Differencing the data

A question that arises in practice is whether to difference the data prior to con­struction of a forecasting model. This arises in all the models discussed above, but for simplicity it is discussed here in the context of a pure AR model. If one knows a priori that there is in fact a unit autoregressive root, then it is efficient to impose this information and to estimate the model in first differences. Of course, in practice this is not known. If there is a unit autoregressive root, then estimates of this root (or the coefficients associated with this root) are generally biased towards zero, and conditionally biased forecasts can obtain. However, the order of this bias is 1/ T, so for short horizon forecasts (h fixed) and T sufficiently large, this bias is negligible, so arguably the decision of whether to difference or not is unimportant to first-order asymptotically.

The issue of whether or not to difference the data, or more generally of how to treat the long-term dependence in the series, becomes important when the fore­cast horizon is long relative to the sample size. Computations in Stock (1996) suggest that these issues can arise even if the ratio, h/T, is small, .1 or greater. Conventional practice is to use a unit root pretest to make the decision about whether to difference or not, and the asymptotic results in Stock (1996) suggest that this approach has some merit when viewed from the perspective of mini­mizing either the maximum or integrated asymptotic risk, in a sense made pre­cise in that paper. Although Dickey-Fuller (1979) unit root pretests are most common, other unit root tests have greater power, and tests with greater power produce lower risk for the pretest estimator. Unit root tests are surveyed in Stock

(1994) and in Chapter 29 in this volume by Bierens.

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