Frequentist and Bayesian Unit Root Tests in Stochastic Volatility Models

Abstract

In stochastic volatility models, the unit root test on the time series of the unobserved log-volatilities may be performed by applying the commonly usedfrequentist unit root tests. For instance, augmented Dickey Fuller tests based on the log-squared meancorrected returns can be used. The log-squared meancorrected returns have the same second order properties as that of an autoregressive moving average process. However, we observed that the moving average parameter of the resulting process (based on the log-squared meancorrected returns) is typically close to the autoregressive parameter. For this reason,the unit root tests applied to stochastic volatility models tend to reject theunit root in finite samples. We propose a method for performing thefrequentist unit root tests in stochastic volatility models based on the finite sampling distribution of the well known test statistics. In addition to the frequentist testing procedures, Bayesian unit root testscan be used to test for a unit root in stochastic volatility models as well. A Bayesian test based on the Bayes factor has been suggested by So and Li (1999). In this approach, they work with the mean corrected returns instead of thelog-squared mean corrected returns. They treat the unobserved log-volatilitiesas missing observations. The prior densities they use for the autoregressiveparameter are continuous densities defined on an interval that does not include the value beingtested. Such prior densities for the autoregressive parameterare not suitable where one's main concern is to test for a unit root inlog-volatilities. We introduce a new prior density for this parameter that puts a positive mass on thepoint being tested. We also consider continuous prior densities defined on an interval that includes thepoint one. These prior densities allow us to use the posterior interval ofthe autoregressive parameter as a testing criterion.The advantage of our method is that it is simple and useful.The performance of these tests are demonstrated by a simulation study. We illustrate thetesting procedures by applying them to four sets of exchange rates.