Autoregressive Heteroscedasticity Model and Its Variants 215
Although the EMWA approach for forecasting future volatility is supe-
rior to the equally weighted average, there is no model that provides a
structure for the expected behavior of volatility. In looking for this type of
structure for the variance in the case of the returns on financial assets, prac-
titioners can be guided by two stylized facts: (1) volatility tends to be time
varying and (2) volatility exhibits clustering. By time varying volatility, it is
meant that there are turbulent periods (i.e., periods of high volatility) and
tranquil periods (i.e., periods of low volatility). By volatility clustering it is
meant that when volatility is low it tends to remain low and when volatility
is high it tends to remain high.
The two statistical models described in this chapter, ARCH and GARCH
models (and its variants), are derived by imposing a structure on volatility
that is consistent with observations about the volatility observed in a mar-
ket. For example, a GARCH model asserts that the best predictor for future
volatility for the next period is made up of three components: (1) a long-run
variance which is constant, (2) a forecast of volatility in the prior period,
and (3) new information not available when the prior forecast of volatility
was made. (The last component is obtained in the square of the prior fore-
casts residual.) The many extensions of the GARCH model involve adapt-
ing models to the structure of the behavior of the variance that has been
observed. For example, in the stock market it is observed that bad news
tends to be more important than good news in terms of its influence on
price. This is referred to as the leverage effect. An extension of the GARCH
model to incorporate this is the threshold GARCH model.
With this understanding of the objective of forecasting future volatility
by providing a structure for the variance, we now move on to describing
ARCH and GARCH models.
ARCH Behavior
Autoregressive conditional heteroscedasticity models, referred to as ARCH
models, are used in financial econometrics to represent time-varying conditional
volatility. Considered a major achievement of modern econometrics, ARCH
models were first described by Robert Engle, who was awarded the 2003 Nobel
Memorial Prize in Economic Sciences for his work in time series econometrics.^3
(^3) Robert F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of
the Variance of United Kingdom Inflation,” Econometrica 50, no. 4 (1982): 987–
- Engle’s development of the ARCH model was only one of his major contribu-
tions to time series econometrics. The corecipient of the award that year was Clive
Granger who jointly with Robert Engle formulated the cointegration technique that
we describe in Chapter 10.