Maximum-likelihood estimation of the limited-dependent variable model with endogenous explanatory variable

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1 Maxmum-lkelhood estmaton of the lmted-dependent varable model wth endogenous explanator varable Deon Flmer The World Bank, US Mchael Lokshn The World Bank, US Abstract. Ths artcle descrbes the probtv command, whch mplements the maxmum lkelhood method to estmate the lmted-dependent varable model wth endogenous varable. Kewords: endogenous varables, maxmum lkelhood, lmted-depended varables 1 Introducton In ths artcle we descrbe the mplementaton of the maxmum lkelhood algorthm to estmate the bnar-dependent varable model wth an endogenous explanator varable. The econometrc problem of estmatng a model wth a bnar dependent varable wth a contnuous rght hand sde varable that ma be correlated wth the error term arses n a varet of settngs: Jacob (1997) estmates a model of take-up of a nutrton supplement program n Jamaca as a functon of total household expendtures. Because of measurement error and preference heterogenet the latter are potentall endogenous. The author estmates a two-stage model usng varables that prox household ncome or wealth are used as nstruments for total expendtures. Elbadaw and Sambans (00) estmate whether an ndex of democrac has an effect on whether a countr experences an epsode of cvl war. The authors are concerned that reverse-causalt mght affect ther results. The estmate a twostage model wth two-perod lagged measures of democrac as nstruments for the current degree of democrac. McGranahan (000) estmates a model for the determnants of chartable gvng n 17 th centur England. The model for whether or not a gft was gven to the poor ncludes the number of faml members mentoned n the wll whch s found to ncrease the probablt of chartable gvng. The author tests whether ths s caused b the endogenet that testators that are smpl more phlanthropc and are smpl more lkel to care about the poor as well as others b usng nstruments derved from the parsh n whch the testator lved.

2 These three examples use the approach descrbed n Rvers and Vuong (1988) for estmatng a two-stage model that elds consstent estmates. 1 Such an approach requres potentall cumbersome adjustments to derve consstent standard-errors. The probtv command mplements Full Informaton Maxmum Lkelhood to estmate the bnar and contnuous parts of the model smultaneousl whch wll eld consstent standard-errors. Lke that n Rvers and Vuong (1988) the approach reles of jont normalt of the error terms n the bnar and contnuous equatons. Methods Consder the followng two equaton smultaneous model (ths model s smlar to Model 4 n Maddala 1983 p. 10): * where = γ = β x + β x + ε 1 + ε (1.1) (1.) = 1 = 0 f * > 0 otherwse and x 1 and x are vectors of weakl exogenous varables. The dentfcaton condtons n ths model are that dsturbance terms ε 1 and ε are ndependent, or else there s at least one varable n x 1 that s not ncluded n x. Assume that (x1, ε 1, ε ) are..d, and ε 1 and ε havng, condtonal on x 1, a jont normal dstrbuton wth mean zero and postve defnte covarance matrx: 1 = σ σ 1 Ω 1 σ Gven the assumpton wth respect to the dstrbuton of the dsturbance terms, the logarthmc lkelhood functon for the sstem of equaton (1.1-1.) s: ln L = 1 w (ln[ F( η )] + ln[ f ( η )]) + (1 1 ) w (ln[1 F( η )] + ln[ f ( η )]) () where F s a cumulatve normal dstrbuton functon, f s a normal denst dstrbuton functon, ρ s a correlaton between ε 1 and ε, w s an optonal weght for observaton and η ( β x + ρ( 1 ρ β x 1 = ) / σ ) 1 Smth and Blundell (1988) derve a smlar estmaton procedure for the Tobt model.

3 In the maxmum lkelhood estmaton, σ and ρ are not drectl estmated. Drectl estmated are ln σ and atanh ρ: atanh ρ = ρ ln 1 ρ 3 The probtv command 3.1 Sntax probtv s mplemented as d evaluator that calculates the over log lkelhood along wth ts frst and second dervatves. The command allows for weghts, robust estmaton, as well as the full set of optons assocated wth Stata s maxmum lkelhood procedures. The generc sntax for the command s: probtv depvar varlst [weght] [f exp] [n range], nstruments(depvar_s = varlst_s)[ robust cluster(varname) maxmze_optons] 3. Optons nstrument(depvar_s=varlst_s) gves the specfcaton of the equaton for Y. varlst_s conssts of the exogenous varables n the sstem and ncludes the set of nstruments that help dentf the model. It s an ntegral part of specfng the probtv model and s not optonal. robust specfes that the Huber/Whte/sandwch estmator of the varance s to be used n place of the conventonal MLE varance estmator. robust combned wth cluster() further allows observatons whch are not ndependent wthn cluster (although the must be ndependent between clusters). If ou specf pweghts, robust s mpled. See [U] 3.14 Obtanng robust varance estmates. cluster(varname) specfes that the observatons are ndependent across groups (clusters) but not necessarl wthn groups. varname specfes to whch group each observaton belongs; e.g., cluster(persond) n data wth repeated observatons on ndvduals. cluster() affects the estmated standard errors and varance-covarance matrx of the estmators (VCE), but not the estmated coeffcents. cluster() can be used wth pweghts to produce estmates for unstratfed cluster-sampled data. Specfng cluster() mples robust. maxmze_optons control the maxmzaton process; see maxmze. Wth the possble excepton of terate(0) and trace, ou should onl have to specf them f the model s unstable.

