3.7 The GARCH-M Model
In finance, the return of a security may depend on its volatility. To model such a phenomenon, one may consider the GARCH-M model, where M stands for GARCH in the mean. A simple GARCH(1,1)-M model can be written as
where μ and c are constants. The parameter c is called the risk premium parameter. A positive c indicates that the return is positively related to its volatility. Other specifications of risk premium have also been used in the literature, including rt = μ + cσt + at and 
.
The formulation of the GARCH-M model in Eq. (3.23) implies that there are serial correlations in the return series rt. These serial correlations are introduced by those in the volatility process 
. The existence of risk premium is, therefore, another reason that some historical stock returns have serial correlations.
For illustration, we consider a GARCH(1,1)-M model with Gaussian innovations for the monthly excess returns of the S&P 500 index from January 1926 to December 1991. The fitted model is
where the standard errors for the two parameters in the mean equation are 0.0023 ...
Get Analysis of Financial Time Series, Third Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
