Multi-step forecast error variances for periodically integrated time seires

A periodically integrated (PI) time series process assumes that the stochastic trend can be removed using a seasonally varying differencing filter. In this paper the multi-step forecast error variances are derived for a quarterly PI time series when low-order periodic autoregressions adequately describe the data. The forecast error variances display seasonal variation, indicating that observations in some seasons can be forecast more precise than those in others. Two examples illustrate the empirical relevance of calculating forecast error variances. A by-product of the analysis is an expression for the seasonally varying impact of the stochastic trend.