COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY

Abstract

This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.

Keywords:

• Common features
• Seasonality
• Codependence
• JEL Classification: C32, C52

ACKNOWLEDGMENTS

Previous drafts of this paper were presented at the ESEM 98 in Berlin, and the XXXIX Meeting of the Italian Statistical Society in Sorrento. Financial support from the Italian Ministry of the Universities and Scientific Research (MURST) is gratefully acknowledged. Thanks are due to Silvano Bordignon, Claudio Lupi, as well as an associate editor and two anonymous referees for useful comments. The usual disclaimers apply.

Notes

¹A set of stationary time-series is codependent if there exist some linear combinations of these series following a vector moving-average (VMA) of lower order than others. The codependence order is q if these linear combinations follow a VMA(q) process. See Gouriéroux and Paucelle (1989), and Vahid and Engle (1997) for discussions on codependence.

²This approach is relevant to the empirical analysis of theoretical ideas concerning the response of the economy to deterministic seasonal perturbations in preferences and technology. See Miron (1996) for a detailed survey of this class of models.

³This assumption is formulated just in order to simplify the algebra and it is not necessary for deriving the results presented in this paper.

⁴In general, the use of seasonally adjusted data induce several serious problems in applied economic analysis; see, among others, Ericsson et al (1994), Ghysels et al (1996), and Maravall (1997) and the references therein included.

⁵The tests results are available from the author upon request.

⁶The weighting matrix for the optimal GMM estimator is computed following West (1997). Particularly, the MA(3) model for the non-optimal GMM estimate of the codependence relation is estimated by the method in Galbraith and Walsh (1994), choosing an approximating AR model of order equal to

⁷A Wald test for restricting to zero the coefficients of the redundant trigonometric functions produces a χ² (6) statistic equal to 3.33.