214 The Basics of financial economeTrics
variance of each one and their covariances. Thus a time series tool for esti-
mating conditional covariances is needed in portfolio construction. Fore-
casts of conditional covariances can be done using multivariate ARCH and
multivariate GARCH models which we very briefly describe at the end of
this chapter.
Estimating and Forecasting Volatility
The simplest approach for measuring historical volatility involves using a
sample of prices or returns observed over a recent, short time period to cal-
culate the variance and standard deviation. Variants of historical volatility
depend on how much weight is given to each observation. Assigning each
sample observation equal weight means that the most recent observations
are given the same influence as the observations at the beginning of the
time period. This approach is referred to as the equally weighted average
approach. The drawback of this approach is that more recent observations
may contain more information regarding future volatility than more distant
observations. To overcome this drawback, the exponentially weighted mov-
ing average (EWMA) approach can be used. This approach assigns more
weight to more recent observations. By assigning more weight to recent
observations, extreme observations that occurred in the past are given less
importance in the calculation of the variance. This approach involves the
selection of a process for weighting observations so as to give more weight
to recent observations and less weight to the more distant observations. An
exponentially weighted scheme is used for this purpose and hence the use
of the term in describing this approach. The user must specify the weight-
ing scheme (i.e., how much weight should be assigned to recent and distant
observations).^2
The drawbacks of the two approaches just described are that they are
historical volatility or realized volatility measures and therefore not neces-
sarily a measure of expected future volatility. The use of a “realized” volatil-
ity measure for forecasting future volatility is based on the assumption that
volatility will remain constant (unchanged) in the future from what it was
during the sample time period. In addition, because it is a sample estimate,
it is subject to sampling error. This means that historical volatility depends
on the sample time period used.
(^2) For a detailed explanation of how to apply estimate and forecast volatility using
the equally weighted average and EWMA approaches, see Carol Alexander, “Mov-
ing Average Models for Volatility and Correlation, and Covariance Matrices,” in
Handbook of Finance, ed. Frank J. Fabozzi, vol. 3 (Hoboken, NJ: John Wiley &
Sons, 2008): 711–724.