13.2 Residual analysis

If the model is correct and the parameters are estimated well, the residuals \(\hat z_t\) (the estimated shocks, computed by the recursion of Section 12) should behave like white noise. The checks are those of regression, adapted to time order:

• plot the residuals against time, looking for trends, changes in spread and outliers;
• compute the residual ac.f., \(r_k(\hat z)\), and see whether any values lie outside \(\pm 2/\sqrt n\);
• look at a histogram or normal plot, since prediction intervals assume normal shocks.

A large residual autocorrelation at some lag means the model has missed a dependence at that lag. Two cautions: with \(20\) lags examined, about one value outside the band is expected by chance; and for the first few lags the band \(\pm 2/\sqrt n\) is too wide for residuals, because fitting the model removes some of their autocorrelation, so a value near the band at lag \(1\) or \(2\) deserves attention.

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