where m$_{1}$ ( X ) = 1 + X$_{1}$ + X$_{2}$ , u (1) and u (0) are independent normal random variables with mean zero and variance 0 . 1. The observed outcome is Y = DY (1)+(1 − D ) Y (0), and the true ATE is 1. Although these outcome equations are not necessary to assess the size and power properties of the tests, they can be used to assess the utility of our proposed tests to distinguish between “good” and “bad” estimates of the ATE . Let X = (1 ,X$_{1}$,X$_{2}$ ) $^{′}$. For DGP 1 − DGP 5, the H$_{0}$ considered is where Φ( · ) is the cumulative distribution function (CDF) of the standard normal distribution. We estimate θ$_{0}$ using the Probit ML, i.e. Clearly, DGP 1 falls under H$_{0}$ , whereas DGP 2 − DGP 5 fall under H$_{1}$ . Note that D follows a heteroskedastic Probit model in DGP 4 and DGP 5. The simulation results are presented in Table 1. We report empirical rejection frequencies at the 5% significance level. Results for 10% and 1% significance levels are similar and are available upon request. We also report the bias of the normalized IPW estimator and the average length and coverage of its estimated 95% confidence interval based on its asymptotically normal approximation (assuming that the propensity score model is correctly specified). We first analyze the size of our test. From the results of DGP 1, we find that the actual finite sample size of both KS$_{n}$ and CvM$_{n}$ tests is close to their nominal size, even when the sample size is as small as 100. The same holds for the other ICM-type tests. On the other hand, we find that Shaikh et al. (2009)’s test is, in general, conservative, and sensitive to the choice of bandwidth. For instance, when c = 0 . 15, the empirical size is close to zero even with n = 1 , 000. On the other hand, with c = 0 . 01, the empirical size of Shaikh et al. (2009)’s test is closer to the nominal value. In terms of traditional balancing tests, we find that Wald test and Bonferroni-corrected t -test do not control size. Such a drawback is due to the random denominator being relatively close to zero: when we trim observations with estimated propensity score outside the [0 . 05 , 0 . 95] range, we find that classical balancing tests can control size. We 20 ̂ ATE$_{n}$ = 1 n n ∑ i =1 ( w$_{1}$$_{,i}$ ¯ w$_{1}$$_{,n}$ − w$_{0}$$_{,i}$ ¯ w$_{0}$$_{,n}$ ) Y$_{i}$, (4.3) ˆ θ$_{n}$ = arg max θ ∈ Θ n ∑ i =1 D$_{i}$ ln ( Φ ( X ′ $_{i}$θ )) + (1 − D$_{i}$ ) ln ( 1 − Φ ( X ′ $_{i}$θ )) . H$_{0}$ : ∃ θ$_{0}$ = ( β$_{0}$,β$_{1}$,β$_{2}$ ) ′ ∈ Θ : E [ D | Φ ( X $^{′}$θ$_{0}$ )] = Φ ( X $^{′}$θ$_{0}$ ) a.s.,