Non-linear exponential smoothing and positive data

We consider the properties of nonlinear exponential smoothing state space models under various assumptions about the innovations, or error, process. Our interest is restricted to those models that are used to describe non-negative observations, because many series of practical interest are so constrained. We first demonstrate that when the innovations process is assumed to be Gaussian, the resulting prediction distribution may have an infinite variance beyond a certain forecasting horizon. Further, such processes may converge almost surely to zero; an examination of purely multiplicative models reveals the circumstances under which this condition arises. We then explore effects of using an (invalid) Gaussian distribution to describe the innovations process when the underlying distribution is lognormal. Our results suggest that this approximation causes no serious problems for parameter estimation or for forecasting one or two steps ahead. However, for longer-term forecasts the true prediction intervals become increasingly skewed, whereas those based on the Gaussian approximation may have a progressively larger negative component. In addition, the Gaussian approximation is clearly inappropriate for simulation purposes. The performance of the Gaussian approximation is compared with those of two lognormal models for short-term forecasting using data on the weekly sales of over three hundred items of costume jewelry.