infty }W_{i,j}^{\ast
}(\delta _{h}\left( t_{a},t_{b}\right) )=W_{i,j}^{\ast }(U_{a}t_{a}),$ if $%
\left\Vert t_{a}\right\Vert <\infty ;$

\item $\lim_{\left\Vert t_{a}\right\Vert \rightarrow \infty }W_{i,j}^{\ast
}(\delta _{h}\left( t_{a},t_{b}\right) )=x_{0}$ either if $\left\Vert
t_{b}\right\Vert <\infty $ or if $\left\Vert t_{b}\right\Vert \rightarrow
\infty $, where $x_{0}\sim {\cal N}(0,1).$
\end{enumerate}
\end{lemma}

In Lemma \ref{linalgebra_aux}$\ $we consider paths of the form $\delta
_{h}\left( t_{a},t_{b}\right) =U_{a}t_{a}+U_{b}t_{a}$ because the
eigenvectors are chosen such that they are orthogonal to each other and
therefore they form a basis of ${\Bbb R}^{m_{h}}$. Since $\left\Vert \delta
_{h}\left( t_{a},t_{b}\right) \right\Vert =\left\Vert t_{a}\right\Vert
+\left\Vert t_{b}\right\Vert $, it follows that $\left\Vert \delta
_{h}\left( t_{a},t_{b}\right) \right\Vert $ goes to infinity in the
following cases: $\left\Vert t_{b}\right\Vert \rightarrow \infty $ and $%
\left\Vert t_{a}\right\Vert <\infty $; $\left\Vert t_{a}\right\Vert
\rightarrow \infty $ and $\left\Vert t_{b}\right\Vert <\infty $; and $%
\left\Vert t_{a}\right\Vert \rightarrow \infty $ and $\left\Vert
t_{b}\right\Vert \rightarrow \infty $. Lemma \ref{linalgebra_aux} shows
that, for fixed $t_{a}$ satisfying $\left\Vert t_{a}\right\Vert <\infty $,
the distribution of $W_{i,j}^{\ast }(\delta _{h}\left( t_{a},t_{b}\right) )$
does not depend on the value of $t_{b}$ and consequently this distribution
is the same whether $\left\Vert t_{b}\right\Vert \rightarrow \infty $ or if $%
\left\Vert t_{b}\right\Vert <\infty $. On the other hand, when $\left\Vert
t_{a}\right\Vert \rightarrow \infty $, $W_{i,j}^{\ast }(\delta _{h}\left(
t_{a},t_{b}\right) )$ converges always to the standard normal distribution.

We prove now the above Lemmata.

\noindent {\bf Proof of Lemma \ref{Lemma_rep}: }First note that for any
matrix $Q$: ${\cal T}\left( \delta _{h}^{\prime }S_{h},\Omega ,Q\right)
=\left( \delta _{h}^{\prime }S_{h}+z^{\prime }C_{\Omega }^{\prime }\right)
Q\left( S_{h}^{\prime }\delta _{h}+C_{\Omega }z\right) =\delta _{h}^{\prime
}S_{h}QS_{h}^{\prime }\delta _{h}+\delta _{h}^{\prime }S_{h}QC_{\Omega
}z+z^{\prime }C_{\Omega }^{\prime }QS_{h}^{\prime }\delta _{h}+z^{\prime
}C_{\Omega }^{\prime }QC_{\Omega }z$. We prove the result by showing that:

\begin{description}
\item[(a)] for $i=1,2:$ ${\cal T}\left( \delta _{h}^{\prime }S_{h},\Omega
,Q_{i}\right) =$ $s_{0}\left( \delta _{h}\right) x_{0}+z^{\prime }C_{\Omega
}^{\prime }Q_{i}C_{\Omega }z$, $s_{0}^{2}\left( \delta _{h}\right) =\delta
_{h}^{\prime }L\delta _{h}$ and $x_{0}\sim {\cal N}(0,1)$.

