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

On the Identification of Misspecified Propensity Scores
Azeem M. Shaikh
Department of Economics
Stanford University
Edward J. Vytlacil
Department of Economics
Columbia University
February 8, 2006
Abstract
Marianne Simonsen
Department of Economics
University of Aarhus
Nese Yildiz
Department of Economics
University of Pittsburgh
Propensity score matching has become a popular method for the evaluation of
treatment effects. In empirical applications, researchers almost always impose a
parametric model for the propensity score. This raises the possibility that the
model is misspecified and therefore the propensity score matching estimator may
be inconsistent. We show that the common practice of calculating estimates of the
densities of the propensity score conditional on the participation decision provides
a means for examining whether the propensity score is misspecified. In particular,
we derive a restriction between the density of the propensity score among participants
versus the density among nonparticipants. We show that this restriction
between the two conditional densities is equivalent to a particular orthogonality
restriction and derive a formal test based upon it. The resulting test is shown to
have dramatically greater power than competing tests for many alternatives. The
principle disadvantage of this approach is loss of power against some alternatives.
Acknowledgements:
The authors wish to thank Jeff Smith for valuable comments and suggestions.

1 Introduction
Propensity score matching is a widely used approach for the estimation of treatment effects.
See Heckman, LaLonde, and Smith [7] for a detailed survey of its use in the treatment effect
literature. The method is based on the following well-known result of Rosenbaum and Rubin
[13]: if selection into the program is independent of the potential outcomes of interest con-
ditional on a vector of covariates, then selection into the program is also independent of the
potential outcomes conditional on the propensity score, where the propensity score denotes the
probability of selection into the program conditional on the same vector of covariates. The high
dimensionality of the vector of observed covariates often forces researchers to adopt a finite-
dimensional approximation to the propensity score, in lieu of a more flexible nonparametric
model. Imposing such restrictions on the functional form for the propensity score raises the
possibility that the propensity score model is misspecified. If the propensity score model is mis-
specified, then in general the estimator of the propensity score will be inconsistent for the true
propensity score and the resulting propensity score matching estimator may be inconsistent
for the treatment effect of interest.
First suggested by Heckman, Ichimura, Smith, and Todd [5], it is now common to calculate
estimates of the densities of the propensity score conditional on the participation decision.
These estimates are calculated in order to determine the region of common support on which
to perform matching. We show that these conditional densities provide a means for examining
whether the propensity score is misspecified. In particular, we derive a restriction between
the density of the propensity score model among participants and the density among nonpar-
ticipants. Failure of this restriction to hold for the estimated conditional densities provides
evidence that the propensity score model is misspecified. We show that this restriction between
the two conditional densities is equivalent to a particular orthogonality restriction and derive
a formal test based upon it. Unlike other tests for correct specification of the propensity score,
the resulting test does not require nonparametric estimation of high dimensional objects. As
a result, our test does not suffer from the curse of dimensionality and has dramatically greater
power than competing tests for many alternatives. We provide striking evidence of this fact
in a limited Monte Carlo study. The principle drawback of this approach is that our test does
not have power against certain alternatives, but we argue that these alternatives are rather
exceptional.
Following [12], applied researchers often employ heuristic balancing score tests for misspeci-
fication of the propensity score to guide the choice of specification of the parametric propensity
score. Specifically, [12] suggests examining whether the observable characteristics of the pop-
ulation are independent of participation conditional on the propensity score. See e.g. [3], [8],
[15], and [17] for applications of such balancing score tests. The formal misspecification test
2

presented in this paper may be considered a complement to this practice.
Our paper proceeds as follows. In Section 2, we first provide a summary of the method of
propensity score matching. We then present in Section 3 the restriction upon which our test
is based. In Section 4, we use this result to derive an asymptotic test of misspecification and
examine its finite sample properties in a Monte Carlo study. Finally, Section 5 concludes.
2 A Review of Matching
Before proceeding, it is useful to review the key theoretical underpinnings of matching as a
means of program evaluation. There are two groups of individuals, participants and nonpar-
ticipants, in a program of interest. Participation in the program of interest is denoted by the
dummy variable D, with D = 1 if the individual chooses to participate and D = 0 otherwise.
Individuals in each of these two groups are associated with observed characteristics X. Two
commonly used metrics for evaluating the effect of participation in a program are the average
treatment effect, given by E[Y1 −Y0], and the average treatment effect on the treated, given by
E[Y1 − Y0|D = 1], where Y1 is the potential outcome in the case of participation and Y0 is the
potential outcome in the case of nonparticipation. The difficulty with estimation of this object
lies with the following missing data problem: the econometrician never observes the counter-
factual outcome Y1−D, which precludes direct estimation of E[Y0|D = 1] and E[Y1|D = 0].
The method of matching resolves this difficulty by matching each participant with a non-
participant that is similar in terms of observed characteristics X. As described in Rosenbaum
and Rubin [13], matching formally requires that:
A1. (Y0, Y1) ⊥ D | X.
A2. 0 < P (x) < 1 where P (x) = Pr[D = 1|X = x] for all x ∈ supp(X).
Note that a consequence of A2 is that E[Y0|D = 1, X = x] and E[Y1|D = 0, X = x] are
well defined for all x ∈ supp(X). Hence, matching suggests estimation of E[Y0|D = 1] and
E[Y1|D = 0] by a two-step procedure in which E[Y0|D = 1, X] and E[Y1|D = 0, X] are first
estimated by exploiting A1, and then integrated with respect to the empirical distribution of
X in order to obtain an estimate of E[Y1 − Y0] or integrated with respect to the empirical
distribution of X conditional on D = 1 to obtain an estimate of E[Y1 − Y0|D = 1]. Following
Heckman, Ichimura, Smith, and Todd [5], it is clear that one can relax the first condition to
only require mean independence instead of full independence. Using this procedure, it is in
3

