Introduction

This chapter introduces inferential procedures for seasonal unit roots. Seasonality is a unique feature of time series, but it was ignored in the previous chapters, which implicitly assumed that there is no seasonality in the data. However, when it is present in the data, it is important to know how we can find an appropriate model for the data and how we perform inference on the chosen model. A popular model for seasonal time series is the seasonal ARIMA model due to Box and Jenkins (1976). One of the key elements of this model is seasonal differencing, which is required when seasonal unit roots are present. In this sense, testing for seasonal unit roots is an essential step in Box and Jenkin's modeling of seasonal time series. Albeit less popular than the seasonal ARIMA model, periodic autoregressive (PAR) models are also useful for modeling seasonal time series. Properties of the PAR model depend on the presence or absence of a unit root, so that testing for a unit root and for the null hypothesis of stationarity is important for the PAR model.

This chapter starts from Dickey, Hasza, and Fuller (1984; DHF hereafter), who use the AR(S) model (S denotes the number of seasons) to test for seasonal unit roots, and it discusses its extensions. Then, the testing procedures of HEGY and their extensions are introduced. HEGY's advantage over DHF is that it can test the presence of positive, negative, and complex unit roots separately. Seasonal stationarity tests that complement the seasonal unit root tests are introduced next. Seasonal unit root and stationarity tests under structural changes are also discussed. In addition, this chapter introduces several methods that can be used to test for a unit root in the PAR model. Last, empirical studies examining seasonal unit roots are discussed.

Testing for Seasonal Unit Roots

A time series {yt}, observed at S equally spaced time intervals per year, is said to have seasonal unit roots if {(1 − BS)yt} is a stationary process.