4 4 Example We wll llustrate the use of the probtv command b the problem of estmatng the mpact of the dstance to the nearest prmar school on the probablt of school enrollment. A tpcal specfcaton mght be the followng: (Enrolled) = α + β (Dstance to nearest prmar school) + δ1 (Age Dumm Varables) + δ (Male) + ε (1) Where (Enrolled) s observed as a bnar varable of whether a chld s enrolled n school or not. Other varables n the model mght nclude age and gender of the chld. An objecton sometmes rased for such a model s that the dstance to the nearest prmar school could be endogenous because schools mght be placed where the are most needed, that s, where the underlng probablt of enrollment s low. Ths postve correlaton between dstance and ε means that schools would tpcall be located close to chldren who are less lkel to enroll, whch would result n an upwardl based estmate of β. The Probt Instrumental Varables approach smultaneousl estmates equaton (1) wth an equaton for the dstance to the nearest school. For the sake of llustraton, consder the use of a set of dumm varables for regon of resdence as legtmate nstruments for dstance to the nearest school. Ths would lead to estmatng: (Dstance to nearest prmar school) = γ1 (Age Dumm Varables) + γ (Male) + λ (Regon Dumm Varables) + µ () Applng a standard Probt model n a dataset of 6 to 14 ear olds for rural areas n a poor Afrcan countr elds the followng:. probt schn d_prm zage7-zage14 male Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton : log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Probt estmates Number of obs = 3379 LR ch(10) = Prob > ch = Log lkelhood = Pseudo R = schn Coef. Std. Err. z P> z [95% Conf. Interval] d_prm One mght argue the valdt of ths nstrument set on the bass that the decson to buld schools s made at the regonal level, but condtonal on the dstance to the nearest school the regon of resdence does not affect enrollment.

5 zage zage zage zage zage zage zage zage male _cons That s, the coeffcent on dstance to the nearest prmar school s and s sgnfcantl dfferent from zero: n ths sample a greater dstance to school s assocated wth lower probablt of enrollment. The margnal effect of a change n dstance on the probablt a chld s enrolled n school s: df/dx = Allowng for endogenous school placement and usng probtv wth regon dumm varables as dentfng nstruments gves the followng.. probtv schn d_prm zage7-zage14 male, nstr(d_prm zage7-zage14 male zregon-zregon6) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton : log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Probt IV Number of obs = 3379 Wald ch(10) = Log lkelhood = Prob > ch = Coef. Std. Err. z P> z [95% Conf. Interval] schn d_prm zage zage zage zage zage zage zage zage male _cons d_prm zage zage zage zage zage zage zage zage male zregon

6 zregon zregon zregon zregon _cons /theta /lnsg rho sgma LR test of ndep. eqns. (rho = 0): ch(1) = 7.49 Prob > ch = The results for Equaton () are reported n the secton of the output wth the d_prm headng. Ths s the analogue of the frst stage regresson n tradtonal nstrumental varables regresson. The results of the model of nterest are reported n the secton of the output wth the schn headng. Note that the parameter estmates cannot tpcall be compared to those of the smple Probt estmates. Ths s because Probt (and Probt IV) estmates are scaled b the standard error of the error term and ths mght mplctl dffer between the two models. One can, however, compare margnal effects. The margnal effect mpled b the probtv estmates s df/dx = -.117, almost a full order of magntude dfferent than that mpled b the probt estmates. The correlaton coeffcent between the two equatons (rho) equals The lkelhood rato test for whether ths s sgnfcantl dfferent from zero s reported n the last lne of the output. In ths case the p-value s Snce rho s postve and sgnfcantl dfferent from zero the model suggests that schools are located closer to chldren who are less lkel to be enrolled (.e. both dstance to school and ε are small). The sgnfcance of ths correlaton coeffcent s a test of the exogenet of the varable n queston (note that the valdt of the test rests on the valdt of the nstruments). The varables sgma, /theta, and /lnsg are ancllar parameters used n the maxmum lkelhood procedure. Sgma s the square-root of the varance of the resduals of the regresson part of the model and lnsg s ts log. Theta s a transformaton of the correlaton between the errors from the two equatons. 5 References Elbadaw, Ibrahm and Ncholas Sambnas (00) How Much War Wll We See? Explanng the Prevalence of Cvl War Journal of Conflct Resoluton 46(3): Jacob, Hanan G. (1997). Self-Selecton and the Redstrbutve Impact of n-knd Transfers: An Econometrc Analss Journal of Human Resources 3():33-49 Maddala, G., (1983) Lmted-Dependent and Qualtatve Varables n Econometrc, Econometrc Socet Monographs No. 3, Cambrdge Unverst Press, New York

7 McGranahan, Lesle (000) Chart and the bequests motve: evdence from seventeenth centur wlls The Journal of Poltcal Econom 108(6): Rvers, D., and H. Vuong (1988) Lmted Informaton Estmators and Exogenet Tests for Smultaneous Probt Models Journal of Econometrcs, Vol. 39: Smth, R., and R. Blundell (1986) An Exogenet Test for a Smultaneous Equaton Tobt Model wth and Applcaton to Labor Suppl Econometrca, Vol. 54(3):