principle possible to construct a √ n-normal estimator of the parameter of interest without
imposing any finite-dimensional restrictions. 1
An alternative approach for estimating E[Y0|D = 1] and E[Y1|D = 0] is propensity score
matching, which relies upon a celebrated result of Rosenbaum and Rubin [13]. They show that
the assumptions underlying matching conditional on X, A1 and A2, imply:
A1*. (Y0, Y1) ⊥ D | P (X).
A2*. 0 < Pr[D = 1|P (X) = p] < 1 for all p ∈ supp(P (X)).
This result can be restated as follows: if matching on X is valid, so is matching based on the
propensity score P (X). This result motivates propensity score matching, in which one first
estimates the propensity score in a first step and then perform matching, as described above,
using the estimated propensity score.
In practice, both matching on X and matching on P (X) suffer from the so-called “curse
of dimensionality”. While matching on X requires the researcher to estimate E[Y0|D = 1, X],
matching on P (X) requires the researcher to estimate E[D|X], an equally high dimensional
object. Thus, if X has more than a few dimensions, nonparametric procedures for estimating
these objects are undesirable due to sizable finite sample bias. This difficulty leads to the
common practice of implementing propensity score matching with a parametric specification
for the propensity score. More specifically, in practice propensity score matching involves
estimating a parametric model for the propensity score by maximum likelihood in a first step,
and then using kernel regression or some other comparable nonparametric procedure to perform
matching on the basis of the estimated propensity scores. 2
If the propensity score is estimated using a parametric model, then it is possible that the
model is misspecified and that the resulting estimator of P (X) is inconsistent. Let Q(X)
denote the probability limit of the estimator of P (X). If P (X) = Q(X), as will likely be
the case if the model for P (X) is misspecified, then in general we cannot expect (Y0, Y1) to
be independent (or even mean independent) of D conditional on Q(X). In this case, the
propensity score matching will in general provide inconsistent estimates of both E[Y1 −Y0] and
E[Y1 − Y0|D = 1].
1 See Heckman, Ichimura, and Todd [6], Hahn [4], and Abadie and Imbens [1].
2 Since binary variables are bounded, there is potentially less room for misspecification of a
parametric model for binary outcome as compared to a continous one, see [2].
4

3 The Restriction
In order to ensure that the appropriate conditional expectations exist, propensity score match-
ing is only valid over the region of common support of the densities of the propensity score con-
ditional on the participation decision. 3 Following the influential work of Heckman, Ichimura,
Smith, and Todd [5], researchers therefore compute estimates of both of these densities to de-
termine the appropriate region over which to perform matching. The test for misspecification
that we develop in the following section will exploit a restriction that must hold between these
two densities.
Throughout the following we will assume that the propensity score P (X) has a density with
respect to Lebesgue measure. Let f(p) denote this density, f1(p) the density of the propensity
score conditional on participation, and f0(p) the density of the propensity score conditional on
nonparticipation. Using this notation, we have the following result:
Lemma 3.1. Let α = Pr[D=0]
Pr[D=1]
p ∈ supp(P ) we have that
and assume 0 < Pr[D = 0] < 1. Then for all 0 < p < 1 and
f1(p) = α p
1 − p f0(p) .
P: Consider 0 < p < 1 and p ∈ supp(P ). For such p, by the Law of Iterated Expectations,
we have that
Bayes’ Theorem implies further that
Similarly, we have that
Pr[D = 1|P (X) = p] = E[E[D|X]|P (X) = p] = p .
f1(p) Pr[D = 1] = Pr[D = 1|P (X) = p]f(p) = pf(p) > 0 .
f0(p) Pr[D = 0] = (1 − p)f(p) > 0 .
Combining these two implications, we have that
f1(p) p
= α
f0(p) 1 − p ,
from which the asserted conclusion follows immediately.
Lemma 3.1 suggests that to the extent that the parametric model for P (X) is correctly
specified, we would expect the stated relationship between the estimates of the conditional
densities of the propensity score to hold approximately. The property can be easily checked
3 Propensity score matching is only valid for the common support for the population but
in practice, all we can make conclusions about is the common support in the sample. As the
sample size gets larger, of course, the two sets of common support converge.
5

given estimates ˆ f1,n(·) and ˆ f0,n(·) by graphing ˆ f1,n(p) along with the function ˆαn p
1−p ˆ f0,n(p),
where ˆαn is the sample analogue of α. Dramatic departures of one graph from another should
be taken as evidence of misspecification of the propensity score.
We now provide an illustration of this heuristic approach to detecting misspecifcation of
the propensity score. Define
X1 = Z1
X2 = Z1 + Z2
√
2
(1a)
, (1b)
where Z1 and Z2 are independent standard normal variables. Consider the model
D ∗ = 1 + X1 + X2 + X1X2 − ǫi, ǫ ∼ N(0, σ 2 ) (2a)
D = 1 [D ∗ > 0] , (2b)
where ǫ ⊥ (X1, X2), and 1[·] is the logical indicator function. We consider the results of fitting
a misspecified model
Q(X, β) = Φ(β0 + β1X1 + β2X2) (3)
that only differs from the true model by omitting the quadratic term.
We generate a sample of 10, 000 observations (Di, X1i, X2i, ǫi) according to the model given
by (2a) and (2b). We then estimate using maximum likelihood the propensity score under the
incorrectly specified model (3). Estimates of the densities of the propensity score conditional
on the participation decision are then constructed using kernel density estimation. 4
Figure 1 depicts the resulting estimates of ˆ f1,n(p) and ˆαn p
1−p ˆ f0,n(p). We see that graphs
of these two objects deviate substantially from each other, which can be taken as evidence of
misspecification.
[FIGURE 1 ABOUT HERE]
[FIGURE 2 ABOUT HERE]
We also consider the case where the true data generating process is given by
D ∗ = 1 + X1 + X2 − ǫ, ǫ ∼ U(−1, 1) (4)
and we again fit the misspecified model (3). The estimated densities from this exercise are
displayed in Figure 2. We find that the two graphs lie very close to one another, suggesting
4 We employ the biweight kernel and choose the bandwidth using Silverman’s rule of thumb,
see [16].
6

