There are two basic goals in using VAR’s to analyze a set of series:

•To test formally theories which imply particular behavior for the vector autoregression.

•To learn more about the historical dynamics of the economy.

We are somewhat limited in what we can test formally using simple VAR’s: most of the interesting hypotheses boil down to multivariate generalizations of the Granger-Sims causality tests. Because the hypotheses will usually be joint restrictions across equations, a simple EXCLUDE instruction will not suffice. We use the special instruction RATIO which tests such cross-equation restrictions. You can also test for the existence of cointegrating (long-run) relationships. In addition, the instruction CVMODEL can be used to identify, estimate and test models for the contemporaneous relationship among the innovations in the VAR.

 

Mostly, VAR's are used to study the dynamics of the data. The coefficients in the estimated VAR are of little use themselves. Instead, we use a set of instructions which provide equivalent information in a more palatable form. The three key instructions are:

 

 

All of these can be handled using the Time Series—VAR (Forecast/Analyze) wizard.

 

Moving Average Representation

IMPULSE, ERRORS and HISTORY are all based upon converting

\begin{equation} {\bf{y}}_t = {\bf{X}}_t \beta + \sum\limits_{s = 1}^p {\Phi _s {\bf{y}}_{t - s} } + {\bf{u}}_t \,\,\,\,\,\,\,\,\,\,\,E\left( {{\bf{u}}_t {\bf{u'}}_t } \right) = \Sigma \label{eq:VARBasic} \end{equation}

into the moving average representation:

\begin{equation} {{\bf{y}}_t} = {{\bf{\hat y}}_t} + \sum\limits_{s = 0}^\infty {{\Psi _s}{{\bf{u}}_{t - s}}} \label{eq:VARBasicMAR} \end{equation}

where

•\(\bf{y}\) is an \(N\)-variate stochastic process

•\({{\bf{\hat y}}_t}\) is the deterministic part of \(\bf{y}_t\)

•\(\bf{u}_t\) is an \(N\)-variate white noise process: if \(t \ne s\), \(\bf{u}_t\) and \(\bf{u}_s\) are uncorrelated. This is called an innovation process for \(\bf{y}\).

It's possible in \eqref{eq:VARBasicMAR} for \(\bf{y}\) to have dimension \(M>N\) if there are some identities that define other variables, but handling those is relatively straightforward.

 

Whether the infinite sum in \eqref{eq:VARBasicMAR} actually converges in any meaningful sense is only important in a few cases where (infinitely) long-run behavior is of interest—in practice, we only need to compute this for a fixed finite number of terms.

 

The mapping from \eqref{eq:VARBasic} to \eqref{eq:VARBasicMAR} is a messy algebraic tangle if you try to do it symbolically, but can be computed fairly easily numerically (given \eqref{eq:VARBasic} with actual values for the \(\beta\) and \(\Phi\) coefficients) by repeatedly solving \eqref{eq:VARBasic} for particular choices for the lagged values for \(\bf{y}_t\) and current and future values for \(\bf{u}_t\). We show below the type of manipulation that underlies the analysis of the moving average representation.

Non-Orthogonal/Forecast Error Innovations

There are many equivalent representations for this model in \eqref{eq:VARBasicMAR}—for any non-singular matrix \({\bf{G}}\), \({{\Psi}_s}\) can be replaced by \({\Psi}_s{{\bf{G}}^{ - 1}}\) and \({\bf{u}}\) by \({\bf{Gu}}\) since that doesn't alter the fact that the innovation series is uncorrelated across time. A particular version is obtained by choosing some normalization.

 

If \({\Psi _0}\) is normalized to be the identity matrix, each component of \({{\bf{u}}_t}\) is the error that results from the one step forecast of the corresponding component of \({{\bf{y}}_t}\). These are the non-orthogonal innovations in the components of \({\bf{y}}\); non-orthogonal because, in general, the covariance matrix \(\Sigma  = E\left( {{\bf{u}}_t{\bf{u'}}_t } \right)\) is not diagonal. While these are the most "natural" set of innovations, it turns out that they aren't particularly helpful for analyzing the dynamics of the model beyond that first forecast step, precisely because they are correlated within a single time period (contemporaneously correlated). Instead, there are a number of methods proposed to choose an innovation process where the innovations are orthogonal (or uncorrelated) contemporaneously.

VAR to MAR

Write the expected value of some random variable \(z\) given data information through time \(t-1\) as \(E(z|t - 1)\) for short. If we look at \eqref{eq:VARBasic}, and look its expected value given \(t-1\), everything in it is known except \({\bf{u}}_t \): \({\bf{X}}_t \) because it's deterministic, the lags of \({\bf{y}}\) because there are dated \(t-1\) and earlier. Thus

\begin{equation} {\bf{y}}_t - E({\bf{y}}_t |t - 1) = {\bf{u}}_t \label{eq:var_onesteperror} \end{equation}

Now shift \eqref{eq:VARBasic} forward to time \(t+1\) and pull the \({\bf{y}}_t \) out of the sum of the lags:

\begin{equation} {\bf{y}}_{t + 1} = {\bf{X}}_{t + 1} \beta + \Phi _1 {\bf{y}}_t + \sum\limits_{s = 2}^p {\Phi _s {\bf{y}}_{t + 1 - s} } + {\bf{u}}_{t + 1} \label{eq:var_twostep} \end{equation}

If we take the expected value of this given \(t-1\) (which would be a two-step out forecast) we have

\begin{equation} E({\bf{y}}_{t + 1} |t - 1) = {\bf{X}}_{t + 1} \beta + \Phi _1 E({\bf{y}}_t |t - 1) + \sum\limits_{s = 2}^p {\Phi _s {\bf{y}}_{t + 1 - s} } \label{eq:var_twostepforecast} \end{equation}

The two-step forecast error can then be simplified to

\begin{equation} {\bf{y}}_{t + 1} - E({\bf{y}}_{t + 1} |t - 1) = \Phi _1 \left( {{\bf{y}}_t - E({\bf{y}}_t |t - 1)} \right) + {\bf{u}}_{t + 1} = \Phi _1 {\bf{u}}_t + {\bf{u}}_{t + 1} \label{eq:var_twostepforecasterror} \end{equation}

We can continue this as many steps as we want. Note that the deterministic terms drop out of the forecast error, as do the actual data from \(t-1\) and earlier; we are left only with the "out-of-sample" values of \(\bf{u}\). By focusing just on forecast errors for a finite number of steps, we can avoid having to deal with the fact that \eqref{eq:VARBasicMAR} might not exist as an infinite sum.

Copyright © 2025 Thomas A. Doan