This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs present in the principal domain of the cumulant spectrum. The test uses a frame (window) averaged sample estimate of the second order cumulant spectrum to build a test statistic that has an asymptotic complex standard normal distribution. We derive the test statistic, study the size and power properties of the test, and demonstrate its implementation with intraday stock market return data. The test has conservative size properties and good power to detect varying variance and unit root in the presence of varying variance.
The space of nonstrictly stationary processes is vast and, therefore, in this manuscript we focus the evaluation of the power of the PHR test on a few alternative hypotheses. We do not study PHR's power against the alternative of a process with periodicities in the mean and random variation in the periodic structure, as in @cite_0 . In the space of tests for strict stationarity in the time domain, @cite_14 proposed a test for strict stationarity and compared its power to the KPSS test of level stationarity. Similarly to @cite_14 we compare the power of the PHR test to the power of the KPSS test. We calculate the power for the alternative hypotheses of unit root, varying variance, and their combination. Our PHR test excels in detecting stationarity violations due to second moment.
@cite_0 @cite_14
2146041186 171502498
We develop a test of the null hypothesis that an observed time series is a realization of a strictly stationary random process. Our test is based on the result that the kth value of the discrete Fourier transform of a sample frame has a zero mean under the null hypothesis. The test that we develop will have considerable power against an important form of nonstationarity hitherto not considered in the mainstream econometric time-series literature, that is, where the mean of a time series is periodic with random variation in its periodic structure. The size and power properties of the test are: investigated and its applicability to real-world problems is demonstrated by application to three: economic data sets. We introduce a test for strict stationarity based on the fluctuations of the quantiles of the data, and we show that this test has power against the alternative hypothesis of unconditional heteroskedasticity while other tests for first order (level) stationarity as the KPSS test (, 1992) and, its robust version, the IKPSS test (de , 2007) have low power against this alternative of time-varying variance. Moreover, our test has power against the alternative hypothesis of time-varying kurtosis, while the test for second order (covariance) stationarity introduced by Xiao and Lima (2007) has power close to size against this alternative.