228 The Basics of financial economeTrics
volatility at time t is a weighted average of past squared volatilities and past
squared returns. Note that the square is a symmetric function while we have
just remarked on asymmetric impact of returns on volatility. Therefore, the
first three models in Table 11.1 replace the square with nonlinear asymmetric
functions that give more weight to negative returns than to positive returns.
Thus, returns at time t – 1 produce larger volatility at time t if they are nega-
tive than if they are positive even if their absolute value is the same.
In all models described thus far, volatility is mean reverting to a long-
term value. However, for some financial time series this property might not
be applicable. In these cases, one might use the IGARCH model, which
makes volatility shocks persistent.
How different are volatility predictions in different GARCH models?
To gain an understanding of the differences, let’s look at the predictions on
Monday, December 17, 2012, for one-month average volatility and one-day
return volatility for the return on the stock of Oracle Corporation, Coca
Cola Corporations, and Caterpillar using the standard GARCH model,
EGARCH model, and GJR-GARCH model. The estimated volatilities are
shown in Table 11.2. The predictions for the three models reported in the
table were obtained from the Web site of V-Lab at Stern School of Business
of New York University.^9
(^9) We thank V-Lab for letting us show their predictions. The V-Lab site can be accessed
at http://vlab.stern.nyu.edu/analysis/VOL.KO:US-R.EGARCH.
tABLe 11.2 Prediction of Monthly Volatility Using GARCH, EGARCH, and
GJR-GARCH Models for Oracle Corporation, Coca Cola Corporations,
and Caterpillar, Monday, December 17, 2012
Model
GARCH
Model
EGARCH
Model
GJR-GARCH
ORACLE Corp
Average volatility (1 Month)
1-day forecast
19.56%
19.18%
18.49%
18.80%
20.60%
20.42%
Coca Cola Corp
Average volatility (1 Month)
1-day forecast
13.66%
14.87%
15.03%
14.04%
15.16%
14.05%
Caterpillar
Average volatility (1 Month)
1-day forecast
25.33%
23.89%
26.60%
24.23%
26.23%
24.62%
Predictions are from the site of V-Lab at Stern University The V-Lab site can be
accessed at: http://vlab.stern.nyu.edu/analysis/VOL.KO:US-R.EGARCH