Testing for mean reversion in heteroskedastic data based on Gibbs-sampling-augmented randomization

Previous work reported that heteroskedasticity did not affect the sampling distribution of the variance ratio, or had assumed that the investigator knew a priori the pattern of heteroskedasticity. This paper uses the Gibbs sampling approach in the context of a three state Markov-switching model to show how heteroskedasticity affects inference and suggest two strategies for valid inference. We find that test procedures which ignore the pattern of heteroskedasticity are biased, rejecting the null hypothesis of no mean reversion too often. We present a resampling strategy that standardizes historical returns, using the Gibbs sampling approach to allow for uncertainty in parameters and states while conditioning on the information in the data. A second strategy is to estimate the VR from standardized data. Again, Gibbs sampling makes appropriate use of the information in the data. For CRSP stock returns 1926-86 we find that evidence of mean reversion is substantially weakened. Gibbs sampling estimates of VRs are closer to unity, and the pattern shifts to shorter lags. There is weak evidence of mean reversion in returns on equal-weighted portfolios and essentially none for the value-weighted. Thus, when returns from high variance periods are appropriately standardized, the sample evidence for mean reversion changes in pattern and significance.