Bayesian analysis of moving average stochastic volatility models: modeling in-mean effects and leverage for financial time series

Abstract

We propose a moving average stochastic volatility in mean model and a moving average stochastic volatility model with leverage. For parameter estimation, we develop efficient Markov chain Monte Carlo algorithms and illustrate our methods, using simulated and real data sets. We compare the proposed specifications against several competing stochastic volatility models, using marginal likelihoods and the observed-data Deviance information criterion. We also perform a forecasting exercise, using predictive likelihoods, the root mean square forecast error and Kullback-Leibler divergence. We find that the moving average stochastic volatility model with leverage better fits the four empirical data sets used.

Keywords:

• In-mean effects
• leverage
• Markov chain Monte Carlo
• moving average
• stochastic volatility

Notes

1 Another choice of prior for $\varphi$ is the (shifted, scaled) beta distribution, as in Nakajima and Omori (2012). Although the beta prior requires no truncation, the normal distribution offers easier interpretability of its hyperparameters. In addition, when using the beta prior, the simulation results remain essentially the same; see Online Appendix.

2 We thank a referee for pointing out this issue.

3 This data set has been used in the textbook of Martin et al. (2012) and can be downloaded from that textbook’s website: http://www.cambridge.org/features/econmodelling/exercises.htm.

4 The parameter λ has also been found negative in other studies of stock returns (Koopman and Hol Uspensky, 2002; Abanto-Valle et al. 2011).

5 In the Online Appendix, we also rerun the MASVM model using the joint prior on ($\varphi ,\lambda$), without observing any substantial changes in the results.

6 This empirical finding is in agreement with the results from the simulated study, according to which the MA part increases the model fit more than the in-mean effect or leverage, when the true models are the two proposed ones.

7 In the Online Appendix, we also run the MASVM model using the joint prior on ($\varphi ,\lambda$), for all the empirical data sets, without observing any substantial changes in the results.

8 We thank a referee for pointing out this issue.