We study a panel data model with heterogeneous effects, allowing slopes to vary across individuals and time. To reduce dimensionality, we assume these slopes follow a factor structure, so slope matrices can be estimated via low-rank regularized regression. We propose a multi-step estimation procedure incorporating sample splitting and partialing-out to enable valid inference after penalized estimation. We establish the asymptotic normality of the resulting estimator, facilitating inference for individualtime- specific effects and their cross-sectional averages. The method’s performance is illustrated through simulations and an empirical application.

We develop a general theory of omitted variable bias for a wide range of common causal parameters, including average treatment effects, average causal derivatives, and policy effects from covariate shifts. We show how plausibility judgments on the maximum explanatory power of omitted variables are sufficient to bound the bias, facilitating sensitivity analysis in otherwise complex models. Finally, we provide statistical inference methods that can leverage modern machine learning algorithms for estimation. These results allow empirical researchers to perform sensitivity analyses in a flexible class of machine-learned causal models using very simple tools. Empirical examples demonstrate the utility of our approach.