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Posts Tagged ‘orthogonalized impulse-response functions’

## 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…