  We consider the problem of estimating the counterfactual joint distribution
of multiple quantities of interests (e.g., outcomes) in a multivariate causal
model extended from the classical difference-in-difference design. Existing
methods for this task either ignore the correlation structures among dimensions
of the multivariate outcome by considering univariate causal models on each
dimension separately and hence produce incorrect counterfactual distributions,
or poorly scale even for moderate-size datasets when directly dealing with such
multivariate causal model. We propose a method that alleviates both issues
simultaneously by leveraging a robust latent one-dimensional subspace of the
original high-dimension space and exploiting the efficient estimation from the
univariate causal model on such space. Since the construction of the
one-dimensional subspace uses information from all the dimensions, our method
can capture the correlation structures and produce good estimates of the
counterfactual distribution. We demonstrate the advantages of our approach over
existing methods on both synthetic and real-world data.
