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Covariance-based orthogonality tests for regressors with unknown persistence

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Alex Maynard and Katsumi Shimotsu

*Econometric Theory*, Vol. 25, Issue 01, February 2009, pp 63-116.
This paper develops a new test of orthogonality based on a zero
restriction on the covariance between the dependent variable and the
predictor. The test provides a useful alternative to regression-based
tests when conditioning variables have roots close or equal to
unity. In this case standard predictive regression tests can suffer
from well-documented size distortion. Moreover, under the alternative
hypothesis, they force the dependent variable to share the same order
of integration as the predictor, whereas in practice the dependent
variable often appears stationary and the predictor may be
near-nonstationary. By contrast, the new test does not enforce the
same orders of integration and is therefore capable of detecting a
rich set of alternatives to orthogonality that are excluded by the
standard predictive regression model. Moreover, the test statistic has
a standard normal limit distribution for both unit root and
local-to-unity conditioning variables, without prior knowledge of the
local-to-unity parameter. If the conditioning variable is stationary,
the test remains conservative and consistent. Simulations suggest good
small-sample performance. As an empirical application, we test for
the predictability of stock returns using two persistent predictors,
the dividend-price ratio and short-term interest rate.