Analysis of Financial Time Series, Third Edition

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

(3.23)

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 ...