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

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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] 12 ,
5 Conclusion [TABLE 8 ABOUT HERE] [TABLE 9 ABOUT HERE] In this paper, we have shown that the commonly computed estimates of the densities of the propensity score conditional on participation provide a means of examining whether the para- metric model is correctly specified. In particular, correct specification of the propensity score implies a certain restriction between the estimated densities. We have shown further that this restriction is equivalent to an orthogonality restriction, which can be used as the basis of a formal test for correct specification. While our test does not have power against all forms of misspecification of the propensity score, we argue that for a large class of alternatives our test will perform better in finite samples than existing tests that have power against all forms of misspecification. Our Monte Carlo study of the finite sample behavior of our test corroborates this hypothesis. The study shows further that our test performs well against many types of misspecification. Since our test is also easily implemented, it is our hope that this work will persuade researchers to examine the specification of their model for the propensity score, as its validity is essential for consistency of their estimates of the treatment effect. A Proof of Theorem 4.1: We start by proving the first claim, namely, that n hn ˆVn d → N(0, Σ). (10) We start proving this result by first showing that ˆVn − Vn0 = oP (1), (11) where ˆVn := with ˆεi = Di − Q(Xi, ˆ θ), and Vn0 := 1 n(n − 1) 1 n(n − 1) nh 1/2 n n 1 Q(Xi, K ˆ θ) − Q(Xj, ˆ θ) ˆεiˆεj, h hn i=1 j=i n 1 Q(Xi, θ0) − Q(Xj, θ0) K hn i=1 j=i 13 hn εi0εj0,
Page 1 and 2: On the Identification of Misspecifi
Page 3 and 4: presented in this paper may be cons
Page 5 and 6: 3 The Restriction In order to ensur
Page 7 and 8: that misspecification is not very s
Page 9 and 10: where ˆεi := Yi − Q(Xi, ˆ θn)
Page 11: Throughout the simulations presente
Page 15 and 16: Ichimura, and Todd [6]. In addition
Page 17 and 18: To check the second condition of th
Page 19 and 20: where M = sup θ∈Θ[D−Q(x, θ)]
Page 21 and 22: [15] S, B. (2004): “An Evaluation
Page 23 and 24: FIGURE 2 E D P S Note: ----- ˆα
Page 25 and 26: TABLE 2: P c=0.05 Proportion of Rej

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