\item[(b)] for $j=3,4:{\cal T}\left( \delta _{h}^{\prime }S_{h},\Omega
,Q_{j}\right) =s_{0}^{2}\left( \delta _{h}\right) +2s_{j}\left( \delta
_{h}\right) x_{j}+z^{\prime }C_{\Omega }^{\prime }Q_{j}C_{\Omega }z$, $%
s_{3}^{2}\left( \delta _{h}\right) =\delta _{h}^{\prime }L\Omega _{h}L\delta
_{h},$ $s_{4}^{2}\left( \delta _{h}\right) =s_{0}^{2}\left( \delta
_{h}\right) -s_{3}^{2}\left( \delta _{h}\right) \geq 0$ and $x_{j}\sim {\cal %
N}(0,1).$
\end{description}

We start by proving (a). Let us consider ${\cal T}\left( \delta _{h}^{\prime
}S_{h},\Omega ,Q_{i}\right) $. Note that for $Q=Q_{1}$ and $Q=Q_{2}$ we have 
$\delta _{h}^{\prime }S_{h}QS_{h}^{\prime }\delta _{h}=0$, $z^{\prime
}C_{\Omega }^{\prime }QS_{h}^{\prime }\delta _{h}=0$ and $\delta
_{h}^{\prime }S_{h}QC_{\Omega }z=\delta _{h}^{\prime }P_{h}\Omega
_{hg}P_{g}S_{g}C_{\Omega }z$ because $S_{g}S_{h}^{\prime }=0$ and $%
S_{h}S_{h}^{\prime }=I_{m_{h}}$. Therefore for $i=1,2$: ${\cal T}\left(
\delta ,\Omega ,Q_{i}\right) =\delta _{h}^{\prime }P_{h}\Omega
_{hg}P_{g}S_{g}C_{\Omega }z+z^{\prime }C_{\Omega }^{\prime }Q_{i}C_{\Omega
}z $. Let $s_{0}^{2}\left( \delta _{h}\right) ={\rm var}\left( \delta
_{h}^{\prime }P_{h}\Omega _{hg}P_{g}S_{g}C_{\Omega }z\right) $. Note that $%
s_{0}^{2}\left( \delta _{h}\right) =\delta _{h}^{\prime }P_{h}\Omega
_{hg}P_{g}S_{g}\Omega S_{g}^{\prime }P_{g}\Omega _{hg}^{\prime }P_{h}\delta
_{h}=\delta _{h}^{\prime }L_{0}\delta _{h}\geq 0$ and let $x_{0}\sim {\cal N}%
(0,1)$. Hence for $i=1,2$ we have ${\cal T}\left( \delta _{h}^{\prime
}S_{h},\Omega ,Q_{i}\right) =$ $s_{0}\left( \delta _{h}\right)
x_{0}+z^{\prime }C_{\Omega }^{\prime }Q_{i}C_{\Omega }z$. To see this note
that if $s_{0}^{2}\left( \delta _{h}\right) =0$, $\delta _{h}^{\prime
}P_{h}\Omega _{hg}P_{g}S_{g}C_{\Omega }z=0$ and $s_{0}\left( \delta
_{h}\right) x_{0}=0$ and if $s_{0}\left( \delta _{h}\right) >0$, then we can
define $x_{0}=\delta _{h}^{\prime }P_{h}\Omega _{hg}P_{g}S_{g}C_{\Omega
}z/s_{0}\left( \delta _{h}\right) $ and since $z\sim {\cal N}(0,I_{m})\,$it
follows that $x_{0}\sim {\cal N}(0,1)$.

We prove now (b). Note that for $Q=Q_{3}$ and $Q=Q_{4}$ we have $\delta
_{h}^{\prime }S_{h}QS_{h}^{\prime }\delta _{h}=\delta _{h}^{\prime
}L_{0}\delta _{h}=s_{0}^{2}\left( \delta _{h}\right) $ because $%
S_{g}S_{h}^{\prime }=0$ and $S_{h}S_{h}^{\prime }=I_{m_{h}}$. Additionally,
by symmetry we have for $Q=Q_{3}$ and $Q=Q_{4}$: $\delta _{h}^{\prime
}S_{h}QC_{\Omega }z+z^{\prime }C_{\Omega }^{\prime }QS_{h}^{\prime }\delta
_{h}=2\delta _{h}^{\prime }S_{h}QC_{\Omega }z$.

Now note that $\delta _{h}^{\prime }S_{h}Q_{3}C_{\Omega }z=\delta
_{h}^{\prime }P_{h}\Omega _{hg}P_{g}\Omega _{hg}^{\prime
}P_{h}S_{h}C_{\Omega }z$ and consequently $s_{3}^{2}\left( \delta
_{h}\right) ={\rm var}\left( \delta _{h}^{\prime }P_{h}\Omega
_{hg}P_{g}\Omega _{hg}^{\prime }P_{h}S_{h}C_{\Omega }z\right) =\delta
_{h}^{\prime }L_{0}\Omega _{h}L_{0}\delta _{h}\geq 0$.

