Inference for Synthetic Controls via Refined Placebo Tests
arxiv(2024)
摘要
The synthetic control method is often applied to problems with one treated
unit and a small number of control units. A common inferential task in this
setting is to test null hypotheses regarding the average treatment effect on
the treated. Inference procedures that are justified asymptotically are often
unsatisfactory due to (1) small sample sizes that render large-sample
approximation fragile and (2) simplification of the estimation procedure that
is implemented in practice. An alternative is permutation inference, which is
related to a common diagnostic called the placebo test. It has provable Type-I
error guarantees in finite samples without simplification of the method, when
the treatment is uniformly assigned. Despite this robustness, the placebo test
suffers from low resolution since the null distribution is constructed from
only N reference estimates, where N is the sample size. This creates a
barrier for statistical inference at a common level like α = 0.05,
especially when N is small. We propose a novel leave-two-out procedure that
bypasses this issue, while still maintaining the same finite-sample Type-I
error guarantee under uniform assignment for a wide range of N. Unlike the
placebo test whose Type-I error always equals the theoretical upper bound, our
procedure often achieves a lower unconditional Type-I error than theory
suggests; this enables useful inference in the challenging regime when α
< 1/N. Empirically, our procedure achieves a higher power when the effect size
is reasonably large and a comparable power otherwise. We generalize our
procedure to non-uniform assignments and show how to conduct sensitivity
analysis. From a methodological perspective, our procedure can be viewed as a
new type of randomization inference different from permutation or rank-based
inference, which is particularly effective in small samples.
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