This paper considers predictive regressions, where $y_t$ is predicted by all $p$ lags of $x_t,$ here with $x_t$ being autoregressive of order $q$, $PR(p,q)$. The literature considers model properties in the cases where $p=q$. We demonstrate that the current augmented regression method can still reduce the bias in predictive coefficients, but its efficiency depends on correctly specifying both $p$ and $q$. We propose an estimation framework for the predictive regression, $PR(p,q)$, with a data-driven auto-selection of $p$ and $q$ to achieve the best bias reduction in predictive coefficients. The corresponding hypothesis testing procedure is also derived. The efficiency of the proposed method is demonstrated with simulations. Empirical applications to equity premium prediction illustrate the substantial difference between the estimates of our method and those obtained by the common predictive regressions with $p=q$.

本文考虑了预测回归，其中 ${ÿ}_{Ť}$ 被所有人预测 $p$ 滞后 $X_{Ť}，$ 在这里 $X_{Ť}$ 顺序自回归 $q$， $P\mathrm{[R}（p，q）$。文献考虑了以下情况下的模型属性：$p=q$。我们证明了当前的增强回归方法仍可以减少预测系数的偏差，但是其效率取决于正确指定两者$p$ 和 $q$。我们为预测回归提出了一个估计框架，$P\mathrm{[R}（p，q）$，并通过数据驱动自动选择 $p$ 和 $q$以获得最佳的预测系数偏差减少。还推导了相应的假设检验程序。仿真结果证明了该方法的有效性。股权溢价预测的经验应用表明，我们的方法的估计值与通过普通预测回归得到的估计值之间存在显着差异。$p=q$。