Let us consider now $\delta _{h}^{\prime }S_{h}Q_{4}C_{\Omega }z$. Note that 
$\delta _{h}^{\prime }S_{h}Q_{4}C_{\Omega }z$ $=\delta _{h}^{\prime
}P_{h}\Omega _{hg}P_{g}(K_{h}-K_{g})^{\prime }C_{\Omega }z$, because $%
S_{g}S_{h}^{\prime }=0$ and and $S_{h}S_{h}^{\prime }=I_{m_{h}}$. Let $%
s_{4}^{2}\left( \delta _{h}\right) ={\rm var}\left( \delta _{h}^{\prime
}P_{h}\Omega _{hg}P_{g}(K_{h}-K_{g})^{\prime }C_{\Omega }z\right) $. After
some lengthy, but simple algebra we can show that $s_{4}^{2}\left( \delta
_{h}\right) =s_{0}^{2}\left( \delta _{h}\right) -s_{3}^{2}\left( \delta
_{h}\right) $. Since $s_{4}^{2}\left( \delta _{h}\right) \geq 0$ it follows
that $s_{0}^{2}\left( \delta _{h}\right) \geq s_{3}^{2}\left( \delta
_{h}\right) $. Consequently ${\cal T}\left( \delta _{h}^{\prime
}S_{h},\Omega ,Q_{j}\right) =s_{0}^{2}\left( \delta _{h}\right)
+2s_{j}\left( \delta _{h}\right) x_{j}+z^{\prime }C_{\Omega }^{\prime
}Q_{j}C_{\Omega }z$ for $j=3,4$ if $s_{j}^{2}\left( \delta _{h}\right) \geq
0 $, $s_{0}\left( \delta _{h}\right) \geq 0$ using the same arguments
adopted in the proof of (a).{%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
}

\noindent {\bf Proof of Lemma \ref{linalgebra_aux}: }Since $L_{0}$ is a real
symmetric matrix it can be factorized as $L_{0}=U\Lambda U^{\prime }$, where 
$UU^{\prime }=U^{\prime }U=I_{m_{h}}$ and $\Lambda $ is a diagonal matrix
whose entries are the eigenvalues of $L_{0}$. Note that $L_{0}=U\Lambda
U^{\prime }=U_{a}\Lambda _{\ast }U_{a}^{\prime }$, where $\Lambda _{\ast }$
is a $(r\times r)$ matrix with only the $r$ positive eigenvalues in the
diagonal. Now notice that $U_{b}t_{b}$ $\in N(L_{0})$ as $%
AU_{b}t_{b}=U_{a}\Lambda _{\ast }U_{a}^{\prime }U_{b}t_{b}=0$, because $%
U_{a}^{\prime }U_{b}=0$. Also $\left( U_{a}t_{a}\right) ^{\prime
}U_{b}t_{a}=t_{a}U_{a}^{\prime }U_{b}t_{b}=0$, as $U_{a}^{\prime }U_{b}=0$.
Hence $\left\Vert \delta _{h}\left( t_{a},t_{b}\right) \right\Vert
=\left\Vert t_{a}\right\Vert +\left\Vert t_{b}\right\Vert $.

