We derive the variance of the Hirano, Imbens and Ridder (Econometrica 66, 315--31, 2003) average treatment effects estimator when the true propensity score is known. This variance is used in the derivation of the variance of a similar two-step estimator, where a M-estimator is used in the first step to estimate the propensity score.Average treatment effects; efficiency bound; two-step estimator;

Gabriel Montes-Rojas

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
A note on the variance of average treatment effects estimators 
  
 
 
Gabriel Montes-Rojas  
City University London 
Abstract 
We derive the variance of the Hirano, Imbens and Ridder (Econometrica 66, 315--31, 2003) average treatment effects 
estimator when the true propensity score is known. This variance is used in the derivation of the variance of a similar 
two-step estimator, where a M-estimator is used in the first step to estimate the propensity score.
I am in debt to Anil Bera and to an anonymous referee for valuable comments and suggestions. 
Citation: Gabriel Montes-Rojas, (2009) ''A note on the variance of average treatment effects estimators'', Economics Bulletin, Vol. 29 no.4 
pp. 2937-2943. 
Submitted: Sep 04 2009.   Published: November 24, 2009. 
 
     1 Introduction
Following the work of Hahn (1998) that derived the asymptotic semipara-
metric eciency bound for average treatment eects (ATE) estimators, Hi-
rano, Imbens and Ridder (2003) proved that weighting by the inverse of a
nonparametric series estimator of the propensity score, rather than the true
propensity score, lead to an ecient estimator of the ATE. Their estimator
is rather simple when compared to alternative estimators in the literature,
such as propensity score matching or treatment regression models. However,
the fact that a particular nonparametric estimator of the propensity score
is needed, which is rather computationally intensive, determine that their
estimator has not been extensively used. For instance Wooldridge (2002,
p.617) states that \as a practical matter, series estimation [of the propen-
sity score] is not ideal, because for a binary response, it is identical to a
linear probability model in functions of x. Plus, it is dicult to estimate the
asymptotic variance of the resulting estimators." Chen, Hong and Tarozzi
(2008) prove that dierent combinations of nonparametric and parametric
estimates of the propensity score have to be specically derived to achieve
the eciency bounds. In practice, the propensity score is estimated using
parametric models (such as logit or probit), and this estimate is used to
construct other consistent estimators of ATE. Moreover, its variance is gen-
erally computed using bootstrap methods because, in many cases, an explicit
derivation of the small sample or even asymptotic variance is dicult.
The goal of this paper is to study the Hirano et al. (2003) estimator
when the propensity score is estimated by an M-estimator (e.g. maximum
likelihood when the propensity score is correctly specifed) and to provide
an explicit expression for the asymptotic variance of this estimator using the
delta-method. This is a particular case of the Chen et al. (2008) inverse
probability weighting based GMM estimators.
As an intermediate step we also derive the variance of the Hirano et
al. (2003) estimator when the propensity score is known. Of course, in
many empirical situations, the propensity score is not known. However,
this has pedagogical importance, provided that the statement above seems
paradoxical (Hahn 1998 proves that the propensity score is an ancilliary
statistic to the ATE estimation). In fact, this answers the question: If the
propensity score were known, what is the cost of using it in order to avoid
its nonparametric series estimator?
Although the derivation developed in this paper is quite simple, its method
1can be applied to more complicated treatment eects estimators. For in-
stance, similar steps can be used to obtain the asymptotic variance of the
Firpo (2007) quantile treatment eects (QTE) estimator, where the condi-
tional mean and variances are replaced by those of the inuence function
of the quantile functions, and the propensity score is estimated by a M-
estimator instead of the nonparametric series estimator.
This paper is organized as follows. Section 2 discusses the assumptions
used in the treatment eects literature. Section 3 derives the asymptotic
variance of the Hirano et al. (2003) estimator when the true propensity score
is known. Section 4 uses this variance to derive the asymptotic variance of an
ATE estimator with a parametric estimate of the propensity score. Section
5 concludes.
2 Notation, denitions and assumptions
We follow the standard notation in Imbens (2004). Consider N individuals
indexed by i = 1;2;:::;N who may receive a \treatment", indicated by the
binary variable Wi = 0;1. Each individual has a pair of potential outcomes
(Y0i;Y1i) that corresponds to the outcome with and without the treatment
eect respectively. The fundamental problem, of course, is the inability to
observe at the same time the same individual both with and without treat-
ment eects. That is, we only observe Yi = Wi  Y1i + (1   Wi)  Y0i and
a set of exogenous variables Xi. Moreover dene the propensity score as
p(X) = P[W = 1jX] and dene j(X) = E[YjjX] and 2
j(X) = V AR[YjjX]
for j = 0;1.
We are interested in estimating the ATE of the W-treatment, dened as
 = E[Y1 Y0] and following Hirano et al. (2003) we propose the estimator:
^  =
1
N
N X
i=1
Wi
p(Xi)
Yi  
1
N
N X
i=1
1   Wi
1   p(Xi)
Yi: (1)
The standard assumptions in the treatment eects literature are
Assumption 1. (Y0;Y1)?WjX
Assumption 2. For c 2 (0;1), c < p(X) < 1   c
Assumption 3. E[Y 2
0 ] < 1, E[Y 2
1 ] < 1
23 ATE estimator with known true propensity
score
Under these assumptions, the unbiasedness of this estimator can be easily
proved following the results of Hirano et al. (2003), and therefore we omit
that proof. We are more interested in showing that this estimator does not
achieve the semiparametric eciency bound (dened in Hahn 1998) and we
explicitly quantify the loss of eciency.
First note that by the i.i.d. set-up,
lim
N!1
V AR[
p
N(^    )] = V AR

