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\title{{\huge{Supplementary Online Appendix} \\
\Large{to \\
``Determining the Cointegration Rank in Heteroskedastic VAR Models of Unknown Order" \\
by \\
Giuseppe Cavaliere, Luca De Angelis, Anders Rahbek
and A.M.Robert Taylor}}}

\maketitle

%\pagebreak

\section*{Summary of Contents}

\large{This Supplement contains the Tables of results from the
Monte Carlo experiments detailed in Sections 3 and 5 of our paper
``Determining the Cointegration Rank in Heteroskedastic VAR
Models of Unknown Order".

In particular, the results reported in Table 1 relate to the
impact that the value of the lag length used has on the ability of
information-based and PLR based procedures to determine the
cointegration rank by reporting the frequencies with which the
wild bootstrap PLR based procedure and a BIC-based procedure,
which minimises the quantity BIC($k, r$) defined in (3.5) over $r
= 0, 1, ..., p$, select the true cointegration rank, in each
case fixing the lag length used at each of $k = 1, ..., 6$.  The
results in Table 2 relate to the behaviour of the information
criteria IC($k,p$) defined in (3.10) in determining the
autoregressive lag length $k$.  The results in Tables 3.1, 3.2 and
3.3 relate to cointegration rank determination for the methods
considered in section 3 of the paper for the cases of no
cointegration ($r_0=0$) and GJR-GARCH innovations (Case A),
Autoregressive Stochastic Volatility (Case B) and a single
volatility break (Case C), respectively.  Finally, the results in
Tables 4.1, 4.2 and 4.3 relate to cointegration rank
determination for these methods for the case of a single
cointegration vector ($r_0=1$) in Cases A, B and C, respectively.

In all of the tables, the frequencies with which the various
procedures select the true cointegration rank or true lag length
are highlighted in bold text.}


\pagebreak

\begin{table}[htbp]
TABLE 1: Cointegration rank determination for different values of lag length $k$. VAR(2) model,  i.i.d.\ Gaussian errors. %[Case A].

\footnotesize{
% Table generated by Excel2LaTeX from sheet 'iid+break'
\begin{tabular}{rrrrrrrrr|rrrrrr}
\hline
      &       &       & \multicolumn{6}{c|}{\footnotesize{BIC($k,r$)}} & \multicolumn{6}{c}{\footnotesize{$Q^{\ast}_{r,k,T}$}} \\
\hline
      & $T$     & $\gamma$ & $k = 1$ & \textbf{2} & 3     & 4     & 5     & 6     & $k = 1$ & \textbf{2} & 3     & 4     & 5     & 6 \\
\hline
   $r_0 = 0$ &        100 &        0.2 &       97.7 & {\bf 98.9} &       98.5 &       97.7 &       96.7 &       94.9 &       89.8 & {\bf 95.7} &       95.8 &       95.9 &       95.9 &       96.0 \\

           &            &        0.5 &       62.5 & {\bf 98.5} &       97.9 &       96.9 &       94.7 &       91.8 &       42.8 & {\bf 95.6} &       95.8 &       95.8 &       95.8 &       95.7 \\

           &            &        0.8 &        1.4 & {\bf 96.0} &       93.8 &       90.6 &       85.8 &       80.0 &        0.9 & {\bf 95.1} &       94.4 &       94.7 &       94.2 &       94.5 \\

           &        400 &        0.2 &       99.9 &  {\bf 100} &        100 &        100 &        100 &        100 &       89.6 & {\bf 95.2} &       95.0 &       95.2 &       95.2 &       95.4 \\

           &            &        0.5 &       86.4 &  {\bf 100} &        100 &        100 &        100 &        100 &       40.8 & {\bf 95.2} &       95.3 &       95.4 &       95.5 &       95.2 \\

           &            &        0.8 &        5.9 &  {\bf 100} &        100 &        100 &       99.9 &       99.9 &        0.9 & {\bf 95.2} &       95.4 &       95.4 &       95.2 &       95.2 \\

   $r_0 = 1$ &        100 &        0.2 &        8.5 & {\bf 26.7} &       14.6 &       11.0 &        9.2 &        9.6 &       28.5 & {\bf 40.0} &       23.1 &       15.4 &       10.7 &        8.0 \\

           &            &        0.5 &       22.2 & {\bf 76.1} &       43.4 &       26.5 &       18.4 &       16.2 &       42.2 & {\bf 73.2} &       43.4 &       25.3 &       15.8 &       10.8 \\

           &            &        0.8 &       43.3 & {\bf 95.5} &       89.4 &       66.2 &       43.2 &       33.6 &       44.2 & {\bf 94.2} &       78.5 &       48.1 &       26.7 &       16.0 \\

           &        400 &        0.2 &       99.8 & {\bf 99.9} &        100 &       99.4 &       94.5 &       82.2 &       91.5 & {\bf 95.3} &       95.4 &       95.4 &       95.5 &       94.7 \\

           &            &        0.5 &       89.2 &  {\bf 100} &        100 &        100 &       99.8 &       97.9 &       58.0 & {\bf 95.5} &       95.6 &       95.5 &       95.5 &       95.4 \\

           &            &        0.8 &       18.5 & {\bf 99.9} &       99.9 &       99.9 &       99.9 &       99.9 &        7.4 & {\bf 95.5} &       95.4 &       95.5 &       95.5 &       95.4 \\

   $r_0 = 2$ &        100 &        0.2 &        3.2 & {\bf 21.0} &        6.3 &        2.6 &        1.3 &        1.1 &       25.7 & {\bf 40.6} &       19.4 &       10.2 &        5.1 &        2.6 \\

           &            &        0.5 &        2.2 & {\bf 78.7} &       36.1 &       13.7 &        5.7 &        3.3 &       18.6 & {\bf 77.9} &       43.8 &       21.8 &        9.6 &        4.7 \\

           &            &        0.8 &       10.1 & {\bf 94.9} &       89.2 &       60.2 &       27.4 &       14.2 &       23.9 & {\bf 94.3} &       82.3 &       49.7 &       21.4 &        9.7 \\

           &        400 &        0.2 &       99.3 & {\bf 99.8} &       99.8 &       99.7 &       99.2 &       94.3 &       93.1 & {\bf 95.3} &       95.6 &       95.5 &       95.5 &       95.2 \\

           &            &        0.5 &       90.6 & {\bf 99.8} &       99.7 &       99.7 &       99.7 &       99.6 &       73.4 & {\bf 95.5} &       95.5 &       95.3 &       95.3 &       95.3 \\

           &            &        0.8 &       40.0 & {\bf 99.6} &       99.7 &       99.7 &       99.7 &       99.6 &       26.0 & {\bf 95.4} &       95.4 &       95.1 &       95.3 &       95.2 \\
\hline
\end{tabular}  }
\end{table}

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\begin{table}[htbp]
%  \centering
TABLE 2: Sequential determination of lag length using IC($k,p)$: $\hat{k}_{\text{IC}}$.
VAR(2) model with rank $r_0 = 1$. \\
 \small{   % Table generated by Excel2LaTeX from sheet 'r=0 iidN_1 comp'
\begin{tabular}{rrrrrrrr|rrrrrr|rrrrrr}
\hline
           &            &             \multicolumn{ 3}{c}{HQC($k,p$)} &             \multicolumn{ 3}{c}{BIC($k,p$)} &             \multicolumn{ 3}{c}{HQC($k,p$)} &             \multicolumn{ 3}{c}{BIC($k,p$)} &             \multicolumn{ 3}{c}{HQC($k,p$)} &             \multicolumn{ 3}{c}{BIC($k,p$)} \\
\hline
           &            &                              \multicolumn{ 6}{c}{GJR-GARCH errors [Case A]} &          \multicolumn{ 6}{c}{Autoregressive Stochastic Volatility [Case B]} &                       \multicolumn{ 6}{c}{Single Volatility Break [Case C]} \\
\hline
   $\gamma$ &    {\it T} &      $k = 1$ &          2 &    3,4,5,6 &      $k = 1$ &          2 &    3,4,5,6 &      $k = 1$ &          2 &    3,4,5,6 &      $k = 1$ &          2 &    3,4,5,6 &      $k = 1$ &          2 &    3,4,5,6 &      $k = 1$ &          2 &    3,4,5,6 \\
\hline
       0.0 &         50 & {\bf 99.2} &        0.6 &        0.3 & {\bf 100.0} &        0.0 &        0.0 & {\bf 89.9} &        4.6 &        5.5 & {\bf 99.0} &        0.8 &        0.3 & {\bf 35.6} &        9.9 &       54.5 & {\bf 95.4} &        3.7 &        0.9 \\

           &        100 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 93.7} &        4.6 &        1.7 & {\bf 99.5} &        0.5 &        0.0 & {\bf 80.9} &       10.5 &        8.7 & {\bf 99.6} &        0.4 &        0.0 \\

           &        200 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 90.4} &        7.2 &        2.4 & {\bf 98.9} &        1.0 &        0.1 & {\bf 92.7} &        5.6 &        1.6 & {\bf 100.0} &        0.1 &        0.0 \\

           &        400 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 83.2} &       11.5 &        5.3 & {\bf 97.5} &        2.2 &        0.2 & {\bf 96.3} &        3.3 &        0.4 & {\bf 100.0} &        0.0 &        0.0 \\

       0.3 &         50 &       76.2 & {\bf 22.7} &        1.1 &       98.4 &  {\bf 1.7} &        0.0 &       57.6 & {\bf 31.6} &       10.8 &       90.6 &  {\bf 8.8} &        0.6 &       16.4 & {\bf 18.5} &       65.1 &       78.5 & {\bf 18.7} &        2.8 \\

