234 The Basics of financial economeTrics
■ (^) The simplest approach for measuring historical volatility involves cal-
culating the variance from a sample of prices or returns observed over
a recent short-time period.
■ (^) Historical volatility can be computed by assigning an equal weight to
each observation or assigning different weights such that more recent
observations are given a greater weight than more distant observations.
■ (^) The drawback of using historical volatility as a forecast of future vola-
tility is that it is based on the assumption that volatility will remain
unchanged in the future from what it was during the sample time period
used to calculate the variance.
■ (^) The approach commonly used in financial econometrics for predicting
future volatility is to impose a structure on the conditional variance based
on stylized facts observed about the variance in the market: (1) volatility
tends to be time varying and (2) volatility exhibits clustering.
■ (^) A time series is called conditionally heteroscedastic if its variance can be
predicted based on past values of the series.
■ (^) The autoregressive conditional heteroscedasticity (ARCH) model pre-
dicts the conditional variance as a linear combination of past squared
returns.
■ (^) The generalized autoregressive conditional heteroscedasticity (GARCH)
model extends ARCH to include past squared returns and past vari-
ances.
■ (^) Several extensions of ARCH/GARCH models have been proposed, in
particular to account for asymmetries in the effect of positive and nega-
tive returns on future variance.
■ (^) Multivariate extensions of ARCH/GARCH models have been proposed
but drastic simplifications are needed to reduce the number of param-
eters of the model that must be estimated.