Modeling and Forecasting Stochastic Seasonality: Are Seasonal Autoregressive Integrated Moving Average Models Always the Best Choice?

In this paper, we study models for stochastic seasonality and compare the well‐known SARIMA models to Seasonal Autoregressive Unit Root Moving Average (SARUMA) models. SARUMA models assume that the polynomial of the stationarizing differencing operator has roots on the unit circle at some seasonal frequencies, while SARIMA models impose roots on all of them. We also compare them with near‐nonstationary ARMA models. We study the covariance structure of SARUMA models and the induced properties of seasonal patterns. SARUMA and SARIMA models exhibit in the medium run a stability of the seasonal patterns, which, however, have increasing amplitudes and variability, as opposed to near‐nonstationary ARMA models; SARUMA and near‐nonstationary ARMA models allow for better control of the regularity of the seasonal pattern. We also study the variance of the forecast errors when the fitted model is misspecified. Theoretical calculations and a simulation study show that if a SARIMA model suffers from over‐differencing, its forecasting performance deteriorates. The variance of the forecast errors will be inflated, especially in the very short run. Augmenting with ARMA terms can reduce variance inflation without always eliminating it. SARUMA models, deciding on the basis of the HEGY test which roots to assume on the unit circle, perform clearly better. JEL Classification: C22, C53