Symmetrically Censored GMM Estimation for Tobit Models with Endogenous Regressors. Haizheng Li

Transcription

1 Symmetrically Cesored GMM Estimatio for Tobit Models with Edogeous Regressors Haizheg Li School of Ecoomics Georgia Istitute of Techology Atlata, GA Phoe: (404) Fax: (404) I would like to thak Jeffrey Zax, Jusha Bai, Shiichi Sakata, Whitey Newey, Doald Waldma, Robert McNow, Nicholas Flores, B.J. Lee, Sogia Che, Buhog Zheg, Thomas Dee, Ly Yag, ad especially two aoymous referees for their commets. Semiar participats at the 996 North America Summer Meetigs of the Ecoometric Society, Ohio Sate Uiversity, Uiversity of Colorado-Boulder, McGill Uiversity, Georgia Istitute of Techology, ad Uiversity of Missouri-Columbia have bee geerous with commets.

2 Symmetrically Cesored GMM Estimatio for Tobit Models with Edogeous Regressors Abstract I this study, a ew semi-parametric estimatio method for cesored models, symmetrically cesored geeralized method of momets (SCGMM) estimator, is proposed. The SCGMM estimator ca be applied to cesored models whe some regressors are correlated with the regressio error. Moreover, it does ot impose the idepedece assumptio for data, ad thus ca be applied to time-series ad pael data models with depedet ad heterogeeous observatios. Furthermore, the SCGMM does ot require ay o-parametric techiques for estimatio, ad is relatively simple to implemet. The SCGMM estimator is also robust to a fairly wide variety of o-ormal ad o-idetical but symmetric disturbace distributios. This estimator adopts a ew approach to idetifyig the true parameter by restrictig the parameter space i the estimatio. The SCGMM estimator is show to be cosistet ad asymptotically ormal for ear epoch depedet fuctios of mixig processes. Thus, it ca be applied to models with mixig processes, ad models i which the disturbace term is a ARMA process of fiite order (with roots outside the uit circle), or a ifiite o-statioary MA process. A heteroskedasticity ad autocorrelatio cosistet estimator of the covariace matrices is also provided. Simulatio results show that the SCGMM estimator performs very well. Keywords: Semi-parametric, cesored model, GMM, ear epoch depedece, edogeeity.

3 I. Itroductio Regressio models with cesored depedet variables are commo i empirical work. It is kow that the commoly used Tobit maximum likelihood estimatio procedure for cesored models is sesitive to the error term distributio. If the uderlyig distributio is ot both ormal ad homoskedastic, the Tobit procedure will geerally be icosistet (Arabmazar ad Schmidt 982, Goldberger 983). Several semi-parametric or o-parametric estimators have bee proposed to relax these assumptios, for example, Powell (984), Horowitz (986), Powell (986), ad Hooré ad Powell (994). These estimators are robust to heteroskedasticity ad o-ormality of the disturbace distributio. However, these semi-parametric estimators require that all regressors i a cesored model be ucorrelated with (or idepedet of) the regressio error. If some explaatory variables are correlated with the regressio error, these estimators are geerally icosistet. This restrictio rules out edogeeity ad measuremet error i ay regressor. I additio, they also assume idepedece for the data i the model. For may cesored models, such as labor supply fuctios ad some demad fuctios (e.g., for tickets, cigarettes, etc.), edogeous explaatory variables, for example wages ad prices, are commo. Furthermore, for cesored models with time-series or pael data, observatios are ulikely to be idepedet. Therefore, it is desirable to have a semi-parametric estimator that ca be applied to models with edogeous regressors ad depedet observatios. Lewbel (998) proposes a oparametric estimator for the latet variable model with edogeous regressors. Hog ad Tamer (2003) exteded the cesored LAD estimator proposed i Power (984) to cesored models with edogeous regressors. These estimators require o-parametric estimatios of distributio fuctios. Additioally, they also maitai the idepedece assumptio for the data. I this study, a ew semi-parametric estimatio method, symmetrically cesored geeralized method of momets (SCGMM) estimator, is proposed. The SCGMM estimator ca be applied to cesored models whe some regressors are correlated with the regressio error.

