A vector autoregression (VAR) models several time series jointly, letting each variable depend on its own past values and the past values of every other variable in the system. It is the multivariate generalisation of the AR model. When the system is stable it has a convergent vector moving-average representation, from which impulse response functions — the effect of a shock to one variable on all the others over time — are read.
How to read these notes
These notes follow the multivariate part of a time-series econometrics course. They assume you are comfortable with the univariate AR model and the ideas of stationarity and white noise. The step to a VAR is conceptually small — replace scalars with vectors and matrices — but it opens up the analysis of how shocks propagate through a whole system, which is the core of modern macroeconometrics.
1. Why model variables jointly
A univariate AR model treats one series in isolation: GDP growth as a function of its own past, say. But economic variables are interdependent. Inflation depends on past interest rates; interest rates respond to past inflation and output. Studying each series alone throws away the cross-variable dynamics that are often exactly what we care about. A VAR keeps them.
A vector autoregression stacks several time series into a vector \(Y_t\) and lets the whole vector depend on its own past. Every variable is regressed on the lags of every variable, so the model captures the full set of dynamic feedbacks between them.
2. The VAR(1)
The simplest case is a first-order VAR. Collect \(k\) variables into the vector \(Y_t = (Y_{1t}, \dots, Y_{kt})'\). The VAR(1) is
This looks exactly like a univariate AR(1) — but now \(\alpha\) is a vector of intercepts, \(\Phi_1\) is a \(k \times k\) matrix of coefficients, and \(\varepsilon_t\) is a vector of shocks. Writing out a bivariate (two-variable) system makes the structure clear:
The off-diagonal coefficients \(\phi_{12}\) and \(\phi_{21}\) are what make it multivariate: \(\phi_{12}\) lets last period's value of variable 2 affect this period's variable 1, and \(\phi_{21}\) does the reverse. Set them to zero and the system collapses back into two separate AR(1)s. A VAR(p) simply adds more lag matrices \(\Phi_1, \dots, \Phi_p\).
3. Vector white noise
The error term \(\varepsilon_t\) is now a vector, so "white noise" needs a multivariate definition.
\(\varepsilon_t\) is vector white noise if it has mean zero, a constant variance-covariance matrix \(\Sigma\) each period, and no correlation across time. The matrix \(\Sigma\) is allowed to have non-zero off-diagonal entries, so the shocks to different variables can be contemporaneously correlated even though there is no correlation across different time periods.
That contemporaneous correlation is important and slightly subtle: the shock hitting inflation this quarter and the shock hitting output this quarter can be related, which has consequences for how impulse responses are interpreted.
4. Stationarity of a VAR
Just as the univariate AR(1) needs \(|\phi_1| < 1\) to be stationary, a VAR needs a stability condition on its coefficient matrix. The condition is the matrix analogue: it is stated in terms of the eigenvalues of \(\Phi_1\) (or, for higher orders, the roots of the characteristic equation built from all the lag matrices).
When the system is stable in this sense, \(Y_t\) is weakly stationary: it has constant means, variances and covariances that do not drift over time, and shocks die out rather than accumulating. This is the multivariate counterpart of requiring the AR(1) coefficient to be inside the unit circle, and it is the first thing to check for any VAR.
5. The moving-average (VMA) representation
A stable VAR can be inverted — exactly as a stationary AR can be written as an MA(∞) — into a vector moving-average representation, expressing \(Y_t\) as an infinite weighted sum of current and past shocks:
Here each \(\Theta_s\) is a \(k \times k\) matrix of coefficients, obtained from the VAR coefficient matrices, and \(\mu\) is the vector of unconditional means. For a stable system the \(\Theta_s\) shrink towards zero as \(s\) grows, so shocks from the distant past matter less and less. This representation is more than algebra — it is the form in which the dynamic effect of shocks becomes readable.
6. Impulse response functions
The single most useful output of a VAR is the impulse response function. It answers a concrete question: if one variable receives a one-off unit shock today, what happens to every variable in the system today, next period, the period after, and so on?
The matrices \(\Theta_s\) in the VMA representation are the impulse responses. The \((i,j)\) entry of \(\Theta_s\) is the effect on variable \(i\) at horizon \(s\) of a one-unit shock to variable \(j\) at time 0. Plotting these entries against the horizon \(s\) traces out how a shock propagates and decays through the system.
For example, in a bivariate VAR you can read off the path of a shock to variable 1 on both variables over horizons \(s = 0, 1, 2, \dots\), and separately the path of a shock to variable 2. In a stable VAR these paths converge back to zero, showing how long it takes the system to absorb a disturbance. Impulse responses are the way economists summarise statements like "a monetary policy shock raises unemployment for several quarters before fading", and they are the main reason VARs are central to empirical macroeconomics.
7. Estimating a VAR
A practical attraction of VARs is that they are easy to estimate. Because every equation has the same right-hand-side variables — the lags of all the variables — each equation can be estimated by OLS equation by equation, and the result is consistent and efficient. Lag length is usually chosen with the same information criteria (AIC, BIC) used for univariate models. The univariate AR model is just the one-variable special case of all this, so everything you know about AR estimation carries over.
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Companion videos: the Time Series Econometrics playlist on @economaths covers stationarity, ARIMA and applied time series.