The Stochastic Volatility in Mean Model: Empirical Evidence from International Stock Markets
The Stochastic Volatility (SV) models we present in this chapter are a practical alternative to the Generalised Autoregressive Conditional Heteroskedasticity (GARCH) type models that have been used so widely in empirical financial research and which have relied on simultaneous modelling of the first and second moment1. For certain financial time series such as stock index returns, which have been shown to display high positive first-order autocorrelations, this constitutes an improvement in terms of efficiency; see Campbell, Lo and MacKinlay (1997, Chapter 2). The volatility of daily stock index returns has been estimated with SV models but usually results have relied on extensive pre-modelling of these series, thus avoiding the problem of simultaneous estimation of the mean and variance 2. New estimation techniques now enable us to include explanatory variables in the mean equation and estimate their coefficients simultaneously with the parameters of the volatility process3. One of the explanatory variables in our model is the variance process itself, hence its name: Stochastic Volatility in Mean (SVM).
KeywordsStock Market Excess Return Stochastic Volatility Stock Index GARCH Model
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- 1.The material in this chapter was previously published in the Journal of Applied Econometrics (2002) 17, 667–689, as “The Stochastic Volatility in Mean Model: Empirical evidence from international stock markets” by S.J. Koopman and E. Hol Uspensky. Reprinted with permission of John Wiley & Sons Limited.Google Scholar
- 2.The same seasonally adjusted S&P Composite stock index series (Gallant, Rossi and Tauchen, 1992) has been used in a number of studies, see for example: Jacquier, Polson and Rossi (1994), Danielsson (1994), Sandmann and Koopman (1998), Fridman and Harris (1998) and Chib, Nardari and Shephard (1998).Google Scholar
- 3.Also see Fridman and Harris (1998) and Chib et al. (1998).Google Scholar
- 4.See, e.g., for the US stock market French et al. (1987) and Campbell and Hentschel (1992), who observe a positive relation, whereas Glosten, Jagannathan and Runkle (1993) present evidence of a negative relation, as does Nelson (1991) with his EGARCH model. Poon and Taylor (1992) who study the UK stock market report a weak positive relationship.Google Scholar
- 5.Even when the firm has no or little debt it is likely to have operating leverage as its expenses initially remain constant as income falls.Google Scholar
- 6.Campbell and Hentschel (1992) find evidence of both volatility feedback and leverage effects, whereas Bekaert and Wu (2000) present results which strongly favour the volatility feedback hypothesis.Google Scholar
- 7.Jacquier, Polson and Rossi (2001) estimate corr(εt, ηt) and observe a convincing negative re¬lationship between contemporaneous unexpected stock index returns and unexpected volatil¬ity. Harvey and Shephard (1996), on the other hand, estimate corr(εt,ηt+1) and Watanabe (1999) develops an SV model which includes the lagged shock to the return process as an explanatory variable in the variance equation allowing for an asymmetric response. Both studies report negative coefficients for the relation between current unexpected returns and future volatility.Google Scholar
- 9.Both the data series and the program used for the estimation of the SVM models which llows for alternative definitions of the mean equation can be found on the Internet at also see Appendix C.Google Scholar
- 10.Fridman and Harris (1998) did however not allow for a constant in the mean and the likelihood ratio test for d = 0 amounted to a value of 0.14.Google Scholar
- 11.Also see Campbell et al. (1997, pp. 497–498).Google Scholar
- 12.We also estimated the GARCH-M model with a normally distributed error term and ob¬served extremely high values for the normality statistic.Google Scholar
- 13.For SV models with alternative error distributions see: Fridman and Harris (1998), Sand¬mann and Koopman (1998) and Liesenfeld and Jung (2000).Google Scholar