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An Instrumental Variables Probit Estimator Using Gretl View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Papers in Economics An Instrumental Variables Probit Estimator using gretl Lee C. Adkins Professor of Economics, Oklahoma State University, Stillwater, OK 74078 [email protected] Abstract. The most widely used commercial software to estimate endogenous probit models offers two choices: a computationally simple generalized least squares estimator and a maximum likelihood estimator. Adkins [1, -. com%ares these estimators to several others in a Monte Carlo study and finds that the 1LS estimator performs reasonably well in some circumstances. In this paper the small sample properties of the various estimators are reviewed and a simple routine us- ing the gretl software is given that yields identical results to those produced by Stata 10.1. The paper includes an example estimated using data on bank holding companies. 1 Introduction 4atchew and Griliches +,5. analyze the effects of various kinds of misspecifi- cation on the probit model. Among the problems explored was that of errors- in-variables. In linear re$ression, a regressor measured with error causes least squares to be inconsistent and 4atchew and Griliches find similar results for probit. Rivers and 7#ong [14] and Smith and 8lundell +,9. suggest two-stage estimators for probit and tobit, respectively. The strategy is to model a con- tinuous endogenous regressor as a linear function of the exogenous regressors and some instruments. Predicted values from this regression are then used in the second stage probit or tobit. These two-step methods are not efficient, &ut are consistent. Consistent estimation of the standard errors is not specifically considered and these estimators are used mainly to test for endogeneity of the regressors–not to establish their statistical significance. Newey +,-. looked at the more generic problem of endogeneity in limited dependent variable models (which include probit and tobit). He proposed what is sometimes called Amemiya’s Generalized Least Squares (AGLS) estimator as a way to efficiently estimate the parameters of probit or tobit when they include a continuous endogenous regressor. This has become one of the standard ways to estimate these models and is an option (twostep= in Stata 10.0 when the MLE is difficult to obtain. The main benefit of using this estimator is that it yields a consistent estimator of the variance-covariance matrix that and can easily &e used for subsequent hypothesis tests about the parameters. 9 Lee C. Adkins Adkins +,. compares the AGLS estimator to several alternatives, which in3 clude a maximum likelihood estimator. The A1LS estimator is simple to com3 pute and yields signi0cance tests that are close in size to the nominal level when samples are not very lar$e <e.g., n=200). The other plug-in estimators are con- sistent for the parameters &ut not the standard errors, making it unlikely that they will perform satisfactorily in hypothesis testing. "he latter problem is taken #% by Adkins +A. who uses a Murphy and "opel +,,. correction to obtain consistent standard errors with some success. Others have explored limited dependent variable models that have discrete endogenous regressors. ;icoletti and Peracchi +,A. look at binary response mod- els with sample selection, Kan and Kao +, . consider a simulation approach to modeling discrete endogenous regressors, and Arendt and Holm +B. extends Nicoletti and Peracchi +,A. to include multiple endogenous discrete variables. Iwata +5. uses a very simple approach to dealing with errors-in-variables for probit and tobit. He shows that simple recentering and rescaling of the observed dependent variable may restore consistency of the standard 27 estimator if the true dependent variable and the 27?s are jointly normally distributed. His /onte Carlo simulation shows evidence that the joint normality may not be necessary to obtain improved results. However, the results for tobit were quite a bit &etter than those for probit. >e compares this estimator to a linear instrumental vari- able estimator that uses a consistent estimator of standard errors. This estimator is considered by Adkins +,. in his comparison. Blundell and Powell +9. develop and implement semiparametric methods for estimating binary dependent variable models that contain continuous en- dogenous regressors. Their paper “extends existin$ results on semiparametric estimation in single-inde* binary response models to the case of endogenous regressors. It develops an approach to account for endogeneity in triangular and fully simultaneous binary response models." 