that misspecification is not very severe. This feature is perhaps not too surprising when one
considers the fact that the only source of misspecification is the distribution for ǫ and the
assumed distribution shares many features with the true distribution. In particular, both are
symmetric about zero.
These examples shows that misspecification of the propensity score may lead to noticeable
departures from the restriction of Lemma 3.1. In order to develop a formal test based on this
restriction, it will be useful to restate it in terms of an equivalent orthogonality condition. To
this end, we have the following result.
Lemma 3.2. Let α = Pr[D=0]
Pr[D=1] and assume that 0 < Pr[D = 1] < 1. Let Q be a random
variable on the unit interval with density with respect to Lebesgue measure. Denote by g1(·) the
density of Q conditional on D = 1 and by g0(·) the density of Q conditional on D = 0. Then,
g1(q) = α q
1−q g0(q) for all q ∈ supp(Q) such that 0 < q < 1 if and only if E[D − Q|Q = q] = 0
for all q ∈ supp(Q) such that 0 < q < 1.
P: First consider necessity. Consider q ∈ supp(Q) such that 0 < q < 1. E[D − Q|Q =
q] = 0 is equivalent to Pr[D = 1|Q = q] = E[D|Q = q] = q. For such q, Pr[D = 1|Q = q] = q
and Bayes’ Theorem imply together that
g1(q) Pr[D = 1] = Pr[D = 1|Q = q]g(q) = qg(q) > 0 ,
where g(q) is the density of the Q. Similarly, we have that
g0(q) Pr[D = 0] = (1 − q)g(q) > 0 .
Combining these two implications, we have that
g1(q) q
= α
g0(q) 1 − q ,
from which the asserted conclusion follows. Now consider sufficiency. Assume that g1(q) =
α q
1−q g0(q) for all q ∈ supp(Q) such that 0 < q < 1. By Bayes’ Theorem,
E[D|Q = q] =
g1(q) Pr[D = 1]
= q
Pr[D = 0]g0(q) + Pr[D = 1]g1(q)
where the last equality follows from plugging in g1(q) = α q
1−q g0(q). It follows that E[D−Q|Q =
q] = q, as desired.
It is important to observe that this characterization of the restriction only involves low
dimensional conditional expections. We will show in the following section that a consequence
of this fact is that, unlike more conventional nonparametric tests for correct specification of
7

the propensity score, our test will not suffer from a curse of dimensionality and will therefore
have much greater power in finite samples against many alternatives. The principle drawback
of this approach is that there will be certain alternatives for which we will not have power.
To examine this issue further, consider the following example. Suppose X = (X1, X2), and
E[D|X1, X2] = E[D|X1]. Let Q(X) = E[D|X1]. Thus, Q(X) = P (X), and yet Q(X) will
satisfy E[D − Q|Q] = 0. More generally, this restriction will not be able to detect the omission
of covariates provided that the conditional expectation of D given the included covariates is
correctly specified. Of course, this example is rather exceptional, since Q(X) is in this case
correctly specified for a subvector of X.
4 A Formal Test
We now propose and develop a formal test for misspecification based on Lemma 3.2, which
allows us to draw upon the large literature on testing orthogonality constraints. In particular,
we adapt the testing methodology of Zheng [18] for our problem. Zheng considers testing if
the conditional expectation E[Y |X] is correctly specified. Under his null hypothesis, E[Y |X]
is assumed to belong to a parametric family of real valued functions Q(X, θ) on R m ×Θ, where
Θ ⊆ R l . Concretely, his null and alternative hypotheses are
H0: ∃θ0 ∈ Θ s.t. Pr[E[Y |X] = Q(X, θ0)] = 1.
H1: Pr[E[Y |X] = Q(X, θ)] < 1 ∀θ ∈ Θ.
Note that any θ0 satisfying the null hypothesis also solves minθ∈Θ E[Y − Q(X, θ)] 2 .
Zheng’s test is based on the following idea. Let ǫ = Y − Q(X, θ0) and let fX(·) denote the
density of X. Then, under the null hypothesis,
while under the alternative hypothesis
E[ǫE[ǫ|X]fX(X)] = 0 ,
E[ǫE[ǫ|X]fX(X)] = E[[E[ǫ|X]] 2 fX(X)] > 0 .
The last inequality follows because under the alternative hypothesis (E[ǫ|X]) 2 > 0 with positive
probability. Based on this observation, he uses the sample analogue of E[ǫE[ǫ|X]fX(X)] to
form his test. The test statistic is given by
ˆVn :=
1
n(n − 1)
n
1
h
i=1 j=i
m n
8
Xi − Xj
K
ˆεiˆεj ,
hn

where ˆεi := Yi − Q(Xi, ˆ θn), ˆ θn is a √ n-consistent estimator of arg minθ∈Θ E[Y − Q(X, θ)] 2 , h
is a smoothing parameter, and K(·) is a kernel.
Zheng shows that under certain assumptions
where Σ can be consistently estimated by
ˆΣn :=
2
n(n − 1)
nh m/2 ˆVn n
d → N(0, Σ)
n
1
h
i=1 j=i
m n
K 2
Xi − Xj
hn
ˆε 2 i ˆε 2 j .
We would like to test whether E[D|Q] = Q, where Q(X) = Q(X, θ0). Our null and
alternative hypotheses are therefore given by
H0: ∃θ0 ∈ Θ s.t. Pr[E[D|Q(X, θ0)] = Q(X, θ0)] = 1
H1: Pr[E[D|Q(X, θ)] = Q(X, θ)] < 1 ∀θ ∈ Θ
Note that this differs from the framework of Zheng [18] in that the variable that is conditioned
on is not observed but has to be estimated. Nevertheless, by analogy with the test statistic
above, we consider testing based upon
n
1 1 Q(Xi,
ˆVn :=
K
n(n − 1)
ˆ θn) − Q(Xj, ˆ
θn)
ˆεiˆεj
hn
i=1 j=i
where ˆεi = Di − Q(Xi, ˆ θn). Unfortunately, the analysis of Zheng [18] does not apply directly
because in our case the estimated Q appears in the kernel function as well. Thus, we require
the following extension of Zheng’s analysis:
Theorem 4.1. Make the following assumptions:
(i) {Di, Xi} n i=1 is a random sample;
(ii) Q is Lipschitz continuous in (x, θ). Q(·, θ0) is continuously differentiable. Moreover, it
has density with respect to Lebesgue measure on [0,1] that is continuously differentiable;
(iii) K(·) is a nonnegative, bounded, symmetric and Lipschitz continuous function such that
K(u)du = 1;
(iv) The bandwidth sequence and the sequence of positive numbers {an} are such that:
9
hn

