Long‐run predictability tests are even worse than you thought
推导了长期最小二乘估计量及其t统计量的渐近性质，发现即使预测变量平稳，内生性也会导致检验规模扭曲，且这种偏差不同于已知的Stambaugh偏差，增加了长期预测回归推断的难度。
Summary We derive asymptotic results for the long‐horizon ordinary least squares (OLS) estimator and corresponding ‐statistic for stationary autoregressive predictors. The ‐statistic—formed using the correct asymptotic variance—together with standard‐normal critical values result in a correctly‐sized test for exogenous predictors. For endogenous predictors, the test is size distorted regardless of the persistence in the predictor and adjusted critical values are necessary. The endogeneity problem stems from the long‐run estimation and is distinct from the ordinary persistence‐dependent “Stambaugh” bias. The bias for fully stationary predictors appears not to have been previously noted and adds further difficulty to inference in long‐run predictive regressions.