4 Moreover, it does ot impose the idepedece assumptio for data, ad thus ca be applied to time-series ad pael data models with depedet ad heterogeeous observatios. Furthermore, the SCGMM does ot require ay o-parametric techiques i the estimatio, ad is relatively simple to implemet. As with other semi-parametric estimators, the SCGMM is also robust to a fairly wide variety of o-ormal ad o-idetical but symmetric disturbace distributios. The SCGMM method geeralizes the symmetrical trimmig procedure i the symmetrically cesored least squares (SCLS) estimator proposed by Powell (986) for cesored models with oly exogeous regressors. This estimatio method ca be directly applied to models with liear or o-liear edogeous explaatory variables. It does ot require costructig reduced form equatios. The SCGMM estimator is show to be cosistet ad asymptotically ormal uder very geeral coditios. Furthermore, a cosistet estimator of the asymptotic covariace matrices is also established i the presece of heteroskedasticity ad autocorrelatio of ukow form. The SCGMM estimatio makes use of istrumets i the estimatio. I order to accommodate over-idetifyig istrumets, it adopts a geeralized method of momets (GMM) framework. However, the stadard asymptotic theory of GMM is ot applicable i this case because of the o-smoothess of the SCGMM criterio fuctio. Nevertheless, uder suitable regularity coditios outlied i the followig sectios, the cosistecy ad asymptotic ormality of the SCGMM estimator ca be established. I order to esure uique idetifiability, the SCGMM estimator adopts a ew approach to idetificatio: istead of costructig a special criterio fuctio to idetify the true parameter, the SCGMM estimator is defied i a costraied parameter space, i which the true parameter is the uique solutio to the momet coditios. Thus, the SCGMM estimator is a costraied optimizatio estimator, ad mathematical programmig methods are eeded for estimatio. No-parametric estimatio geerally requires choosig smoothig parameters, ad o geeral satisfactory solutio to this problem exists. 2

5 Furthermore, i order to fit a geeral depedece structure, the SCGMM estimator exteds Hase's GMM estimator (982) from statioary ad ergodic sequeces to ear epoch depedet (NED) processes. Hece it ca be applied to models with m-depedet or mixig processes, ad models i which the disturbace term is a ARMA process of fiite order (with roots outside the uit circle), or a ifiite o-statioary MA process (uder appropriate mild coditios o the MA weights). Sice it does ot require a statioarity assumptio for depedet processes, the SCGMM estimatio allows for both coditioal ad ucoditioal heteroskedasticity. The simulatio results show that the SCGMM estimator performs very well, much better tha the SCLS i the presece of edogeous regressors. Although the SCGMM estimatio requires a stroger assumptio o the istrumets, such istrumets are ot ucommo i empirical applicatios. Moreover, the SCGMM estimator is robust to a variety of cesored models, especially whe some explaatory variables are correlated with the disturbace term. I additio, it icludes the SCLS estimatio as a special case. The rest of the paper is orgaized as follows. I ext sectio, the SCGMM estimatio procedure is itroduced. Sectios III ad IV provide regularity coditios for cosistecy ad asymptotic ormality. Cosistet estimatio of the asymptotic covariace matrices is discussed i Sectio V. I sectio VI, a small scale simulatio is coducted, ad sectio VII cocludes. Techical proofs are give i the appedix. II. The SCGMM Estimator Cosider a true uderlyig liear regressio model (2.) y t* = x t β 0 + u t * ( t =,2,..., ), where t idexes observatios, y t* is a depedet variable, x t is a k vector of exogeous ad edogeous regressors, β is a k vector of ukow costat coefficiets, ad u t* is a 3

