SEASONALITY AND THE ORDER OF INTEGRATION FOR CONSUMPTION*
| Published date | 01 November 1988 |
| Author | A. P. L. Chui,C. R. Birchenhall,Denise R. Osborn,Jeremy P. Smith |
| Date | 01 November 1988 |
| DOI | http://doi.org/10.1111/j.1468-0084.1988.mp50004002.x |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,4(1988)
0305-9049 $3.00
SEASONALITY AND THE ORDER OF
INI'EGRATION FOR CONSUMPTION*
Denise R. Osborn, A. P. L. Chu4 Jeremy P. Smith and C. R. Birchenhall
I. INTRODUCTION
Cointegration has aroused considerable interest in the recent literature of
dynamic modelling for economic time series, with a special issue of this
Journal devoted to the topic. Nevertheless, despite the importance of
seasonal movements as a feature of economic data, seasonality has received
relatively little attention in the cointegration analysis. A notable exception to
this lack of attention, however, is the paper by Engle et al. (1987).
In a companion paper, Birchenhall et al. (1987), we use cointegration as
the basis for a dynamic analysis of non-durable consumption in the UK.
There we take the view that consumer preferences are seasonal, which
implies the equilibrium relationship between consumption and its explana-
tory variables has a seasonal structure. The equilibrium used, in its non-
seasonal form, is based on
C=KY(1+t)ß(W1) (1)
where C is real non-durable consumers' expenditure, Y is real personal
disposable income, r is the rate of inflation and W is the end-of-period real
liquid assets. All analysis is carried out with the data in logarithms, converting
(1) to a linear function. This equilibrium equation is a generalization of that
used by Davidson et al. (1978) and Hendry and von Ungern-Sternberg
(1981) for modelling UK consumption.
Prior to examining the relationships between variables, however, univariate
time series properties need to be established. This is important because if this
equation is to represent a long-term relationship, or cointegrating regression,
the left- and right-hand sides of (1) must have compatible long-run pro-
perties: that is, they must be integrated to the same order (see Granger, 1981).
The purpose of this paper is to investigate these univariate properties in the
light of the pronounced seasonality in consumption.
*This work is based on research funded by the Economic and Social Research Council
(ESRC), reference number B00232 124. We are grateful to our project colleague Robin Bladen-
Hovell, who suggested the form of the equilibrium equation used here. Our thanks also go to
the editors of the BULLETIN, particularly David Hendry, for their helpful comments. Jeremy
Smith is now at the Australian National University.
361
362 BULLETIN
In Section lIthe concept of integration is broadened to allow for a mixture
of one-period (or unit) and seasonal differencing. A number of tests, includ-
ing that recently proposed by Engle et al. (1987), are used to decide on the
combination required for each of the variables in (1). An apparent conflict
arises here between the levels of integration of consumption and the right-
hand side variables, with both unit and seasonal differences being indicated
for the former, but only one level of differencing is needed for each of the
latter. Conventional differencing does not, however, capture the dynamics of
consumption, since seasonal preferences generally imply seasonally varying
coefficients. Section III then examines the non-stationary structure of
consumption in the context of a periodic or seasonally-varying specification,
and defines periodic integration. We conclude that consumption is periodi-
cally integrated of order one. This classification of consumption allows us to
maintain (1) as an equilibrium relationship, albeit in a periodic form.
II. INTEGRATION AND SEASONALITY
To study the degree of integration for strongly seasonal variables, such as
consumption, we amend the standard terminology (used by Granger, 1986,
Engle and Granger, 1987, and others) as follows:
Definition. A non-deterministic series X, is said to be integrated of order
(d, D), denoted X, I( d, D), if the series has a stationary, invertible ARMA
representation after one-period differencing d times and seasonally dif-
ferencing D times.
This terminology is directly in the seasonal time series modelling tradition of
Box and Jenkins.
Seasonality, as considered in this paper, may have both deterministic and
stochastic components. Thus, if the observed series is Y,, then it is assumed
that
Yt=Xt+kq (2)
where X, is purely stochastic and kq is the deterministic seasonal component
for season q. We remove deterministic seasonality by a prior regression of the
levels series on four quarterly dummy variables; the residuals for this regres-
sion are then treated in the subsequent analysis as if they are the true X,. This
procedure is justified by Dickey, Hasza and Fuller (1984) for testing a unit
root at a seasonal lag and by Dickey, Bell and Miller (1986) for testing a non-
seasonal unit root in the presence of deterministic seasonality.
It is important that deteiiiiinistic seasonality be considered when testing
whether the stochastic seasonal autoregressive polynomial includes a unit
root; otherwise, seasonal differencing may be indicated simply because of the
failure to eliminate deterministic seasonal movements. This is discussed, in a
related context, by Wallis (1976).
The details of our testing for I( d, D) integration are presented below.
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