  A common statistical problem in econometrics is to estimate the impact of a
treatment on a treated unit given a control sample with untreated outcomes.
Here we develop a generative learning approach to this problem, learning the
probability distribution of the data, which can be used for downstream tasks
such as post-treatment counterfactual prediction and hypothesis testing. We use
control samples to transform the data to a Gaussian and homoschedastic form and
then perform Gaussian process analysis in Fourier space, evaluating the optimal
Gaussian kernel via non-parametric power spectrum estimation. We combine this
Gaussian prior with the data likelihood given by the pre-treatment data of the
single unit, to obtain the synthetic prediction of the unit post-treatment,
which minimizes the error variance of synthetic prediction. Given the
generative model the minimum variance counterfactual is unique, and comes with
an associated error covariance matrix. We extend this basic formalism to
include correlations of primary variable with other covariates of interest.
Given the probabilistic description of generative model we can compare
synthetic data prediction with real data to address the question of whether the
treatment had a statistically significant impact. For this purpose we develop a
hypothesis testing approach and evaluate the Bayes factor. We apply the method
to the well studied example of California (CA) tobacco sales tax of 1988. We
also perform a placebo analysis using control states to validate our
methodology. Our hypothesis testing method suggests 5.8:1 odds in favor of CA
tobacco sales tax having an impact on the tobacco sales, a value that is at
least three times higher than any of the 38 control states.