To prove 1 note that since $U_{b}t_{b}\in N(L_{0})$, it follows that $%
U_{b}t_{b}\in N(L_{1})$ and $U_{b}t_{b}\in N(L_{0}-L_{1})$ because $L_{0}$, $%
L_{1}$ and $L_{0}-L_{1}$are positive semidefinite by Lemma \ref{Lemma_rep}
(see Abadir and Magnus, 2005, solution of Exercise, 8.41, p. 227).
Consequently $s_{0}^{2}\left( \delta _{h}\left( t_{a},t_{b}\right) \right)
=t_{a}U_{a}^{\prime }L_{0}U_{a}t_{a}$, $s_{3}^{2}\left( \delta _{h}\left(
t_{a},t_{b}\right) \right) =t_{a}U_{a}^{\prime }L_{1}U_{a}t_{a}$ and $%
s_{4}^{2}\left( \delta _{h}\left( t_{a},t_{b}\right) \right)
=t_{a}U_{a}^{\prime }\left( L_{0}-L_{1}\right) U_{a}t_{a}$ and the result
follows from Lemma \ref{Lemma_rep}. The result 2 is a consequence of 1
because $W_{i,j}^{\ast }(U_{a}t_{a})$ does not depend on $t_{b}.$ To prove 3
we only need to prove that $\lim_{\left\Vert t_{a}\right\Vert \rightarrow
\infty }s_{0}^{2}\left( \delta _{h}\left( t_{a},t_{b}\right) \right)
=+\infty $ as in this case $\lim_{\left\Vert t_{a}\right\Vert \rightarrow
\infty }s_{j}\left( \delta _{h}\left( t_{a},t_{b}\right) \right)
/s_{0}^{2}\left( \delta _{h}\left( t_{a},t_{b}\right) \right) =0$ due to the
fact that $0\leq s_{j}\left( \delta _{h}\left( t_{a},t_{b}\right) \right)
/s_{0}^{2}\left( \delta _{h}\left( t_{a},t_{b}\right) \right) \leq
s_{0}^{-1}\left( \delta _{h}\left( t_{a},t_{b}\right) \right) $, $j=3,4$ for 
$s_{0}\left( \delta _{h}\left( t_{a},t_{b}\right) \right) >0$ by \ref%
{Lemma_rep}. Now since $U_{a}^{\prime }U_{a}=I_{r}$ we have $s_{0}^{2}\left(
\delta _{h}\left( t_{a},t_{b}\right) \right) =t_{a}U_{a}^{\prime
}L_{0}U_{a}t_{a}=t_{a}U_{a}^{\prime }U_{a}\Lambda _{\ast }U_{a}^{\prime
}U_{a}t_{a}=t_{a}\Lambda _{\ast }t_{a}\geq c\left\Vert t_{a}\right\Vert ,$
where $c$ is a positive constant. Hence $s_{0}^{2}\left( \delta _{h}\left(
t_{a},t_{b}\right) \right) \rightarrow \infty $ as $\left\Vert
t_{a}\right\Vert \rightarrow \infty $.{%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}%
%BeginExpansion
\endproof%
%EndExpansion
}

\renewcommand{\thesection}{SM3}

\section{Monte Carlo study (additional results)}

Tables SM1 and SM2 present the empirical sizes and powers for the tests
based on the non-nested test statistics computed with the exponential
tilting (ET) estimator for the sample sizes $200$ and $400$. The nominal
level for all tests reported is $0.05$. \ We report the results for ${\cal S}%
_{\alpha }$, ${\cal \tilde{S}}_{\alpha }$, ${\cal LM}_{\alpha }$ and ${\cal J%
}_{\alpha }$ for $\alpha \rightarrow 0,$ $\alpha =1,1.5,2,3$ and for the
Ramalho and Smith (2002) statistics based on ET. We use the notation ${\cal S%
}_{\text{{\sc rs}}}$, ${\cal \tilde{S}}_{\text{{\sc rs}}}$, ${\cal LM}_{%
\text{{\sc rs}}}$ and ${\cal J}_{\text{{\sc rs}}}$ for the Ramalho and Smith
(2002) statistics. We also present in the Tables SM1 and SM2 the results for
the tests for overidentifying restrictions based on the likelihood ratio
statistic computed with ET [Kitamura and Stutzer (1997) and Imbens {\it et
al.} (1998)], which is labelled ${\cal LR}_{\text{{\sc et}}}$. ET is
calculated using the Broyden--Fletcher--Goldfarb--Shanno (BFGS) algorithm.

%TCIMACRO{\TeXButton{\footnotesize}{\footnotesize}}%
%BeginExpansion
\footnotesize%
%EndExpansion
%TCIMACRO{%
%\TeXButton{linespace1\normalsize}{\renewcommand{\baselinestretch}{0.95}\footnotesize}}%
%BeginExpansion
\renewcommand{\baselinestretch}{0.95}\footnotesize%
%EndExpansion
\setlength{\tabcolsep}{3.5pt}

