%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%	miscelaneous commands                    %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\renewcommand{\hat}{\widehat}
\def\singlespace{\def\baselinestretch{1}\@normalsize}
\def\endsinglespace{}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Abbreviations in math mode                   %%        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \newcommand{\AR}{\mbox{AR}}
\newcommand{\AR}{{\rm AR}}
% \newcommand{\MA}{\mbox{MA}}
\newcommand{\MA}{{\rm MA}}
% \newcommand{\ARMA}{\mbox{ARMA}}
\newcommand{\ARMA}{{\rm ARMA}}
\newcommand{\AAR}{{\rm AAR}}
% \newcommand{\ARIMA}{\mbox{ARIMA}}
\newcommand{\ARIMA}{{\rm ARIMA}}
% \newcommand{\FAR}{\mbox{FAR}}
\newcommand{\FAR}{{\rm FAR}}
% \newcommand{\ARCH}{\mbox{ARCH}}
\newcommand{\ARCH}{{\rm ARCH}}
% \newcommand{\GARCH}{\mbox{GARCH}}
\newcommand{\GARCH}{{\rm GARCH}}
\newcommand{\Beta}{\mbox{Beta}}
\newcommand{\IRSC}{\mbox{IRSC}}
% \newcommand{\IID}{\mbox{IID}}
\newcommand{\IID}{{\rm IID}}
\newcommand{\LS}{{\mbox{\scriptsize LS}}}
\newcommand{\MISE}{\mbox{MISE}}
\newcommand{\MSE}{\mbox{MSE}}
\newcommand{\RSS}{\mbox{RSS}}
\newcommand{\WN}{\mbox{WN}}
\newcommand{\argmin}{\mbox{argmin}}
\newcommand{\bias}{\mbox{Bias}}
\newcommand{\cov}{\mbox{Cov}}
\newcommand{\cor}{\mbox{Corr}}
\newcommand{\diag}{\mbox{diag}}
\newcommand{\etal}{\mbox{\em et al.}}
\newcommand{\sgn}{\mbox{sgn}}
\newcommand{\sopt}{\mbox{{\scriptsize opt}}}
\newcommand{\tr}{\mbox{tr}}
\newcommand{\CV}{\mbox{CV}}
\newcommand{\GCV}{\mbox{GCV}}
\newcommand{\var}{\mbox{Var}}
\newcommand{\diff}{\nabla}

\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bOmega}{\boldsymbol{\Omega}}
\newcommand{\bomega}{\boldsymbol{\omega}}
\newcommand{\bom}{{\boldsymbol{\omega}}}
\newcommand{\bSigma}{\boldsymbol{\Sigma}}
\newcommand{\bga}{\boldsymbol{\gamma}}
\newcommand{\bve}{\mbox{\boldmath$\varepsilon$}}
\newcommand{\bpi}{\mbox{\boldmath$\pi$}}
\newcommand{\brho}{\mbox{\boldmath$\rho$}}
\newcommand{\bTheta} {\boldsymbol{\Theta}}
\newcommand{\bRho} {\boldsymbol{\Rho}}
\newcommand{\btheta} {{\boldsymbol{\theta}}}
\newcommand{\bbeta}{\boldsymbol{\beta}}
\newcommand{\balpha} {{\boldsymbol{\alpha}}}
\newcommand{\bpsi} {{\boldsymbol{\psi}}}
\newcommand{\bxi} {\boldsymbol{\xi}}
\newcommand{\bmu} {\boldsymbol{\mu}}
\newcommand{\bnu} {\boldsymbol{\nu}}
\newcommand{\bzeta} {\boldsymbol{\zeta}}
\newcommand{\bGamma} {\boldsymbol{\Gamma}}
\newcommand{\bLambda} {\boldsymbol{\Lambda}}
\newcommand{\bC}{{\mathbf C}}
\newcommand{\bD}{{\mathbf D}}
\newcommand{\bT}{{\mathbf T}}
\newcommand{\bN}{{\mathbf N}}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  boldface in math model                         %%        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\bA}{{\mathbf A}}
\newcommand{\bH}{{\mathbf H}}
\newcommand{\bB}{{\mathbf B}}
\newcommand{\bF}{{\mathbf F}}
\newcommand{\bG}{{\mathbf G}}
\newcommand{\bI}{{\mathbf I}}
\newcommand{\bR}{{\mathbf R}}
\newcommand{\bP}{{\mathbf P}}
\newcommand{\bS}{{\mathbf S}}
\newcommand{\bV}{{\mathbf V}}
\newcommand{\bW}{{\mathbf W}}
\newcommand{\bX}{{\mathbf X}}
\newcommand{\bU}{{\mathbf U}}
\newcommand{\bY}{{\mathbf Y}}
\newcommand{\bZ}{{\mathbf Z}}

\newcommand{\ba}{{\mathbf a}}
\newcommand{\bh}{{\mathbf h}}
\newcommand{\bb}{{\mathbf b}}
\newcommand{\bc}{{\mathbf c}}
\newcommand{\be}{{\mathbf e}}
\newcommand{\bff}{{\mathbf f}}
\newcommand{\bm}{{\mathbf m}}
\newcommand{\br}{{\mathbf r}}
\newcommand{\bt}{{\mathbf t}}
\newcommand{\bu}{{\mathbf u}}
\newcommand{\bv}{{\mathbf v}}
\newcommand{\bw}{{\mathbf w}}
\newcommand{\bx}{{\mathbf x}}
\newcommand{\by}{{\mathbf y}}
\newcommand{\bz}{{\mathbf z}}


\newcommand{\bdelta} {\boldsymbol{\delta}}
\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bepsilon}{\bolsymbol{\varepsilon}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Other mathematical symbols                     %%        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\bbox}{\hfill $\Box$}
\newcommand{\inte}{\int_{-\infty}^{+\infty}}
\newcommand{\rr}{\hbox{$\hskip .1em R\hskip -1.0em\relax I\hskip 0.50em$}}
\newcommand{\sI}{{\cal I}}
\newcommand{\calI}{{\cal I}}
\newcommand{\calJ}{{\cal J}}
\newcommand{\calX}{{\cal X}}
\newcommand{\calY}{{\cal Y}}
\newcommand{\calK}{{\cal K}}
\newcommand{\calZ}{{\cal Z}}
\newcommand{\calF}{{\cal F}}
\newcommand{\calH}{{\cal H}}
\newcommand{\calA}{{\cal A}}
\newcommand{\calB}{{\cal B}}
\newcommand{\calC}{{\cal C}}

\newcommand{\wcon}{\stackrel{D}{\longrightarrow}}
\newcommand{\scon}{\stackrel{a.s.}{\longrightarrow}}
\newcommand{\pcon}{\stackrel{P}{\longrightarrow}}
\newcommand{\vs}{\Longleftrightarrow}
\newcommand{\qed}{\hfill $\blacksquare$ \vspace{3mm}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Aliases added by QY                            %%        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\ve}{{\varepsilon}}
\newcommand{\la}{{\lambda}}
\def\ga{\gamma}
\def\om{\omega}
\def\hga{{\widehat \gamma}}
\def\bga{\mbox{\boldmath$\gamma$}}
\def\askip{\vspace{0.15in}}

\def\endp{{\hfill \vrule width 5pt height 5pt\par}}
\def\wh{\widehat}
\def\wt{\widetilde}
\def\askip{\vspace{0.08in}}