WY
p(X)
 
(1   W)Y
1   p(X)

:
Expressing the variance as the sum of the variance of the conditional
expectation and the expectation of the conditional variance, and using the
assumptions above, it becomes
E
"
2
1(X)
p(X)
+
2
0(X)
1   p(X)
+ (1   p(X))p(X)

1(X)
p(X)
+
0(X)
1   p(X)
2#
+V AR[1(X)   0(X)]:
Therefore, the asymptotic variance can be written as
lim
N!1
V AR
hp
N(^    )
i
= B + L; (2)
where
B = E[
2
1(X)
p(X)
+
2
0(X)
1   p(X)
+ (
(X)   
)
2]
is the semiparametric eciency bound in Hahn (1998), (X) = 1(X)  
0(X), and L is the loss of eciency for using the true propensity score and
not a nonparametric series estimator,
L = E
2
4
 s
1   p(X)
p(X)
1(X) +
s
p(X)
1   p(X)
0(X)
!23
5;
3which is nonnegative. Note that even if all the variables are constants in X,
i.e. p(X) = p and 1(X) = 0(X) = , L = 2

2 +
(1 p)2+p2
p(1 p)

 0, which
has a minimum for p = 1=2 with L = 42. Therefore, the incurred loss is
potentially big if  is.
4 Two-step ATE estimator
Following Hirano et al. (2003), ^  can be seen as a M-estimator with estimat-
ing equation
 (Y;W;X;p(X);) =
WY
p(X)
 
(1   W)Y
1   p(X)
  ; (3)
and therefore,
lim
N!1
V AR[
p
N(^    
)] = E[ (Y;W;X;p(X);
)
2] (4)
Of course, after some algebra, the variance in eq. (4) is the same as that in
eq. (2).
Now assume that p(X) = (X;) and that  is unknown to the econo-
metrician.  is used to denote a distribution function (not necessarily the
normal c.d.f.). Moreover, assume that a
p
N-consistent estimator of  is ^ ,
which satisfy
p
N(^    
) = N
 1=2
N X
i=1
s(Wi;Xi;
) + op(1):
Any M-estimator of  can be framed in these terms under standard regularity
conditions, including, of course, the maximum likelihood estimator.
Moreover, ^  satises
p
N(^    
) = N
 1=2
N X
i=1
(Yi;Wi;Xi;
;
) + op(1);
4where  (:;p(:);) = (:;;). Therefore, the asymptotic variance of ^ ^ , the
two-step estimator of  where ^  is used to construct the propensity score
instead of , can be obtained using the delta-method (see Wooldridge 2002,
ch.12),
lim
N!1
V AR[
p
N(^ ^  
)] = V AR[(Y;W;X;
;
)+H
(W;X;
;
)s(W;X;
)];
where
H
(
;
) = E[r(Y;W;X;
;
)] =  E

