\def\bx{\bm{x}}
\def\bX{\bm{X}}
\def\xbar{\overline{x}}
\def\xmean{\overline{x}}
\def\Xmedian{\widetilde{X}}

\def\Xhat{\widehat{X}}

\def\Zhat{\widehat{Z}}

\def\by{\bm{y}}
\def\bY{\bm{Y}}
\def\ybar{\overline{y}}
\def\ymean{\overline{y}}
\def\zmean{\overline{z}}
\def\Zmean{\overline{Z}}
\def\Ymean{\overline{Y}}

\def\Fhat{\widehat{F}}
\def\Fhatn{\widehat{F}_n}

\def\shat{\widehat{s}}
\def\sehat{\widehat{\text{se}}}
\def\se{\text{se}}


\def\bz{\bm{z}}
\def\bZ{\bm{Z}}
\def\bv{\bm{v}}
\def\br{\bm{r}}
\def\be{\bm{e}}
\def\bb{\bm{b}}
\def\bd{\bm{d}}
\def\bu{\bm{u}}
\def\bw{\bm{w}}

\def\O{\text{O}}
\def\o{\text{o}}

\newcommand\textsclarge[1]{\fontshape{sc}\fontsize{14}{15} \selectfont{#1}}
\newcommand\textscnormal[1]{\fontshape{sc}\fontsize{12}{15} \selectfont{#1}}

\newcommand\argmin[1]{\underset{#1}{\textrm{argmin}}}
\newcommand\argmax[1]{\underset{#1}{\textrm{argmax}}}
\def\omegavec{{\omega}}

\def\lra{\longrightarrow}
\def\Lra{\Longrightarrow}
\def\Xseq{X_1,\ldots,X_m}
\def\Yseq{Y_1,\ldots,Y_n}

\def\thetavec{{\theta}}

\def\leqM{\stackrel{p}{<=}}


\def\Rpackold{\texttt{R}}

\def\Rpacknew{{\rotatebox{-0}{\vspace{0.5 in}\begin{figure}\includegraphics[scale=0.6]{Rlogo.ps}\end{figure}}}}

\def\Rpack{\textcolor[rgb]{0.23,0.49,0.77}{{\Large \textsf{R}}}}

\def\thetahat{\widehat{\theta}}
\def\varthetahat{\widehat{\vartheta}}
\def\betahat{\widehat{\beta}}
\def\phihat{\widehat{\phi}}
\def\betatilde{\widetilde{\beta}}
\def\mutilde{\widetilde{\mu}}

\def\E{\mathbb{E}}
\def\Var{\mathbb{V}ar}
\def\Cov{\mathbb{C}ov}
\def\C{\mathbb{C}}
\def\Corr{\mathbb{C}orr}
\def\P{\mathbb{P}}
\def\R{\mathbb{R}}

\def\IID{\text{IID}}
\def\WN{\text{WN}}
\def\MA{\text{MA}}
\def\AR{\text{AR}}

\def\calJ{\mathcal{J}}
\def\calN{\mathcal{N}}
\def\calL{\mathcal{L}}
\def\calH{\mathcal{H}}
\def\calS{\mathcal{S}}

\def\Fisher{\mathcal{I}}
\def\Ind{\mathds{1}}

\def\Fhatn{\widehat{F}_n}

\def\lra{\longrightarrow}

\def\CiD{\stackrel{d}{\longrightarrow}}
\def\CiP{\stackrel{p}{\longrightarrow}}
\def\Cas{\stackrel{a.s.}{\longrightarrow}}

\def\transposeold{{\sf{T}}}
\def\transpose{\intercal}

\def\Xvec{{\skew0\bm{X}}}
\def\Xveca{{\skew0\bm{X}}_1}
\def\Xvecb{{\skew0\bm{X}}_2}

\def\Yvec{{\skew0\bm{Y}}}
\def\xvec{{\skew0\bm{x}}}
\def\xveca{{\skew0\bm{x}}_1}
\def\xvecb{{\skew0\bm{x}}_2}

\def\avec{{\skew0\bm{a}}}

\def\mvec{{\skew0\bm{m}}}
\def\muvec{{\skew0\bm{\mu}}}
\def\bmu{{\bm{\mu}}}

\def\pitilde{\widetilde{\pi}}
\def\ptilde{\widetilde{p}}
\def\phat{\widehat{p}}
\def\Lhat{\widehat{L}}
\def\What{\widehat{W}}

\def\cvec{{\skew0\mathbf{c}}}

\def\Xbar{\overline{X}}
\def\xbar{\overline{x}}
\def\ybar{\overline{y}}
\def\yhat{\hat{y}}
\def\betahat{\widehat{\beta}}
\def\betavec{\utilde{\beta}}
\def\betahatvec{\skew3\widehat{\utilde{\beta}}}
\def\sigmahat{\widehat{\sigma}}
\def\ehat{\widehat{e}}
\def\zhat{\widehat{z}}
\def\nhat{\widehat{n}}