8lundell and Powell [6], p. 9BB In the following sections a linear model with continuous endogenous re3 gressors and its estimators are considered. Fith respect to models having a dichotomous dependent variable, a relatively sim%le generalized least squares estimator discussed in Newey +,-. is presented and an algorithm for its comp#3 tation in gretl is given. "o give the reader an idea of how this estimator compares to alternatives, including a maximum likelihood estimator (mle), some results from a simulation study conducted by Adkins +1, -. are summarized. "he re- sults from the gretl routine and from Stata , are compared using an example from the banking literature. An Instrumental 7ariables Probit Estimator using $retl 9, 2 Linear Model and Estimators Gollowing the notation in Newey [12], consider a linear statistical model in ∗ which the continuous dependent variable will be calledy t &ut it is not directly observed. Instead, we observey t in only one of two possi&le states. So, ∗ yt =Y tβ+X 1tγ+u t =Z tδ+u t� t=1� . � N <,= � � � th whereZ t = [Yt� X1t]�δ = [β � γ ],Y t is thet observation on an endogenous explanatory variable,X 1t is a 1xs vector of exogenous explanatory variables, andδ is theqx1 vector of regression para meters. The endogenous variable is related to a1Xk vector of instrumental va riables Xt by the equation Yt =X 1tΠ1 +X 2tΠ2 +V t =X tΠ+V t <-= whereV t is a disturbance. Thek−s variables inX 2t are additional exogenous explanatory variables. Equation <-= is the reduced form equation for the endoge- nous explanatory variable. Fithout loss of generality only one endogenous e*3 planatory variable is considered below. See Newey +,-. for notation extendin$ this to additional endogenous variables. ∗ When the continuous variabley t is observed, then one could use either least squares or instrumental variable estimator to estimateδ. Collecting then ob- ∗ th ∗ servations into matricesy ,X, andZ of which thet row isy t ,X t, andZ t, T −1 T ∗ respectively we have the least squares estimator ofδ, δˆols = �Z Z) Z y , which is biased and inconsistent. The instrumental variable estimator uses the orthogonal projection ofZ onto T −1 T the column space ofX, i.e.,P X Z whereP X =X�X X) X . The 27 esti- mator is T −1 T ∗ δliv = �Z PX Z) Z PX y . <A= The (linear) instrumental variable estimator is &iased in finite samples, &#t con- sistent. The heteroskedasticity robust estimator of covariance Davidson and /acK- innon [7], p. AAB is T −1 T T −1 ΣˆHCCME = �Z PX Z) Z PX ΦPˆ X Z�Z PX Z) (4) th 2 where Φˆ is an nxn diagonal matrix with thet diagonal element equal toˆut , the squared 27 residual. 9- Lee C. Adkins 3 Binary Choice Model and Estimators ∗ In some cases,y t is not directly observed. 2nstead, we observe ∗ 1y t >0 yt = <B= �0 otherwise Assuming the errors of the model <,= are normally distributed leads to the probit model. 3.1 Linear, MLE, and Plug-in There are several estimators of this model, some consistent forδ and others not. T −1 T ∗ The first is least squares. The least squares estimator δˆols = �Z Z) Z y is consistent ifZ is exogenous. If any of th e elements ofZ are endogenous then it is not. Still, it is easy to compute and the degree of inconsistency may be small in certain circumstances. The linear instrumental variable estimator <A= is also inconsistent and het3 eroscedastic. Iwata +5. suggests a means of rescaling and recentering <66= the data that can bring about consistency in this case. >owever, in his Monte Carlo the 66 versions of OLS and 27 estimation don?t %erform particularly well for probit (although much better for tobit). The usual probit mle can &e used to estimate the %arameters. However, when any of the regressors are endogenous, then this estimator is also inconsistent <4atchew and Griliches +,9]). "o develop the notation, let the probability thaty t is equal one be denoted pr�yt = 1) =Φ�y t� Ytβ+X 1tγ)=Φ�y t� Ztδ) <9= whereΦ is the normal cumulative density,y t is the observed binary dependent variable, andY tβ+X 1tγ is the (unnormalized) inde* function. As usual, the 2 model is normalized assumingσ = 1. Basically, this equation implies thatY t, andX 1t be included as regressors in the probit model and the log likelihood function is maximized with respect toδ T = [βT � γT ]. Since the endogeneity of Yt is ignored, the mle is inconsistent. Another estimator uses predicted values ofY t from a first stage least squares estimation of equation <-). Denote the first stage as Yˆt =X 1tΠˆ1 +X 2tΠˆ2 = . T T . T XtΠˆ whereX t = [X1t.X2t] and Πˆ = [Πˆ1 .Πˆ2 ]. Then the conditional proba- bility pr�yt = 1) =Φ�y t� Zˆtδ)< 7) An Instrumental 7ariables Probit Estimator using $retl 9A . with Zˆt = [Yˆt.X1t]. The parameters are found by maximizing the conditional likelihood. This is referred to here as 27 probit <27P). Although IVP is consis- tent forδ the standard errors estim ated as the outer product of the gradient are not. This can be easily remedied using a Murphy and "opel +,,. type correction. Another estimator adds the least squares residuals from equation (2), Vˆt = Yt −X tΠˆ to (7).
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