  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.