\def\xbarA{\overline{x}_A}
\def\xbarB{\overline{x}_B}

\def\calR{\mathcal{R}}
\def\calX{\mathcal{X}}
\def\calS{\mathcal{S}}
\def\bhatint{\widehat{\beta}_0}
\def\bhatsl{\widehat{\beta}_1}

\def\ahat{\widehat{a}}
\def\bhat{\widehat{b}}
\def\chat{\widehat{c}}
\def\ghat{\widehat{g}}

\def\lhat{\widehat{l}}
\def\uhat{\widehat{u}}
\def\Ihat{\widehat{I}}
\def\Itilde{\widetilde{I}}
\def\Icheck{\widecheck{I}}

\def\rhohat{\widehat{\rho}}
\def\pihat{\widehat{\pi}}


\def\btilde{\widetilde{b}}
\def\bjack{\widehat{b}_{\text{JACK}}}

\def\One{\mathbf{1}}
\def\Zero{\mathbf{0}}

\newcommand\xbarBi[1]{\overline{x^{\text{\tiny{(B)}}}_{#1}}}
\newcommand\xbarib[1]{\overline{x^{\text{\tiny{(B)}}}_{#1}}}


\def\Fstat{\frac{(SSE_R - SSE_C)/(k-g)}{SSE_C/(n-k-1)}}

\def\approxsim{\sim \! \! \! : \;}
\def\hsp{\vspace{0.1 in}}
\def\fsp{\vspace{0.2 in}}

\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\dint{\mathop{\displaystyle \int}}%
\def\dsum{\mathop{\displaystyle \sum}}%

\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]\ignorespaces
}{%
  \popQED\endtrivlist\@endpefalse
}
\providecommand{\proofname}{Proof}

\def\sectitle{\thesection.~\insertsectionhead}
\def\subsectitle{\thesection.\thesubsection~\insertsubsectionhead}

\def\xhat{\widehat x}
\def\Xhat{\widehat X}
\def\Rhat{\widehat R}

\def\what{\widehat w}

\def\muhat{\widehat \mu}
\def\gammahat{\widehat \gamma}
\def\Gammahat{\widehat \Gamma}
\def\phihat{\widehat \phi}

\def\Phitilde{\widetilde \Phi}
\def\Thetatilde{\widetilde \Theta}



\def\Pred{\widehat P}
\def\Ptil{\widetilde P}

\def\Z{\mathbb{Z}}
\def\ARMA{\text{ARMA}}
\def\ARIMA{\text{ARIMA}}
\def\SARIMA{\text{SARIMA}}
\def\VAR{\text{VAR}}
\def\VARMA{\text{VARMA}}

\def\phivec{\utilde\phi}
\def\rhovec{\rho}

\def\tR{{\scriptscriptstyle \text{R}}}
\def\tF{{\scriptscriptstyle \text{F}}}
\def\tB{{\scriptscriptstyle \text{B}}}

\def\cvec{{\skew0\mathbf{c}}}
\def\rvec{{\skew0\bm{r}}}

\def\Xbar{\overline{X}}
\def\xbar{\overline{x}}
\def\ybar{\overline{y}}
\def\yhat{\hat{y}}
\def\betahat{\widehat{\beta}}
\def\betavec{\utilde{\beta}}
\def\betahatvec{\skew3\widehat{\utilde{\beta}}}
\def\sigmahat{\widehat{\sigma}}
\def\ehat{\widehat{e}}
\def\zhat{\widehat{z}}
\def\nhat{\widehat{n}}

\def\xbarA{\overline{x}_A}
\def\xbarB{\overline{x}_B}

\def\calR{\mathcal{R}}
\def\bhatint{\widehat{\beta}_0}
\def\bhatsl{\widehat{\beta}_1}

\def\bGt{\bm{G}_{t}}
\def\bHt{\bm{H}_{t}}
\def\bdt{\bm{d}_{t}}
\def\bTt{\bm{T}_{t}}
\def\bZt{\bm{Z}_{t}}
\def\beps{\bm{\epsilon}}
\def\balpha{\bm{\alpha}}
\def\ba{\bm{a}}
\def\bP{\bm{P}}
\def\bF{\bm{F}}
\def\bK{\bm{K}}
\def\bL{\bm{L}}
\def\bv{\bm{v}}
\def\bC{\bm{C}}



\def\Fstat{\frac{(SSE_R - SSE_C)/(k-g)}{SSE_C/(n-k-1)}}

\def\approxsim{\sim \! \! \! : \;}
\def\hsp{\vspace{0.1 in}}
\def\fsp{\vspace{0.2 in}}

\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\dint{\mathop{\displaystyle \int}}%
\def\dsum{\mathop{\displaystyle \sum}}%

%\input{tcilatex.tex}
%\input{psfig.tex}

\def\transpose{{\textsf{\tiny{T}}}}
\def\Xvec{{\skew0\bm{X}}}
\def\Xveca{{\skew0\bm{X}}_1}
\def\Xvecb{{\skew0\bm{X}}_2}

