Testing for a Slowly Changing Level with Special Reference to Stochastic Volatility - (Now published in 'Journal of Econometrics', 87 (1998), pp.167-189.)

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. This is done by setting up the autoregression in such a way that it reduces to a constant, instead of a random walk, when the autoregressive parameter goes to ome. The locally best invariant test principle is then applied. The test is then shown to be equivalent to the locally best invariant test for the presence of random walk in noise. The properties of this test have been studied by a number of researchers, and its asymptotic distribution is known to be the von Mises distribution. The test can also be applied if the model contains explanatory variables. The test is compared with some standard tests for serial correlation in the context of a stochastic volatility model and shown to have some attractive properites. The tests are examined for different transformations of the observations and an example is given involving daily exchange rates.