(a) n √ hn → ∞, (nh 3/2
n ) −1 → c < ∞;
(b)
1
anh 2 n
→ a < ∞, na−2 n
log n → ∞, n√hn a2 n
→ b < ∞.
Let ˆ θn be a √ n-consistent estimator of arg minθ∈Θ E[D − Q(X, θ)] 2 , where Θ is a compact
and convex subset of R l . Then,
hn
i=1 j=n
n hn ˆVn d → N(0, Σ)
where Σ can be consistently estimated by
n 2
n(n − 1)
1
K 2
Q(Xi, ˆ θn) − Q(Xj, ˆ
θn)
P: See Appendix A.
Assumptions (i)-(iii) are almost the same as Assumption 1 and 5 of [18] adapted to
our problem. The only difference is that here we require that Q and K are Lipschitz con-
tinuous functions. In addition, the requirement that ˆ θn be a √ n-consistent estimator of
arg minθ∈Θ E[D − Q(X, θ)] 2 is guaranteed by Assumptions 2-4 in [18]. The extra assump-
tions on the bandwidth sequence are there because in our analysis Q(x, ˆ θ) appears in the
argument of the kernel function as well.
As noted earlier, there are alternatives against which Zheng’s test has power tending to
1, but our test does not. However, Zheng’s procedure requires estimation of expectations
conditional on X, which in practice is high dimensional, and therefore suffers from a curse of
dimensionality. Our procedure only requires estimation of expectations conditional on Q and
thus avoids this difficulty. As a result, we would expect our test to perform noticeably better
in finite samples against many alternatives of interest.
We now examine the finite sample properties of this testing procedure in a limited Monte
Carlo study. Our setup will follow Zheng [18] closely. As before, let X1 and X2 be given by
(1a) and (1b). We define now several different data generating processes for D ∗ that will be
used at different points in our Monte Carlo study.
hn
ˆε 2 i ˆε 2 j
D ∗ = 1 + X1 + X2 − ǫ, ǫ ∼ N(0, σ 2 ) (5)
D ∗ = 1 + X1 + X2 + X1X2 − ǫ, ǫ ∼ N(0, σ 2 ) (6)
D ∗ = (1 + X1 + X2) 2 − ǫ, ǫ ∼ N(0, σ 2 ) (7)
D ∗ = 1 + X1 + X2 − ǫ, ǫ ∼ χ 2 1
D ∗ = 1 + X1 + X2 − ǫ, ǫ ∼ U(−1, 1) (9)
For each of these data generating processes, D = 1[D ∗ > 0] and ǫ ⊥ (X1, X2).
10
(8)

Throughout the simulations presented below, we use the normal kernel given by
K(u) = 1
−u2 √ exp .
2π 2
1
− The bandwidth hn is chosen to be equal to cn 5 . In our simulations, we report results for values
of c equal to 0.05, 0.10, and 0.15. We consider samples sizes n equal to 100, 200, 400, 500, 800,
and 1000. The number of replications for each simulation is always 1000. The bandwidth is
chosen to fulfill the requirements from Theorem 4.1. 5
We first examine the size of our test. The data is generated according to (5). We use our
procedure to test the null hypothesis
H0: ∃β0 ∈ R 3 s.t. Pr[E[D|Q(X, β0)] = Q(X, β0)] = 1
H1: Pr[E[D|Q(X, β)] = Q(X, β)] < 1 ∀β ∈R 3 ,
where Q(X, β) = Φ(β0+β1X1+β2X2). These results are summarized in Table 1. Our principle
finding is that the actual size of the test is conservative and sensitive to the choice of c, but,
as expected, the actual size approaches the asymptotic size as the sample size increases.
We now go on to consider the power of our test against certain misspecifications of the
propensity score. In Tables 2 - 5, we consider four different scenarios in which the true data
generating process is given by (6) - (9), respectively. For each scenario, the null hypothesis is
the same as the one above. The test performs admirably when the true data generating process
is given by (6) - (8), showing high power for even moderately sized samples. Recall that (6)
is precisely that of our heuristic from Lemma 3.1 presented in Figure 1. Given the noticeable
departure of the two graphs in Figure 1, it is not surprising that our test performs well. The
test performs less well, however, when the true data generating process is given by (9), which
is again consonant with our earlier findings in Figure 2. This outcome is to be expected due to
the shared features of the distributions for ǫ. On the other hand, as evidenced by Table 4, the
test performs much better when the distribution is assumed to be χ2 1 , which is not symmetric
about zero.
We consider next a scenario in which the true data generating process is (5) and the null
hypothesis is the same as the one above, but now Q(X, β) = β0 + β1X1 + β2X2. Naturally, we
would expect any sensible test to have high power against such alternatives. Q(X, β) is not
5 − The bandwidth employed in [18], c·n 2
5 does not fulfill the assumptions required by theorem
1
− 4.1. However, using this bandwidth instead of c · n 5 does not change the conclusion from the
Monte Carlo analysis presented below. Results are available on request.
11

