Testing for a slowly changing level with special reference to stochastic volatility

A test for the presence of a stationary first-order autoregressive process embedded in white noise is constructed so as to be relatively powerful when the autoregressive parameter is close to one. The test statistic is shown to have a Cramér–von Mises distribution in large samples. A comparison is made with some standard tests for serial correlation in the context of a stochastic volatility model and the proposed test is shown to have relatively high power for the parameter values typically found with daily financial time series. It is recommended that tests on returns be carried out using absolute values.