On seasonality and cointegration

Abstract: This thesis, which consists of four essays, focus on seasonal and periodic cointegration models. These models are tools to describe changing seasonality.Essay 1 "Forecasting performance of seasonal cointegration models", with Johan Lyhagen. Forecasts from two different seasonal cointegration specifications are compared in an empirical forecasting example and in a Monte Carlo study. One of the two specifications include a certain parameter restriction at the annual frequency, wheras the other specification is more general. In the empirical forecasting example we also include a standard cointegration model based on first differences and seasonal dummies and analyze the effects of restricting or not restricting seasonal dummies in the seasonal cointegration models. While the Monte Carlo results favor the general specification, and definitely so if larger sample sizes are considered, we do not find such clear cut evidence in the empirical example.Essay 2 "On forecasting cointegrated seasonal time series", with Philip Hans Franses. In this essay we analyze periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We include both single equation and multiple equations methods for those two classes of models. A VAR model in first differences, with and without cointegration restrictions, and a VAR model in annual differences are also included in the analysis, where they serve as benchmark models. Our empirical results indicate that the VAR model in first differences without cointegration is best if one-step ahead forecasts are considered. For longer forecast horizons however, the VAR model in annual differences is better. When comparing periodic versus seasonal cointegration models, we find that the seasonal cointegration models tend to yield better forecasts. Essay 3 "Size and power of the likelihood ratio test for seasonal cointegration in small samples: A Monte Carlo study", This essay investigates the small sample size and power properties of the likelihood ratio test in the seasonal error correction model. Two specifications of the model at the annual frequency are analyzed. One is more restricted (RS), designed for the particular case of 'synchronous cointegration', whereas the other specification is general (GS). The results indicate that RS has poor size properties in cases where non-synchronous cointegration clearly should play a role. There is a risk of finding 'evidence' of too many cointegrating vectors at the annual frequency when using RS. On the other hand, if the restriction is almost satisfied, the general specification looses power at least for small sample sizes, while tests in RS have good properties. Essay 4 "On seasonal error correction when the processes include different numbers of unit roots", with Johan Lyhagen. We propose a seasonal error correction model (SECM) for quarterly data which includes variables with different numbers of unit roots and thus needs to be transformed in different ways in order to yield stationarity. A Monte Carlo simulation is carried out to investigate the consequences of specifying a SECM with all variables in annual diffrerences in this situation. The SECM in annual differences is compared to the correctly specified model. Pre-testing for unit roots using two different approaches, and where the models are specified according to the unit root test results, is also considered. The results indicate that, in practice, a cointegration model where all variables are transformed with the annual difference filter is more robust than one obtained by pre-testing for a smaller number of unit roots.