Two essays in time series analysis: I. Some issues about time series decomposition and seasonal adjustment. II. Asymptotic distributions of some portmanteau statistics for nonstationary time series

I. Some issues of time series decomposition and seasonal adjustment are revisited and discussed. The focus is on linking two commonly used decomposition procedures: the Census X-12 procedure and the ARIMA-model-based procedure. First, we provide a model-based interpretation for some options of the X-12 procedure and show that each given filter may correspond to multiple overall models. In practice, we can choose the overall model which fits the data best. However, it is hard to define the whole class of possible overall models and the forms of overall models are very complicated. Second, we provide a guideline on how to select an X-12 filter. It is well-known that the model-based procedure has the identification problem. Each overall model from the data corresponds to multiple component models and each observationally equivalent component model is of equal claim. It motivates us to propose a new criterion based on the average of observationally equivalent component models. In particular, we can select an X-12 filter such that the component estimates from the Census procedure will be closest to the average component models. Empirical results based on an observed series which follows the airline model are presented. Finally, we discuss the problem of forecasting future values of the unobserved components.;II. The goodness-of-fit of a time series model can be tested by some portmanteau statistics. However, existing portmanteau statistics are derived under the stationarity assumption. The results of this paper justify the use of these statistics when the time series is nonstationary. The portmanteau statistic considered here is still asymptotic distributed as chi-squared, but the degrees of freedom have to be adjusted to accommodate the existence of nonstationary characteristic roots. This is because when the time series is nonstationary, the estimates of the characteristic roots on the unit circle converge at a faster rate so that no loss in degrees of freedom is encountered. Simulation studies and a real example show that the consequence of failing to adjust the degrees of freedom can be severe. We also discuss how pretesting the number of the nonstationary roots will affect the portmanteau tests.