Prediction intervals

In some cases, the object of forecasting is not to produce a point forecast but rather to produce a range within which yt+h has a prespecified probability of falling. Even if within the context of point forecasting, it is useful to provide users of forecasts with a measure of the uncertainty of the forecast. Both ends can be accomplished by reporting prediction intervals.

In general, the form of the prediction interval depends on the underlying distribution of the data. The simplest prediction interval is obtained by assuming that the data are conditionally homoskedastic and normal. Under these assump­tions and regularity conditions, a prediction interval with asymptotic 67 percent coverage is given by уши ± Zh, where Zh = SSRh/(T – p), where SSRh is the sum of squared residuals from the h-step ahead regression (27.1) and T – p are the degrees of freedom of that regression.

If the series is conditionally normal but is conditionally heteroskedastic, this simple prediction error formula must be modified and the conditional variance can be computed using, for example, an ARCH model. If the series is condition­ally nonnormally distributed, other methods, such as the bootstrap, can be used to construct asymptotically valid prediction intervals.

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