\begin{gather*}
\text{{\bf Table SM1:} Rejection frequencies of the tests under the null.} \\
\begin{tabular}{c|c|ccc|ccc|ccc|ccc}
\hline\hline
\multicolumn{2}{c|}{$n$} & \multicolumn{6}{|c|}{$200$} & \multicolumn{6}{|c}{%
$400$} \\ \hline
\multicolumn{2}{c|}{Statistic} & \multicolumn{3}{|c|}{original} & 
\multicolumn{3}{|c|}{Shi-type} & \multicolumn{3}{|c|}{original} & 
\multicolumn{3}{|c}{Shi-type} \\ \hline
Estimator & $\rho _{u_{g},w_{h2}}$ & $0.0$ & $0.2$ & $0.4$ & $0.0$ & $0.2$ & 
$0.4$ & $0$ & $0.2$ & $0.4$ & $0$ & $0.2$ & $0.4$ \\ \hline
& ${\cal LR}_{\text{{\sc et}}}$ & $5.8$ & $5.6$ & $6.2$ & $-$ & $-$ & $-$ & $%
5.6$ & $5.3$ & $5.7$ & $-$ & $-$ & $-$ \\ \cline{2-14}
& ${\cal S}_{0}$ & $18.0$ & $15.6$ & $14.5$ & $1.8$ & $1.7$ & $1.9$ & $16.3$
& $13.6$ & $12.5$ & $1.6$ & $1.4$ & $1.7$ \\ 
& ${\cal \tilde{S}}_{0}$ & $5.5$ & $6.1$ & $8.6$ & $4.6$ & $6.4$ & $7.9$ & $%
4.9$ & $4.9$ & $7.3$ & $4.0$ & $4.9$ & $6.2$ \\ 
& ${\cal LM}_{0}$ & $7.8$ & $6.4$ & $7.8$ & $4.5$ & $4.2$ & $4.4$ & $6.4$ & $%
5.8$ & $6.6$ & $4.3$ & $3.3$ & $3.5$ \\ 
& ${\cal J}_{0}$ & $5.7$ & $4.8$ & $6.0$ & $3.5$ & $3.1$ & $3.0$ & $5.5$ & $%
5.1$ & $5.7$ & $3.8$ & $2.8$ & $2.8$ \\ \cline{2-14}
& ${\cal S}_{1}$ & $18.2$ & $16.0$ & $15.8$ & $2.0$ & $1.8$ & $1.8$ & $16.4$
& $13.9$ & $14.2$ & $1.6$ & $1.3$ & $1.3$ \\ 
& ${\cal \tilde{S}}_{1}$ & $6.1$ & $5.9$ & $8.2$ & $4.4$ & $5.8$ & $7.1$ & $%
5.4$ & $5.1$ & $6.5$ & $3.9$ & $4.6$ & $5.4$ \\ 
& ${\cal LM}_{1}$ & $7.5$ & $6.4$ & $7.8$ & $4.7$ & $4.3$ & $4.2$ & $6.3$ & $%
5.7$ & $6.5$ & $4.3$ & $3.3$ & $3.4$ \\ 
& ${\cal J}_{1}$ & $5.4$ & $4.8$ & $6.2$ & $3.9$ & $3.3$ & $2.7$ & $5.4$ & $%
4.9$ & $5.7$ & $3.8$ & $2.8$ & $2.7$ \\ \cline{2-14}
& ${\cal S}_{1.5}$ & $18.2$ & $16.3$ & $16.0$ & $2.2$ & $2.0$ & $1.9$ & $%
16.4 $ & $14.1$ & $14.5$ & $1.8$ & $1.4$ & $1.5$ \\ 
& ${\cal \tilde{S}}_{1.5}$ & $6.4$ & $6.0$ & $8.0$ & $4.2$ & $5.5$ & $6.7$ & 
