Autoregressive Heteroscedasticity Model and Its Variants 227
similar models have been proposed. Table 11.1 lists some of the most com-
mon variants of GARCH models and their general properties.
The first three models shown in Table 11.1 address the leverage
effect—initially identified by Fischer Black—by which negative returns have
a larger impact on volatility than positive returns.^8 This asymmetric impact
of returns on volatility has been subsequently confirmed in many studies.
These variants use various nonlinear specifications of the GARCH model.
In the classical GARCH model, equation (11.9) shows that the squared
(^8) Fischer Black,“Studies of Stock Price Volatility Changes,” in Proceedings of the
1976 American Statistical Association, Business and Economical Statistics Section
(1976), 177–181.
tABLe 11.1 Univariate Extensions of GARCH Models
Model About the Model
Normal GARCH
(NGARCH)
Also called nonlinear
GARCH
Introduced by Engle and Ng,a is a non linear
asymmetric GARCH specification where negative
and positive returns have different effects on future
volatility.
Exponential GARCH
(EGARCH)
Introduced by Nelson,b models the logarithm of the
conditional variance. It addresses the same leverage
effect as the NGARCH; that is, a negative return affects
volatility more than a positive return.
GJR-GARCH/
Threshold GARCH
(TGARCH)
The GARCH model by Glosten, Jagannathan, and
Runkle (and bearing their initials)c and by Zakoian,d
models the asymmetries in the effects of positive and
negative returns.
Integrated GARCH
(IGARCH)
A specification of GARCH models in which conditional
variance behaves like a random walk and in which
shocks to the variance are therefore permanent.e
a^ Robert Engle and Victor K. Ng, “Measuring and Testing the Impact of News on
Volatility,” Journal of Finance 48 (1993): 1749–78.
b (^) Daniel B. Nelson, “Conditional Heteroscedasticity in Asset Returns: A New
Approach,” Econometrica 59 (1991): 347–70.
c (^) Lawrence R. Glosten, Ravi Jagannathan, and David E. Runkle, “On the Relation
between the Expected Value and the Volatility of the Nominal Excess Return on
Stocks,” Journal of Finance 48 (1993): 1779–1801.
d (^) Jean-Michele Zakoian, “Threshold Heteroscedastic Models,” Journal of Economic
Dynamics and Control 18 (1994): 931–55.
e (^) Robert F. Engle and Tim Bollerslev, “Modeling the Persistence of Conditional Vari-
ances,” Econometric Reviews 5 (1986): 1–50.