We propose an efficient moment estimator for the probit model with a continuous endogenous regressor. The estimation can be readily implemented using a standard statistical package that can estimate a non-linear system two-stage least squares (instrumental variable) estimator.Probit, Continuous endogenous regressor, Moment estimation

Daiji Kawaguchi

Hisahiro Naito

English

Research Papers in Economics

 
Hi-Stat
 
 
 
 
 
Discussion Paper Series 
 
 
No.106 
 
 
The efficient moment estimation of the probit 
model with an endogenous continuous regressor 
 
Daiji Kawaguchi and Hisahiro Naito 
 
 
June 2005 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Hitotsubashi University Research Unit 
for Statistical Analysis in Social Sciences 
A 21st-Century COE Program 
 
Institute of Economic Research   
Hitotsubashi University 
Kunitachi, Tokyo, 186-8603 Japan 
http://hi-stat.ier.hit-u.ac.jp/ The eﬃcient moment estimation of the probit
model with an endogenous continuous regressor
Daiji Kawaguchi∗
Faculty of Economics, Hitotsubashi University
Hisahiro Naito†
Graduate School of Humanities and Social Sciences
University of Tsukuba
June 29, 2005
Abstract
We propose an eﬃcient moment estimator for the probit model
with a continuous endogenous regressor. The estimation can be read-
ily implemented using a standard statistical package that can estimate
a non-linear system two-stage least squares (instrumental variable) es-
timator.
JEL Classiﬁcation: C25
Keywords: Probit, Continuous endogenous regressor, Moment estima-
tion.
∗Tel: +81-48-580-8851, Address: Naka 2-1, Kunitachi, Tokyo 186-8601, E-mail:
kawaguch@econ.hit-u.ac.jp
†Tel: +81-29-853-7432, Address: Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8573, E-mail:
naito@dpipe.tsukuba.ac.jp1 Introduction
Consider the following probit model with a continuous endogenous regressor.
y∗
1 = αy2 + z1β + u
y1 = 1 if y∗
1 > 0
y1 = 0 if y∗
1 ≤ 0
y2 = zγ + v
E(v|z) = 0
V ar(v|z) = σ2
v
u = ρv + e
e|z ∼ N(0,1)
where z1 is a (1 × k) vector of exogenous explanatory variables, y2 is a
continuous explanatory variable and z is a (1 × l) vector of instrumental
variables that includes z1 as a subset. The system of equations is assumed
to be just or over-identiﬁed (i.e., k + 1 ≤ l and the part of γ that corre-
sponds to the excluded variables includes at least a non-zero element). The
probit model with a continuous endogenous regressor is typically estimated
by the maximum likelihood method under the assumption of multivariate
normality of u and v. However, researchers often feel that this assumption
is restrictive and attempt to estimate the above model that does not impose
multivariate normality of u and v. Rivers and Vuong (1988) proposed a
two-step maximum likelihood estimation that estimates γ by ordinary least
squares (OLS) in the ﬁrst stage and introduces y2 − zˆ γ as an additional re-
gressor in the second-stage probit. This approach is widely used by applied
researchers due to its simplicity, but the standard errors are not correctly
calculated when ρ 6= 0.1 Although the procedure to correct standard errors
1Because the estimation error associated with γ does not aﬀect the asymptotic distri-
bution of the second-stage probit estimator under the null of ρ = 0, the test of endogeneity
of y2 can be implemented by testing H0 : ρ = 0.
1is well established, it involves rather cumbersome matrix algebra.2
The purpose of this note is to propose a moment estimator for the above
probit model that is eﬃcient in the class of moment estimators. A general-
ized moment method (GMM) estimator of a probit model with continuous
endogenous regressors was originally suggested by Grogger (1990), but Da-
genais (1999) and Lucchetti (2002) have shown the inconsistency of the
proposed GMM estimator. The proposed estimator in this note is a consis-
tent estimator and correct standard errors are obtained without corrections.
It can be readily implemented using a standard statistical package that
can estimate a system of non-linear equations by the instrumental variable
method.
2 Moment conditions and the optimal instruments
By substitutions, the conditional expectation of y1 on y2 and z is given as:
E(y1|y2,z) = Φ(αy2 + z1β + ρ(y2 − zγ)). (1)
From this conditional expectation, the residual function
r1(w;θ) = y1 − Φ(αy2 + z1β + ρ(y2 − zγ)), (2)
where w = [y1 y2 z] and θ = [α,β,ρ,γ], is orthogonal to any function of z
and y2. The conditional expectation of y2 on z is:
E(y2|z) = zγ. (3)
From the above conditional expectation, the second residual function
r2(w;θ) = y2 − zγ (4)
is orthogonal to any function of z.
2This is probably one reason why applied researchers continue to rely on the linear
probability model when regressors include an endogenous regressor.
2For the above residual functions, the following moment condition holds:
E[r1(w;θ0)|z,y2] = E[r2(w;θ0)|z] = 0, (5)
where θ0 is the true parameter value. Then, the natural question is what
combination of z and y2 should be used as instruments.
The optimal instrument matrix (2 × (2 + k + l)) that attains minimum
estimator variance under the moment restriction of Eq. (5) is given as:
Z∗(z,y2) ≡

