Seasonal Variability in a Distributed Lag Model

Abstract

To explain the kind of problem that this paper deals with it seems easiest to begin with an illustration. Consider a lending institution that approves applications for new home loans which are subsequently either cancelled or drawn. Suppose that typically a loan is not fully drawn immediately on approval; rather the sequence of payments is distributed over time and takes the form of periodic progress payments to builders. Further suppose that the time unit we are dealing with is a month. Then the total loan drawings in any one month would be a function of current and past approvals net of cancellations, i.e. a standard distributed lag model. Now, however, consider the additional complication that the dependence between the current loan drawings and any previous month's approvals is not fixed but varies in a systematic way with the month (or season) we are dealing with. In other words we wish to allow the distributed lag weights themselves to vary seasonally. Such seasonal variation in the distributed lag weights may be induced by the seasonal pattern of work and completion of building including the incidence of public holidays, or by random fluctuations in weather patterns which in turn affect building activity, or by seasonal variation in the technology of payments. Other reasons could also be thought of. It is emphasised that we are considering not just a seasonal intercept shift but also changes in the lag weights themselves. This paper considers a variety of specifications of such a model and tackles the estimation problem from a Bayesian viewpoint. But first we shall develop a suitable notation and framework for analysing this problem. This will highlight the bi-dimensional nature of the estimation problem. We shall then corsider several alternative representations of prior information regarding the lag coefficients in either the lag or the seasonal dimension. In particular we shall argue that exchangeable priors and smoothness priors which have been previously developed in other contexts are useful notions for representing prior information. Moreover it will be argued that such Bayesian priors can be usefully combined and not simply used separately. It is the emphasis on and systematic exploitation of the bi-dimensional nature of the specification and estimation within a Bayesian framework that constitutes the main novel feature of the paper. The main body of the paper, Section 2, shows how the results of previous work of Lindley and Smith (1972) on Bayesian estimation of a linear model can be adapted and extended in such a way as to yield a family of shrinkage estimators for the seasonally varying distributed lag model. Note however thatLindley and Smith are essentially dealing with a one-dimensional problem and we wish to deal with a model in which there are two. It would be necessary therefore to adapt their ideas to our situation. The analysis of Section 2 is based on the assumption of known prior variances. The final section