$5.6$ & $5.2$ & $6.9$ & $3.8$ & $4.0$ & $5.2$ \\ 
& ${\cal LM}_{1.5}$ & $7.3$ & $6.5$ & $7.5$ & $4.7$ & $4.5$ & $4.2$ & $6.2$
& $5.6$ & $5.8$ & $4.2$ & $3.4$ & $3.3$ \\ 
ET & ${\cal J}_{1.5}$ & $5.3$ & $5.1$ & $5.6$ & $3.8$ & $3.4$ & $2.8$ & $5.4$
& $4.8$ & $5.1$ & $3.8$ & $3.0$ & $2.7$ \\ \cline{2-14}
& ${\cal S}_{2}$ & $18.2$ & $16.3$ & $15.9$ & $2.4$ & $2.3$ & $2.4$ & $16.4$
& $14.1$ & $14.3$ & $1.9$ & $1.7$ & $1.7$ \\ 
& ${\cal \tilde{S}}_{2}$ & $6.7$ & $6.2$ & $7.9$ & $4.2$ & $5.2$ & $6.1$ & $%
5.8$ & $5.2$ & $7.2$ & $3.8$ & $3.6$ & $4.9$ \\ 
& ${\cal LM}_{2}$ & $7.1$ & $6.5$ & $6.9$ & $4.7$ & $4.7$ & $4.3$ & $6.2$ & $%
5.5$ & $5.9$ & $4.2$ & $3.5$ & $3.6$ \\ 
& ${\cal J}_{2}$ & $5.3$ & $4.8$ & $5.2$ & $3.8$ & $3.6$ & $3.0$ & $5.3$ & $%
4.7$ & $5.0$ & $3.8$ & $2.9$ & $3.0$ \\ \cline{2-14}
& ${\cal S}_{3}$ & $18.2$ & $16.3$ & $15.3$ & $3.4$ & $3.2$ & $3.5$ & $16.4$
& $13.9$ & $13.7$ & $2.3$ & $2.1$ & $2.5$ \\ 
& ${\cal \tilde{S}}_{3}$ & $6.9$ & $6.8$ & $8.4$ & $4.1$ & $4.5$ & $4.9$ & $%
6.3$ & $5.6$ & $7.6$ & $3.7$ & $3.2$ & $4.9$ \\ 
& ${\cal LM}_{3}$ & $7.0$ & $6.5$ & $7.1$ & $4.8$ & $5.0$ & $5.0$ & $6.0$ & $%
5.4$ & $6.4$ & $4.2$ & $3.7$ & $4.2$ \\ 
& ${\cal J}_{3}$ & $5.2$ & $4.8$ & $5.5$ & $4.0$ & $3.9$ & $3.7$ & $5.2$ & $%
4.5$ & $5.5$ & $3.9$ & $3.2$ & $3.6$ \\ \cline{2-14}
& ${\cal S}_{\text{{\sc rs}}}$ & $16.2$ & $15.0$ & $15.1$ & $2.1$ & $1.0$ & $%
1.4$ & $15.9$ & $13.6$ & $12.0$ & $1.6$ & $1.1$ & $0.8$ \\ 
& ${\cal \tilde{S}}_{\text{{\sc rs}}}$ & $5.7$ & $5.3$ & $7.4$ & $3.4$ & $%
5.8 $ & $7.6$ & $5.5$ & $5.4$ & $7.5$ & $3.5$ & $5.2$ & $7.1$ \\ 
& ${\cal LM}_{\text{{\sc rs}}}$ & $7.1$ & $6.9$ & $7.4$ & $4.1$ & $4.3$ & $%
3.9$ & $6.4$ & $6.6$ & $5.6$ & $3.7$ & $4.1$ & $2.9$ \\ 
& ${\cal J}_{\text{{\sc rs}}}$ & $5.2$ & $5.1$ & $5.5$ & $3.2$ & $3.0$ & $%
2.4 $ & $5.6$ & $5.6$ & $4.8$ & $3.4$ & $3.5$ & $2.4$ \\ \hline\hline
\end{tabular}%
\end{gather*}

