14.5 Vector autoregressions
Sims (1980) argued that the exclusion restrictions that identify structural macroeconomic models are often “incredible”: there is rarely a convincing reason why a variable should appear in one equation and not another. His alternative was to model every variable as depending on lags of all of them.
Definition 14.5.1. A vector autoregression of order \(p\), VAR(\(p\)), for a \(G \times 1\) vector \(y_t\) is \[ y_t = c + A_1y_{t-1} + \dots + A_py_{t-p} + \varepsilon _t, \] where the \(A_j\) are \(G \times G\) matrices and \(\varepsilon _t\) are independent with mean zero and covariance matrix \(\Sigma \). The variable \(y_2\) does not Granger-cause \(y_1\) if the coefficients of all lags of \(y_2\) in the equation for \(y_1\) are zero, so that past \(y_2\) does not help to forecast \(y_1\) once past \(y_1\) is known.
- (i)
- The VAR is stable (stationary) if and only if all eigenvalues of the \(Gp \times Gp\) companion matrix \[ C = \begin {bmatrix} A_1 & A_2 & \cdots & A_{p-1} & A_p\\ I & 0 & \cdots & 0 & 0\\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & I & 0 \end {bmatrix} \] lie strictly inside the unit circle.
- (ii)
- Least squares applied to each equation separately is identical to SUR on the whole system, and, conditionally on the first \(p\) observations and under normality, to maximum likelihood.
- (iii)
- A stable VAR has the moving average representation \(y_t = \mu + \sum _{h=0}^\infty \Psi _h\varepsilon _{t-h}\), with \(\Psi _0 = I\) and \[ \Psi _h = \sum _{j=1}^{\min (h, p)}A_j\Psi _{h-j}, \qquad h \geq 1 . \]
Proof. (i) Stack \(Y_t = (y_t', y_{t-1}', \dots , y_{t-p+1}')'\). Then \(Y_t = \tilde c + CY_{t-1} + E_t\), with \(E_t = (\varepsilon _t', 0, \dots , 0)'\), a VAR(1). Iterating, \(Y_t\) depends on \(C^hE_{t-h}\), and \(C^h \to 0\) if and only if every eigenvalue of \(C\) has modulus less than one, which is the condition for the infinite sum to converge, as for the AR(1) scheme in Theorem 4.5.2.
(ii) Every equation has the same regressors, a constant and \(p\) lags of all the variables, so Theorem 14.4.1(ii) applies. With normal \(\varepsilon _t\) the conditional likelihood is that of a SUR system, which is maximised by GLS, here least squares.
(iii) Substitute \(y_{t-j} = \mu + \sum _h\Psi _h\varepsilon _{t-j-h}\) into the VAR and match the coefficients of \(\varepsilon _{t-h}\) on both sides: the coefficient of \(\varepsilon _{t-h}\) on the right is \(\sum _jA_j\Psi _{h-j}\), with \(\Psi _m = 0\) for \(m < 0\), and on the left it is \(\Psi _h\). □
The matrix \(\Psi _h\) gives the impulse responses: its \((i, j)\) element is the effect on \(y_{i,t+h}\) of a unit shock to the \(j\)th disturbance at time \(t\). But the disturbances of different equations are correlated, so a shock to one rarely occurs without the others. The usual remedy is to orthogonalise: with the Cholesky factorisation \(\Sigma = LL'\), \(L\) lower triangular, the shocks \(\eta _t = L^{-1}\varepsilon _t\) are uncorrelated with unit variance, and the responses to them are \(\Psi _hL\). The price is an ordering assumption: the first variable’s shock may affect all the others within the period, while the last variable’s shock affects only itself. Different orderings can give different answers, and the ordering should be justified by economic timing. Granger causality is tested by the \(F\) test of the relevant exclusion, and the lag order is chosen by an information criterion, \[ \text {AIC}(p) = \log |\hat \Sigma _p| + \frac {2K_p}{n}, \qquad \text {BIC}(p) = \log |\hat \Sigma _p| + \frac {K_p\log n}{n}, \] where \(K_p\) is the total number of coefficients, all the models being fitted to the same sample.
Granger causality is a statement about prediction, not about causation in the sense of Section 13. A variable that is chosen by forward-looking agents, such as a share price or an interest rate, Granger-causes the variables it anticipates even if it has no effect on them.
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