st: Matching, bootstrapping, sub-sampling

[Joachim Wagner <wagner@leuphana.de>]

Dear List:

This is both a Stata related and a statistics question.

Short version:
If bootstrapping is invalid for estimating the standard errors for 
the ATT after nearest neighbor matching, does sub-sampling help, and 
if so, how?

Long version:
psmatch2 is rather popular among many of us (according to download 
statistics). Although the help file warns that it is "unclear whether 
the bootstrap is valid in this context"  bootstrapping is popular to 
estimate the standard errors of the Average Treatment Effect on the 
Treated (ATT), too. But the times they are (expected to be) 
a-changin' : In the November 2008 issue of the Econometrica Alberto 
Abadie and Guido Imbens published a paper entitled "On the failure of 
the bootstrap for Matching Estimators" arguing that bootstrap 
standard errors are not valid as a basis for inference with simple 
nearest-neighbor matching estimators with replacement and a fixed 
number of neighbors. This result is popularized in a recent survey by 
Imbens and Jeffrey Wooldridge (Recent developments in the 
econometrics of program evaluation, published in the Journal of 
Economic Literature in March 2009). (For those of you who are working 
in different fields let me add that both journals are among the top 
journals in economics/econometrics.)

What is to be done? One suggestion found in both articles goes like 
this (Imbens and Wooldridge, p. 42): "In cases where bootstrapping is 
not valid, often subsampling (..) remains valid, but this has not 
been applied in practice." The authors refer to Dimitris N. Politis 
et al., Subsampling, New York: Springer 1999. Subsampling means using 
only a fraction, say, 75 percent, of the sample for a bootstrap draw.

Contrary to what Imbens and Wooldridge say there are some (working) 
papers using sub-sampling and bootstrapping to compute the standard 
errors of the ATT.  They use ca. 75 percent of the sample in doing 
so. Nobody (as yet) told me why - the authors argue that others do so 
as well, or they do not reveal the somewhat secret formula, or rule 
of thumb, applied.

Two questions:

1. Can someone please explain in (more or less) plain English why 
subsampling is a solution?
2. How large should the subsamples be, and why?

Many thanks in advance for any comments etc.

Joachim




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