Z∗
1(z,y2)
Z∗
2(z)

≡ Ω0(z,y2)−1R0(z,y2), (6)
where
Ω0(z,y2) ≡

V ar(r1|z,y2) Cov(r1,r2|z)
Cov(r1,r2|z) V ar(r2|z)

=

E[r2
1|z,y2] 0
0 E[r2
2|z]

and
R0(z,y2) ≡

E[∇θr1|z,y2]
E[∇θr2|z]

.
This setting is slightly diﬀerent from the usual setting of the optimal instru-
ment. It uses diﬀerent information set for diﬀerent equations, contrary to
the standard optimal instrument (Wooldridge (2001): pp. 439–442), but we
can show that the above instrument attains the minimum estimator vari-
ance.3 Using this optimal instrument, we can calculate the eﬃcient moment
estimator ˆ θ by solving the following equation:
n X
i=1
Z∗(z,y2)0r(w; ˆ θ) = 0, (7)
where r(w; ˆ θ) ≡ [r1(w; ˆ θ) r2(w; ˆ θ)]0. The above moment condition in our
case is
n X
i=1

 
 

φ(.)y2
Φ(.)(1−Φ(.))(y1 − Φ(.))
φ(.)
Φ(.)(1−Φ(.))z1(y1 − Φ(.))
φ(.)
Φ(.)(1−Φ(.))(y2 − zγ)(y1 − Φ(.))
z
σ2
v(y2 − zγ)

 
 