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,

with εi0 = Di − Q(Xi, θ0). Then we use the analysis in [18] to argue that
so that combining (11) and (12) gives us (10).
n hnVn0 d → N(0, Σ), (12)
To prove (11) we are going to rely on Lemma 3 given on page 286 of Heckman, Ichimura,
and Todd [6]. To use that lemma, we define 6
Θn := {θ
∈ Θ : an||θ − θ0|| ≤ ǫθ}
Ψn :=
ψn0(Di,Xi,Dj,Xj) :=
ˆψn(Di,Xi,Dj,Xj) :=
so that n √ hn ˆVn =
i
ψn : ψn(Di, Xi, Dj, Xj) = εiεj
(n−1) √ hn
Q(Xi,θ0)−Q(Xj,θ0)
εi0εj0
(n−1) √ hn K
ˆεi ˆεj
(n−1) √ hn K
hn
Q(Xi, ˆ θ)−Q(Xj, ˆ θ)
hn
j=i ˆ ψn(i, j) and n √ hnVn0 =
will verify two conditions: (i) that the U-process
i
i
K
Q(Xi,θ)−Q(Xj,θ)
hn
, θ ∈ Θn
j=i ψn0(i, j). To show (11), we
j=i ψn(i, j) is equicontinuous over Ψn
in a neighborhood of ψn0, and (ii) that, with probability approaching to 1, ˆ ψn lies in that
neighborhood.
To show equicontinuity, we will rely on the equicontinuity lemma which requires a sym-
metric and degenerate process. The process
i
j=i ψn(i, j) is symmetric, if the kernel is
symmetric, but it is not degenerate. By repeatedly adding and subtracting certain terms,
however, we see that
i,j ψn(i, j) equals
εiεj
(n − 1) √
Q(Xi, θ) − Q(Xj, θ)
εiεj
K
− E
hn
hn
(n − 1) √
Q(Xi, θ) − Q(Xj, θ)
K
|Dj, Xj
hn
hn
i
+ n√ hn
n−1
j=i
−
j
+
j
εi0εj0 j E hn K
n √ hnεj0
n − 1 E
Q(Xi ,θ)−Q(X j ,θ)
hn
hn
|Dj,Xj
εi0
−[Q(Xj,θ)−Q(Xj,θ0)]E hn K
hn
Q(Xi ,θ)−Q(X j ,θ)
[Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
K
|Dj, Xj
n √ hn[Q(Xj, θ) − Q(Xj, θ0)]
E
n − 1
hn
hn
hn
|Dj,Xj
(13)
(14)
[Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
K
|Dj, Xj
By the law of iterated expectations, the expression on the second line is zero. Furthermore,
the expressions in (13) and (14) are respectively second and first order symmetric degenerate
U-processes, thus, their asymptotic properties can be analyzed using Lemma 3 of Heckman,
6 an is an increasing sequence of numbers that satisfies certain conditions.
14
(15)

Ichimura, and Todd [6]. In addition, the expression in (15) can be further broken into
n √ hn
n − 1
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
E
K
|Dj, Xj
j
hn
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
−E
K
hn
(16)
+n
[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)] Q(Xi, θ) − Q(Xj, θ)
hnE
K
.
hn
hn
(17)
(16) is a degenerate order 1 process, and hence could be analyzed using the same lemma. We
study (17) at the end.
Our first step is verifying the conditions of the equicontinuity lemma for {ψn(i, j) −
2
E[ψn(i, j)|j]}. Note that since |D − Q(x, θ)| ≤ 1 for all x, θ,
(n−1) √
Q(Xi,θ)−Q(Xj,θ)
K
is
hn
hn
an envelope for this process. Moreover,
lim sup
4
E K
n→∞
2
Q(Xi, θ)− Q(Xj, θ)
= lim sup
Q(Xi, θ)− Q(Xj, θ)
.
n→∞
i
j
(n − 1) 2 hn
In addition,
1
E K
hn
2
Q(Xi, θ)− Q(Xj, θ)
=
hn
1
E K
hn
2
Q(Xi, θ)− Q(Xj, θ)
hn
1 Q(Xi, θ0)−Q(Xj, θ0)
+E
.
K
hn
2
hn
hn
hn
4
E K
hn
2
−K 2
hn
hn
Q(Xi, θ0)−Q(Xj, θ0)
Since K and Q are Lipschitz, and K is bounded, for some constant C, the first expression is
bounded by C/(anh 2 n ), which has a finite limit. Moreover, since Q(X, θ0) is assumed to have
a Lebesgue density, denoted by fQ, we have
Q(Xi, θ0)− Q(Xj, θ0)
=
1
E K
hn
2
hn
hn
K 2 (u)fQ(Q(Xj, θ0) + uhn)dufQ(Q(Xj, θ0))dQ.
But boundedness of K implies that this last expression has a finite limit as well. Thus, the first
condition of the equicontinuity lemma is satisfied for our process. To
verify the second condi-
1
tion, we note that for each δ > 0, if the kernel function is bounded, 1
(n−1) √ hn K hn
equals 1 for only finitely many n because n √ hn → ∞. On the other hand, by the dominated
convergence theorem, we have
limn→∞ E 1
= E
limn→∞
(n−1) 2
i
i
j
j 1
hn K2
1
(n−1) 2 hn K2
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
hn
15
1 1
(n−1) √ hn K
1 1
(n−1) √ hn K
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
hn
Q(Xi,θ)−Q(Xj,θ)
> δ
> δ
= 0.
> δ

Thus, the second condition of the equicontinuity lemma is also satisfied.
Next, we verify the same conditions for the process given in (14). First, we observe that
we can rewrite each term of that summation as
n √ hnεj0
E
an(n − 1)hn
an[Q(Xi, θ) − Q(Xi, θ0)]K
Q(Xi, θ) − Q(Xj, θ)
|Xj .
Since Q is Lipschitz with Lipschitz constant LQ, each of these terms is bounded by
n √ hn|εj0|LQǫθ
an(n − 1) E
1
hn
K
Q(Xi, θ) − Q(Xj, θ)
|Xj .
hn
Then using the Lipschitz continuity of K and Q, we have
n √ hn|εj0|LQǫθ
an(n − 1) E
1
hn
K
Q(Xi, θ) − Q(Xj, θ)
|Xj
hn
= n|εj0|LQǫθ
an(n − 1) √
Q(Xi, θ) − Q(Xj, θ) Q(Xi, θ0) − Q(Xj, θ0)
E K
−K
|Xj +
hn
hn
hn
n|εj0|LQǫθ
an(n − 1) √
Q(Xi, θ0) − Q(Xj, θ0)
E K
|Xj
hn
hn
nC
≤
a2 nh 3/2
n (n − 1) |εj0| + n√hn|εj0|LQǫθ an(n − 1) E
1
hn
K
Q(Xi, θ0) − Q(Xj, θ0)
|Xj ,
hn
where C = 2LKL2 gǫ2 θ . Then using a change of variables operation, we see that the second term
in the last expression equals
n √ hn|εj0|LQǫθ
an(n − 1)
hn
|K (u)| fQ Q(Xj, θ0) + uhn du.
Since K is bounded the integrals in this last expression are all finite. Therefore,
n √ hn
an(n − 1) |εj0|
1
,
anh2 C + C1
n
where C1 is the appropriate constant, is an envelope for (14). To verify the first condition of
the equicontinuity lemma, consider
n
lim sup
n→∞
j
2hn a2 n(n − 1) 2 E ε 2
j0
1
anh2 2 nhn
C + C1 ≤ lim sup
n
n→∞ a2
1
n anh2 2 C + C1 .
n
Sufficient conditions for this to be finite are that limn→∞ nhna −2
n < ∞, and limn→∞ 1
anh 2 n
< ∞,
which are implied by our assumptions. Thus the first condition of the equicontinuity lemma
is satisfied.
16

