10 - Seasonal Integration and Cointegration

Summary

This paper develops tests for roots in linear time series which have a modulus of one but which correspond to seasonal frequencies. Critical values for the tests are generated by Monte Carlo methods or are shown to be available from Dickey–Fuller or Dickey–Hasza–Fuller critical values. Representations for multivariate processes with combinations of seasonal and zero-frequency unit roots are developed leading to a variety of autoregressive and error-correction representations. The techniques are used to examine cointegration at different frequencies between consumption and income in the U.K.

INTRODUCTION

The rapidly developing time-series analysis of models with unit roots has had a major impact on econometric practice and on our understanding of the response of economic systems to shocks. Univariate tests for unit roots were first proposed by Fuller (1976) and Dickey and Fuller (1979) and were applied to a range of macroeconomic data by Nelson and Plosser (1982). Granger (1981) proposed the concept of cointegration which recognized that even though several series all had unit roots, a linear combination could exist which would not. Engle and Granger (1987) present a theorem giving several representations of cointegrated series and tests and estimation procedures. The testing is a direct generalization of Dickey and Fuller to the hypothesized linear combination.

All of this work assumes that the root of interest not only has a modulus of one, but is precisely one.