  Motivated by a recent literature on the double-descent phenomenon in machine
learning, we consider highly over-parameterized models in causal inference,
including synthetic control with many control units. In such models, there may
be so many free parameters that the model fits the training data perfectly. We
first investigate high-dimensional linear regression for imputing wage data and
estimating average treatment effects, where we find that models with many more
covariates than sample size can outperform simple ones. We then document the
performance of high-dimensional synthetic control estimators with many control
units. We find that adding control units can help improve imputation
performance even beyond the point where the pre-treatment fit is perfect. We
provide a unified theoretical perspective on the performance of these
high-dimensional models. Specifically, we show that more complex models can be
interpreted as model-averaging estimators over simpler ones, which we link to
an improvement in average performance. This perspective yields concrete
insights into the use of synthetic control when control units are many relative
to the number of pre-treatment periods.