           &        100 &       39.1 & {\bf 60.9} &        0.0 &       89.6 & {\bf 10.4} &        0.0 &       32.3 & {\bf 63.2} &        4.5 &       74.2 & {\bf 25.6} &        0.2 &       17.6 & {\bf 60.7} &       21.7 &       68.5 & {\bf 31.2} &        0.3 \\

           &        200 &        2.0 & {\bf 98.0} &        0.0 &       47.9 & {\bf 52.1} &        0.0 &        6.4 & {\bf 87.3} &        6.2 &       35.9 & {\bf 63.6} &        0.4 &        1.8 & {\bf 89.4} &        8.8 &       28.3 & {\bf 71.6} &        0.1 \\

           &        400 &        0.0 & {\bf 100.0} &        0.0 &        0.7 & {\bf 99.3} &        0.0 &        0.4 & {\bf 88.4} &       11.3 &        5.7 & {\bf 93.2} &        1.1 &        0.0 & {\bf 96.0} &        4.0 &        0.6 & {\bf 99.4} &        0.0 \\

       0.5 &         50 &       11.2 & {\bf 86.3} &        2.5 &       53.2 & {\bf 46.5} &        0.3 &       11.1 & {\bf 73.0} &       15.9 &       44.5 & {\bf 54.3} &        1.2 &        2.8 & {\bf 24.5} &       72.7 &       36.3 & {\bf 56.6} &        7.1 \\

           &        100 &        0.0 & {\bf 100.0} &        0.0 &        1.3 & {\bf 98.7} &        0.0 &        0.5 & {\bf 93.8} &        5.7 &        6.5 & {\bf 93.1} &        0.4 &        0.1 & {\bf 75.8} &       24.1 &        4.0 & {\bf 95.2} &        0.8 \\

           &        200 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 93.4} &        6.6 &        0.1 & {\bf 99.3} &        0.6 &        0.0 & {\bf 91.0} &        9.0 &        0.0 & {\bf 99.9} &        0.1 \\

           &        400 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 88.0} &       12.0 &        0.0 & {\bf 98.8} &        1.2 &        0.0 & {\bf 95.9} &        4.1 &        0.0 & {\bf 100.0} &        0.0 \\

       0.9 &         50 &        0.0 & {\bf 95.9} &        4.1 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 76.5} &       23.5 &        0.1 & {\bf 97.8} &        2.1 &        0.0 & {\bf 17.3} &       82.7 &        0.0 & {\bf 86.2} &       13.8 \\

           &        100 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 92.9} &        7.1 &        0.0 & {\bf 99.4} &        0.6 &        0.0 & {\bf 73.9} &       26.1 &        0.0 & {\bf 99.1} &        0.9 \\

           &        200 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 91.5} &        8.5 &        0.0 & {\bf 99.1} &        0.9 &        0.0 & {\bf 90.6} &        9.4 &        0.0 & {\bf 99.9} &        0.1 \\

           &        400 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 84.6} &       15.4 &        0.0 & {\bf 98.1} &        1.9 &        0.0 & {\bf 95.8} &        4.2 &        0.0 & {\bf 100.0} &        0.0 \\
\hline
\end{tabular}  }
\end{table}%

\pagebreak

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\begin{table}[htbp]
TABLE 3.1: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 0$, GJR-GARCH errors [Case A] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 & {\bf 61.8} &       29.3 &        9.0 & {\bf 95.4} &        4.6 &        0.0 & {\bf 50.6} &       25.1 &       24.2 & {\bf 61.8} &       29.2 &        8.9 & {\bf 80.9} &       11.7 &        7.4 & {\bf 95.4} &        4.6 &        0.0 \\

           &        100 & {\bf 81.3} &       15.9 &        2.7 & {\bf 99.4} &        0.6 &        0.0 & {\bf 75.3} &       20.8 &        4.0 & {\bf 78.4} &       17.6 &        4.0 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 \\

           &        200 & {\bf 93.1} &        4.8 &        2.2 & {\bf 98.9} &        1.1 &        0.0 & {\bf 85.9} &       12.5 &        1.6 & {\bf 86.9} &       11.6 &        1.6 & {\bf 100.0} &        0.0 &        0.0 & {\bf 100.0} &        0.0 &        0.0 \\

           &        400 & {\bf 92.7} &        6.8 &        0.4 & {\bf 99.7} &        0.3 &        0.0 & {\bf 92.6} &        7.0 &        0.4 & {\bf 92.7} &        6.9 &        0.4 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 &        0.0 \\

       0.3 &         50 & {\bf 33.6} &       39.8 &       26.6 & {\bf 80.5} &       15.2 &        4.3 & {\bf 19.8} &       25.2 &       55.0 & {\bf 34.6} &       41.5 &       23.9 & {\bf 56.3} &       22.9 &       20.8 & {\bf 77.9} &       16.9 &        5.2 \\

           &        100 & {\bf 65.1} &       27.2 &        7.8 & {\bf 95.7} &        3.7 &        0.6 & {\bf 66.4} &       26.0 &        7.6 & {\bf 53.0} &       35.6 &       11.4 & {\bf 97.2} &        2.2 &        0.6 & {\bf 96.6} &        2.9 &        0.6 \\

           &        200 & {\bf 80.7} &       15.9 &        3.4 & {\bf 97.7} &        2.3 &        0.0 & {\bf 85.3} &       12.5 &        2.2 & {\bf 77.0} &       18.6 &        4.5 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 &        0.0 \\

           &        400 & {\bf 92.0} &        7.1 &        0.9 & {\bf 99.7} &        0.3 &        0.0 & {\bf 91.9} &        7.2 &        0.9 & {\bf 92.1} &        7.1 &        0.9 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 &        0.0 \\

       0.5 &         50 & {\bf 28.4} &       35.0 &       36.5 & {\bf 64.2} &       27.4 &        8.4 & {\bf 14.0} &       19.1 &       66.9 & {\bf 11.4} &       35.8 &       52.8 & {\bf 45.5} &       21.0 &       33.5 & {\bf 70.3} &       20.8 &        8.9 \\

           &        100 & {\bf 69.6} &       25.8 &        4.7 & {\bf 97.8} &        1.9 &        0.3 & {\bf 65.0} &       28.0 &        7.1 & {\bf 67.1} &       26.2 &        6.8 & {\bf 96.9} &        2.8 &        0.3 & {\bf 98.4} &        1.3 &        0.3 \\

           &        200 & {\bf 84.7} &       12.8 &        2.5 & {\bf 99.7} &        0.3 &        0.0 & {\bf 84.0} &       13.4 &        2.6 & {\bf 84.7} &       12.8 &        2.6 & {\bf 99.4} &        0.6 &        0.0 & {\bf 99.7} &        0.3 &        0.0 \\

           &        400 & {\bf 91.5} &        7.4 &        1.1 & {\bf 99.7} &        0.3 &        0.0 & {\bf 91.6} &        7.2 &        1.2 & {\bf 91.5} &        7.4 &        1.1 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 &        0.0 \\

       0.9 &         50 &  {\bf 3.6} &        8.5 &       88.0 & {\bf 56.8} &       24.9 &       18.3 &  {\bf 0.6} &        3.2 &       96.2 &  {\bf 3.6} &        9.2 &       87.2 & {\bf 16.3} &       11.0 &       72.6 & {\bf 51.4} &       23.0 &       25.6 \\

           &        100 & {\bf 28.5} &       31.6 &       39.9 & {\bf 92.8} &        5.3 &        1.9 & {\bf 22.1} &       28.8 &       49.1 & {\bf 26.7} &       29.8 &       43.5 & {\bf 87.1} &        9.2 &        3.7 & {\bf 93.2} &        6.2 &        0.6 \\

           &        200 & {\bf 59.6} &       29.9 &       10.5 & {\bf 99.3} &        0.7 &        0.0 & {\bf 58.0} &       31.5 &       10.6 & {\bf 59.6} &       29.9 &       10.5 & {\bf 98.5} &        1.6 &        0.0 & {\bf 99.4} &        0.6 &        0.0 \\

           &        400 & {\bf 81.9} &       16.7 &        1.5 & {\bf 99.9} &        0.1 &        0.0 & {\bf 81.6} &       16.7 &        1.7 & {\bf 81.8} &       16.7 &        1.5 & {\bf 99.9} &        0.1 &        0.0 & {\bf 99.9} &        0.1 &        0.0 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 & {\bf 59.9} &       31.8 &        8.2 & {\bf 96.3} &        3.7 &        0.0 & {\bf 92.1} &        6.5 &        1.4 & {\bf 96.3} &        3.4 &        0.3 & {\bf 96.2} &        3.5 &        0.3 &            &            &            \\

           &        100 & {\bf 80.3} &       17.4 &        2.3 & {\bf 99.0} &        1.0 &        0.0 & {\bf 94.2} &        5.4 &        0.4 & {\bf 94.7} &        4.7 &        0.7 & {\bf 94.5} &        4.9 &        0.6 &            &            &            \\

           &        200 & {\bf 86.0} &       13.4 &        0.6 & {\bf 100.0} &        0.0 &        0.0 & {\bf 95.2} &        4.8 &        0.0 & {\bf 95.2} &        4.8 &        0.0 & {\bf 95.2} &        4.8 &        0.0 &            &            &            \\

           &        400 & {\bf 92.9} &        6.5 &        0.6 & {\bf 100.0} &        0.0 &        0.0 & {\bf 94.0} &        5.2 &        0.8 & {\bf 94.2} &        5.1 &        0.7 & {\bf 94.1} &        5.1 &        0.8 &            &            &            \\

