I am trying to understand the simple surrogate index estimator from Athey, Chetty, Imbens, and Kang (2016) in Section 5.1 of v2 of their paper.
The setup is that you have an experiment that alters tumor size/test scores/browsing behavior and you are interested in learning how that translates into a long-run outcome of mortality/wages/revenue, but you are not willing to wait long enough to observe the long-run outcomes from the experiment. The paper shows that given some assumptions, you can learn something about the long-run outcome from the experimentally induced change in the statistical surrogates.
More concretely, there are two cross-sectional datasets that have no overlap. The first one is observational and contains data on the long-run outcome of interest ($y$), a vector of statistical surrogates ($s$), and some pre-treatment covariates ($x$). The second is an experimental dataset that contains information on the treatment, and the many surrogates.
ACIK suggest estimating an OLS regression for the outcome of interest on all the surrogates and the pre-treatment covariates in the observational data:
$$y = \gamma_0+\gamma_s's + \gamma_x'x$$
The treatment effect is then
$$\hat \tau ^E=\hat \gamma_s'\hat \tau_s,$$
where $\hat \tau_s$ is the vector of effects on the surrogates from the experimental regression.
My question concerns estimating standard errors for this estimator. With $k$ surrogate variables, you effectively have a sum of $k$ products:
$$\hat \tau ^E=\hat \gamma_{s1}\cdot\hat \tau_{s1}+...+\hat \gamma_{sk}\cdot\hat \tau_{sk}.$$
You can get the covariance among the $\gamma$s from the observational regression. You can get the covariance among the $\tau$s from the experimental regression. Assuming all the $\tau -\gamma$ covariances are zero since the samples don't overlap, then you can just use the delta method on the products and then their sum. I think you could easily do this in a SUR setting.
Is this sensible? Is there something better?