= 0, (8)
3The proof is in the appendix for refereeing purposes.
3where the arguments of φ(.) and Φ(.) are αy2 + z1β + ρ(y2 − zγ).4
The expression of the optimal instrument includes unknown parameter
values and it should be estimated by the ﬁrst-stage estimation. The detailed
estimation procedure is explained in the following section.
3 Estimation procedure
The actual procedure for the estimation involves the following steps.
1. Run OLS regression of y2 on z, keep the residual ˆ v and calculate
ˆ σ2
v = (1/n)
Pn
i=1 ˆ v2
i .
2. Run probit regression of y1 on y2, z1, and ˆ v and keep ˆ α, ˆ β, ˆ ρ. Calculate
the predicted value of the linear index as ˆ ind = ˆ αy2 + z1ˆ β + ˆ ρˆ v2.
3. Calculate the optimal instruments for the ﬁrst residual function as
˜ y2 = −[φ( ˆ ind)/{Φ( ˆ ind)(1 − Φ( ˆ ind))}]y2 , ˜ z1 = −[φ( ˆ ind)/{Φ( ˆ ind)(1 −
Φ( ˆ ind))}]z1, and ˜ v = −[φ( ˆ ind)/{Φ( ˆ ind)(1−Φ( ˆ ind))}]ˆ v. Calculate the
optimal instrument for the second residual function as ˜ z = −z/ˆ σ2
v.
4. Estimate the system of equations
y1 = Φ(αy2 + z1β + ρ(y2 − zγ)) + r1 (9)
y2 = zγ + r2 (10)
using [˜ y2 ˜ z1 ˜ v ˜ z] as instruments for a non-linear instrumental variable
(IV) estimation procedure, assuming that r1 and r2 are not correlated.5
The instruments generated do not aﬀect the asymptotic distribution of
estimators, and thus standard errors are correctly calculated. This estimator
is an eﬃcient estimator because the optimal instruments are used for the
estimation. From a practical viewpoint, all the estimation procedures used
4The derivation is in the appendix for refereeing purposes.
5This is the system 2SLS estimation in Limdep.
4above are readily available in statistical packages such as Limdep, Eviews or
TSP. Thus, applied researchers can easily obtain an eﬃcient estimator with
correct standard errors.
4 Example: Smoking during pregnancy and
family income
We consider estimation of the eﬀect of family income on a pregnant mother’s
smoking behavior. The structural model for the latent variable is:
smoke∗ = β0 + β1motheduc + β2white + αlog(faminc) + u, (11)
smoke = 1 if smoke∗ > 0 and smoke = 0 if smoke∗ ≤ 0. The vari-
ables smoke takes value one if the mother smokes during her pregnancy,
motheduc is the mother’s years of education, white is the dummy variable
that takes one if the mother is white, log(faminc) is the natural log of fam-
ily income. We are interested in α. However, family income may include
the mother’s income during pregnancy, and log(faminc) and u may show
positive correlation, given that cigarettes are normal goods. To deal with
this possible endogeneity, we instrument log(faminc) by father’s years of
education, fatheduc. The equation for log(faminc) is:
log(faminc) = γ0 + γ1motheduc + γ2white + γ3fatheduc + v. (12)
The above model is estimated by following three methods: (1) probit
estimation assuming that family income is exogenous; (2) two-step probit
estimation according to Rivers and Vuong (1988); and (3) the moment es-
timation proposed in this paper. The data set used for estimations is taken
from Wooldrige (2001), which was originally taken from Mullahy (1997).
The descriptive statistics of the analysis sample are shown in Table 1.
Column 1 of Table 2 reports the results of probit regression, assuming
the exogeneity of log(faminc). This result implies that higher family in-
5come reduces the probability of the mother smoking during her pregnancy;
however, this estimate could be upward biased.
Column 2 reports the ﬁrst-stage OLS result according to the Rivers and
Vuong (1988) procedure. The coeﬃcient for father’s education is statistically
signiﬁcant and this implies that father’s education serves as an instrument
