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 and 0.818, respectively, and those for the parameters in the volatility equation are 2.51 × −5, 0.0205, and 0.0196, respectively. The estimated risk premium for the index return is positive but is not statistically significant at the 5% level. Here the result is obtained using S-Plus. Other forms of GARCH-M specification in S-Plus are given in Table 3.2. The idea of risk premium applies to other GARCH models.

Table 3.2 GARCH-M Models Allowed in S-Plus: Mean Equation Is rt = μ + cg(σt) + at

g(σt)	Command
	var.in.mean
σt	sd.in.mean
lnσt2	logvar.in.mean

S-Plus Demonstration