       0.3 &         50 & {\bf 36.4} &       42.3 &       21.4 & {\bf 82.7} &       15.8 &        1.5 & {\bf 84.6} &       14.0 &        1.4 & {\bf 82.1} &       15.6 &        2.3 & {\bf 81.7} &       15.8 &        2.5 &            &            &            \\

           &        100 & {\bf 55.4} &       35.6 &        9.0 & {\bf 92.1} &        7.5 &        0.4 & {\bf 93.0} &        6.4 &        0.7 & {\bf 88.1} &       11.0 &        1.0 & {\bf 83.2} &       15.0 &        1.8 &            &            &            \\

           &        200 & {\bf 64.1} &       32.8 &        3.1 & {\bf 96.0} &        4.1 &        0.0 & {\bf 94.7} &        4.6 &        0.7 & {\bf 94.6} &        4.7 &        0.8 & {\bf 86.7} &       12.0 &        1.3 &            &            &            \\

           &        400 & {\bf 73.6} &       21.8 &        4.6 & {\bf 98.6} &        1.4 &        0.0 & {\bf 93.6} &        5.3 &        1.2 & {\bf 93.7} &        5.2 &        1.1 & {\bf 93.6} &        5.3 &        1.1 &            &            &            \\

       0.5 &         50 & {\bf 11.4} &       32.0 &       56.6 & {\bf 44.8} &       40.4 &       14.8 & {\bf 86.6} &       11.8 &        1.6 & {\bf 79.2} &       16.3 &        4.5 & {\bf 54.5} &       34.7 &       10.7 &            &            &            \\

           &        100 & {\bf 20.0} &       41.8 &       38.2 & {\bf 62.7} &       32.3 &        5.0 & {\bf 95.2} &        4.4 &        0.4 & {\bf 96.8} &        2.9 &        0.4 & {\bf 93.6} &        5.4 &        0.9 &            &            &            \\

           &        200 & {\bf 29.8} &       44.9 &       25.3 & {\bf 70.3} &       29.0 &        0.6 & {\bf 93.5} &        6.4 &        0.0 & {\bf 93.4} &        6.5 &        0.0 & {\bf 93.5} &        6.4 &        0.0 &            &            &            \\

           &        400 & {\bf 34.0} &       43.8 &       22.2 & {\bf 85.4} &       13.6 &        1.0 & {\bf 93.8} &        5.1 &        1.1 & {\bf 93.9} &        5.0 &        1.1 & {\bf 93.9} &        4.9 &        1.2 &            &            &            \\

       0.9 &         50 &  {\bf 0.0} &        0.5 &       99.5 &  {\bf 0.0} &        1.7 &       98.3 & {\bf 81.1} &       15.6 &        3.3 & {\bf 87.5} &       10.1 &        2.4 & {\bf 92.4} &        6.0 &        1.6 &            &            &            \\

           &        100 &  {\bf 0.0} &        0.8 &       99.3 &  {\bf 0.0} &        2.2 &       97.8 & {\bf 90.5} &        6.7 &        2.8 & {\bf 93.7} &        4.6 &        1.7 & {\bf 93.7} &        4.5 &        1.8 &            &            &            \\

           &        200 &  {\bf 0.0} &        0.6 &       99.4 &  {\bf 0.0} &        3.5 &       96.5 & {\bf 93.8} &        4.9 &        1.3 & {\bf 94.8} &        4.0 &        1.3 & {\bf 94.5} &        4.2 &        1.3 &            &            &            \\

           &        400 &  {\bf 0.0} &        1.8 &       98.2 &  {\bf 0.0} &        3.7 &       96.3 & {\bf 94.6} &        4.5 &        0.9 & {\bf 95.0} &        4.1 &        0.9 & {\bf 94.9} &        4.3 &        0.9 &            &            &            \\

\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table}[htbp]
TABLE 3.2: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 0$, Autoregressive Stochastic Volatility [Case B] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 & {\bf 45.5} &       34.5 &       20.0 & {\bf 83.4} &       14.6 &        2.0 & {\bf 26.8} &       25.3 &       47.9 & {\bf 46.6} &       35.6 &       17.9 & {\bf 54.1} &       24.7 &       21.3 & {\bf 78.5} &       16.1 &        5.3 \\

           &        100 & {\bf 60.1} &       29.4 &       10.6 & {\bf 91.2} &        8.2 &        0.7 & {\bf 52.4} &       32.1 &       15.6 & {\bf 60.4} &       29.1 &       10.5 & {\bf 85.9} &       12.3 &        1.8 & {\bf 90.2} &        8.9 &        0.9 \\

           &        200 & {\bf 68.0} &       25.8 &        6.2 & {\bf 94.5} &        5.3 &        0.2 & {\bf 64.7} &       27.9 &        7.4 & {\bf 67.5} &       26.1 &        6.4 & {\bf 93.7} &        6.1 &        0.3 & {\bf 94.0} &        5.8 &        0.2 \\

           &        400 & {\bf 75.6} &       21.0 &        3.5 & {\bf 97.1} &        2.9 &        0.1 & {\bf 75.3} &       21.1 &        3.6 & {\bf 74.6} &       22.0 &        3.4 & {\bf 97.4} &        2.6 &        0.1 & {\bf 97.0} &        2.9 &        0.0 \\

       0.3 &         50 & {\bf 29.9} &       37.1 &       33.0 & {\bf 69.9} &       24.5 &        5.6 & {\bf 13.2} &       22.0 &       64.7 & {\bf 31.1} &       39.2 &       29.7 & {\bf 38.3} &       29.2 &       32.6 & {\bf 62.7} &       25.5 &       11.8 \\

           &        100 & {\bf 51.4} &       34.6 &       13.9 & {\bf 84.6} &       13.8 &        1.7 & {\bf 45.2} &       35.6 &       19.2 & {\bf 46.2} &       37.2 &       16.6 & {\bf 83.6} &       14.2 &        2.2 & {\bf 85.6} &       12.7 &        1.7 \\

           &        200 & {\bf 65.9} &       27.6 &        6.6 & {\bf 93.4} &        6.3 &        0.3 & {\bf 63.0} &       29.1 &        7.9 & {\bf 61.7} &       30.3 &        8.0 & {\bf 94.0} &        5.8 &        0.3 & {\bf 94.2} &        5.6 &        0.2 \\

           &        400 & {\bf 76.4} &       20.2 &        3.5 & {\bf 97.3} &        2.6 &        0.0 & {\bf 76.2} &       20.3 &        3.5 & {\bf 75.1} &       21.3 &        3.6 & {\bf 97.6} &        2.4 &        0.0 & {\bf 97.5} &        2.5 &        0.0 \\

       0.5 &         50 & {\bf 20.3} &       33.5 &       46.3 & {\bf 56.8} &       31.1 &       12.1 &  {\bf 8.7} &       15.7 &       75.6 & {\bf 13.9} &       34.8 &       51.3 & {\bf 30.0} &       25.6 &       44.4 & {\bf 54.1} &       27.0 &       19.0 \\

           &        100 & {\bf 49.9} &       34.8 &       15.3 & {\bf 87.2} &       11.4 &        1.4 & {\bf 42.0} &       35.8 &       22.2 & {\bf 47.0} &       34.9 &       18.1 & {\bf 82.3} &       15.1 &        2.6 & {\bf 86.8} &       11.6 &        1.6 \\

           &        200 & {\bf 65.3} &       27.9 &        6.8 & {\bf 94.9} &        5.0 &        0.2 & {\bf 61.8} &       29.8 &        8.5 & {\bf 65.0} &       28.1 &        6.8 & {\bf 93.6} &        6.1 &        0.3 & {\bf 94.5} &        5.3 &        0.2 \\

           &        400 & {\bf 76.4} &       20.1 &        3.4 & {\bf 97.4} &        2.6 &        0.0 & {\bf 76.0} &       20.4 &        3.6 & {\bf 75.8} &       20.6 &        3.6 & {\bf 97.6} &        2.4 &        0.0 & {\bf 97.5} &        2.5 &        0.0 \\

       0.9 &         50 &  {\bf 2.9} &        8.9 &       88.2 & {\bf 42.8} &       28.7 &       28.5 &  {\bf 0.7} &        2.2 &       97.1 &  {\bf 3.5} &       11.1 &       85.4 &  {\bf 9.1} &        9.2 &       81.7 & {\bf 32.5} &       22.8 &       44.8 \\

           &        100 & {\bf 20.3} &       31.0 &       48.7 & {\bf 75.7} &       19.6 &        4.7 & {\bf 14.3} &       25.8 &       59.9 & {\bf 20.6} &       31.5 &       47.9 & {\bf 63.4} &       25.1 &       11.5 & {\bf 73.6} &       20.4 &        6.0 \\

           &        200 & {\bf 47.6} &       34.8 &       17.5 & {\bf 90.7} &        8.6 &        0.7 & {\bf 42.7} &       35.8 &       21.4 & {\bf 47.7} &       35.0 &       17.4 & {\bf 88.6} &       10.5 &        0.9 & {\bf 90.4} &        8.9 &        0.8 \\

           &        400 & {\bf 69.6} &       25.2 &        5.2 & {\bf 96.8} &        3.2 &        0.0 & {\bf 68.5} &       25.6 &        5.9 & {\bf 69.0} &       25.7 &        5.3 & {\bf 97.0} &        2.9 &        0.1 & {\bf 96.9} &        3.1 &        0.0 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 & {\bf 46.8} &       35.7 &       17.5 & {\bf 83.6} &       14.6 &        1.9 & {\bf 86.2} &       11.8 &        2.0 & {\bf 91.6} &        7.1 &        1.4 & {\bf 94.8} &        4.5 &        0.7 &            &            &            \\

