Long-Horizon Regression Test of Mean Reversion: A Finite Sample Analysis

In recent two decades, long-horizon regression has become a popular choice for testing mean reversion in stock prices. Due to overlapping and multi-period return summation, finite sample analysis of long-horizon regression coefficient is complicated and largely unavailable in the literature. Empirical studies almost entirely rely on asymptotic tests, which can have serious size distortions and be highly unreliable in finite samples. In this paper, we bridge the gap by providing a finite sample analysis under the assumption that returns follow a multivariate elliptical distribution. First, we derive analytical expressions for the first two moments of long-horizon regression slope coefficient, and provide simple formulas that approximate the mean and variance extremely well. Second, we develop efficient numerical procedures to compute the exact distribution which allows us to perform exact test. In addition, we propose a simple and reliable approximate test assuming regression coefficient follows a normal distribution. Third, we analyze the size and power of the exact test under various popular alternative hypotheses. We find the exact test has better size properties than asymptotic test but long-horizon regression tests in general have lower power detecting slowly mean reverting process. Finally, when applied to empirical data, the exact test lends less support of mean reversion than asymptotic tests. Even such moderate evidence is mainly due to the pre-1941 data.