This paper studies linear time series regressions with many regressors. Weak exogeneity is the most used identifying assumption in time series. Weak exogeneity requires the structural error to have zero conditional expectation given the present and past regressor values, allowing errors to correlate with future regressor realizations. We show that weak exogeneity in time series regressions with many controls may produce substantial biases and even render the least squares (OLS) estimator inconsistent. The bias arises in settings with many regressors because the normalized OLS design matrix remains asymptotically random and correlates with the regression error when only weak (but not strict) exogeneity holds. This bias's magnitude increases with the number of regressors and their average autocorrelation. To address this issue, we propose an innovative approach to bias correction that yields a new estimator with improved properties relative to OLS. We establish consistency and conditional asymptotic Gaussianity of this new estimator and provide a method for inference.

翻译：本文研究含多个回归变量的线性时间序列回归问题。弱外生性是时间序列分析中最常用的识别假设，它要求结构误差在当前及过往回归变量取值条件下的条件期望为零，但允许误差项与未来回归变量实现值相关。研究表明，在包含大量控制变量的时间序列回归中，弱外生性假设可能导致显著偏误，甚至使得最小二乘（OLS）估计量不一致。当回归变量数量众多时，由于仅在弱外生性（而非严格外生性）成立条件下，标准化OLS设计矩阵仍保持渐近随机性并与回归误差相关，从而产生偏误。该偏误幅度随回归变量数量及其平均自相关性的增加而增大。针对这一问题，本文提出创新的偏误校正方法，由此得到较OLS具有更优统计性质的新型估计量。我们证明了该估计量的一致性和条件渐近高斯性，并给出了统计推断方法。