           &        100 & {\bf 60.4} &       29.1 &       10.5 & {\bf 91.1} &        8.2 &        0.7 & {\bf 89.7} &        8.9 &        1.4 & {\bf 93.7} &        5.4 &        0.8 & {\bf 94.8} &        4.6 &        0.6 &            &            &            \\

           &        200 & {\bf 67.5} &       26.0 &        6.5 & {\bf 94.4} &        5.4 &        0.2 & {\bf 91.6} &        7.5 &        0.9 & {\bf 94.0} &        5.4 &        0.7 & {\bf 94.5} &        4.9 &        0.6 &            &            &            \\

           &        400 & {\bf 74.6} &       22.0 &        3.4 & {\bf 97.0} &        3.0 &        0.1 & {\bf 93.4} &        5.9 &        0.8 & {\bf 94.2} &        5.1 &        0.7 & {\bf 94.6} &        4.8 &        0.6 &            &            &            \\

       0.3 &         50 & {\bf 32.9} &       39.7 &       27.5 & {\bf 69.6} &       24.8 &        5.6 & {\bf 83.0} &       14.8 &        2.2 & {\bf 81.9} &       15.4 &        2.7 & {\bf 84.3} &       13.5 &        2.2 &            &            &            \\

           &        100 & {\bf 47.0} &       36.5 &       16.6 & {\bf 81.5} &       16.3 &        2.1 & {\bf 88.7} &        9.7 &        1.5 & {\bf 89.2} &        9.3 &        1.5 & {\bf 86.4} &       11.9 &        1.7 &            &            &            \\

           &        200 & {\bf 56.6} &       33.1 &       10.3 & {\bf 88.3} &       10.8 &        1.0 & {\bf 91.8} &        7.2 &        1.0 & {\bf 93.2} &        6.0 &        0.8 & {\bf 91.0} &        8.0 &        0.9 &            &            &            \\

           &        400 & {\bf 64.5} &       29.4 &        6.1 & {\bf 93.1} &        6.7 &        0.2 & {\bf 93.6} &        5.7 &        0.7 & {\bf 94.3} &        5.2 &        0.6 & {\bf 94.1} &        5.4 &        0.5 &            &            &            \\

       0.5 &         50 & {\bf 12.3} &       35.8 &       51.9 & {\bf 39.8} &       39.9 &       20.3 & {\bf 83.4} &       14.0 &        2.6 & {\bf 78.6} &       16.8 &        4.6 & {\bf 68.3} &       24.2 &        7.5 &            &            &            \\

           &        100 & {\bf 21.4} &       40.4 &       38.3 & {\bf 55.3} &       33.8 &       11.0 & {\bf 89.3} &        9.1 &        1.7 & {\bf 92.7} &        6.2 &        1.2 & {\bf 89.9} &        8.1 &        2.0 &            &            &            \\

           &        200 & {\bf 28.7} &       43.1 &       28.2 & {\bf 67.0} &       27.2 &        5.8 & {\bf 91.9} &        7.1 &        1.0 & {\bf 94.0} &        5.2 &        0.8 & {\bf 94.4} &        4.9 &        0.7 &            &            &            \\

           &        400 & {\bf 35.3} &       43.4 &       21.3 & {\bf 76.4} &       21.0 &        2.6 & {\bf 93.4} &        6.0 &        0.7 & {\bf 94.2} &        5.2 &        0.6 & {\bf 94.2} &        5.3 &        0.5 &            &            &            \\

       0.9 &         50 &  {\bf 0.0} &        0.5 &       99.5 &  {\bf 0.0} &        2.4 &       97.6 & {\bf 77.9} &       18.3 &        3.8 & {\bf 77.6} &       17.5 &        4.9 & {\bf 87.4} &        8.9 &        3.7 &            &            &            \\

           &        100 &  {\bf 0.0} &        0.9 &       99.1 &  {\bf 0.1} &        3.2 &       96.8 & {\bf 83.9} &       13.7 &        2.4 & {\bf 89.7} &        8.7 &        1.6 & {\bf 91.5} &        7.2 &        1.3 &            &            &            \\

           &        200 &  {\bf 0.0} &        1.2 &       98.8 &  {\bf 0.1} &        4.9 &       94.9 & {\bf 90.0} &        8.6 &        1.4 & {\bf 92.3} &        6.6 &        1.1 & {\bf 92.9} &        6.1 &        1.0 &            &            &            \\

           &        400 &  {\bf 0.0} &        1.7 &       98.3 &  {\bf 0.4} &        7.6 &       92.1 & {\bf 93.6} &        5.8 &        0.6 & {\bf 94.2} &        5.3 &        0.5 & {\bf 94.3} &        5.2 &        0.5 &            &            &            \\
\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table}[htbp]
TABLE 3.3: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 0$, Single Volatility Break [Case C] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 & {\bf 14.4} &       26.9 &       58.7 & {\bf 76.1} &       20.4 &        3.6 &  {\bf 1.2} &       12.0 &       86.8 & {\bf 26.9} &       39.2 &       33.9 & {\bf 16.6} &       40.9 &       42.5 & {\bf 33.3} &       33.6 &       33.1 \\

           &        100 & {\bf 41.5} &       39.6 &       19.0 & {\bf 90.3} &        9.1 &        0.6 & {\bf 17.6} &       39.8 &       42.5 & {\bf 43.2} &       39.7 &       17.1 & {\bf 68.9} &       26.5 &        4.5 & {\bf 84.6} &       13.5 &        1.9 \\

           &        200 & {\bf 56.4} &       33.0 &       10.6 & {\bf 96.5} &        3.4 &        0.1 & {\bf 41.5} &       39.3 &       19.2 & {\bf 57.0} &       32.6 &       10.4 & {\bf 91.0} &        8.6 &        0.4 & {\bf 95.9} &        4.0 &        0.2 \\

           &        400 & {\bf 66.8} &       27.1 &        6.1 & {\bf 98.8} &        1.2 &        0.0 & {\bf 58.3} &       32.4 &        9.4 & {\bf 67.0} &       26.9 &        6.1 & {\bf 97.8} &        2.1 &        0.1 & {\bf 98.7} &        1.3 &        0.0 \\

       0.3 &         50 &  {\bf 6.8} &       21.7 &       71.5 & {\bf 52.3} &       36.5 &       11.2 &  {\bf 0.6} &        8.2 &       91.2 & {\bf 13.6} &       38.0 &       48.4 & {\bf 11.4} &       36.6 &       52.0 & {\bf 19.0} &       36.5 &       44.5 \\

           &        100 & {\bf 30.3} &       42.0 &       27.7 & {\bf 73.2} &       23.5 &        3.3 & {\bf 15.5} &       39.2 &       45.3 & {\bf 25.0} &       45.5 &       29.5 & {\bf 65.8} &       28.7 &        5.5 & {\bf 72.6} &       23.1 &        4.4 \\

           &        200 & {\bf 51.9} &       34.7 &       13.4 & {\bf 92.0} &        7.4 &        0.6 & {\bf 39.5} &       40.1 &       20.4 & {\bf 44.8} &       38.0 &       17.2 & {\bf 91.3} &        8.1 &        0.6 & {\bf 94.3} &        5.4 &        0.4 \\

           &        400 & {\bf 65.0} &       28.1 &        6.9 & {\bf 98.6} &        1.4 &        0.0 & {\bf 57.7} &       32.8 &        9.5 & {\bf 64.7} &       28.4 &        6.9 & {\bf 97.9} &        2.0 &        0.1 & {\bf 98.5} &        1.5 &        0.0 \\

       0.5 &         50 &  {\bf 3.5} &       15.0 &       81.5 & {\bf 40.0} &       40.0 &       20.0 &  {\bf 0.2} &        4.6 &       95.1 &  {\bf 5.3} &       27.6 &       67.1 &  {\bf 7.6} &       30.4 &       62.1 & {\bf 14.6} &       30.7 &       54.7 \\

           &        100 & {\bf 29.7} &       40.4 &       29.9 & {\bf 83.0} &       15.0 &        2.1 & {\bf 13.7} &       36.9 &       49.4 & {\bf 29.7} &       40.8 &       29.5 & {\bf 63.2} &       30.4 &        6.4 & {\bf 76.1} &       20.0 &        3.8 \\

           &        200 & {\bf 50.0} &       35.8 &       14.3 & {\bf 95.3} &        4.5 &        0.2 & {\bf 37.4} &       40.6 &       22.0 & {\bf 50.3} &       35.7 &       14.0 & {\bf 90.1} &        9.2 &        0.7 & {\bf 94.2} &        5.5 &        0.3 \\

           &        400 & {\bf 63.6} &       29.1 &        7.3 & {\bf 98.5} &        1.5 &        0.0 & {\bf 56.6} &       33.4 &       10.0 & {\bf 63.6} &       29.2 &        7.3 & {\bf 97.7} &        2.3 &        0.1 & {\bf 98.4} &        1.6 &        0.0 \\

       0.9 &         50 &  {\bf 0.2} &        2.0 &       97.8 & {\bf 29.3} &       33.8 &       36.9 &  {\bf 0.0} &        0.1 &       99.9 &  {\bf 1.5} &        9.2 &       89.3 &  {\bf 0.6} &        5.3 &       94.1 &  {\bf 3.7} &        9.8 &       86.5 \\

           &        100 & {\bf 12.0} &       30.8 &       57.3 & {\bf 70.0} &       24.5 &        5.5 &  {\bf 3.0} &       16.9 &       80.1 & {\bf 13.8} &       33.4 &       52.8 & {\bf 39.2} &       39.2 &       21.6 & {\bf 58.1} &       30.2 &       11.7 \\

