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 $\frac{J}{T}$ 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.

与 Stambaugh (1999) 类似，本文推导了J水平预测回归中估计量的小样本偏差，在预测变量的样本大小、范围和持久性方面提供了一个封闭形式的解决方案。对于大J，偏差是线性的$\frac{J}{吨}$斜率取决于预测变量的持久性。该论文提供了许多其他有用的结果，包括 (i) 原始 Stambaugh (1999) 设置的重要扩展，(ii) 流行的替代长期估计量的闭式偏差公式，(iii) 样本外分析有和没有偏差调整，以及对样本外统计的新解释，以及 (iv) 基于 Hansen 和 Hodrick (1980) 以及 Newey 和 West ( 1987）。小样本偏差调整大大降低了可预测性的长期估计量。