6 uobservable error. For a cesored regressio, oly x t ad y t = max {0, y t* } are observable. 2 Thus the error term is also cesored as u t = max {u t*, x t β 0 }. I this case, eve if all elemets of x t are exogeous, the coditioal expectatio of the cesored error u t is ozero, i.e., E(u t x t ) 0. I this case, OLS estimatio is icosistet. A commo alterative is the Tobit procedure, where it is ecessary to specify completely the distributio of u t* i order to adopt the likelihood-based approach. If the distributio of u t* is ukow, or if it is heteroskedastic with ukow form, the Tobit procedure will be icosistet. If all elemets i x t are exogeous ad u t*, coditioal o x t, is distributed symmetrically * aroud zero, the SCLS proposed by Powell (986) ca be applied whe the distributio of u t is ukow ad is heteroskedastic of ukow form. More specifically, i the SCLS estimatio, u t is cesored from the upper tail by x t β 0, ad thus the symmetry of the distributio is restored. As a result, the ew cesored error becomes u ts = mi {u t, x t β 0 }. As the values of y t* below zero s have bee cesored to zero, this procedure cesors the values of y t* above 2x t β 0 to 2x t β 0, i.e., y t = mi{y t, 2x t β 0 }, to elimiate the asymmetry of the distributio. Thus it results i the followig desirable momet coditio: (2.2) E((x t β 0 >0) u ts x t ) = 0. I this case, the OLS estimatio ca be applied without ay further assumptios. However, if some elemets of x t are correlated with the regressio error, the symmetrically cesored error u ts will ot satisfy the momet coditios (2.2). Sice the trimmig procedure depeds o x, hece the trimmig itself is also edogeous. Therefore, the SCLS estimator caot be applied. Moreover, i this case, the usual istrumetal variable estimatio caot be simply used either. More specifically, suppose there exists a vector of istrumets w t ad E(u t* w t )=0, ad that coditioal o w, u t* is distributed symmetrically aroud zero. The expectatio of the symmetrically cesored error u ts o w t, i.e., E((x t β 0 >0) u ts w t ), will deped 2 The cesorig poit here is assumed to be zero, though this is oly a coveiet ormalizatio. 4

7 o the joit distributio of x t, u t* ad w t, ad will geerally be o-zero. Therefore, the usual momet coditio eeded for IV estimatio, E((x t β 0 >0) u ts w t ) =0, is ot satisfied. 3 Nevertheless, suppose that x t = (x t, x 2t ), ad x t is k vector of edogeous variables, x 2t is k 2 vector of exogeous variables, ad β 0 =(β 0, β 02 ) The model (2.) ca be writte as y t* = x t β 0 + x 2t β 02 + u t * ( t =,2,..., ). Assume that there exists a l vector of istrumets w t that is correlated with x t but ucorrelated with u t*, ad l k, w t =(w t, w 2t, x 2t ), where w t is a k vector of istrumets ad w 2t is a (l-k -k 2 ) vector of istrumets. I this case, E(u t* w t )=0. Furthermore, for istrumet w t, assume the followig coditio holds, (2.3) x t β 0 w t β 0, for t=, 2,,. Defie that istrumet z iclude all exogeous regressors ad the subset of istrumets that satisfies the coditio (2.3), i.e., z t =(w t, x 2t ). Clearly, equatio (2.3) implies: (2.4) x t β 0 z t β 0 for t=, 2,,. Give the existece of such istrumets, the trimmig procedure ca be performed by z * istead of x. Therefore, a ew error ca be obtaied by cesorig u t below z t β 0 to z t β 0 ad * symmetrically cesor u t above z t β 0 to z t β 0, i.e., (2.5) u tc = mi{max{u t*, z t β 0 }, z t β 0 }. The coditio (2.4) ca be viewed as a requiremet that -z t β 0 falls ito the regio of u t* that is ot cesored by x t β 0. 3 Newey (985b) exteds the SCLS to models with edogeous regressors ad proposes a Two-Stage Istrumetal Variable (2SIV) estimatio. I the first stage, the OLS estimatio is applied to produce the predicted value for the edogeous regressors, ad i the secod stage, the SCLS procedure is applied with the edogeous regressors replaced by their predicted values. The 2SIV estimator is based o a early versio of the SCLS estimator ad the criterio fuctio is icorrect. Thus the asymptotic theory of the 2SIV estimator eeds to be re-established. Moreover, i the 2SIV estimatio, it essetially requires that E[( x ˆt β 0 >0) mi{u t, x ˆt β 0 } w t ]=0. Sice the predicted value of x t depeds o both w t ad x t, this momet coditio may also deped o joit distributio of x t, w t, ad u t. Therefore, it seems that more justificatios are eeded for the 2SIV estimator. 5

