"Frontmatter". In: Analysis of Financial Time Series

THE GARCH-M MODEL 101two models. The unconditional variance of a t , hence that of r t , is not defined underthe above IGARCH(1, 1) model. This seems hard to justify for an excess returnseries. From a theoretical point of view, the IGARCH phenomenon might be causedby occasional level shifts in volatility. The actual cause of persistence in volatilitydeserves a careful investigation.When α 1 + β 1 = 1, repeated substitutions in Eq. (3.16) giveσ 2 h (l) = σ 2 h (1) + (l − 1)α 0, l ≥ 1,where h is the forecast origin. Consequently, the effect of σh 2 (1) on future volatilitiesis also persistent, and the volatility forecasts form a straight line with slope α 0 .Nelson (1990) studies some probability properties of the volatility process σt2 underan IGARCH model. The process σt2 is a martingale for which some nice results areavailable in the literature. Under certain conditions, the volatility process is strictlystationary, but not weakly stationary because it does not have the first two moments.The case of α 0 = 0 is of particular interest in studying the IGARCH(1, 1) model.In this case, the volatility forecasts are simply σh 2 (1) for all forecast horizons. Thisspecial IGARCH(1, 1) model is the volatility model used in RiskMetrics, which isan approach for calculating Value at Risk; see Chapter 7.3.6 THE GARCH-M MODELIn 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 GARCHin mean. A simple GARCH(1, 1)-M model can be written asr t = µ + cσt 2 + a t , a t = σ t ɛ t ,σt 2 = α 0 + α 1 at−1 2 + β 1σt−1 2 , (3.21)where µ and c are constant. The parameter c is called the risk premium parameter.A positive c indicates that the return is positively related to its past volatility. Otherspecifications of risk premium have also been used in the literature, including r t =µ + cσ t + a t .The formulation of the GARCH-M model in Eq. (3.21) implies that there areserial correlations in the return series r t . These serial correlations are introducedby those in the volatility process {σt 2 }. 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 for the monthly excessreturns of S&P 500 index from January 1926 to December 1991. The fitted model isr t = 0.0028 + 1.99σ 2t + a t , σ 2t = 0.00016 + 0.1328a 2 t−1 + 0.8137σ 2t−1 ,where the standard errors for the two parameters in the mean equation are 0.0022and 0.7425, respectively, and those for the parameters in the volatility equation are