Shrinkage estimators for periodic autoregressions

A periodic autoregression [PAR] is a seasonal time series model where the autoregressive parameters vary over the seasons. A drawback of PAR models is that the number of parameters increases dramatically when the number of seasons gets large. Hence, one needs many periods with intra-seasonal data to be able to get reliable parameter estimates. Therefore, these models are rarely applied for weekly or daily observations. In this paper we propose shrinkage estimators which shrink the periodic autoregressive parameters to a common value determined by the data. We derive the asymptotic properties of these estimators in case of a quadratic penalty and we illustrate the bias–variance trade-off. Empirical illustrations show that shrinkage improves forecasting with PAR models.