To check the second condition of the lemma, let δ > 0,
limn→∞ j
n 2 hn
a2 n (n−1)2
1
anh2 2
C+C1 E ε
n
2 j01 =limn→∞ hn
a2
1
n anh2 C+C1
n
≤limn→∞ hn
a 2 n
1
anh2 C+C1
n
2
j E
ε2 j01 2
j E
n √ hn|εj0 |
an(n−1)
n √ hn|εj0 |
an(n−1)
ε2 j01 √
n hn
an(n−1)
1
anh2
C+C1 >δ
n
1
anh2 C+C1
n
1
anh2 C+C1
n
>δ
>δ ,
where we use the fact that |εj0| ≤ 1 to write the last inequality. Note that limn→∞[1/(anh2 n )] <
∞ implies that 1/(a2 nh 3/2
n ) → 0. This in turn implies that for each δ > 0 only finitely many
terms in the last summation will be non-zero, and therefore the second condition of the lemma
is satisfied as well.
On the other hand, the same arguments show that the same conditions are satisfied for the
process
n[Q(Xj, θ) − Q(Xj, θ0)] Q(Xi, θ) − Q(Xi, θ0) Q(Xi, θ) − Q(Xj, θ)
E √ K
|Xj
n − 1
hn
hn
j
n[Q(Xj, θ) − Q(Xj, θ0)][Q(Xi, θ) − Q(Xi, θ0)]
− E
(n − 1) √
Q(Xi, θ) − Q(Xj, θ)
K
hn
hn
To verify the third condition note that since Θ is a compact subset of R l , its L 2 packing
number, D2(τ, Θ), is less than or equal to 2lτ −l . If K(·) is bounded and Lipschitz, Q(·, ·) is
Lipschitz, and n−1h −3/2
n = O(1), then D2(τ, Ψn) ≤ C · τ −l . Thus, the third condition of the
lemma is also satisfied.
We know that ψn0 ∈ Ψn. In addition, if
√ n( ˆ θ − θ0) d → N(0, Σθ),
then for any {an} sequence such that an/ √ n → 0, we will have an( ˆ θ − θ0) P → 0, and ˆ ψn will
belong to Ψn with probability approaching to 1.
j
The arguments given so far allow us to argue that
ˆψn(i, j) − E[
i,j
ˆ
ψn(i, j)|j] − ψn0(i, j) − E[ψn0 (i, j)|j]
=oP (1),
√
hnεj0
n − 1 E
(εi0 − ˆεi) 1
Q(Xi,
K
hn
ˆ θ) − Q(Xj, ˆ
θ)
|Xj
hn
=oP (1), and
n √ hn(εj0 −ˆε j )
n−1
E
(εi0 −ˆε i )
hn
i,j
K
Q(Xi , ˆ θ)−Q(X j , ˆ θ)
hn
|Xj
−E
17
n(εj0 −ε j )(ε i0 −ˆε i )
(n−1) √ hn
K
Q(Xi , ˆ θ)−Q(X j , ˆ θ)
hn
=oP (1).

The last thing we need to show is that
lim
n→∞ E
n(εj0 − ˆεj)(εi0 − ˆεi) Q(Xi,
√ K
hn
ˆ θ) − Q(Xj, ˆ
θ)
= 0.
hn
But this expression equals
an[Q(Xj, ˆ θ) − Q(Xj, θ0)]an[Q(Xi, ˆ
θ) − Q(Xi, θ0)] Q(Xi,
K
ˆ θ) − Q(Xj, ˆ
θ)
.
n √ hn
a2 E
n
Since n√ hn
a 2 n
lim
n→∞ E
hn
= O(1), the above expression will go to 0 if we can show that
an[Q(Xj, ˆ θ) − Q(Xj, θ0)]an[Q(Xi, ˆ
θ) − Q(Xi, θ0)] Q(Xi,
K
ˆ θ) − Q(Xj, ˆ
θ)
= 0.
hn
To show (18) we will first argue that
hn
hn
(18)
an|| ˆ θ − θ0|| → 0 a.s. (19)
Then using this almost sure convergence result and the Lipshitz continuity of Q and K we are
going to argue that
an[Q(Xj, ˆ θ) − Q(Xj, θ0)]an[Q(Xi, ˆ
θ) − Q(Xi, θ0)] Q(Xi,
K
ˆ θ) − Q(Xj, ˆ
θ)
hn
is almost surely bounded by an integrable function. Then the result will follow from Theorem
1.3.6 of [14].
Since our estimator ˆ θ for θ0 is defined as the minimizer of
ˆϕn(θ) := 1
n
n
[Dj − Q(Xj, θ)] 2
j=1
over a compact parameter space Θ, the first step towards showing (19) is to show that
where
1
a −1
n
sup | ˆϕn(θ) − ϕ0(θ)| → 0 a.s., (20)
θ∈Θ
ϕ0(θ) := E[(Dj − Q(Xj, θ)) 2 ].
To show (20) we rely on Theorem 37 from Pollard [10]. We apply that theorem on the following
family of functions
Φn := {(D − Q(x, θ)) 2 /M : θ ∈ Θ}
18
hn