           &        200 & {\bf 32.7} &       40.2 &       27.1 & {\bf 90.4} &        9.0 &        0.6 & {\bf 22.2} &       39.4 &       38.4 & {\bf 33.2} &       40.3 &       26.5 & {\bf 81.8} &       16.0 &        2.2 & {\bf 89.0} &       10.0 &        1.0 \\

           &        400 & {\bf 51.5} &       35.5 &       13.0 & {\bf 97.5} &        2.5 &        0.1 & {\bf 44.6} &       39.0 &       16.4 & {\bf 51.5} &       35.5 &       13.0 & {\bf 95.5} &        4.3 &        0.2 & {\bf 97.2} &        2.7 &        0.1 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 & {\bf 28.5} &       40.1 &       31.4 & {\bf 76.5} &       20.1 &        3.3 & {\bf 82.2} &       15.8 &        2.0 & {\bf 72.1} &       23.6 &        4.2 & {\bf 91.2} &        7.1 &        1.7 &            &            &            \\

           &        100 & {\bf 43.4} &       39.6 &       17.0 & {\bf 90.3} &        9.1 &        0.6 & {\bf 74.9} &       21.4 &        3.7 & {\bf 88.0} &        9.9 &        2.1 & {\bf 93.6} &        5.5 &        0.9 &            &            &            \\

           &        200 & {\bf 57.0} &       32.6 &       10.4 & {\bf 96.5} &        3.4 &        0.1 & {\bf 84.8} &       13.2 &        2.1 & {\bf 92.9} &        6.4 &        0.7 & {\bf 93.8} &        5.6 &        0.6 &            &            &            \\

           &        400 & {\bf 67.0} &       26.9 &        6.1 & {\bf 98.8} &        1.2 &        0.0 & {\bf 90.9} &        7.8 &        1.2 & {\bf 94.5} &        4.9 &        0.6 & {\bf 94.7} &        4.7 &        0.6 &            &            &            \\

       0.3 &         50 & {\bf 16.9} &       41.2 &       42.0 & {\bf 52.8} &       36.2 &       11.0 & {\bf 82.7} &       15.3 &        2.0 & {\bf 67.4} &       27.4 &        5.3 & {\bf 72.7} &       22.7 &        4.6 &            &            &            \\

           &        100 & {\bf 25.4} &       46.0 &       28.6 & {\bf 67.0} &       28.5 &        4.5 & {\bf 75.4} &       20.9 &        3.8 & {\bf 79.0} &       17.7 &        3.3 & {\bf 76.9} &       20.2 &        3.0 &            &            &            \\

           &        200 & {\bf 33.8} &       44.4 &       21.8 & {\bf 77.7} &       20.2 &        2.1 & {\bf 85.1} &       12.8 &        2.1 & {\bf 91.1} &        7.7 &        1.2 & {\bf 85.4} &       12.7 &        1.9 &            &            &            \\

           &        400 & {\bf 39.8} &       43.4 &       16.8 & {\bf 85.2} &       13.7 &        1.0 & {\bf 91.1} &        7.7 &        1.3 & {\bf 94.0} &        5.4 &        0.7 & {\bf 94.1} &        5.3 &        0.6 &            &            &            \\

       0.5 &         50 &  {\bf 5.9} &       31.6 &       62.5 & {\bf 24.3} &       46.1 &       29.6 & {\bf 82.7} &       15.2 &        2.1 & {\bf 68.7} &       26.2 &        5.1 & {\bf 57.4} &       32.3 &       10.4 &            &            &            \\

           &        100 &  {\bf 8.7} &       39.1 &       52.2 & {\bf 32.9} &       47.2 &       19.9 & {\bf 76.0} &       20.6 &        3.4 & {\bf 84.2} &       13.2 &        2.6 & {\bf 86.5} &       10.5 &        3.0 &            &            &            \\

           &        200 & {\bf 11.9} &       41.5 &       46.7 & {\bf 42.6} &       43.4 &       14.0 & {\bf 85.1} &       12.8 &        2.1 & {\bf 91.8} &        7.1 &        1.0 & {\bf 93.1} &        6.1 &        0.8 &            &            &            \\

           &        400 & {\bf 14.0} &       44.7 &       41.3 & {\bf 51.7} &       38.7 &        9.6 & {\bf 90.7} &        8.1 &        1.3 & {\bf 94.0} &        5.4 &        0.7 & {\bf 94.2} &        5.2 &        0.6 &            &            &            \\

       0.9 &         50 &  {\bf 0.0} &        0.8 &       99.2 &  {\bf 0.0} &        3.5 &       96.5 & {\bf 75.7} &       20.5 &        3.7 & {\bf 62.0} &       31.1 &        6.9 & {\bf 72.3} &       19.8 &        7.9 &            &            &            \\

           &        100 &  {\bf 0.0} &        0.9 &       99.1 &  {\bf 0.0} &        3.4 &       96.5 & {\bf 72.5} &       23.6 &        3.9 & {\bf 79.7} &       17.3 &        3.0 & {\bf 89.3} &        9.2 &        1.5 &            &            &            \\

           &        200 &  {\bf 0.0} &        1.0 &       99.0 &  {\bf 0.0} &        3.8 &       96.2 & {\bf 84.9} &       12.8 &        2.4 & {\bf 91.1} &        7.8 &        1.1 & {\bf 92.7} &        6.4 &        0.9 &            &            &            \\

           &        400 &  {\bf 0.0} &        1.2 &       98.8 &  {\bf 0.1} &        5.2 &       94.7 & {\bf 90.3} &        8.4 &        1.3 & {\bf 93.9} &        5.4 &        0.7 & {\bf 94.2} &        5.1 &        0.6 &            &            &            \\
\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}

\pagebreak

\begin{table}[htbp]
TABLE 4.1: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 1$, GJR-GARCH errors [Case A] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 &       37.5 & {\bf 44.4} &       18.2 &       89.4 & {\bf 10.3} &        0.3 &       30.1 & {\bf 39.1} &       30.8 &       37.2 & {\bf 44.6} &       18.3 &       78.4 & {\bf 13.9} &        7.7 &       88.8 & {\bf 10.7} &        0.6 \\

           &        100 &       10.7 & {\bf 74.0} &       15.3 &       78.1 & {\bf 21.3} &        0.6 &       14.4 & {\bf 68.9} &       16.7 &       15.2 & {\bf 69.7} &       15.1 &       73.7 & {\bf 25.5} &        0.8 &       76.0 & {\bf 23.2} &        0.8 \\

           &        200 &        0.0 & {\bf 86.3} &       13.7 &        8.3 & {\bf 91.7} &        0.0 &        0.0 & {\bf 86.3} &       13.7 &        0.0 & {\bf 86.3} &       13.7 &        9.3 & {\bf 90.7} &        0.0 &        8.3 & {\bf 91.7} &        0.0 \\

           &        400 &        0.0 & {\bf 90.3} &        9.7 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 90.3} &        9.7 &        0.0 & {\bf 90.4} &        9.6 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 \\

       0.3 &         50 &       37.7 & {\bf 36.3} &       25.9 &       85.7 & {\bf 12.4} &        1.9 &       15.1 & {\bf 31.1} &       53.8 &       42.1 & {\bf 36.9} &       21.0 &       51.9 & {\bf 28.4} &       19.7 &       76.2 & {\bf 20.5} &        3.3 \\

           &        100 &       15.8 & {\bf 66.7} &       17.6 &       86.3 & {\bf 13.3} &        0.4 &        6.2 & {\bf 73.8} &       20.1 &       24.4 & {\bf 52.5} &       23.1 &       56.8 & {\bf 42.3} &        0.9 &       63.5 & {\bf 35.7} &        0.9 \\

           &        200 &        0.0 & {\bf 86.2} &       13.8 &       27.8 & {\bf 71.3} &        1.0 &        0.0 & {\bf 86.2} &       13.9 &        0.0 & {\bf 79.3} &       20.7 &        5.7 & {\bf 94.0} &        0.3 &        5.4 & {\bf 93.9} &        0.7 \\

           &        400 &        0.0 & {\bf 90.9} &        9.1 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 91.0} &        9.0 &        0.0 & {\bf 91.0} &        9.0 &        0.1 & {\bf 99.6} &        0.3 &        0.0 & {\bf 99.7} &        0.3 \\

       0.5 &         50 &       10.7 & {\bf 38.3} &       51.1 &       56.9 & {\bf 36.1} &        7.1 &        4.6 & {\bf 27.1} &       68.3 &       11.7 & {\bf 39.2} &       49.0 &       36.5 & {\bf 37.7} &       25.8 &       52.6 & {\bf 37.0} &       10.4 \\

           &        100 &        0.6 & {\bf 77.2} &       22.2 &       25.5 & {\bf 71.8} &        2.8 &        0.6 & {\bf 75.9} &       23.5 &        0.7 & {\bf 76.2} &       23.1 &       23.2 & {\bf 73.8} &        3.1 &       23.1 & {\bf 74.1} &        2.8 \\

           &        200 &        0.0 & {\bf 85.2} &       14.8 &        0.0 & {\bf 98.9} &        1.1 &        0.0 & {\bf 84.7} &       15.3 &        0.0 & {\bf 84.4} &       15.6 &        0.3 & {\bf 99.1} &        0.6 &        0.0 & {\bf 99.7} &        0.3 \\

           &        400 &        0.0 & {\bf 90.7} &        9.3 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 90.5} &        9.5 &        0.0 & {\bf 90.6} &        9.4 &        0.1 & {\bf 99.6} &        0.3 &        0.0 & {\bf 99.7} &        0.3 \\

       0.9 &         50 &        0.3 &  {\bf 8.9} &       90.8 &        1.7 & {\bf 56.6} &       41.6 &        0.3 &  {\bf 2.5} &       97.3 &        0.3 &  {\bf 8.8} &       91.0 &        0.8 & {\bf 25.9} &       73.2 &        1.4 & {\bf 54.5} &       44.1 \\

