  This paper provides estimation and inference methods for a conditional
average treatment effects (CATE) characterized by a high-dimensional parameter
in both homogeneous cross-sectional and unit-heterogeneous dynamic panel data
settings. In our leading example, we model CATE by interacting the base
treatment variable with explanatory variables. The first step of our procedure
is orthogonalization, where we partial out the controls and unit effects from
the outcome and the base treatment and take the cross-fitted residuals. This
step uses a novel generic cross-fitting method we design for weakly dependent
time series and panel data. This method "leaves out the neighbors" when fitting
nuisance components, and we theoretically power it by using Strassen's
coupling. As a result, we can rely on any modern machine learning method in the
first step, provided it learns the residuals well enough. Second, we construct
an orthogonal (or residual) learner of CATE -- the Lasso CATE -- that regresses
the outcome residual on the vector of interactions of the residualized
treatment with explanatory variables. If the complexity of CATE function is
simpler than that of the first-stage regression, the orthogonal learner
converges faster than the single-stage regression-based learner. Third, we
perform simultaneous inference on parameters of the CATE function using
debiasing. We also can use ordinary least squares in the last two steps when
CATE is low-dimensional. In heterogeneous panel data settings, we model the
unobserved unit heterogeneity as a weakly sparse deviation from Mundlak
(1978)'s model of correlated unit effects as a linear function of
time-invariant covariates and make use of L1-penalization to estimate these
models. We demonstrate our methods by estimating price elasticities of
groceries based on scanner data. We note that our results are new even for the
cross-sectional (i.i.d) case.
