One of the most important problem of misspeciﬁcation in the probit model is the correlation between regressors and error term. To deal with this problem, some commercial software gives a solution such as Stata. For the famous R language the ivprobit gives the users the way to estimate the instrumental probit model

Zaghdoudi, Taha

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
ivporbit:An R package to estimate the
probit model with continuous
endogenous regressors
Taha Zaghdoudi
University of Jendouba, Faculty of Law Economics and Management
of Jendouba
25 September 2014
Online at https://mpra.ub.uni-muenchen.de/72383/
MPRA Paper No. 72383, posted 6 July 2016 06:59 UTC
ivporbit:An R package to estimate the
probit model with continuous endogenous
regressors
Taha Zaghdoudi∗
25-10-2014
Abstract
One of the most important problem of misspecification in the probit model is
the correlation between regressors and error term. To deal with this problem, some
commercial software gives a solution such as Stata. For the famous R language the
ivprobit gives the users the way to estimate the instrumental probit model.
1 Introduction
Many econometrics study are focused in various kinds of misspecification in the
limited-dependent variable models such as correlation between regressors and error
term which produce inconsistent results. To avoid this problem we apply the instru-
mental variable method in which we use the correlated variables as instruments. A
two-stage method are used by some authors Blundell and Powell [2004] to fit the
probit model but it produce non efficient result.
Otherwise, Newey [1987] expose an efficient way to estimate limited-dependent
variable model by using the Amemiya’s Generalized Least Squares estimators Amemiya
[1978]. The idea is to include in the two-stage model a continuous endogenous re-
gressor. This method is used in Stata when the MLE fail to estimate the model.
For the ivprobit we apply the same method as used in Stata.
In the following sections we expose the Newey method to estimate the instru-
mental probit model then we give a simple example how to use the ivprobit Package.
2 Estimation method
Generally the model is:
y∗1i = y2iβ + x1iγ + µi
y2i = x1iΠ1 + x2iΠ2 + υi.
∗University of Jendouba, Faculty of Law, Economics and Management of Jendouba
1
Where i = 1, .., N , y2i is a 1 × p vector of endogenous variables, x1i is a 11 vector
of exogenous variable, x2i is a 1 × k2 vector of additional instruments, and the
equation for y2i is written in reduced form. By assumption, (υi, υi)˜N(0,Σ), where
σ11 is normalized to one to identify the model. β and γ are vectors of structural
parameters, and Π1 and Π2 are matrices of reduced-form parameters. This is a
recursive model: y2i appears in the equation for y
∗
1i, but y
∗
1i does not appear in the
equation for y2i. We do not observe y
∗
1i; instead, we observe:
y1i = {0 y
∗
1i≥0
1 y∗1i≤1
The order condition for identification of the structural parameters requires that
k2 ≥. Presumably, is not block diagonal between µi and υi; otherwise, y2i would
not be endogenous.
To obtain the two-step estimates,Newey [1987] use minimum chi-squared esti-
mator and the model is:
y∗1i=ziδ+µi y2i=xiΠ + υi (1a)
(1b)
where zi=(y2i,x1i),xi=(x1i, x2i), δ=(β
′,δ′)’, andΠ=(Π′1,Π′2)’.
The reduced-form equation for y∗1i is:
y∗1i = (xiΠ + υi)β + x1iγ + µi = xiα+ υiβ + µi = xiα+ νi
where
νi = υiβ + µi. Because µi and υi are jointly normal,i is also normal. Note that:
α = {Π1Π2}β + {I0}γ = D(Π)δ
where D = (Π) = (Π, I1) and I1 is defined such that xiIi. Letting ẑi =
(xiΠ̂, x1i), ẑi = xiD(Π̂)δ, where D(Π̂) = D(Π̂, I1). Thus one estimator of α is
D(Π̂)δ; donate this estimator by D̂δ.
α could also be estimated directly as the solution to:
maxα,λ =
∑N
i=1 l(y1i, xiα+ υ̂iλ)
where l(.) is the log likelihood for probit. Denote this estimator by αˆ. The
inclusion of the term υ̂iλ follows because the multivariate normality of (µi, υi) im-
plies that, conditional on y2i, the expected value of µi is nonzero. Because υi is
unobservable, the least-squares residuals from fitting (1b) are used.
Amemiya [1978] shows that the estimator of δ:
max
δ
(α˜− Dˆδ)’Ωˆ−1(α˜− Dˆδ) (2)
whereΩˆ is a consistent estimator of the covariance of
√
N(α˜− Dˆδ), is asymptot-
ically efficient relative to all other estimators that minimize the distance between α˜
ans D(Πˆ) δ. Thus an efficient estimator of δ is :
δˆ=(Dˆ′Ωˆ−1Dˆ) −1(Dˆ′Ωˆ−1α˜) (3)
