A wide variety of economic models includes expectational variables among the list of variables determining behavior. In this paper it is shown that for a large class of time series, expectations about the future of observed series or about unobserved components of economic time series may lead to rational lag distributions. Speciﬁcally it is shown that rational distributed lags arise whenever the series (or the ρ-th diﬀerence of the series) have autoregressive, moving average representations and linear least squares forecasts are calculated. The orders of the lag distributions are also given. In Section 4 an example is given that shows the application of these results to an inventory adjustment model. One of the diﬀiculties with distributed lag models with a rational function of the lag operator is that the order of the operator is frequently not known a priori, though in some cases one may need this information in order to identify the structural parameters of interest. The results presented here show that the orders of the operators depend in a simple way upon the structure of the series being forecast or estimated, and, thus, the results should be useful in the formulation and estimation of distributed lag models.
Grether, David M., "Distributed Lags, Prediction, and Signal Extraction" (1969). Cowles Foundation Discussion Papers. 512.