Biases in long-horizon predictive regressions

Analogous to Stambaugh (1999), this paper derives the small sample bias of estimators in J-horizon predictive regressions, providing a closed-form solution in terms of the sample size, horizon and persistence of the predictive variable. For large J, the bias is linear in JT with a slope that depends on the predictive variable's persistence. The paper offers a number of other useful results, including (i) important extensions to the original Stambaugh (1999) setting, (ii) closed-form bias formulas for popular alternative long-horizon estimators, (iii) out-of-sample analysis with and without bias adjustments, along with new interpretations of out-of-sample statistics, and (iv) a detailed investigation of the bias of the overlapping estimator's standard error based on the methods of Hansen and Hodrick (1980) and Newey and West (1987). The small sample bias adjustments substantially reduce the magnitude of long-horizon estimates of predictability.