\begin{gather*}
\text{{\bf Table SM2:} Rejection frequencies of the tests under the
alternative.} \\
\begin{tabular}{c|c|cccccc|cccccc}
\hline\hline
\multicolumn{2}{c|}{$n$} & \multicolumn{6}{|c|}{$200$} & \multicolumn{6}{|c}{%
$400$} \\ \hline
\multicolumn{2}{c|}{Statistic} & \multicolumn{3}{|c}{original} & 
\multicolumn{3}{|c|}{Shi-type} & \multicolumn{3}{|c}{original} & 
\multicolumn{3}{|c}{Shi-type} \\ \hline
Estimator & $\omega $ & $3$ & $4$ & $5$ & \multicolumn{1}{|c}{$3$} & $4$ & $%
5 $ & $3$ & $4$ & $5$ & \multicolumn{1}{|c}{$3$} & $4$ & $5$ \\ \hline
& ${\cal LR}_{\text{{\sc et}}}$ & $35.3$ & $53.1$ & $66.3$ & 
\multicolumn{1}{|c}{$-$} & $-$ & $-$ & $34.6$ & $55.6$ & $73.9$ & 
\multicolumn{1}{|c}{$-$} & $-$ & $-$ \\ \cline{2-14}
& ${\cal S}_{0}$ & $43.7$ & $57.0$ & $62.7$ & \multicolumn{1}{|c}{$11.9$} & $%
19.9$ & $27.2$ & $40.7$ & $55.9$ & $66.8$ & \multicolumn{1}{|c}{$10.7$} & $%
19.8$ & $30.6$ \\ 
& ${\cal \tilde{S}}_{0}$ & $10.3$ & $19.6$ & $33.6$ & \multicolumn{1}{|c}{$%
5.1$} & $10.4$ & $19.2$ & $12.5$ & $26.2$ & $42.9$ & \multicolumn{1}{|c}{$%
5.7 $} & $14.2$ & $27.6$ \\ 
& ${\cal LM}_{0}$ & $43.6$ & $61.2$ & $73.3$ & \multicolumn{1}{|c}{$34.5$} & 
$52.4$ & $65.3$ & $43.8$ & $63.2$ & $79.2$ & \multicolumn{1}{|c}{$33.4$} & $%
53.9$ & $72.8$ \\ 
& ${\cal J}_{0}$ & $37.9$ & $55.5$ & $68.0$ & \multicolumn{1}{|c}{$27.7$} & $%
44.3$ & $57.4$ & $40.6$ & $60.3$ & $77.0$ & \multicolumn{1}{|c}{$29.8$} & $%
49.7$ & $68.6$ \\ \cline{2-14}
& ${\cal S}_{1}$ & $48.9$ & $62.1$ & $67.6$ & \multicolumn{1}{|c}{$12.7$} & $%
20.9$ & $28.7$ & $47.7$ & $64.3$ & $75.4$ & \multicolumn{1}{|c}{$11.1$} & $%
21.4$ & $33.3$ \\ 
& ${\cal \tilde{S}}_{1}$ & $18.4$ & $32.1$ & $49.0$ & \multicolumn{1}{|c}{$%
10.9$} & $20.8$ & $35.8$ & $20.1$ & $35.8$ & $54.5$ & \multicolumn{1}{|c}{$%
11.1$} & $24.0$ & $40.5$ \\ 
& ${\cal LM}_{1}$ & $40.2$ & $58.9$ & $71.9$ & \multicolumn{1}{|c}{$30.4$} & 
$49.2$ & $62.9$ & $40.4$ & $60.6$ & $77.5$ & \multicolumn{1}{|c}{$29.0$} & $%
50.1$ & $70.1$ \\ 
& ${\cal J}_{1}$ & $34.6$ & $53.0$ & $66.2$ & \multicolumn{1}{|c}{$23.3$} & $%
40.9$ & $54.2$ & $38.0$ & $57.7$ & $75.2$ & \multicolumn{1}{|c}{$25.7$} & $%
45.4$ & $65.6$ \\ \cline{2-14}
& ${\cal S}_{1.5}$ & $50.1$ & $63.9$ & $69.4$ & \multicolumn{1}{|c}{$14.0$}
& $22.9$ & $30.7$ & $49.2$ & $66.5$ & $77.9$ & \multicolumn{1}{|c}{$12.4$} & 
$23.4$ & $36.0$ \\ 
& ${\cal \tilde{S}}_{1.5}$ & $23.1$ & $37.3$ & $54.5$ & \multicolumn{1}{|c}{$%
15.3$} & $26.3$ & $42.6$ & $23.8$ & $40.1$ & $59.1$ & \multicolumn{1}{|c}{$%
14.6$} & $28.8$ & $46.4$ \\ 
& ${\cal LM}_{1.5}$ & $37.2$ & $56.4$ & $70.2$ & \multicolumn{1}{|c}{$27.3$}
