A Nonparametric Test of the Predictive Regression Model

Abstract

This article considers testing the significance of a regressor with a near unit root in a predictive regression model. The procedures discussed in this article are nonparametric, so one can test the significance of a regressor without specifying a functional form. The results are used to test the null hypothesis that the entire function takes the value of zero. We show that the standardized test has a normal distribution regardless of whether there is a near unit root in the regressor. This is in contrast to tests based on linear regression for this model where tests have a nonstandard limiting distribution that depends on nuisance parameters. Our results have practical implications in testing the significance of a regressor since there is no need to conduct pretests for a unit root in the regressor and the same procedure can be used if the regressor has a unit root or not. A Monte Carlo experiment explores the performance of the test for various levels of persistence of the regressors and for various linear and nonlinear alternatives. The test has superior performance against certain nonlinear alternatives. An application of the test applied to stock returns shows how the test can improve inference about predictability.

KEY WORDS:

• Finance
• Kernel regression
• Near unit root
• Time series

Notes

For an introduction to kernel regression and nonparametric regression, see Härdle (1989a).

In particular, the partial sums of (w_t, ε_t)^⊤ satisfy a multivariate invariance principal from Assumption 1. I thank an anonymous referee for suggesting a more lucid exposition.

We thank an anonymous referee for pointing out the divergence and the potential problem with the resulting bandwidth.

The results are available upon request.

We use T = 100 for all of the power figures.

The long yield is the Moody’s seasoned Aaa corporate bond yield.

From Table 1, we see that if the null hypothesis of no relationship between y_t and x_{t − 1} is true, we would still reject the null hypothesis around 40% of the time.

I also test the null of linearity using the U-statistic proposed in Zheng (1996). The test is based on residuals from a linear model and uses the same bandwidth as the nonparametric tests proposed in this article. The variable used is the yield spread and find a p-value of 0.0985, so that there is some evidence that the relationship is nonlinear.