\(\newcommand{\epsb}{{\boldsymbol{\epsilon}}}

\newcommand{\mub}{{\boldsymbol{\mu}}}

\newcommand{\thetab}{{\boldsymbol{\theta}}}

\newcommand{\Thetab}{{\boldsymbol{\Theta}}}

\newcommand{\etab}{{\boldsymbol{\eta}}}

\newcommand{\Sigmab}{{\boldsymbol{\Sigma}}}

\newcommand{\Phib}{{\boldsymbol{\Phi}}}

\newcommand{\Phat}{\hat{{\bf P}}}\)Vector autoregression (VAR) is a useful tool for analyzing the dynamics of multiple time series. VAR expresses a vector of observed variables as a function of its own lags.

**Simulation**

Let’s begin by simulating a bivariate VAR(2) process using the following specification,

\[

\begin{bmatrix} y_{1,t}\\ y_{2,t}

\end{bmatrix}

= \mub + {\bf A}_1 \begin{bmatrix} y_{1,t-1}\\ y_{2,t-1}

\end{bmatrix} + {\bf A}_2 \begin{bmatrix} y_{1,t-2}\\ y_{2,t-2}

\end{bmatrix} + \epsb_t

\]

where \(y_{1,t}\) and \(y_{2,t}\) are the observed series at time \(t\), \(\mub\) is a \(2 \times 1\) vector of intercepts, \({\bf A}_1\) and \({\bf A}_2\) are \(2\times 2\) parameter matrices, and \(\epsb_t\) is a \(2\times 1\) vector of innovations that is uncorrelated over time. I assume a \(N({\bf 0},\Sigmab)\) distribution for the innovations \(\epsb_t\), where \(\Sigmab\) is a \(2\times 2\) covariance matrix.

I set my sample size to 1,100 and Read more…