  Error-in-variables regression is a common ingredient in treatment effect
estimators using panel data. This includes synthetic control estimators,
counterfactual time series forecasting estimators, and combinations. We study
high-dimensional least squares with correlated error-in-variables with a focus
on these uses. We use our results to derive conditions under which the
synthetic control estimator is asymptotically unbiased and normal with
estimable variance, permitting inference without assuming time-stationarity,
unit-exchangeability, or the absence of weak factors. These results hold in an
asymptotic regime in which the number of pre-treatment periods goes to infinity
and the number of control units can be much larger $(p \gg n)$.