\def\Yvec{{\skew0\bm{Y}}}
\def\yvec{{\skew0\bm{y}}}
\def\xvec{{\skew0\bm{x}}}
\def\xveca{{\skew0\bm{x}}_1}
\def\xvecb{{\skew0\bm{x}}_2}

\def\mvec{{\skew0\bm{m}}}
\def\muvec{{\skew0\bm{\mu}}}

\def\cvec{{\skew0\mathbf{c}}}
\def\dvec{{\skew0\mathbf{d}}}

\def\Xbar{\overline{X}}
\def\xbar{\overline{x}}

\def\gbar{\overline{g}_T}

\def\thetahat{\skew1\widehat{\theta}_n}
\def\phihat{\skew1\widehat{\phi}_n}
\def\alphahat{\skew0\widehat{\alpha}_n}
\def\betahat{\skew2\widehat{\beta}_n}
\def\epsilonhat{\skew2\widehat{\epsilon}}

\def\Ahat{\skew3\widehat{A}}
\def\Bhat{\widehat{B}}

\def\CiP{\stackrel{p}{\longrightarrow}}
\def\CiL{\stackrel{\mathfrak{L}}{\longrightarrow}}
\def\Cas{\stackrel{a.s.}{\longrightarrow}}

\def\thtrue{\theta_0}

\def\ldot{\skew2\dot{l}}
\def\lddot{\skew3\ddot{l}}

\def\FI{\mathcal{I}}
\def\FIinv{\mathcal{J}}
\newcommand\Ident[1]{\bm{I}_{#1}}
\newcommand\Zerovec{\bm{0}}
\newcommand\zerovec{\bm{0}}

\newcommand\ScoreFunc[2]{S_{#1}(#2)}

\def\OI{I_n}

\def\O{\text{O}}

\def\Asim{\stackrel{\text{\tiny $\mathcal{A}$}}{\sim}}

\def\vsp{\vspace{0.1 in}}

\def\Xstar{X^\star}
\def\xstar{x^\star}

\def\Uvec{{\skew0\bm{U}}}
\def\Zvec{{\skew0\bm{Z}}}

\newcommand\PsiA[1]{\Psi_{#1}^A}
\newcommand\PsiB[1]{\Psi_{#1}^B}

\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}

\def\bmtheta{{\bm{\theta}}}
\def\bmthetahat{{\bm{\skew2\widehat{\theta}}}}
\def\bmbeta{\bm \beta}
\def\bmone{\bm 1}
\def\bmzero{\bm 0}
\def\bmA{\bm A}
\def\bmB{\bm B}
\def\bmD{\bm D}
\def\bmG{\bm G}
\def\bmI{\FI}
\def\bmR{\bm R}
\def\bmV{\bm V}
\def\bmY{\bm Y}
\def\bmmu{\bm \mu }
\def\v{Var}
\def\c{Cov}
\def\s2{\sigma^2}
\def\hs2{\hat{\sigma}^2}

\def\thetahatT{\widehat{\theta}_T}

\def\rT{{\!{\textsf{\tiny{T}}}}}
\def\matrixI{{\!\scriptscriptstyle -1}}
\newcommand\smallwidehat[1]{{\!\scriptscriptstyle \widehat{#1}}}

\def\dprod{\mathop{\displaystyle \prod }}

\def\shsp{\hspace{0.01in}}
\def\bhsp{\hspace{0.02in}}

\def\Shat{\widehat{S}}
\def\Fhat{\widehat{F}}
\def\phihat{\widehat{\phi}}
\def\pihat{\widehat{\pi}}

\def\brho{\bm{\rho}}
\def\bGamma{\bm{\Gamma}}
\def\bPhi{\bm{\Phi}}
\def\bOmega{\bm{\Omega}}
\def\bSigma{\bm{\Sigma}}
\def\bPsi{\bm{\Psi}}
\def\bUps{\bm{\Upsilon}}
\def\bphi{\bm{\phi}}
\def\bth{\bm{\theta}}
\def\bpi{\bm{\pi}}

\def\bTh{\bm{\Theta}}
\def\bA{\bm{A}}
\def\bB{\bm{B}}

\def\lra{\longrightarrow}
\def\iif{\Longleftrightarrow}

\def\Yfilt{\mathcal{Y}}
\def\alone{\alpha_1}
\def\beone{\beta_1}

\def\alzero{\alpha_0}
\def\bezero{\beta_0}

\def\ali{\alpha_i}
\def\bei{\beta_i}

\def\alpt{\alpha_t}

\def\sigt{\sigma_t}
\def\yt{y_t}
\def\Yt{Y_t}

\def\Yfilt{\mathcal{Y}}

\def\epst{Z_t}
%\def\etat{\eta_t}
\def\etat{U_t}

\def\GARCH{{\text{GARCH}}}