This paper studies tests for covariance stationarity under conditions which permit failure in the existence of fourth order moments. The problem is important because many econometric diagnostics such as tests for parameter constancy, constant variance and ARCH and GARCH eﬀects routinely rely on fourth moment conditions. Moreover, such tests have recently been extensively employed with ﬁnancial and commodity market data, where fourth moment conditions may well be quite tenuous and are usually untested. This paper considers several tests for covariance stationarity including sample split prediction tests, cusum of squares tests and modiﬁed scaled range tests. When fourth moment conditions fail we show how the asymptotic theory for these tests involves functionals of an asymmetric stable Levy process, in place of conventional standard normal or Brownian bridge asymptotics. An interesting outcome of the new asymptotics is that the power of these tests depends critically on the tail thickness in the data. Thus, for data with no ﬁnite second moment, the above mentioned tests are inconsistent. Some new tests for heterogeneity are suggested that are consistent in the inﬁnite variance case. These are easily implemented and rely on standard normal asymptotics. A consistent estimator of the maximal moment exponent of a distribution is also proposed. Again this estimator is easily implemented, has standard normal asymptotics and leads to a simple test for the existence of moments up to a given order. An empirical application of these methods to the monthly stock return data recently studied in Pagan and Schwert (1989a, 1989b) and the daily returns of the Standard and Poors 500 stock index is presented.
Phillips, Peter C.B. and Loretan, Mico, "Testing Covariance Stationarity under Moment Condition Failure with an Application to Common Stock Returns" (1990). Cowles Foundation Discussion Papers. 1190.