           &        100 &        0.0 & {\bf 41.4} &       58.6 &        0.0 & {\bf 89.8} &       10.2 &        0.0 & {\bf 39.3} &       60.8 &        0.0 & {\bf 41.4} &       58.6 &        0.0 & {\bf 88.1} &       11.9 &        0.0 & {\bf 89.9} &       10.1 \\

           &        200 &        0.0 & {\bf 72.8} &       27.2 &        0.0 & {\bf 97.7} &        2.3 &        0.0 & {\bf 72.7} &       27.3 &        0.0 & {\bf 72.7} &       27.3 &        0.0 & {\bf 97.7} &        2.3 &        0.0 & {\bf 97.7} &        2.3 \\

           &        400 &        0.0 & {\bf 84.9} &       15.1 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 84.5} &       15.5 &        0.0 & {\bf 85.0} &       15.0 &        0.0 & {\bf 99.7} &        0.3 &        0.0 & {\bf 99.7} &        0.3 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 &       40.3 & {\bf 39.8} &       19.9 &       90.1 &  {\bf 9.4} &        0.5 &       87.7 & {\bf 11.1} &        1.3 &       90.3 &  {\bf 8.8} &        1.0 &       90.8 &  {\bf 8.6} &        0.6 &            &            &            \\

           &        100 &       12.4 & {\bf 70.8} &       16.8 &       76.7 & {\bf 23.0} &        0.3 &       55.9 & {\bf 40.7} &        3.4 &       56.6 & {\bf 40.2} &        3.2 &       56.8 & {\bf 40.1} &        3.1 &            &            &            \\

           &        200 &        0.0 & {\bf 84.7} &       15.3 &        9.9 & {\bf 90.1} &        0.0 &        2.2 & {\bf 91.7} &        6.1 &        2.1 & {\bf 91.9} &        6.0 &        2.1 & {\bf 92.0} &        5.9 &            &            &            \\

           &        400 &        0.0 & {\bf 90.2} &        9.8 &        0.0 & {\bf 99.9} &        0.1 &        0.0 & {\bf 94.4} &        5.6 &        0.0 & {\bf 94.6} &        5.4 &        0.0 & {\bf 94.6} &        5.4 &            &            &            \\

       0.3 &         50 &       37.6 & {\bf 43.2} &       19.2 &       86.0 & {\bf 12.3} &        1.7 &       83.4 & {\bf 15.5} &        1.2 &       83.7 & {\bf 14.7} &        1.6 &       86.4 & {\bf 12.1} &        1.5 &            &            &            \\

           &        100 &       28.4 & {\bf 49.3} &       22.4 &       88.8 & {\bf 10.4} &        0.9 &       51.1 & {\bf 44.6} &        4.3 &       53.0 & {\bf 42.9} &        4.1 &       64.9 & {\bf 29.8} &        5.4 &            &            &            \\

           &        200 &        0.6 & {\bf 73.9} &       25.6 &       49.4 & {\bf 49.4} &        1.2 &        1.1 & {\bf 93.7} &        5.2 &        0.8 & {\bf 94.0} &        5.3 &        5.6 & {\bf 83.6} &       10.8 &            &            &            \\

           &        400 &        0.0 & {\bf 76.0} &       24.0 &        0.0 & {\bf 97.9} &        2.1 &        0.1 & {\bf 95.0} &        4.9 &        0.0 & {\bf 94.8} &        5.2 &        0.0 & {\bf 94.7} &        5.3 &            &            &            \\

       0.5 &         50 &       20.6 & {\bf 46.3} &       33.2 &       59.1 & {\bf 33.6} &        7.3 &       82.2 & {\bf 17.0} &        0.9 &       76.0 & {\bf 21.5} &        2.5 &       66.3 & {\bf 29.6} &        4.2 &            &            &            \\

           &        100 &       23.2 & {\bf 39.0} &       37.8 &       72.5 & {\bf 24.4} &        3.1 &       26.9 & {\bf 68.7} &        4.4 &       26.9 & {\bf 69.3} &        3.8 &       26.9 & {\bf 68.9} &        4.2 &            &            &            \\

           &        200 &        0.6 & {\bf 50.8} &       48.6 &       64.9 & {\bf 27.0} &        8.1 &        0.3 & {\bf 94.3} &        5.4 &        0.0 & {\bf 95.2} &        4.8 &        0.0 & {\bf 95.2} &        4.8 &            &            &            \\

           &        400 &        0.0 & {\bf 46.8} &       53.2 &        0.0 & {\bf 86.1} &       13.9 &        0.0 & {\bf 94.7} &        5.3 &        0.0 & {\bf 94.5} &        5.5 &        0.0 & {\bf 94.6} &        5.5 &            &            &            \\

       0.9 &         50 &        0.5 &  {\bf 6.9} &       92.6 &        1.2 & {\bf 16.9} &       81.9 &       50.9 & {\bf 44.4} &        4.7 &       17.4 & {\bf 77.0} &        5.6 &       17.0 & {\bf 78.2} &        4.8 &            &            &            \\

           &        100 &        0.2 &  {\bf 9.7} &       90.1 &        1.2 & {\bf 22.6} &       76.2 &        1.7 & {\bf 90.4} &        7.8 &        0.2 & {\bf 94.2} &        5.7 &        0.1 & {\bf 94.2} &        5.7 &            &            &            \\

           &        200 &        0.0 &  {\bf 1.2} &       98.8 &        1.8 & {\bf 25.7} &       72.5 &        0.0 & {\bf 94.8} &        5.2 &        0.0 & {\bf 95.3} &        4.7 &        0.0 & {\bf 95.0} &        5.0 &            &            &            \\

           &        400 &        0.0 &  {\bf 0.6} &       99.4 &        0.6 &  {\bf 1.8} &       97.6 &        0.0 & {\bf 95.3} &        4.7 &        0.0 & {\bf 95.5} &        4.6 &        0.0 & {\bf 95.4} &        4.6 &            &            &            \\
\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[htbp]
TABLE 4.2: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 1$, Autoregressive Stochastic Volatility [Case B] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 &       29.7 & {\bf 39.6} &       30.8 &       75.5 & {\bf 20.2} &        4.3 &       18.0 & {\bf 30.2} &       51.7 &       30.3 & {\bf 40.4} &       29.4 &       51.3 & {\bf 29.8} &       19.0 &       71.5 & {\bf 21.4} &        7.1 \\

           &        100 &       14.1 & {\bf 54.9} &       31.0 &       63.2 & {\bf 32.2} &        4.7 &       13.9 & {\bf 52.6} &       33.6 &       13.9 & {\bf 55.1} &       31.1 &       61.7 & {\bf 33.1} &        5.2 &       62.0 & {\bf 33.0} &        5.1 \\

           &        200 &        1.4 & {\bf 68.7} &       29.9 &       18.3 & {\bf 76.1} &        5.6 &        3.4 & {\bf 66.8} &       29.8 &        1.0 & {\bf 68.6} &       30.3 &       30.3 & {\bf 65.2} &        4.4 &       19.0 & {\bf 75.4} &        5.6 \\

           &        400 &        0.2 & {\bf 75.2} &       24.6 &        0.9 & {\bf 95.6} &        3.5 &        0.7 & {\bf 76.0} &       23.3 &        0.1 & {\bf 74.6} &       25.3 &       10.4 & {\bf 86.9} &        2.8 &        1.6 & {\bf 95.0} &        3.5 \\

       0.3 &         50 &       27.8 & {\bf 36.1} &       36.1 &       72.6 & {\bf 22.4} &        5.0 &       10.1 & {\bf 24.6} &       65.3 &       31.9 & {\bf 36.9} &       31.1 &       36.4 & {\bf 34.9} &       28.7 &       58.4 & {\bf 29.1} &       12.5 \\

           &        100 &       13.9 & {\bf 51.4} &       34.7 &       70.4 & {\bf 25.0} &        4.7 &        8.6 & {\bf 52.6} &       38.8 &       20.6 & {\bf 45.8} &       33.7 &       50.1 & {\bf 42.8} &        7.1 &       52.4 & {\bf 40.7} &        6.9 \\

           &        200 &        0.8 & {\bf 69.5} &       29.8 &       22.9 & {\bf 71.3} &        5.9 &        1.9 & {\bf 67.4} &       30.8 &        0.9 & {\bf 67.2} &       31.9 &       20.7 & {\bf 74.4} &        4.9 &       11.8 & {\bf 82.5} &        5.7 \\

           &        400 &        0.1 & {\bf 77.0} &       22.9 &        0.6 & {\bf 96.1} &        3.3 &        0.2 & {\bf 77.2} &       22.5 &        0.1 & {\bf 76.1} &       23.8 &        4.8 & {\bf 92.4} &        2.8 &        1.0 & {\bf 95.9} &        3.2 \\

       0.5 &         50 &        8.5 & {\bf 35.2} &       56.4 &       47.7 & {\bf 39.2} &       13.1 &        3.5 & {\bf 19.4} &       77.1 &       10.3 & {\bf 36.5} &       53.2 &       24.2 & {\bf 36.8} &       39.0 &       38.5 & {\bf 40.5} &       21.0 \\

           &        100 &        2.7 & {\bf 56.9} &       40.4 &       27.4 & {\bf 63.8} &        8.8 &        3.5 & {\bf 53.1} &       43.5 &        3.0 & {\bf 56.0} &       41.0 &       29.5 & {\bf 61.1} &        9.5 &       24.8 & {\bf 65.6} &        9.6 \\

