Exponomial forecasts of nonstationary time series

Abstract

This article is concerned with the problem of forecasting nonstationary time series when the data at hand are finite. One way to tackle this problem is to choose a deterministic linear model, however intuitive, and fit it to the data by generalized least squares. With some additional assumptions best linear unbiased predictors can be constructed. Here we do just that, in fitting an exponential polynomial, termed exponomial in [3], to a discrete nonstationary time series, and show how to construct optimal predictors. Exponomials are particularly appealing, as the best linear unbiased one-step predictors are constructed in a most straightforward manner. The extrapolation of infinite time series when the underlying model is an exponomial was first considered in [3] and later in [l]. Although we consider essentially the same problem, we shall be concerned mainly with statistical aspects given a finite nonstationary discrete time series. For the paper to be self-contained we first give a brief account of generalized least squares. Next we consider the prediction problem followed by two important special cases. We also suggest a problem concerning the generalized Vandermonde matrix. By an exponomial we mean [3] a function p(x) such that