Maximum-likelihood estimation of endogenous switching regression models

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1 Mamum-lkelhood estmaton of endogenous swtchng regresson models Mchael Lokshn The World Bank, US urab Saaa The World Bank, US Abstract. Kewords: Ths artcle descrbes the movesta STATA command, whch mplements the mamum lkelhood method to estmate the endogenous swtchng regresson model. ndogenous varables, Mamum lkelhood, Lmted-dependent varables, Swtchng regresson. ntroducton n ths artcle we descrbe the mplementaton of the mamum lkelhood ML algorthm to estmate the endogenous swtchng regresson model. n ths model a swtchng equaton sorts ndvduals over two dfferent states wth one regme observed. The econometrc problem of estmatng a model wth endogenous swtchng arses n a varet of settngs n labor economcs, the modelng of housng demand, and the modelng of markets n dsequlbrum. For eample: The unon-nonunon model of Lee 978 nvestgates the ont determnaton of the etent of unonsm and the effects of unons on wage rates. The propenst to on a unon depends on the net wage gans that mght result from trade unon membershp. The paper eplctl models the nterdependence between the wage gan equaton and the unon membershp equaton. Adamchk and Bed 000 use data from Poland to eamne whether there are an wage dfferentals of workers n the publc and prvate sectors. The paper nterprets sectoral wage dfferentals n terms of epected benefts and the desrablt of workng n a partcular sector. Thorst 977 models the housng-demand problem b eamnng the ependtures on housng servces n owner-occuped and rental housng. The stud models the ndvdual decson to own or rent a house and the amount spent on housng servces. Models wth endogenous swtchng can be estmated one equaton at a tme ether b two-step least square or mamum lkelhood estmaton. However, both of these estmaton methods are neffcent. n addton, these approaches requre potentall cumbersome adustments to derve consstent standard errors. The movesta command, on the other hand, mplements the full nformaton ML method FML to smultaneousl estmate bnar and contnuous parts of the model n order to eld

2 consstent standard errors. Ths approach reles on ont normalt of the error terms n the bnar and contnuous equatons. Methods Consder the followng model, whch descrbes the behavor of an agent wth two regresson equatons, and a crteron functon that determnes whch regme the agent faces : f + u 0 f + u Regme: Regme : > 0 0 X X + ε + ε f f Here, are the dependent varables n the contnuous equatons, X and X are vectors of weakl eogenous varables, and,, and are vectors of parameters. Assume that u, ε and ε have a trvarate normal dstrbuton, wth mean vector zero and covarance matr: u Ω where u s a varance of the error term n the selecton equaton, and and are varances of the error terms n the contnuous equatons. s a covarance of u and ε and 3 s a covarance of u and ε. The covarance between ε and ε s not defned as and are never observed smultaneousl. We can assume that u s estmable onl up to a scalar factor. The model s dentfed b constructon through nonlneartes. Gven the assumpton wth respect to the dstrbuton of the dsturbance terms, the logarthmc lkelhood functon for the sstem of equaton.-.3 s: ln L { w [ln F η + ln f ε / / + w [ln F η + ln f ε / / ]}.4 where F s a cumulatve normal dstrbuton functon, f s a normal denst dstrbuton functon, w s an optonal weght for observaton and η + ε /, The dscusson n ths secton draws from Maddala 983 p. 3-4.

3 3 where u s the coeffcent of correlaton between ε and u and 3 u are the coeffcents of correlaton between ε and u. To make sure that estmated, are bounded between and and estmated, are alwas postve, the mamum lkelhood drectl estmates ln, ln and atanh : + ln atanh After estmatng the model s parameters the followng condtonal and uncondtonal epectatons could be calculated:.0 0,.9,.8 0,.7, Condtonal epectatons :.6.5 Uncondtonal epectatons : F f F f F f F f The movesta command 3. Snta movesta s mplemented as a d ML evaluator that calculates the overall 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 mamum lkelhood procedures. The generc snta for the command s: movesta depvar [] varlst [depvar varlst] [weght] [f ep] [n range], selectdepvar_s varlst_s[ robust clustervarname mamze_optons] pweghts, fweghts and weghts are allowed. n cases when the eplanator varables n the regressons are the same and there s onl one dependent varable, onl one equaton need be specfed. Alternatvel, when the set of eogenous varables n the frst regresson s dfferent from the set of eogenous varables n the second regresson and/or the dependent varables are dfferent between the two regressons, both equatons must be specfed.