where M = sup θ∈Θ[D−Q(x, θ)] 2 . Since Θ is a compact subset of a finite dimensional Euclidean
space, the covering number condition needed to apply the theorem is satisfied by Theorem 4.8
of Pollard [11], and applying Theorem 37 of Pollard [10] with δn = M and αn = a −1
n
the desired result (20) as long as nM2a −2
n
log n
= M 2 na −2
n
log n
gives us
→ ∞. Then, a small modification of the
consistency proof in [9] yields an|| ˆ θ − θ0|| → 0 a.s.. This in turn implies that for ǫθ > 0 and
sufficiently large n, we almost surely have
an[Q(Xj,
ˆ θ) − Q(Xj, θ0)]an[Q(Xi, ˆ
θ) − Q(Xi, θ0)] Q(Xi,
K
hn
ˆ θ) − Q(Xj, ˆ
θ)
hn
K
Q(Xi, ˆ θ) − Q(Xj, ˆ
θ)
≤ ǫ3 θLKLQ
K
Q(Xi, θ0) − Q(Xj, θ0)
.
≤ ǫ2 θ
hn
Since lim 1
anh 2 n
[14]. Thus,
hn
< ∞ and E 1
K hn
anh 2 n
Q(Xi,θ0)−Q(Xj,θ0)
hn
+ ǫ2 θ
hn
n
hn
ˆVn − Vn0 = oP (1).
hn
< ∞, (18) follows from Theorem 1.3.6 of
Combining this with Lemma 3.3a of [18] completes the proof for the first part of Theorem 4.1.
Next we argue that Σ = 2 K2 (u)du [E(D2 i |Qi0)] 2f 2 Q (Qi0)dQi0
, and Σ can be consistently
estimated by
n 2 1
K
n(n − 1)
2
Q(Xi, ˆ θ) − Q(Xj, ˆ
θ)
ˆε 2 i ˆε 2 j.
hn
i=1 j=n
We argue this in two steps again. First, note that if K is symmetric, bounded and Lipschitz,
then so is K2 . Thus, under the same conditions as before the analysis above yields that,
n
2 1
K
n(n − 1)
2
Q(Xi, ˆ θ) − Q(Xj, ˆ
θ)
ˆε 2 i ˆε 2 j −K 2
Q(Xi, θ0) − Q(Xj, θ0)
ε 2 i ε 2
j = op(1).
hn
i=1 j=i
hn
Second, the arguments in the proof of Lemma 3.3e of [18] show that
hn
i=1 j=i
hn
n 2 1
K
n(n − 1)
2
Q(Xi, θ0) − Q(Xj, θ0)
Combining these gives us the second result in Theorem 4.1.
References
hn
hn
ε 2 i ε 2 j = Σ + oP (1).
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for Average Treatment Effect,” mimeo, UC-Berkeley and Harvard University.
19
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[2] B, D. A., J. A. S (2004): “How Robust is the Evidence on the Effects of
College Quality? Evidence from Matching,” Journal of Econometrics, 121, 99—124.
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20

[15] S, B. (2004): “An Evaluation of the Swedish System of Active Labor Market
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365—375.
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mation Techniques,” Journal of Econometrics, 75(2), 263—289.
21

FIGURE 1
E D P S
Note: ––––- ˆαn p
1−p ˆ f0,n(p), - - - - - - - - ˆ f1,n(p)
DGP: D ∗ = 1 + X1 + X2 + X1X2 − ǫi, ǫ ∼ N(0, σ 2 ),
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)
Density estimated using the biweight kernel and
Silverman’s rule of thumb, see Silverman[16]

FIGURE 2
E D P S
Note: ––––- ˆαn p
1−p ˆ f0,n(p), - - - - - - - - ˆ f1,n(p)
DGP: D ∗ = 1 + X1 + X2 + −ǫi, ǫ ∼ U(−1, 1),
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)
Density estimated using the biweight kernel and
Silverman’s rule of thumb, see Silverman[16]

TABLE 1: S
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.011 0.046 0.089
n=200 0.006 0.037 0.082
n=400 0.003 0.031 0.081
n=500 0.006 0.044 0.101
n=800 0.006 0.040 0.088
n=1000
c=0.10
0.005 0.038 0.086
n=100 0.006 0.043 0.101
n=200 0.002 0.024 0.070
n=400 0.004 0.021 0.060
n=500 0.004 0.027 0.074
n=800 0.004 0.035 0.085
n=1000
c=0.15
0.005 0.022 0.070
n=100 0.005 0.027 0.079
n=200 0.003 0.011 0.057
n=400 0.003 0.008 0.049
n=500 0.004 0.012 0.061
n=800 0.008 0.024 0.070
n=1000 0.005 0.019 0.053
DGP: D ∗ = 1 + X1 + X2 − ǫi, ǫ ∼ N(0, σ 2 ),
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)

TABLE 2: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.061 0.146 0.218
n=200 0.273 0.432 0.518
n=400 0.787 0.886 0.918
n=500 0.916 0.964 0.975
n=800 0.997 0.999 0.999
n=1000
c=0.10
1.000 1.000 1.000
n=100 0.094 0.189 0.237
n=200 0.417 0.559 0.618
n=400 0.910 0.962 0.978
n=500 0.977 0.990 0.993
n=800 1.000 1.000 1.000
n=1000
c=0.15
1.000 1.000 1.000
n=100 0.101 0.185 0.241
n=200 0.459 0.597 0.664
n=400 0.948 0.976 0.984
n=500 0.984 0.993 0.996
n=800 1.000 1.000 1.000
n=1000 1.000 1.000 1.000
DGP: D ∗ = 1 + X1 + X2 + X1X2 − ǫi, ǫ ∼ N(0, σ 2 ),
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)

