###### Job Market Paper

“Estimation of Fixed Effects Models with Missing Covariate Data, with an Application to Valuing Local Water Quality”

### This paper considers estimation of a linear fixed effects model in which covariate values may be missing. Two inverse probability weighted (IPW) estimators are proposed. The main assumption is a missing at random assumption (MAR) which allows missingness (observation) to be related to the outcome and its shocks, but requires that the probability of observation is not related to the missing values. The inverse of the estimated probability of observation is used to re-weight the estimating equations, which are then estimated in a second stage by either computationally simple pooled OLS or more asymptotically efficient GMM. Both of the proposed estimators are consistent and root-n asymptotically normal, and the asymptotic variance is derived. The main results are developed for the classical linear fixed effects model under strict exogeneity, and the approach generalizes to many panel models, including dynamic linear unobserved effects models.

### As an application, the proposed estimator is applied to a hedonic housing price model in which the willingness to pay for local water quality is reflected in house prices. Water quality, the main regressor of interest, is missing for many houses in many time periods. Empirical evidence suggests that, in line with the MAR assumption, observation is related to house prices but not water quality itself. The results suggest that accounting for the missing mechanism is important, as the estimated willingness to pay differs in magnitude and statistical significance between the proposed two-step IPW estimator and the more commonly used un-weighted estimator which drops incomplete observations.