Vector autoregression—simulation, estimation, and inference in Stata
\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…