Covariance-based orthogonality tests for regressors with unknown persistence

This paper develops a new covariance-based test of orthogonality that may be attractive when regressors have roots close or equal to unity. In this case standard regression-based orthogonality tests can suffer from (i) size distortions and (ii) uncertainty regarding the appropriate model in which to frame the alternative hypothesis. The new test has good size and power against a wide range of reasonable alternatives for stationary, non-stationary, and local to unity regressors, while avoiding non-standard limiting distributions, size correction, and unit root pre-tests. Asymptotic results are derived and simulations suggest good small sample performance. As an empirical application we test for the predictability of stock returns using two persistent regressors, the dividend-price-ratio and short-term interest rate. The recent literature highlights the role of size distortions in traditional tests using these predictors. On the other hand, while often overturning these rejections, recently employed size-corrected regression-based tests may restrict power to alternatives that become less plausible the more persistent the regressor. The covariance-based tests, which have correct size without restricting power, also show considerably weaker evidence against orthogonality than do traditional regressions. Nevertheless, even allowing for near-unit root behavior, in many cases we still reject orthogonality at long horizons using the dividend yield and at short to medium horizons using the one-month treasury bill rate