Abstract
In predictive regressions with variables of unknown persistence, the use of extended IV (IVX) instruments leads to asymptotically valid inference. Under highly persistent regressors, the standard normal or chi-squared limiting distributions for the usual t and Wald statistics may, however, differ markedly from the actual finite-sample distributions which exhibit in particular noncentrality. Convergence to the limiting distributions is shown to occur at a rate depending on the choice of the IVX tuning parameters and can be very slow in practice. A characterization of the leading higher-order terms of the t statistic is provided for the simple regression case, which motivates finite-sample corrections. Monte Carlo simulations confirm the usefulness of the proposed methods.
Highlights
A common inferential task of practical relevance is to decide whether a potential predictor variable does forecast another variable of interest
This effect of recursive adjustment is very much in the spirit of the proposal of Kostakis et al (2015), who point out that not demeaning the instrument zt−1 reduces the finite-sample correlation between the numerator and the denominator of the IVX t statistic
The tvWx statistic does not behave too well in each tail taken alone, as it tends to overreject to the right and to underreject to the left
Summary
A common inferential task of practical relevance is to decide whether a potential predictor variable does forecast another variable of interest. Predictors such as dividend yields or earnings-price ratios are often quite persistent, even if still mean-reverting (typically captured by a value of ρ close to unity), and its shocks are contemporaneously correlated with the variable to be predicted (see Phillips, 2015, for a recent review) This biases the OLS estimator of the slope. Should the regressor xt be highly persistent with localization parameter c close to zero, the IVX-based test of no predictability may still be seriously distorted in finite samples, even if less so than the OLS-based test This is clearly the case when choosing η too close to unity or a too close to zero, and the difference in terms of persistence between the instrument zt and the regressor xt becomes small: for example, the rule of thumb proposed by Kostakis et al (2015), which sets = 1 − 1/T0.95, is equivalent to a near unit root with localizing coefficient cbetween 1 and 2 for sample sizes between T = 100 and T = 10,000. The technical details of the proofs can be found in the Appendix and in an Online Supplement, which contains additional simulation results pertaining to conditional heteroskedasticity
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