Quality & Quantity 32: 155–163, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Relationship between the Mean and the Constant in
a Box–Jenkins Time Series Model
, JAIME ARNAU
& PILAR JARA
Area de Metodología, Departamento de Psicología, Universidad ‘Jaime I’, Campus Carretera de
Borriol, Aptdo. 224, 12080 Castellón, Spain
University of Barcelona
Abstract. In some publications the mean is identiﬁed with the constant of a Box–Jenkins time series
model. In this paper the relation between both terms is demonstrated. Furthermore, by means of an
example, the errors which may be made if one does not use each term adequately are shown.
Key words: time series, Box–Jenkins models.
Since Box and Jenkins published their text about time series analysis in 1970
(Box & Jenkins, 1970, 1976), it has been adapted by several different authors in
an attempt to make its contents more comprehensible to graduates researching in
the human and social sciences (Glass, Wilson & Gottman, 1975; Gottman, 1981;
McCleary & Hay, 1982; Gregson, 1983; Vandaele, 1983; Arnau, 1984; Bowerman
& O’Connell, 1987; Wei, 1989; Kendall & Ord, 1990; Mills, 1991). The implicit
error of many of the previous adaptations is that they identify the constant with the
mean of the differenced series, because Box and Jenkins did not demonstrate the
relationship between these two parameters in a general time series equation. This
is due to the fact that many of the statistical properties (variance, autocovariance,
autocorrelation, etc.) of a time series do not vary when a constant is added to the
model. In the following pages we will establish the relationship between the mean
and the constant in a Box–Jenkins time series model.
If we take a Y
time variable, with a mean
Y, with the aim of avoiding the
time series constant (δ), the time variable is transformed through its deviation in
relation to its mean (z
Y). The general seasonal multiplicative mixed
ARI MA(p, d, q)(P , D, Q)
model adopts the compact form (Box & Jenkins,
1976, pg. 305):
This research was funded by a grant to the Universidad ‘Jaime I’ (Proyecto P.25.035/93, from
Fundaci´on Caja de Castell´on).