Moving average filters and periodic integration

Abstract

Several quarterly observed macroeconomic time series may contain a stochastic trend which effects the seasonal fluctuations. An example is the unemployment rate which displays seasonality in business cycle expansion periods because of seasonal labor supply, and which shows much less seasonal fluctuations in the contraction periods because the dismissal of employees may be free of seasonal effects. Hence, the seasonal pattern of economic time series can change over time, and these changes may be caused by the stochastic trend. A class of univariate time series models that can describe such time series contains the periodic autoregressions with unit roots [PIAR], see, e.g., [1,2]. The basic assumption underlying seasonal adjustment methods is that, one way or another, seasonality, trend and cycles can be separated. However, when it is found that a PIAR can give an adequate description of a time series, the crucial requirement is violated. Hence, seasonal correction filters may either not remove all seasonal fluctuations or effect trend and cyclical patterns. In this paper, we focus on the effect of one particular filter, i.e. the linear moving average filter (1 + B + B 2 + B 3) on testing for common stochastic trends across periodically integrated time series, where B is the familiar backward shift operator. The outline of this paper is as follows. In Section 2, we discuss a few concepts related to PIAR processes. In Section 3, we discuss the linear moving average filter in relation to a PIAR. In Section 4, we present the results of some Monte Carlo exercises. The main conclusion is that the probalzility of finding true common trends across PIARs is dramatically reduced when moving average filters are used. In Section 5, we conclude with some remarks.