4 The command mspredct can follow movesta to calculate the predctve statstcs. The statstcs could be both n and out of the sample; tpe "mspredct... f esample..." f statstcs are wanted onl for the estmaton sample. mspredct newvarname [f] [n range], statstcs 3. General optons selectdepvar_svarlst_s gves the specfcaton of swtchng equaton for. varlst_s ncludes the set of nstruments that help dentf the model. t s an ntegral part of the movesta estmaton and s not optonal. The selecton equaton s estmated based on all eogenous varables specfed n the contnuous equatons plus nstruments. f there are no nstrumental varables n the model, the depvar_s must be specfed as selectdepvar_s. n that case the model wll be dentfed b non-lneartes and the selecton equaton wll contan all the ndependent varables that enter n the contnuous equatons. robust specfes that the Huber/Whte/sandwch estmator of the varance s to be used n place of the conventonal ML varance estmator. robust combned wth cluster further allows observatons whch are not ndependent wthn cluster although the must be ndependent between clusters. f ou specf pweghts, robust s mpled. See [U] 3.4 Obtanng robust varance estmates. clustervarname specfes that the observatons are ndependent across groups clusters but not necessarl wthn groups. varname specfes to whch group each observaton belongs; e.g., clusterpersond refers to data wth repeated observatons on ndvduals. cluster affects the estmated standard errors and varance-covarance matr of the estmators VC, but not the estmated coeffcents. cluster can be used wth pweghts to produce estmates for unstratfed cluster-sampled data. Specfng cluster mples robust. mamze_optons control the mamzaton process; see mamze. Wth the possble ecepton of terate0 and trace, ou should onl have to specf them f the model s unstable. 3.3 Optons for mspredct One of the followng statstcs can be specfed wth the mspredct command: Psel calculates the probablt of beng n regme. Ths s the default statstc. b calculates the lnear predcton for the regresson equaton n regme. Ths s the uncondtonal predcton referred to n the Methods secton quaton.5. 4

5 b calculates the lnear predcton for the regresson equaton n regme. Ths s the uncondtonal predcton refereed to n the Methods secton quaton.6. c_ calculates the epected value of the dependent varable n the frst equaton condtonal on the dependent varable beng observed quaton.7. c_ calculates the epected value of the dependent varable n the frst equaton condtonal on the dependent varable not beng observed quaton.8. c_ calculates the epected value of the dependent varable n the second equaton condtonal on the dependent varable beng observed quaton.9. c_ calculates the epected value of the dependent varable n the second equaton condtonal on the dependent varable not beng observed quaton.0. mlls and mlls calculate correspondng Mll s ratos for the two regmes. 4 ample We wll llustrate the use of the movesta command b lookng at the problem of estmatng ndvdual earnngs n the publc and prvate sectors. A tpcal specfcaton mght be the followng: ln w ln w * X + ε δ ln w X + ε ln w + + u Here * s a latent varable that determnes the sector n whch ndvdual s emploed; w s the wage of ndvdual n sector ; s a vector of characterstcs that nfluences the decson regardng sector of emploment. X s a vector of ndvdual characterstcs that s thought to nfluence ndvdual wage.,, and are vectors of parameters, and u, ε and ε are the dsturbance terms. The observed dchotomous realzaton of latent varable * of whether the ndvdual s emploed n a partcular sector has the followng form: f * > 0 0 otherwse 4.4 The assumpton that s often made n ths tpe of model s that the sector of emploment s endogenous to wages. Some unobserved characterstcs that nfluence the probablt to choose a partcular sector of emploment could also nfluence the wages the ndvdual receves once he s emploed. Neglectng these selectvt effects s lkel to gve a false pcture of the relatve earnng postons n both the publc and prvate sectors. The smultaneous ML estmaton of equatons corrects for the selecton bas n sectoral wage estmates. 5

