106 The Basics of financial economeTrics
stochastic.^2 Instead of the cyclical term Zt and the disturbance Ut, one some-
times incorporates the so-called irregular term of the form IItt=+φ⋅ − 1 Ut
with 01 <≤φ. That is, instead of equation (5.1), we now have
(^) xTtt=+SItt+ (5.2)
With the coefficient φ, we control how much of the previous time’s irregular
value is lingering in the present. If φ is close to zero, the prior value is less
significant than if φ were close to one or even equal to one.
Note that Ut and It–1 are independent. Since It depends on the prior
value It− 1 scaled by φ and disturbed only by Ut, this evolution of It is referred
to as autoregressive of order one.^3 As a consequence, there is some relation
between the present and the previous level of I. Thus, these two are correlated
to an extent depending on φ. This type of correlation between levels at time
t and different times from the same variable is referred to as autocorrelation.
In Figure 5.2, we present the decomposition of some hypothetical time
series. The straight solid line T is the linear trend. The irregular component I
is represented by the dashed line and the seasonal component S are the two
dash-dotted lines at the bottom of the figure. The resulting thick dash-dotted
line is the time series {x}t obtained by adding all components.
Application to S&P 500 Index Returns
As an example, we use the daily S&P 500 returns from January 2, 1996,
to December 31, 2003. To obtain an initial impression of the data, we plot
them in the scatter plot in Figure 5.3. At first glance, it is kind of difficult
to detect any structure within the data. However, we will decompose the
returns according to equation (5.2). A possible question might be, is there
a difference in the price changes depending on the day of the week? For the
seasonality, we consider a period of length five since there are five trading
days within a week. The seasonal components, St(weekday), for each week-
day (i.e., Monday through Friday) are given below:
Monday –0.4555
Tuesday 0.3814
Wednesday 0.3356
Thursday –0.4723
Friday 0.1759
(^2) The case where all four components of the time series are modeled as stochastic
quantities is not considered here.
(^3) Order one indicates that the value of the immediately prior period is incorporated
into the present period’s value.