1(X)
(X;)
+
0(X)
1   (X;)

r(X;
)

:
Finally, note that
V AR[
p
N(^ ^    )]
= E[(Y;W;X;;)2] + H(;)E[s(W;X;)s(W;X;)0]H(;)0
+2  COV [(Y;W;X;;);H(;)s(W;X;)]
= V AR[
p
N(^    )] + H(;)V AR[
p
N(^    )]H(;)0
+2  COV [(Y;W;X;;);H(;)s(W;X;)]:
Then, the asymptotic variance of the two-step estimator can be re-written
as
V AR[
p
N(^ ^    
)] = B + L + G; (5)
where B and L were dened in Section 3 and G = H(;)V AR[
p
N(^   
)]H(;)0 + 2  COV [(Y;W;X;;);H(;)s(W;X;)]. There-
fore L + G can be seen is the eciency loss arising from using a two-step
estimator of the propensity score instead of the series estimator.
In practice, a consistent estimator of this variance can be obtained by the
OPG method:
Est:V AR[
p
N(^ ^    
)] =
1
N
N X
i=1

(Yi;Wi;Xi; ^ ; ^ ^ ) + ^ H(^ ; ^ ^ )s(Wi;Xi; ^ )
2
;
where
^ H(^ ; ^ ^ ) =  
1
N
N X
i=1

WiYi
2(Xi; ^ )
+
(1   Wi)Yi
(1   (Xi; ^ ))2

r(Xi; ^ )

:
55 Conclusion
This note derives the variance of a simple ATE estimator using the known
true propensity score. Moreover, it also derives its variance if a M-estimator
is used to estimate the propensity score. This ATE estimator, where the
propensity score is estimated by probit or logit models, is widely used in the
empirical literature on treatment eects. However, despite its simple deriva-
tion, no explicit formulation of the asymptotic variance was given elsewhere
in the literature.
References
Chen, X., H. Hong and A. Tarozzi (2008) \Semiparametric eciency in GMM
models with auxiliary data" Annals of Statistics 36, 808{43.
Firpo, S. (2007) \Ecient semiparametric estimation of quantile treatment
eects" Econometrica 75, 259-76.
Hahn, J. (1998) \On the role of the propensity score in ecient semipara-
metric estimation of average treatment eects" Econometrica 66, 315-31.
Hirano, K., G.W. Imbens and G. Ridder (2003) \Ecient estimation of av-
erage treatment eects using the estimated propensity score" Econometrica
71, 1161-90.
Imbens, G.W. (2004) \Nonparametric estimation of average treatment eects
under exogeneity: a review" Review of Economics and Statistics 86, 4-29.
Wooldridge, J.M. (2002) Econometric Analysis of Cross Section and Panel
Data, The MIT Press: Cambridge, Massachusetts.
6

Ecient estimation of average treatment eects using the estimated propensity score&quot;

Ecient semiparametric estimation of quantile treatment eects&quot;

Econometric Analysis of Cross Section and Panel Data,

Nonparametric estimation of average treatment eects under exogeneity: a review&quot;

On the role of the propensity score in ecient semiparametric estimation of average treatment eects&quot;

Semiparametric eciency in GMM models with auxiliary data&quot;

A note on the variance of average treatment effects estimators

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A note on the variance of average treatment effects estimators

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