Linear Regression with Common Unobserved Heterogeneity

Abstract

We study inference on common slope parameters in linear panel models with two-way unobserved heterogeneity. In linear regression, bias arises from unobserved heterogeneity that is shared between the residual and the covariates. Our common unobserved heterogeneity (CUH) estimator targets this common variation directly: it searches over functionals of the residualised outcome for variation that also resides in the covariates, and seeks to further project out only bias-inducing variation. This makes the estimator less sensitive to weak outcome factors while avoiding the complete removal of exogenous variation in covariates that is unrelated to the confounding heterogeneity. We develop a weak-factor theory based on a singular-value strength index and establish $\sqrt{NT}$-asymptotic normality of the cross-fitted CUH estimator. Under homoskedastic errors, its asymptotic variance is no larger than that of a common correlated effects factor estimator. Simulations and empirical applications illustrate the resulting bias--variance trade-off and the gains from selectively removing common variation.