The effect of aggregation on prediction and estimation in the autoregressive model

Suppose we have a time series that follows a p - th order autoregressive system and we aggregate it over m periods to obtain the non-overlapping aggregate sequence. First we ask what structure the aggregate sequence follows. Theorem 1 of Section 2 shows that the aggregate sequence follows an autoregressive system with a moving-average residual. In Section 3 we use the results of Theorem 1 to obtain the mean square errors of 4 predictors of the aggregate variable. They are (1) the optimal linear predictor using the disaggregate observations, (2) the optimal linear predictor using the aggregate observations, (3) the predictor obtained by the least squares regression on the p past aggregate variables, and (4) the predictor obtained by ignoring the residual part of the aggregate structure. Theorem 2 of Section 3 shows that the ratio of the mean square errors of any pair of the four predictors approaches 1 as m goes to infinity for a fixed p. The numerical evaluation of the mean square errors for the cases p = 1 and 2 show that the aggregate predictors, especially the optimal one, perform very well in comparison to the optimal disaggregate predictor. However, Section 4 shows that the estimates of the original parameters obtained from the aggregate data could be considerably inferior to those obtained from the disaggregate data.