6 n our eample, the sector choce ndcator prvate takes value f the ndvdual s emploed n the prvate sector and 0 f she s emploed n the publc sector. The wage equatons estmate log of monthl ndvdual earnngs: lmo_earn. The set of eogenous varables n the wage regressons are based on tpcal Mnser s tpe specfcaton Mnser and Polachek, 974 and ncludes such ndvdual characterstcs as age, age, educatonal, and regonal dummes. n addton to these varables, the sector selecton equaton 4.3 ncludes two varables to mprove dentfcaton. An ndvdual s martal status and the number of obholders n the household are beleved to nfluence ndvdual s choce of the sector of emploment, but not to affect the wages. The ML estmaton of ths specfcaton usng movesta command and the dataset movesta_eample.dta s shown below:. use movesta_eample, clear. local str age age edu3 edu4 edu5 reg reg3 reg4. movesta lmo_wage `str', selectprvate m_s ob_hold Fttng ntal values... teraton 0: log lkelhood teraton output omtted..... teraton 6: log lkelhood ndogenous swtchng regresson model Number of obs 094 Wald ch Log lkelhood Prob > ch Coef. Std. rr. z P>z [95% Conf. nterval] lmo_wage_ age age edu edu edu reg reg reg _cons lmo_wage_0 age age edu edu edu reg reg reg _cons prvate age age edu edu

7 edu reg reg reg m_s ob_hold _cons /lns /lns /r /r sgma_ sgma_ rho_ rho_ LR test of ndep. eqns. : ch Prob > ch The results of the sector selecton equaton are reported n the secton of the output headed prvate. The results of the wage regresson n the prvate sector are reported n the lmo_wage_ secton and the wage regresson n the prvate sector s outputted n the lmo_wage_0 secton. The correlaton coeffcents rho_ and rho_ are both postve, but sgnfcant onl for the correlaton between the sector choce equaton and the publc sector wage equaton. Snce rho_ s postve and sgnfcantl dfferent from zero the model suggests that ndvduals who choose to work n the publc sector earn lower wages n that sector than a random ndvdual from the sample would have earned, and those workng n the prvate sector do no better or worse than a random ndvdual. The lkelhood rato test for ont ndependence of the three equatons s reported n the last lne of the output. The varables sgma, /lns, lns, /r, and /r are ancllar parameters used n the mamum lkelhood procedure. Sgma and Sgma are the square-roots of the varances of the resduals of the regresson part of the model and lnsg s ts log. r and r are the transformaton of the correlaton between the errors from the two equatons. 5 References Adamchk, V., and V. Bed, 000 Wage dfferentals between the publc and the prvate sectors: vdence from an econom n transton. Labour conomcs, Vol. 7: Lee, L., 978. Unonsm and Wage Rates: A Smultaneous quatons Model wth Qualtatve and Lmted Dependent Varables. nternatonal conomc Revew, 9: Maddala, G., 983 Lmted-Dependent and Qualtatve Varables n conometrc, conometrc Socet Monographs No. 3, Cambrdge Unverst Press, New York 7

8 Mncer, J., and S. Polachek, 974 Faml nvestment n Human Captal: arnngs of Women. Journal of Poltcal conom Supplement, Vol. 8: S76-S08 Thorst, R., 977 Demand for Housng: A Model Based on nter-related Choces Between Ownn and Rentng. Ph.D. dssertaton, Unverst of Florda 8