Quality & Quantity 36: 305–310, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
A Note on Seasonal Unit Root Tests
OLIVIER DARNÉ and CLAUDE DIEBOLT
Abstract. The seasonal unit root tests make it possible to determine the nature of the deterministic
and stochastic seasonal ﬂuctuations. In Section 2, we deﬁne the main seasonal time series models
and the seasonal integration notion. Section 3 describes the HEGY test procedure.
Key words: deterministic and stochastic ﬂuctuations, seasonality, seasonal unit root tests, time series
One of the major characteristics of many economic time series is the presence of
seasonal movements. The other main types of movements are the trend, the cycle
and the irregular. For Hylleberg (1992), seasonality is the systematic, although
not necessarily regular, intra-day movement caused by changes of the weather, the
calendar, and timing of decisions, directly or indirectly through the production and
consumption decisions made by the agents of the economy. These decisions are
inﬂuenced by the endowments, the expectations and the preferences of the agents,
and the production techniques available in the economy. An important part in this
deﬁnition shows that seasonal ﬂuctuations can be deterministic because of, for ex-
ample, calendar and weather effects, but they may also be caused by the behaviour
of economic agents and may therefore not be constant.
In general, the study of seasonal ﬂuctuations has a long tradition in the analysis
of economic time series. Historically, seasonal ﬂuctuations have been considered
as a nuisance that obscures the more important components, i.e., the trend, growth
and cyclical components. Consequently, seasonal adjustment procedures have been
implemented to eliminate seasonality.
Recently, a new viewpoint has emerged,
showing that seasonal ﬂuctuations are not necessarily a nuisance. They are an
integral part of economic data and should not be ignored or obscured in economic
analysis. Therefore, the study of the seasonal behaviour in the series is important
for model evaluation and forecasting.
Address all communications to Claude Diebolt, LAMETA/CNRS, Universit
e Montpellier I,
e des Sciences Economiques, Espace Richter, Avenue de la Mer, B.P. 9606, 34054 Mont-
pellier Cedex 1, France. Tel.: 33 (0)220.127.116.11.20 (direct line), Fax.: 33 (0)18.104.22.168.67, E-mail:
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