TABLE 3: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.358 0.479 0.537
n=200 0.818 0.874 0.896
n=400 0.991 0.993 0.994
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000
c=0.10
1.000 1.000 1.000
n=100 0.290 0.401 0.483
n=200 0.821 0.876 0.902
n=400 0.993 0.994 0.995
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000
c=0.15
1.000 1.000 1.000
n=100 0.193 0.283 0.343
n=200 0.737 0.814 0.840
n=400 0.990 0.994 0.994
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000 1.000 1.000 1.000
DGP: D ∗ = (1 + X1 + X2) 2 − ǫi, ǫ ∼ N(0, σ 2 ),
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)

TABLE 4: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.023 0.082 0.150
n=200 0.074 0.169 0.232
n=400 0.321 0.490 0.587
n=500 0.440 0.621 0.698
n=800 0.838 0.917 0.942
n=1000
c=0.10
0.929 0.977 0.986
n=100 0.037 0.076 0.134
n=200 0.131 0.242 0.303
n=400 0.457 0.620 0.691
n=500 0.642 0.776 0.848
n=800 0.935 0.969 0.980
n=1000
c=0.15
0.986 0.997 0.997
n=100 0.039 0.076 0.124
n=200 0.162 0.271 0.331
n=400 0.557 0.689 0.751
n=500 0.707 0.825 0.877
n=800 0.957 0.978 0.986
n=1000 0.993 0.998 0.998
DGP: D ∗ = 1 + X1 + X2 − ǫi, ǫ ∼ χ 2 1 ,
Estimated model: Q(X, β) = Φ(β 0 +β 1 X1+β 2 X2)

TABLE 5: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.008 0.057 0.127
n=200 0.007 0.032 0.078
n=400 0.005 0.048 0.091
n=500 0.009 0.045 0.087
n=800 0.021 0.073 0.126
n=1000
c=0.10
0.033 0.094 0.147
n=100 0.008 0.048 0.117
n=200 0.005 0.021 0.063
n=400 0.009 0.039 0.074
n=500 0.013 0.035 0.071
n=800 0.035 0.075 0.124
n=1000
c=0.15
0.048 0.114 0.169
n=100 0.005 0.038 0.107
n=200 0.007 0.021 0.057
n=400 0.009 0.034 0.066
n=500 0.014 0.035 0.074
n=800 0.036 0.081 0.135
n=1000 0.056 0.110 0.176
DGP: D ∗ = 1 + X1 + X2 − ǫi, ǫ ∼ U(−1, 1)
Estimated model: Q(X, β) = Φ(β 0 +β 1 X1+β 2 X2)

TABLE 6: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.218 0.337 0.426
n=200 0.637 0.781 0.826
n=400 0.984 0.993 0.997
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000
c=0.10
1.000 1.000 1.000
n=100 0.371 0.505 0.592
n=200 0.813 0.904 0.936
n=400 0.998 1.000 1.000
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000
c=0.15
1.000 1.000 1.000
n=100 0.449 0.584 0.662
n=200 0.891 0.952 0.970
n=400 1.000 1.000 1.000
n=500 1.000 1.000 1.000
n=800 1.000 1.000 1.000
n=1000 1.000 1.000 1.000
DGP: D ∗ = 1 + X1 + X2 − ǫi, ǫ ∼ N(0, σ 2 ),
Estimated model: Q(X, β) = β 0 +β 1X1+β 2X2

TABLE 7: P, Z[18]
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.000 0.011 0.053
n=200 0.001 0.016 0.070
n=400 0.005 0.033 0.079
n=500 0.005 0.038 0.106
n=800 0.012 0.059 0.125
n=1000
c=0.10
0.009 0.067 0.131
n=100 0.002 0.030 0.085
n=200 0.005 0.052 0.110
n=400 0.011 0.072 0.126
n=500 0.014 0.082 0.157
n=800 0.041 0.132 0.210
n=1000
c=0.15
0.063 0.175 0.294
n=100 0.002 0.046 0.106
n=200 0.008 0.061 0.115
n=400 0.035 0.109 0.176
n=500 0.043 0.141 0.227
n=800 0.107 0.271 0.370
n=1000 0.166 0.373 0.494
DGP: D ∗ = 1 + X1 + X2 + X1X2 − ǫi, ǫ ∼ N(0, σ 2 )
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2)

TABLE 8: P
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.034 0.079 0.127
n=200 0.049 0.106 0.163
n=400 0.162 0.288 0.364
n=500 0.229 0.377 0.448
n=800 0.495 0.620 0.696
n=1000
c=0.10
0.594 0.751 0.807
n=100 0.038 0.087 0.134
n=200 0.081 0.143 0.190
n=400 0.274 0.387 0.465
n=500 0.356 0.480 0.560
n=800 0.637 0.744 0.797
n=1000
c=0.15
0.768 0.857 0.895
n=100 0.042 0.078 0.128
n=200 0.102 0.170 0.217
n=400 0.320 0.448 0.508
n=500 0.415 0.550 0.619
n=800 0.700 0.789 0.837
n=1000 0.829 0.895 0.924
DGP: D ∗ = 1 + X1 + X2 + X1X2 + X3 − ǫi, ǫ ∼ N(0, σ 2 )
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2 + β3X3)

TABLE 9: P, Z[18]
c=0.05
Proportion of Rejections
1% level 5% level 10% level
n=100 0.000 0.003 0.009
n=200 0.000 0.002 0.009
n=400 0.000 0.002 0.015
n=500 0.000 0.003 0.021
n=800 0.000 0.013 0.041
n=1000
c=0.10
0.000 0.012 0.052
n=100 0.000 0.000 0.014
n=200 0.000 0.009 0.044
n=400 0.003 0.018 0.064
n=500 0.001 0.019 0.079
n=800 0.001 0.034 0.092
n=1000
c=0.15
0.001 0.041 0.114
n=100 0.001 0.011 0.041
n=200 0.002 0.021 0.071
n=400 0.002 0.036 0.096
n=500 0.003 0.037 0.089
n=800 0.003 0.055 0.110
n=1000 0.012 0.064 0.128
DGP: D ∗ = 1 + X1 + X2 + X1X2 + X3 − ǫi, ǫ ∼ N(0, σ 2 )
Estimated model: Q(X, β) = Φ(β 0 +β 1X1+β 2X2 + β3X3)