8 If u t*, coditioal o w t, is distributed symmetrically aroud zero, the followig momet coditio ca be obtaied to idetify the true parameters: (2.6) E((z t β 0 > 0) u c t w t ) = 0. This trimmig procedure is related to the Wisorized Mea estimator for cesored models with oly exogeous regressors proposed by Lee (992). I the Wisorized Mea estimatio, a arbitrary costat (istead of x t β 0 ) is itroduced to perform the trimmig i the SCLS, i order to allow for some asymmetry i the error distributio. 4 The SCLS procedure requires that u * t is symmetric up to ±x t β 0, while the momet coditio (2.6) oly requires it be symmetric up to ±z t β 0. Thus, give the coditio (2.4), the symmetry assumed i equatio (2.6) is weaker tha that i the SCLS. The coditio z t β 0 >0 i equatio (2.6) follows the coditio x t β 0 >0 i SCLS. As discussed i Powell (986), for the SCLS, if x t β 0 0, the most of the distributio of u t* has bee cesored to -x t β 0. Symmetrically cesorig the upper tail of u t will result i cesorig all values of u t to x t β 0, thus it yields o iformatio cocerig the ukow β 0. Therefore, observatios with x t β 0 0 are deleted from the sample. Similarly, to esure the uique idetifiability, z t β 0 is * required to be positive for a substatial proportio of the sample. I practice, because every y t itself is cesored to be above zero, it is reasoable for x t β 0 to be o-egative for a positive fractio of idividuals i the sample. Geerally the value of z t β 0 should be close to the values of x t β 0 because there are very few edogeous regressors i x for most empirical models. Hece, the coditio that z t β 0 is positive for a proportio of the sample will likely to be satisfied. The requiremet that z t β 0 >0 for a proportio of the sample ad that z t β 0 x t β 0 for all t puts a boud for the istrumet. Whe there is oe edogeous regressor, the coditio (2.3) essetially requires that the value of istrumet is smaller or larger tha the edogeous regressor 4 I Lee (992), the arbitrary costat caot be determied i the estimatio. 6

9 i populatio, depedig o the sig of the correspodig elemet i β 0. 5 Such istrumets are ot ucommo i empirical work. For example, Butcher ad Case (994) use the presece of ay sister withi a family as a istrumetal variable for years of schoolig of female workers, Levitt (997) uses the timig of mayoral ad guberatorial electio as istrumets for size of the police force i estimatig the effects of police o city crime rates. Card (995) employs a dummy variable that idicates whether a ma grew up i the viciity of a four-year college as a istrumetal variable for years of schoolig. I these cases, the istrumet takes the values 0 or, ad is certaily smaller tha the correspodig edogeous regressor. Thus, the coditio (2.3) is satisfied. I may time series applicatios, lagged values are used as istrumets; ad lagged values are smaller tha the curret values i some cases, for example, lagged cosumptio or GDP. Therefore, they also meet the requiremet. The momet coditio (2.6) ca geerate the followig l ucoditioal oes: (2.7) E[(z t β 0 > 0) mi{max{u t*, z t β 0 }, z t β 0 } w t ] = 0. 6 Sice z is oly a subset of istrumet w, the above momet coditio has the advatage of accommodatig additioal istrumets. Thus, all iformatio provided by the full set of istrumets w ca be used to improve the efficiecy, while oly some of the istrumets i z eed to satisfy the coditio (2.3). Moreover, with over-idetifyig restrictios, it is possible to test for the model specificatio or the validity of some istrumets. Sice z t β 0 falls ito the ucesored regio of u t*, coditio (2.7) is equivalet to: (2.8) E[(z t β 0 >0) mi{max{u t, z t β 0 }, z t β 0 } w t ] =0, where u t =max {u t*, x t β 0 }, ad u t =y t x t β 0. The sample couterpart of the above populatio orthogoality coditio is defied as: 5 There are cosiderable flexibilities i maipulatig a istrumet to satisfy this requiremet, for example, the egative of a istrumet is still a good istrumet, ad the same is true whe a istrumet plus a costat. Hece, the coditio (2.3) is ot as restrictive as it first appears. 6 The symbol (A) deotes the idicator fuctio of the evet A, i.e., it is a fuctio which takes the value oe if A is true ad is zero otherwise. 7

