Abstract: This paper studies linear time series regressions with many regressors. Weak ex- ogeneity is the most commonly used identifying assumption in time series. Weak exogeneity requires that the structural error has zero expectation conditional on the current and past value of the regressors, but it allows the errors to be corre- lated with future realizations of regressors. We show that weak exogeneity in a time series regression with many controls may produce very large biases and can even lead to inconsistency of the least squares (OLS) estimator. The bias arises in settings with many autocorrelated regressors because the normalized OLS de- sign matrix remains asymptotically random and is correlated with the regression error, when only weak but not strict exogeneity holds. We propose an innovative approach to bias correction that yields a large class of estimators with improved properties relative to OLS. We analyze asymptotic bias and variance for this class of estimators, and suggest bias-aware confidence sets that account for any potential remaining bias in the improved estimators.
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