for family income. Column 3 reports the result for second-stage probit,
which includes the residual of ﬁrst-stage OLS as an additional explanatory
variable. The estimated ρ is 0.61, with standard error of 0.37, which is
marginally signiﬁcant. This positive ˆ ρ implies that u and v are positively
correlated and, after considering this correlation, the estimated coeﬃcient
for log(faminc) is −0.76, which is smaller than the coeﬃcient estimated by
probit. The standard errors reported in column 4 of Table 2 are not adjusted
for the two-step estimation.
Columns 4 and 5 show results for the moment estimation of Eqs. (9) and
(10) using [˜ y2 ˜ z1 ˜ v ˜ z] as instrumental variables. The estimated coeﬃcients
are similar to those obtained using the Rivers and Vuong (1988) procedure,
but all the standard errors are lower, probably due to the eﬃciency gain.
The estimated ρ is closer to zero and is statistically insigniﬁcant. Thus, we
cannot reject the null hypothesis that log(faminc) is exogenous.
All the above estimations were calculated using Limdep 8.0 and can be
similarly implemented using any statistical package that allows non-linear
system 2SLS (IV) estimation.
References
[1] Dagenais, M.G., 1999. Inconsistency of a proposed nonlinear instrumen-
tal variables estimator for probit and logit models with endogenous re-
gressors. Economics Letters, 63(1), pp. 19–21.
[2] Grogger, J., 1990. A simple exogeneity test for probit, logit, and Poisson
regression models. Economics Letters, 33(4), pp. 329–332.
6[3] Lucchetti, R., 2002. Inconsistency of naive GMM estimation for QR
models with endogenous regressors. Economics Letters, 75(2), pp. 197–
185.
[4] Mullahy, J., 1997. Instrumental-Variable Estimation of Count Data Mod-
els: Applications to Models of Cigarette Smoking Behavior. Review of
Economics and Statistics, 79 (4), pp. 596–593.
[5] Rivers, D. and Vuong, Q.H., 1988. Limited information estimators and
exogeneity tests for simultaneous probit models. Journal of Economet-
rics, 39(3), pp. 347–366.
[6] Wooldridge, J., 2001. Econometric Analysis of Cross Section and Panel
Data. MIT Press.
A Optimality of the proposed instrument
This appendix is attached for the purpose of refereeing, not for publication.
The proposed optimal instrument in (6) is not standard since diﬀerent
information sets are used to calculate the conditional expectation for dif-
ferent residual functions. A use of diﬀerent information sets for diﬀerent
residual functions is inevitable in the presence of endogenous regressors in
a nonlinear model. This note shows that our proposed instrument archives
the eﬃciency bound within a class of GMM estimator despite the use of
diﬀerent information sets for diﬀerent residual functions.
Let Z1(y2i,z) and Z2(zi) be arbitrary 1×L, where L ≥ (2+l+k), instru-
ment vectors and let Z0
i be a L×2 matrix where Z0
i = (Z1(y2i,zi)0 Z2(zi)0).
Also deﬁne the 2×1 residual vector ri as r(y2i,zi;θ) ≡ (r1(y2i,zi;θ) r2(zi;θ)).
Let m(y,z,θ) = Z0
ir(y2i,zi;θ). Note that the covariance matrix of GMM es-
timator is (G0ΞG)−1G0ΞΛΞG(G0ΞG)−1 where Ξ is the probability limit of
the weighting matrix; G = E[∂m
∂θ ]; Λ = E[mm0]. Let s∗ be (2 + l + k) × 1
matrix where s∗ = Z∗0r and Z∗0 is deﬁned in (6). Let s be (2 + l + k) × 1
7matrix where s = G0ΞZ0r. If we can show that E[ss∗0] = G0ΞG, then it also
proves that Z∗0 attains the minimum variance with a a class of GMM from
the Lemma 14.1 of Woodridge (2001). By using the deﬁnition of s∗ and s,
E[ss∗0] can be calculated as follows:
E[ss∗0] = E[E[G0ΞZ0rr0Z∗| y2,z]]
= E[G0ΞZ0E[rr0|y2,z]Z∗]
= G0ΞE[Z0E[rr0|y2,z]Ω−1Ro]
From the the deﬁnition of Ω−1 and Ro, the above equations becomes
= G0ΞE[Z0

