Estimating Continuous Treatment Effects in Panel Data using Machine Learning with a Climate Application

Economists often estimate continuous treatment effects in panel data using linear two-way fixed effects models (TWFE). When the treatment-outcome relationship is nonlinear, TWFE is misspecifed and potentially biased for the average partial derivative (APD). We develop an automatic double/de-biased machine learning (ADML) estimator that is consistent for the population APD while allowing additive unit fixed effects, nonlinearities, and high dimensional heterogeneity. We prove asymptotic normality and add two refinements - optimization based de-biasing and analytic derivatives - that reduce bias and remove numerical approximation error. Simulations show that the proposed method outperforms high order polynomial OLS and standard ML estimators. Our estimator leads to significantly larger (by 50%), but equally precise, estimates of the effect of extreme heat on corn yield compared to standard linear models.