10 (2.9) t = [(z t β>0) mi{max{ y t x t β, z t β}, z t β} w t ] = 0. It is difficult to esure idetificatio usig oly these momet coditios, sice multiple solutios to the momet coditio exist. The usual approach is to itegrate back the momet fuctio to get a global miimad to idetify the true parameter. I the SCLS, the criterio fuctio is obtaied by itegratig the just-idetified sample orthogoal system, ad the observatios with x t β 0 0 are pealized by the amout y t2 /2 i the criterio fuctio. I this costructio, the SCLS criterio fuctio avoids the multiplicity of miimum solutios. However, the SCLS structure caot be applied here because the trimmig is performed by z istead of x, ad the umber of istrumets may be larger tha that of edogeous regressors. Nevertheless, followig Hase (982), a quadratic form GMM criterio fuctio based o the momet coditio ca be costructed. By miimizig the GMM type criterio fuctio, the k liear combiatios of the sample orthogoality coditios are made as close as possible to zero. This gives k parameter-defiig mappig equatios, which ca be solved for β. Sice the GMM estimator is defied o the symmetric cesored momet fuctios, this estimator will be called symmetrically cesored GMM estimator (SCGMM). It is clear that the SCGMM estimator icludes the SCLS as a special case for w t =z t =x t. Therefore, the GMM criterio fuctio is defied as (2.0) Q (β) = F (β) A F (β), where F (β) = t = f t (β) ad f t (β) =(z t β>0) mi{max{ y t x t β, z t β}, z t β} w t, ad A is a sequece of arbitrary, symmetric, positive defiite, ad possibly data-depedet weightig matrices. Uder suitable regularity coditios, it ca be show that the GMM criterio fuctio will coverge to the followig limitig fuctio, (2.) Q 0 (β) = M(β) A * M(β), 8

11 where M(β) = t = m t (β) ad m t (β)=e[(z t β>0) mi{max{ y t x t β, z t β}, z t β} w t ], ad A * is a O() sequece of o-stochastic, symmetric, uiformly positive defiite matrices. Ad m t (β) ad M(β) are the expectatios of f t (β) ad F (β), respectively. For the limitig criterio fuctio Q 0 (β), give the momet coditio (2.8) ad positive defiiteess of A *, the miimum zero will be achieved at the true value β 0. However, a particular problem for idetificatio is that the miimum ca also be achieved at other βs that makes z t β 0 for all t. For example, β=0 will always achieve the miimum of Q 0 (β). Give the momet coditio, it is difficult to costruct a criterio fuctio that ca exclude all uwated roots to idetify the true parameter. The SCGMM estimator, istead, is defied as a costraied estimator. I particular, the parameter space is costraied so that all uwated roots are excluded. This approach is differet from a usual extreme estimator, i which the parameter space is ot explicitly costraied. I fact, a costraied optimizatio is more desirable if the researcher believes that the true parameter lies i a proper costraied subset (Amemiya 985). Therefore, the SCGMM estimator β of β 0 is defied as the value of β miimizig Q (β) over a costraied subset of the parameter space. That is (2.2) Q ( β ) = if ( β) β Θ Q s. t. Θ {β Θ: [ (y t >0) z t β ] >0}, t = wheever > 0, for some positive 0. 7 The compact parameter space Θ is a subset of Θ, ad it ca be viewed as a sufficietly large ball that cotais β 0, i which the sample average of z t β for positive y t is positive. Hece, i the parameter space Θ, all uwated roots such as β=0 are ruled out, ad oly the true value β 0 7 I am grateful for the suggestio from a referee to specify the costrait this way. 9

12 satisfies the momet coditio ad miimizes the limitig SCGMM criterio fuctio. Give the regularity coditio discussed i ext sectio, the true parameter ca be idetified i Θ. Clearly, Θ is radom set depedig o the realizatio of z t ad y t. Thus, the SCGMM estimator becomes a extreme estimator costraied to the radom set that coverges i a certai probabilistic sese to a fixed set. The radom set poses o particular problem for provig cosistecy ad asymptotic ormality. I practice, to fid the miimum i a costraied parameter space, a mathematical programmig method is eeded i estimatio. Give the availability of mathematical programmig packages i computer software, the computatio does ot pose a problem. III. Cosistecy of the SCGMM Estimator I this sectio, the cosistecy of the SCGMM estimator is established for processes that are ear epoch depedet (NED) o a mixig process. The followig defiitios are based o Gallat ad White (988): Defiitio 3.: Let {V t : Ω R b } be a sequece of radom variables o a probability space (Ω, F, P), where b N. The triagular array {Z t : t =,...,, N} of radom variables o (Ω, F, P) is ear epoch depedet (NED) o {V t } if E[ Z t 2 ]< for ay t, N ad ν(m) 0 as t m m, where ν(m) sup sup t { Zt E + ( Zt ) : t {,...,}, N}, ad 2 t t m m E + t m ( ) E( F t + m ), t m t m F t + m σ(v t-m,...,v t+m ), ad L- p orm Z p =E /p Z p. Defiitio 3.2: I defiitio 3., suppose ν(m)= O(m λ ) for all λ< a. The ν(m) is said to be of size a. The size for mixig is defied similarly. I the defiitio for NED, ν(m) is basically the root of the worst mea square forecast error whe Z t is predicted by t m E t + m (Z t ); ad ν(m) will ever icrease as m icreases for all t, as more ad more iformatio is used for forecastig. If ν(m) teds to zero at a appropriate rate 0

