Periodic autoregressive models for time series with integrated seasonality

A comprehensive seasonally integrated periodic autoregressive model is suggested which is shown to be flexible enough to include both the stochastic seasonal integrated and random trigonometric polynomial-based models. The demonstration of the equivalence between the two approaches is the objective of two theorems that are stated and proved in some details. A nice advantage of our model building procedure is that it is able to provide the user not only with a detailed model for data description and forecasting purpose but in addition with a hint at the presence of seasonal unit roots. The model which is illustrated in the present paper may be considered parsimonious, i.e. the number of estimated parameters, for a given goodness-of-fit criterion, is taken as low as possible, in two ways. First, by imposing unit roots and seasonal unit roots so that some estimated parameters are replaced by a differencing operator with fixed coefficients, and, second, by adopting a subset periodic autoregressive model, so that some parameters do not need to be estimated as they are constrained to equal zero. The effectiveness of our model is highlighted by an extensive simulation experiment that supports our claim that the model building procedure described here may be of good use as well for checking the existence of seasonal unit roots. Applications to real-world time series data sets are reported, and promising results are obtained that allow us to suggest that the seasonally integrated periodic autoregressive model may be safely used for modelling a wide range of seasonal time series data. In addition, well-known widely used tests, such as HEGY and the Taylor variance ratio test, are shown to provide us with results that generally agree with our findings.