and
Var(δˆ)=(Dˆ′Ωˆ−1Dˆ) −1 (4)
2
To implement this estimator, we need Ωˆ−1. Consider the two-step maximum
likelihood estimator that results from first fitting (1b) by OLS and computing the
residuals υˆi = y2i − xi Πˆ. The estimator is then obtained by solving:
max
δ,λ
=
N∑
i=1
l(y1i, ziδ + υˆiλ)
This is the two-step instrumental variables (2SIV) estimator proposed by Rivers
and Vuong [1988], and its role will become apparent shortly.
From Proposition 5 of [Newey, 1987],
√
N(α˜-Dˆδ) d−→N(0,Ω), where
Ω = J−1αα + (λ− β)′Σ22(λ− β)Q−1
and Σ22 = E{υ′iυi}.J−1αα is simply the covariance matrix of α˜, ignoring that Πˆ is
an estimated parameter matrix. Moreover, Newey shows that the covariance matrix
from an OLS regression of yi2(λˆ − βˆ) on xi is a consistent estimator of the second
term.λˆ can be obtained from solving (2), and the 2SIV estimator yields a consistent
estimate, βˆ.
The basic algorithm proceeds as follows:
1. Each of the endogenous right-hand-side variables is regressed on all the exoge-
nous variables, and the fitted values and residuals are calculated. The matrix
Dˆ = D(Πˆ) is assembled from the estimated coefficients.
2. probit is used to solve (2) and obtain α˜ and λ˜. The portion of the covariance
matrix corresponding to α, J−1αα , is also saved.
3. The 2SIV estimator is evaluated, and the parameters βˆ corresponding to y2i
are collected.
4. yi2(λˆ-βˆ) is regressed on xi. The covariance matrix of the parameters from this
regression is added to J−1αα , yielding Ωˆ.
5. Evaluating (3) and (4) yields the estimates δˆ and Var(δˆ).
6. A Wald test of the null hypothesis H 0 : λ=0, using the 2SIV estimates, serves
as our test of exogeneity.
3 Example
To dwonload the package you can just type install.packages(”ivprobit”) in R or
Rstudio.
# load the package
library(ivprobit)
The data we use in this example represents the Foreign-Exchange Derivatives
Use By Large U.S. Bank Holding Companies from 1996 to 2000.
# load the database
dat<-system.file("data","eco.RData",package="ivprobit")
load(dat)
pro<-library("ivprobit", lib.loc="C:/Program Files/R/R-3.0.3/library")
#load data
dat<-system.file("data","eco.RData",package="ivprobit")
load(dat)
3
The function is ivprobit(y x,y2 z, data):
• y: the dichotomous l.h.s vector
• x: the r.h.s. exogenous variables matrix.
• y2: the r.h.s. endogenous variables vector or matrix
• z: the complete set of instruments (a matrix)
• data: the dataframe
In this example we have d2 the dichotomus l.h.s vector, the r.h.s exogenous variables
are (ltass,roe and div), the r.h.s. endogenous variables matrix is (eqrat,bonus) and
the instrumental variables are (ltass,roe,div,gap,cfa).
# fit the instrumental probit model
pro<-ivprobit(d2~ltass+roe+div,cbind(eqrat,bonus)~ltass+roe+div+gap+cfa,mydata)
# the results summary
summary(pro)
## Coef S.E. t-stat p-val
## (Intercept) -1.6862e+01 9.7317e+00 -1.7327 0.08355 .
## X1ltass 9.4760e-01 6.2070e-01 1.5267 0.12724
## X1roe 6.6492e-02 1.1312e-01 0.5878 0.55684
## X1div 2.0378e-06 4.2745e-06 0.4767 0.63368
## yhateqrat 9.4371e+00 1.2937e+01 0.7295 0.46593
## yhatbonus -1.4173e-06 2.8163e-06 -0.5032 0.61493
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
4 Conclustion
In this paper we introduced the ivprobit to estimate the parameters of an dichoto-
mous choice model that contains endogenous regressors. The routine is simple and
yields the same results as the two-step option in the commercially available Stata
software.
References
Takeshi Amemiya. The estimation of a simultaneous equation generalized probit
model. Econometrica: Journal of the Econometric Society, 46:1193–1205, 1978.
Richard W Blundell and James L Powell. Endogeneity in semiparametric binary
response models. The Review of Economic Studies, 71(3):655–679, 2004.
Whitney K Newey. Efficient estimation of limited dependent variable models with
endogenous explanatory variables. Journal of Econometrics, 36(3):231–250, 1987.
Douglas Rivers and Quang H Vuong. Limited information estimators and exogeneity
tests for simultaneous probit models. Journal of Econometrics, 39(3):347–366,
1988.
4


ivporbit:An R package to estimate the probit model with continuous endogenous regressors

https://mpra.ub.uni-muenchen.de/72383/1/MPRA_paper_72383.pdf

ivporbit:An R package to estimate the probit model with continuous endogenous regressors

Abstract

Similar works

Full text

Available Versions

Munich Personal RePEc Archive