& $46.0$ & $60.4$ & $36.6$ & $56.6$ & $75.5$ & \multicolumn{1}{|c}{$25.6$} & 
$45.7$ & $67.3$ \\ 
ET & ${\cal J}_{1.5}$ & $31.6$ & $49.9$ & $64.0$ & \multicolumn{1}{|c}{$20.1$%
} & $37.8$ & $52.0$ & $34.2$ & $53.6$ & $72.6$ & \multicolumn{1}{|c}{$22.9$}
& $41.1$ & $62.4$ \\ \cline{2-14}
& ${\cal S}_{2}$ & $50.7$ & $64.5$ & $70.2$ & \multicolumn{1}{|c}{$16.2$} & $%
25.6$ & $34.0$ & $50.4$ & $67.9$ & $79.5$ & \multicolumn{1}{|c}{$13.9$} & $%
26.6$ & $39.5$ \\ 
& ${\cal \tilde{S}}_{2}$ & $26.3$ & $41.7$ & $58.0$ & \multicolumn{1}{|c}{$%
18.5$} & $30.9$ & $47.8$ & $26.5$ & $43.1$ & $61.9$ & \multicolumn{1}{|c}{$%
17.4$} & $32.6$ & $50.5$ \\ 
& ${\cal LM}_{2}$ & $33.7$ & $52.6$ & $67.7$ & \multicolumn{1}{|c}{$24.2$} & 
$42.4$ & $57.5$ & $32.4$ & $51.9$ & $71.4$ & \multicolumn{1}{|c}{$22.0$} & $%
41.0$ & $61.6$ \\ 
& ${\cal J}_{2}$ & $28.5$ & $46.2$ & $61.3$ & \multicolumn{1}{|c}{$17.7$} & $%
33.4$ & $48.8$ & $29.9$ & $48.8$ & $67.7$ & \multicolumn{1}{|c}{$19.1$} & $%
36.1$ & $56.6$ \\ \cline{2-14}
& ${\cal S}_{3}$ & $50.2$ & $65.3$ & $71.0$ & \multicolumn{1}{|c}{$21.2$} & $%
32.0$ & $40.6$ & $49.4$ & $67.8$ & $79.7$ & \multicolumn{1}{|c}{$18.8$} & $%
33.1$ & $47.1$ \\ 
& ${\cal \tilde{S}}_{3}$ & $28.2$ & $45.1$ & $62.0$ & \multicolumn{1}{|c}{$%
21.2$} & $36.1$ & $53.6$ & $25.7$ & $42.2$ & $59.5$ & \multicolumn{1}{|c}{$%
18.7$} & $33.8$ & $51.9$ \\ 
& ${\cal LM}_{3}$ & $29.1$ & $46.1$ & $61.3$ & \multicolumn{1}{|c}{$21.2$} & 
$35.3$ & $51.1$ & $27.1$ & $43.8$ & $60.0$ & \multicolumn{1}{|c}{$20.3$} & $%
34.0$ & $51.3$ \\ 
& ${\cal J}_{3}$ & $23.9$ & $39.0$ & $53.8$ & \multicolumn{1}{|c}{$15.5$} & $%
27.3$ & $42.2$ & $24.3$ & $40.1$ & $56.2$ & \multicolumn{1}{|c}{$17.5$} & $%
30.3$ & $46.4$ \\ \cline{2-14}
& ${\cal S}_{\text{{\sc rs}}}$ & $52.6$ & $67.5$ & $74.5$ & 
\multicolumn{1}{|c}{$13.9$} & $29.2$ & $43.7$ & $50.0$ & $71.5$ & $84.8$ & 
\multicolumn{1}{|c}{$11.4$} & $28.1$ & $45.8$ \\ 
& ${\cal \tilde{S}}_{\text{{\sc rs}}}$ & $10.7$ & $29.7$ & 
\multicolumn{1}{c|}{$48.3$} & $7.7$ & $22.9$ & $39.6$ & $5.1$ & $13.4$ & $%
29.9$ & \multicolumn{1}{|c}{$1.9$} & $7.1$ & $21.0$ \\ 
& ${\cal LM}_{\text{{\sc rs}}}$ & $37.7$ & $51.2$ & \multicolumn{1}{c|}{$%
62.5 $} & $27.5$ & $41.6$ & $54.6$ & $40.7$ & $59.4$ & \multicolumn{1}{c|}{$%
68.3$} & $29.2$ & $47.1$ & $58.2$ \\ 
& ${\cal J}_{\text{{\sc rs}}}$ & $32.1$ & $44.8$ & $55.5$ & 
\multicolumn{1}{|c}{$20.0$} & $34.1$ & $46.4$ & $38.2$ & $56.5$ & $64.8$ & 
\multicolumn{1}{|c}{$25.5$} & $42.3$ & $52.9$ \\ \hline\hline
\end{tabular}%
\end{gather*}

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\end{document}