           &        200 &        0.2 & {\bf 69.5} &       30.4 &        2.3 & {\bf 91.5} &        6.2 &        0.8 & {\bf 67.4} &       31.8 &        0.1 & {\bf 69.3} &       30.6 &       10.7 & {\bf 83.6} &        5.7 &        2.8 & {\bf 91.0} &        6.2 \\

           &        400 &        0.0 & {\bf 77.2} &       22.8 &        0.2 & {\bf 96.7} &        3.2 &        0.1 & {\bf 77.2} &       22.7 &        0.0 & {\bf 76.6} &       23.4 &        2.3 & {\bf 94.8} &        2.9 &        0.3 & {\bf 96.5} &        3.1 \\

       0.9 &         50 &        0.1 &  {\bf 9.8} &       90.2 &        3.2 & {\bf 48.6} &       48.2 &        0.0 &  {\bf 3.1} &       96.9 &        0.1 & {\bf 11.9} &       88.0 &        1.5 & {\bf 19.7} &       78.8 &        2.5 & {\bf 40.5} &       57.0 \\

           &        100 &        0.0 & {\bf 35.0} &       65.0 &        0.3 & {\bf 78.7} &       21.0 &        0.1 & {\bf 29.9} &       70.0 &        0.0 & {\bf 35.3} &       64.6 &        3.9 & {\bf 71.0} &       25.2 &        0.4 & {\bf 77.7} &       21.9 \\

           &        200 &        0.0 & {\bf 58.8} &       41.2 &        0.0 & {\bf 91.0} &        9.0 &        0.1 & {\bf 56.0} &       43.9 &        0.0 & {\bf 58.7} &       41.3 &        2.0 & {\bf 88.4} &        9.5 &        0.1 & {\bf 90.9} &        9.0 \\

           &        400 &        0.0 & {\bf 74.6} &       25.4 &        0.0 & {\bf 96.3} &        3.7 &        0.1 & {\bf 74.5} &       25.5 &        0.0 & {\bf 74.3} &       25.8 &        0.3 & {\bf 96.2} &        3.5 &        0.0 & {\bf 96.3} &        3.7 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 &       30.4 & {\bf 40.5} &       29.1 &       75.6 & {\bf 20.3} &        4.2 &       81.8 & {\bf 15.2} &        3.0 &       85.7 & {\bf 11.3} &        3.0 &       89.0 &  {\bf 8.8} &        2.3 &            &            &            \\

           &        100 &       13.9 & {\bf 55.2} &       31.0 &       63.1 & {\bf 32.3} &        4.7 &       62.4 & {\bf 31.9} &        5.7 &       65.0 & {\bf 29.7} &        5.3 &       66.5 & {\bf 28.5} &        5.0 &            &            &            \\

           &        200 &        1.0 & {\bf 68.7} &       30.3 &       18.1 & {\bf 76.3} &        5.6 &       25.0 & {\bf 67.7} &        7.3 &       18.8 & {\bf 74.3} &        6.8 &       19.0 & {\bf 74.6} &        6.5 &            &            &            \\

           &        400 &        0.0 & {\bf 74.6} &       25.4 &        0.7 & {\bf 95.7} &        3.6 &        6.1 & {\bf 87.0} &        6.9 &        1.9 & {\bf 92.0} &        6.1 &        1.5 & {\bf 92.7} &        5.8 &            &            &            \\

       0.3 &         50 &       34.9 & {\bf 37.8} &       27.4 &       73.4 & {\bf 22.1} &        4.5 &       79.7 & {\bf 17.2} &        3.2 &       79.6 & {\bf 16.9} &        3.5 &       84.5 & {\bf 12.9} &        2.5 &            &            &            \\

           &        100 &       26.9 & {\bf 43.0} &       30.1 &       76.9 & {\bf 19.1} &        4.0 &       55.7 & {\bf 37.9} &        6.4 &       57.1 & {\bf 36.3} &        6.5 &       66.6 & {\bf 27.6} &        5.8 &            &            &            \\

           &        200 &        2.7 & {\bf 63.3} &       34.0 &       45.5 & {\bf 47.7} &        6.8 &       17.9 & {\bf 74.7} &        7.4 &       12.6 & {\bf 80.5} &        6.9 &       18.4 & {\bf 73.3} &        8.3 &            &            &            \\

           &        400 &        0.1 & {\bf 69.6} &       30.3 &        2.3 & {\bf 91.1} &        6.6 &        3.1 & {\bf 90.1} &        6.8 &        1.0 & {\bf 93.1} &        5.9 &        0.7 & {\bf 93.3} &        5.9 &            &            &            \\

       0.5 &         50 &       20.4 & {\bf 39.4} &       40.3 &       53.8 & {\bf 34.6} &       11.6 &       77.4 & {\bf 19.3} &        3.3 &       72.0 & {\bf 23.6} &        4.4 &       69.2 & {\bf 25.7} &        5.1 &            &            &            \\

           &        100 &       21.7 & {\bf 37.0} &       41.3 &       64.7 & {\bf 27.8} &        7.5 &       40.5 & {\bf 52.6} &        6.9 &       37.3 & {\bf 56.2} &        6.5 &       37.4 & {\bf 56.0} &        6.6 &            &            &            \\

           &        200 &        3.3 & {\bf 43.9} &       52.8 &       53.8 & {\bf 32.7} &       13.5 &       10.6 & {\bf 82.2} &        7.1 &        4.7 & {\bf 89.2} &        6.2 &        4.3 & {\bf 89.9} &        5.8 &            &            &            \\

           &        400 &        0.1 & {\bf 48.3} &       51.6 &        3.9 & {\bf 78.1} &       18.0 &        1.7 & {\bf 91.7} &        6.6 &        0.5 & {\bf 93.6} &        5.9 &        0.3 & {\bf 93.9} &        5.8 &            &            &            \\

       0.9 &         50 &        0.1 &  {\bf 8.2} &       91.7 &        0.9 & {\bf 17.7} &       81.3 &       60.9 & {\bf 32.9} &        6.2 &       28.3 & {\bf 61.7} &       10.0 &       26.2 & {\bf 64.8} &        8.9 &            &            &            \\

           &        100 &        0.3 &  {\bf 9.7} &       90.0 &        1.6 & {\bf 23.6} &       74.7 &       12.3 & {\bf 77.8} &        9.9 &        2.8 & {\bf 88.9} &        8.2 &        2.3 & {\bf 90.3} &        7.4 &            &            &            \\

           &        200 &        0.1 &  {\bf 1.1} &       98.8 &        2.7 & {\bf 25.5} &       71.8 &        3.5 & {\bf 88.9} &        7.6 &        0.3 & {\bf 93.4} &        6.2 &        0.1 & {\bf 93.9} &        6.0 &            &            &            \\

           &        400 &        0.0 &  {\bf 0.6} &       99.4 &        1.4 &  {\bf 2.3} &       96.3 &        0.5 & {\bf 93.3} &        6.1 &        0.1 & {\bf 94.1} &        5.8 &        0.0 & {\bf 94.2} &        5.7 &            &            &            \\
\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[htbp]
TABLE 4.3: Determination of cointegration rank.
VAR(2) model with rank $r_0 = 1$, Single Volatility Break [Case C] \\
 \small{
\begin{tabular}{rrrrrrrr|rrrrrrrrrrrr}
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($k,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($k,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{HQC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{HQC($\hat{k}_{\text{BIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{AIC}},r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($\hat{k}_{\text{HQC}},r$)}} \\
\hline
 $\gamma$ &       $T$ &     $r=0$ &          1 &      2,3,4 &     $r=0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 &   $r = 0$ &          1 &      2,3,4 \\
\hline
       0.0 &         50 &        6.9 & {\bf 28.6} &       64.5 &       60.6 & {\bf 31.9} &        7.5 &        1.4 & {\bf 14.4} &       84.2 &       12.6 & {\bf 40.8} &       46.7 &       19.0 & {\bf 43.1} &       37.9 &       28.2 & {\bf 39.4} &       32.5 \\

           &        100 &        4.8 & {\bf 54.1} &       41.1 &       46.8 & {\bf 48.9} &        4.4 &        8.3 & {\bf 39.0} &       52.7 &        4.3 & {\bf 55.4} &       40.2 &       53.8 & {\bf 38.6} &        7.7 &       44.2 & {\bf 49.6} &        6.2 \\

           &        200 &        0.1 & {\bf 65.9} &       34.1 &        6.7 & {\bf 90.1} &        3.3 &        3.4 & {\bf 56.9} &       39.7 &        0.0 & {\bf 66.1} &       33.9 &       41.3 & {\bf 55.1} &        3.7 &        7.4 & {\bf 89.2} &        3.4 \\

           &        400 &        0.0 & {\bf 73.1} &       26.9 &        0.0 & {\bf 98.5} &        1.5 &        0.1 & {\bf 68.4} &       31.5 &        0.0 & {\bf 73.2} &       26.8 &        9.4 & {\bf 88.8} &        1.8 &        0.0 & {\bf 98.5} &        1.5 \\

       0.3 &         50 &        4.8 & {\bf 22.0} &       73.2 &       51.6 & {\bf 36.4} &       12.0 &        0.9 & {\bf 11.4} &       87.8 &       11.3 & {\bf 35.1} &       53.6 &       14.2 & {\bf 41.2} &       44.7 &       17.6 & {\bf 40.8} &       41.6 \\

           &        100 &        3.7 & {\bf 48.7} &       47.6 &       49.1 & {\bf 43.7} &        7.2 &        5.8 & {\bf 36.6} &       57.6 &        5.9 & {\bf 46.2} &       47.9 &       46.3 & {\bf 43.4} &       10.3 &       31.1 & {\bf 58.4} &       10.6 \\

