Autoregressive Heteroscedasticity Model and Its Variants 217
random draws from a standard normal distribution. Recall that a standard
normal distribution has a mean equal to zero and a variance equal to one.
The assumption of normal distribution of returns is an approximation which
simplifies simulations. (We will examine this assumption later in this book.)
The main points are (1) the magnitude of returns in the plot varies randomly
and (2) the plot does not seem to follow any regular pattern. In particular, the
magnitude of simulated returns oscillates from a minimum of approximately
–0.3 (–30%) to a maximum of +0.3 (+30%) but we cannot recognize any
extended periods when the absolute value of returns is larger or smaller.
The simulated plot in Figure 11.1 is only an approximation of what
is observed in real-world financial markets. In fact, stock returns are only
approximately i.i.d. variables. Empirically, researchers have found that
although the direction of the fluctuations (i.e., the sign of the returns) is
almost unpredictable, it is possible to predict the absolute value of returns.
In other words, given past returns up to a given time t, we cannot predict
the sign of the next return at time t + 1 but we can predict if the next return
will be large or small in absolute value. This is due to the fact that the aver-
age magnitude of returns, that is, their average absolute value, alternates
between periods of large and small average magnitude.
Consider, for example, Figure 11.2, which represents the returns of
the stock of Oracle Corporation in the period from January 12, 2008, to
December 30, 2011. That period includes 1,000 trading days. The plot in
Figure 11.2 has more structure than the plot in Figure 11.1. By looking at
the plot in Figure 11.2, we can recognize that extended periods where fluc-
tuations are small alternate with extended periods where fluctuations are
large. The average magnitude of unpredictable fluctuations is called volatil-
ity. Volatility is a measure of risk. Therefore we can say that periods of high
risk alternate with periods of lower risk.
Let’s review a few terms that we introduced in our discussion of regres-
sion analysis in Chapter 4 regarding the assumptions of the general linear
model. A sequence of random variables is said to be homoscedastic if all vari-
ables have the same measure of dispersion, in particular the same variance. An
i.i.d. sequence is homoscedastic because it is formed with random variables
that are identical and therefore have the same variance or any other measure
of dispersion. A sequence of random variables is said to be heteroscedastic if
different variables have different variance or other measures of dispersion.
Recall that a time series, that is a sequence of random variables, is called
covariance-stationary or weakly stationary^4 if (1) all first- and second-order
(^4) A covariance stationary series is often called simply “stationary.” A time series is
called “strictly stationary” if all finite distributions are time independent. A strictly
stationary series is not necessarily weakly stationary because the finite distributions
might fail to have first and second moments.