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Abstract Most procedures for detecting stock return predictability rely on linear regression models. When assessing the null hypothesis of no predictive power in a possibly nonlinear model, practitioners essentially have two choices. One could resort to a suitable nonparametric test and be prepared to lose power because of leaving the parametric framework. Since the model is linear under the null of no predictability, one could also conduct inference in a linear model, and be prepared to lose power because of the misspecification under the alternative hypothesis. To help decide which approach to use, the paper focuses on size and local power under the additional difficulty that the persistence of the regressors, as quantified by their largest autoregressive roots, is unknown. Regarding nonparametrics, the statistics employed by Juhl (2014, JBES 32, 387-394) and Kasparis et al. (2015, J. Econometrics 185, 468-494), have \chi^{2} limiting null distributions for both low and high regressor persistence, but are asymptotically dominated in terms of local power by simple linear procedures. We show, theoretically and in simulations, that an overidentified IV testing scheme following Kostakis et al. (2015, Review of Financial Studies 28, 1506-1553) and Breitung and Demetrescu (2015, J. Econometrics 187, 358-375), is particularly well suited for inference in additive predictive models with uncertain predictor persistence. The proposed test is robust to the degree of persistence of the regressors and to time-varying volatility. An analysis of predictability of S&P 500 stock returns finds significant predictability, part of which is nonlinear in nature. For log dividend yields and the long-term rate of return we find a monotonic regression function, while log earnings price ratios exhibit a U-shaped relation. The latter is driven entirely by the 2008 financial crisis, suggesting that, during crises, firm-specific characteristics such as valuation ratios may be inconsistent signals of stock price performance.