           &        200 &        0.1 & {\bf 63.7} &       36.3 &        9.7 & {\bf 85.2} &        5.1 &        1.8 & {\bf 56.6} &       41.6 &        0.1 & {\bf 60.1} &       39.8 &       29.5 & {\bf 66.1} &        4.5 &        4.1 & {\bf 91.6} &        4.2 \\

           &        400 &        0.0 & {\bf 72.0} &       28.0 &        0.0 & {\bf 98.3} &        1.7 &        0.0 & {\bf 67.8} &       32.2 &        0.0 & {\bf 71.8} &       28.2 &        3.8 & {\bf 94.2} &        2.1 &        0.0 & {\bf 98.3} &        1.7 \\

       0.5 &         50 &        1.5 & {\bf 16.0} &       82.4 &       27.7 & {\bf 50.0} &       22.3 &        0.3 &  {\bf 8.3} &       91.4 &        3.2 & {\bf 26.3} &       70.5 &        9.8 & {\bf 37.3} &       53.0 &       10.9 & {\bf 38.1} &       51.0 \\

           &        100 &        0.9 & {\bf 47.3} &       51.9 &       15.5 & {\bf 74.4} &       10.1 &        3.9 & {\bf 34.9} &       61.1 &        0.5 & {\bf 48.7} &       50.8 &       37.1 & {\bf 49.7} &       13.2 &       14.8 & {\bf 72.1} &       13.2 \\

           &        200 &        0.0 & {\bf 61.7} &       38.3 &        0.3 & {\bf 95.1} &        4.6 &        0.8 & {\bf 54.7} &       44.4 &        0.0 & {\bf 61.9} &       38.1 &       19.8 & {\bf 74.6} &        5.6 &        0.6 & {\bf 94.3} &        5.1 \\

           &        400 &        0.0 & {\bf 71.1} &       28.9 &        0.0 & {\bf 98.3} &        1.7 &        0.0 & {\bf 67.0} &       33.0 &        0.0 & {\bf 71.2} &       28.8 &        1.4 & {\bf 96.4} &        2.3 &        0.0 & {\bf 98.3} &        1.7 \\

       0.9 &         50 &        0.0 &  {\bf 2.6} &       97.4 &        2.8 & {\bf 43.7} &       53.5 &        0.0 &  {\bf 0.9} &       99.1 &        0.1 &  {\bf 8.7} &       91.2 &        1.5 & {\bf 13.3} &       85.2 &        0.8 & {\bf 15.9} &       83.3 \\

           &        100 &        0.0 & {\bf 27.8} &       72.2 &        0.1 & {\bf 77.8} &       22.1 &        0.3 & {\bf 17.7} &       82.1 &        0.0 & {\bf 30.2} &       69.8 &       13.3 & {\bf 55.5} &       31.2 &        0.6 & {\bf 71.2} &       28.3 \\

           &        200 &        0.0 & {\bf 49.2} &       50.8 &        0.0 & {\bf 91.6} &        8.4 &        0.0 & {\bf 41.7} &       58.3 &        0.0 & {\bf 49.9} &       50.1 &        3.1 & {\bf 85.3} &       11.6 &        0.0 & {\bf 90.8} &        9.2 \\

           &        400 &        0.0 & {\bf 63.2} &       36.8 &        0.0 & {\bf 97.1} &        2.9 &        0.0 & {\bf 58.4} &       41.6 &        0.0 & {\bf 63.3} &       36.7 &        0.0 & {\bf 96.2} &        3.8 &        0.0 & {\bf 97.0} &        3.0 \\
\hline
           &            & \multicolumn{ 3}{c}{\footnotesize{HQC($1,r$)}} & \multicolumn{ 3}{c}{\footnotesize{BIC($1,r$)}} & \multicolumn{ 3}{|c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{AIC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{HQC}},T}$}} & \multicolumn{ 3}{c}{\footnotesize{$Q^{\ast}_{r,\hat{k}_{\text{BIC}},T}$}} &            &            &            \\
\cline{1-17}
       0.0 &         50 &       13.3 & {\bf 42.0} &       44.7 &       61.1 & {\bf 31.6} &        7.3 &       81.4 & {\bf 16.3} &        2.3 &       66.8 & {\bf 27.4} &        5.8 &       84.4 & {\bf 12.4} &        3.2 &            &            &            \\

           &        100 &        4.4 & {\bf 55.5} &       40.1 &       46.7 & {\bf 49.0} &        4.3 &       58.7 & {\bf 33.3} &        8.0 &       59.2 & {\bf 34.2} &        6.6 &       64.9 & {\bf 30.7} &        4.4 &            &            &            \\

           &        200 &        0.0 & {\bf 66.2} &       33.8 &        6.6 & {\bf 90.1} &        3.3 &       34.2 & {\bf 56.8} &        9.0 &       11.6 & {\bf 82.2} &        6.2 &       11.6 & {\bf 82.6} &        5.8 &            &            &            \\

           &        400 &        0.0 & {\bf 73.2} &       26.8 &        0.0 & {\bf 98.5} &        1.5 &        4.9 & {\bf 87.6} &        7.5 &        0.0 & {\bf 94.4} &        5.6 &        0.0 & {\bf 94.6} &        5.5 &            &            &            \\

       0.3 &         50 &       15.2 & {\bf 39.1} &       45.8 &       54.6 & {\bf 34.8} &       10.6 &       82.1 & {\bf 15.7} &        2.2 &       62.9 & {\bf 30.9} &        6.2 &       70.3 & {\bf 24.2} &        5.4 &            &            &            \\

           &        100 &        8.9 & {\bf 46.5} &       44.7 &       56.7 & {\bf 35.8} &        7.5 &       54.6 & {\bf 37.0} &        8.4 &       45.8 & {\bf 45.3} &        8.9 &       54.3 & {\bf 38.5} &        7.2 &            &            &            \\

           &        200 &        0.2 & {\bf 53.5} &       46.3 &       25.6 & {\bf 63.5} &       11.0 &       26.0 & {\bf 64.4} &        9.6 &        7.1 & {\bf 85.8} &        7.1 &       10.6 & {\bf 80.4} &        9.0 &            &            &            \\

           &        400 &        0.0 & {\bf 56.2} &       43.8 &        0.0 & {\bf 89.5} &       10.5 &        2.2 & {\bf 89.9} &        7.9 &        0.0 & {\bf 94.2} &        5.8 &        0.0 & {\bf 94.3} &        5.7 &            &            &            \\

       0.5 &         50 &        8.5 & {\bf 34.0} &       57.4 &       32.8 & {\bf 44.8} &       22.5 &       82.2 & {\bf 15.5} &        2.3 &       62.2 & {\bf 31.2} &        6.6 &       53.9 & {\bf 36.8} &        9.4 &            &            &            \\

           &        100 &        6.5 & {\bf 35.4} &       58.1 &       38.5 & {\bf 43.9} &       17.6 &       49.1 & {\bf 41.8} &        9.2 &       31.1 & {\bf 59.6} &        9.3 &       33.8 & {\bf 58.7} &        7.4 &            &            &            \\

           &        200 &        0.4 & {\bf 33.1} &       66.5 &       26.5 & {\bf 48.7} &       24.8 &       17.6 & {\bf 72.5} &        9.9 &        1.2 & {\bf 91.7} &        7.1 &        1.0 & {\bf 92.6} &        6.4 &            &            &            \\

           &        400 &        0.0 & {\bf 32.2} &       67.8 &        0.2 & {\bf 68.2} &       31.6 &        0.8 & {\bf 91.1} &        8.1 &        0.0 & {\bf 94.1} &        5.9 &        0.0 & {\bf 94.2} &        5.8 &            &            &            \\

       0.9 &         50 &        0.1 &  {\bf 6.9} &       93.0 &        0.6 & {\bf 17.2} &       82.2 &       77.1 & {\bf 19.5} &        3.4 &       51.9 & {\bf 39.2} &        8.9 &       26.2 & {\bf 61.1} &       12.8 &            &            &            \\

           &        100 &        0.1 &  {\bf 7.6} &       92.3 &        1.0 & {\bf 21.4} &       77.6 &       37.0 & {\bf 51.7} &       11.3 &        2.9 & {\bf 85.2} &       11.8 &        0.8 & {\bf 90.8} &        8.4 &            &            &            \\

           &        200 &        0.1 &  {\bf 1.1} &       98.8 &        1.2 & {\bf 19.4} &       79.4 &        5.8 & {\bf 83.5} &       10.8 &        0.0 & {\bf 92.1} &        7.9 &        0.0 & {\bf 93.0} &        7.0 &            &            &            \\

           &        400 &        0.0 &  {\bf 0.4} &       99.6 &        0.5 &  {\bf 1.9} &       97.6 &        0.0 & {\bf 92.2} &        7.8 &        0.0 & {\bf 94.1} &        5.9 &        0.0 & {\bf 94.3} &        5.7 &            &            &            \\
\cline{1-17}
\end{tabular}  }

\footnotesize{IC($k,r$) denotes the joint IC-based procedure for determining $k$ and $r$;
IC($\hat{k}_{\text{IC}},r$) denotes the sequential IC-based procedure for determining $k$ in the first step and $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained in the first step; IC($1,r$) denotes the semi-parametric IC-based approach by Cheng and Phillips (2009, 2012) for determining $r$ by fixing $k=1$; $Q^{\ast}_{r,\hat{k}_{\text{IC}},T}$ denotes the sequential wild bootstrap PLR test-based procedure for determining $r$ in the second step based on the lag order $\hat{k}_{\text{IC}}(p)$ obtained by an IC-based procedure in the first step.}
\end{table}


\end{landscape}
\end{document}