E[r2
1|y2,z] 0
0 E[r2
2|y2,z]

·

E[r2
1|y2,z] 0
0 E[r2
2|z]
−1 
E[∇θr1|z,y2]
E[∇θr2|z]

]
= G0ΞE[Z0

E[∇θr1|z,y2]
E[r2
2|y2,z]E[r2
2|z]−1E[∇θr2|z]

]
= G0ΞE[Z0

E[∇θr1|z,y2]
E[r2
2|z]E[r2
2|z]−1E[∇θr2|z]

]
= G0ΞG
Thus, E[ss∗0] = G0ΞG. From the Lemma 14.1 of Woodridge (2001), Z∗
archives a minimum variance among instruments deﬁned by Z0
i = (Z1(y2i,zi)0
Z2(zi)0).
B Derivation of the optimal instrument
This appendix includes the derivation of the optimal instruments for refer-
eeing purposes. This appendix is not for publication.
Ω0(z,y2) ≡


E[(y1 − Φ(.))2|z,y2] E[(y1 − Φ(.))(y2 − zγ)|z]
E[(y1 − Φ(.))(y2 − zγ)|z] E[(y2 − zγ)2|z]


=

Φ(.)(1 − Φ(.)) 0
0 σ2
v

,
8because E[(y1 − Φ(.))(y2 − zγ)|z] = E[E[(y1 − Φ(.))(y2 − zγ)|z,y2]|z] =
E[E[(y1 − Φ(.))|z,y2](y2 − zγ)|z] = 0.
R0(z,y2) ≡

E[∂r1/∂α ∂r1/∂β ∂r1/∂ρ ∂r1/∂γ|z]
E[∂r2/∂α ∂r2/∂β ∂r2/∂ρ ∂r2/∂γ|z]

=

−φ(.)y2 −φ(.)z1 −φ(.)(y2 − zγ) φ(.)ρz
0 0 0 −z

.
Thus, the optimal instrument is:
Z∗(w) = Ω0(z)−1R0(z)
=
 
−
φ(.)y2
Φ(.)(1−Φ(.)) −
φ(.)z1
Φ(.)(1−Φ(.)) −
φ(.)(y2−zγ)
Φ(.)(1−Φ(.))
φ(.)ρz
Φ(.)(1−Φ(.))
0 0 0 −z/σ2
v
!
.
The ﬁrst, third and fourth columns in the ﬁrst row are linearly dependent
and we place 0 in the fourth column.
9 10
Table 1: Descriptive statistics 
Variable Mean  Standard 
Deviation 
Minimum Maximum 
Smoking during the pregnancy  0.16  -  0  1 
Family income (1988, $1000)  32.22  17.96  0.5  65 
Log (Family income)  3.28  0.72  -0.69  4.17 
Mother’s years of education    13.13  2.42  2  18 
Father’s years of education  13.19  2.74  1  18 
White 0.84  -  0  1 
Note: N=1191. 
Table 2: The effect of family income on mother’s smoking during the pregnancy 
  (1) (2) (3) (4) (5) 
 Probit  Rivers-Voung  (1988) 
Two Step Estimation 
Efficient Moment Estimation 
Dependent 
Variable 
Smoking Log(family 
income) 
Smoking Log(family 
income) 
Smoking 
Log (family 
income) 
-0.17 
(0.07) 
- -0.76 
(0.37) 
- -0.46 
(0.24) 
Mother’s 
education 
-0.15 
(0.23) 
0.07 
(0.01) 
-0.08 
(0.05) 
0.07 
(0.004) 
-0.08 
(0.03) 
White 0.23 
(0.14) 
0.35 
(0.05) 
0.46 
(0.20) 
0.35 
(0.02) 
0.33 
(0.14) 
Father’s 
education 
- 0.06 
(0.01) 
- 0.06 
(0.003) 
- 
Constant 1.13 
(0.30) 
1.24 
(0.11) 
1.99 
(0.60) 
1.24 
(0.04) 
1.18 
(0.40) 
ρ -  -  0.61 
(0.37) 
- 0.34 
(0.25) 
Note: N=1191. 

A simple exogeneity test for probit, logit, and Poisson regression models.

Inconsistency of a proposed nonlinear instrumental variables estimator for probit and logit models with endogenous regressors.

Inconsistency of naive GMM estimation for QR models with endogenous regressors.

Instrumental-Variable Estimation of Count Data Models: Applications to Models of Cigarette Smoking Behavior.

Let m(y,z,θ) = Z0 ir(y2i,zi;θ). Note that the covariance matrix of GMM estimator is (G0ΞG)−1G0ΞΛΞG(G0ΞG)−1 where Ξ is the probability limit of the weighting matrix; G = E[∂m ∂θ ]; Λ =

Limited information estimators and exogeneity tests for simultaneous probit models.

Z∗ archives a minimum variance among instruments deﬁned by Z0 i = (Z1(y2i,zi)0 Z2(zi)0). B Derivation of the optimal instrument This appendix includes the derivation of the optimal instruments for refereeing purposes. This appendix is not for publication.

The efficient moment estimation of the probit model with an endogenous continuous regressor

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The efficient moment estimation of the probit model with an endogenous continuous regressor

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