Anon

218 The Basics of financial economeTrics

moments exist and (2) they are constant and not time-dependent. That is, a
covariance-stationary series has a constant mean, a constant variance, and
constant autocovariances.
A time series is said to be homoscedastic if it has a constant variance. A
homoscedastic time series is not necessarily stationary because its mean can
be time dependent. A heteroscedastic sequence of random variables is not a
stationary series because the variance of the sequence is not constant in time
(i.e., the variance depends on time).
Now, consider a sequence such as that represented in Figure 11.2. The
sequence exhibits periods in which returns are large in absolute value and
periods in which returns are small. Given that the magnitude of returns in
absolute value is persistent, we can predict, albeit in a probabilistic sense,
when returns will be large in absolute value and when they will be small.
That is, we can predict if returns will be large or small in absolute value
conditionally on previous returns. We say that the sequence is conditionally
heteroscedastic, because the variance of returns at any moment depends
on previous returns. A sequence is conditionally heteroscedastic if we can

FIGure 11.2 Plot of 1,000 Returns of the Oracle Corporation Stock in the Period
from January 12, 2008, to December 30, 2011

0

–0.2

–0.15

–0.1

Returns–0.05

0

0.05

0.1

0.15

200 400 600

Trading Days

800 1,000 1,200