On the Identification of Misspecified Propensity Scores - School of ...

even guaranteed to be between 0 and 1. Fortunately, we find that our test has high power for
even very small sample sizes in this situation. These results are summarized in Table 6.
Finally, we compare our test with the test proposed by Zheng [18]. First, we consider the
same setup of Table 2 above, where the data generating process is given by (6). The results are
presented in Table 7. As noted earlier, Zheng’s test suffers from a curse of dimensionality and
so we might expect that our test would perform better in finite samples for many alternatives.
Indeed, this feature is borne out by our Monte Carlo study: our test rejects the null hypothesis
much more frequently for nearly all samples sizes. To investigate the severity of this problem,
we consider a further model which includes an additional covariate. Define
X3 = Z1 + Z3
√
2
where Z3 is a standard normal random variable independent of Z1 and Z2. The data generating
process for D ∗ in this instance is given by
D ∗ = 1 + X1 + X2 + X1X2 + X3 − ǫ, ǫ ∼ N(0, σ 2 ) ,
where ǫ ⊥ (X1, X2, X3). The null hypothesis is as before but now Q(X, β) = Φ(β0 + β1X1 +
β2X2 + β3X3). The results of this comparison are presented in Tables 8 and 9. In this case,
with the additional covariate, we find that our test retains high power at all sample sizes, but
the highest power of Zheng’s test is approximately 20 percent and often much lower. Thus, we
believe that our test will in practice be of much greater use than Zheng’s, especially when X
is high dimensional. Of course, it should be emphasized again that this advantage of our test
comes at the expense of power against certain alternatives.
[TABLE 1 ABOUT HERE]
[TABLE 2 ABOUT HERE]
[TABLE 3 ABOUT HERE]
[TABLE 4 ABOUT HERE]
[TABLE 5 ABOUT HERE]
[TABLE 6 ABOUT HERE]
[TABLE 7 ABOUT HERE]
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