When are Variance Ratio Tests for Serial Dependence Optimal?
研究一类基于滤波时间序列样本方差与原序列样本方差之比的统计量，证明其在正态性下对白噪声与某些序列依赖备择假设的检验是最优的，并应用于均值回归的方差比检验。
This paper considers a class of statistics that can be written as the ratio of the sample variance of a filtered time series to the sample variance of the original series. Any such statistic is shown to be optimal under normality for testing a null of white noise against some class of serially dependent alternatives. A simple characterization of the class of alternative models is provided in terms of the filter upon which the statistic is based. These results are applied to demonstrate that a variance ratio test for mean reversion is an optimal test for mean reversion and to illustrate the forms of mean reversion it is best at detecting. MANY TEST STATISTICS for time series dependence can be interpreted as variance ratios, and it has long been known that such tests can be optimal tests of certain hypotheses. Durbin and Watson (1950), for example, took T. W. Anderson's (1948) general results on this topic as a starting point in deriving their test for first-order dependence. However, many of these theoretical results on the optimality of variance ratios provide no simple answers to two practical questions. First, given a variance ratio statistic, how does one determine the null and alternative hypotheses that the statistic is best for testing? Second, given a null and alternative, what variance ratio test should one use? The inability to answer questions such as these has complicated the imple- mentation and interpretation of a popular variance ratio test for mean reversion (see, e.g., Cochrane (1988), Cogley (1988), Kim, et. al. (1988), Lo and MacKinlay (1988, 1989), Poterba and Summers (1988), Richardson and Smith (1989)). Uncertainty about the test's optimality in detecting mean reversion has led to questions about the importance of test results. Further, implementation of the test requires selecting a maximum order of serial correlation to consider, and it has been unclear how the optimality properties of the test vary with this choice. This paper sets out and analyzes a class of variance ratio statistics for which many of these issues are easily resolved. Statistics in this class are the ratio of the sample variance of a filtered time series to the sample variance of the original series and will be called filter variance ratio (FVR) statistics. The class includes the variance ratio test for mean reversion and the Durbin-Watson statistic. Two central results are provided. First, each FVR test is shown to share the large sample properties of a likelihood ratio test for a certain null and alternative. Second, a simple relation between the filter a statistic is based on and the forms of dependence against which the test has optimal power is