13 uiformly i t, the Z t depeds essetially o the recet epoch of {V t }. If Z t is idepedet, m- depedet, or mixig, the it is trivially ear epoch depedet of ay size. Let the L- p orm of the radom variables u t*, x t, ad w t be deoted as u t * p =E /p u t p, x t p =E /p x t p, ad wt p = E/p w t p, where xt ad w t are Euclidea orms. The followig assumptios are eeded for cosistecy. Assumptio E: The error terms u t* are cotiuously ad symmetrically distributed about zero coditioally o w t with cotiuous desity fuctio g t (λ), where g t (λ)=g t ( λ), g t (λ)<l 0, ad g t (λ)>ξ 0 i the eighborhood of zero uiformly i t for some L 0 >0, ξ 0 >0. Assumptio E implies the coditio E(u t* w t )=0 ad m t (β 0 )=0. Assumptio MX: {x t, w t } is a mixig sequece such that either φ m is of size -r/(2r-2), r 2 or α m is of size -r/(r-2) with r>2. Assumptio NE: The elemets of sequece {u t* } are ear epoch depedet o {x t, v t } of size -(r-)/(r-2) for r>2, where {v t } is a mixig sequece such that either φ m is of size -r/(2r-2), r 2 or α m is of size -r/(r-2) with r>2. Assumptio DM: The radom variables u t*, x t, ad w t are 2r-itegrable uiformly i t, for t =, 2..., ad r>2, that is, their L- 2r orms are bouded for all t, * ut 2 r <, xt 2 r <, ad wt 2 r <. The mixig assumptio of {x t, w t } geeralizes the idepedece assumptio i SCLS. If the istrumet vector w t icludes iteractio terms or fuctios of some exogeous elemets i x t, the mixig property of these terms is implied by the mixig property of x t, because a fuctio of mixig processes is mixig. It will suffice to assume just α-mixig (strog mixig), because φ- mixig (uiform mixig) implies α-mixig. The domiatio coditio places bouds o some

14 momets greater tha the fourth, which is a stroger requiremet. However, this will allow for much greater depedece ad fairly arbitrary heterogeeity. It would be techically easier if the disturbace term u t* is also a mixig process, which still exteds the GMM from statioary sequeces to heterogeeous sequeces ad makes the SCGMM applicable to models with depedet observatios. For may cases, the mixig assumptio is sufficiet because if u t* oly depeds o a fiite umber of lagged values of some mixig processes, u t* itself is still mixig. However, the mixig assumptio has a drawback. A fuctio of a mixig sequece (or eve a idepedet sequece) that depeds o a ifiite umber of lags of the sequeces is ot geerally mixig. The property of α-mixig (φ-mixig) is ot ecessarily preserved uder trasformatios that ivolve the ifiite past (Adrews, 984). For example, eve if u t* is a simple AR() process o a uderlyig process v t ad v t is mixig or eve idepedet, u t* itself ca fail to be either α-mixig or φ-mixig. Assumptio NE adopts a more geeral depedece structure, ear epoch depedece. Based o the defiitio, a process may deped o the etire history of other processes, but if ν(m) teds to zero at a appropriate rate, the it depeds essetially o the recet epoch ad does ot deped too much o the distat past or future. Therefore, Assumptio NE allows u t* to deped o the ifiite past ad/or future of the uderlyig processes, provided the extet of depedece is cotrolled appropriately. Sice some elemets of x are edogeous, u t* is allowed to deped o these edogeous elemets i the NED structure. It ca be show that, if u t* is a ARMA process of fiite order with zeros lyig outside the uit circle, or if u t* is a ifiite movig average process, u t* is NED. If u t* is a mixig process, it is trivially NED. I additio, we eed ot impose statioarity o u t* but istead may allow a substatial amout of heterogeeity. Sice ear epoch depedet fuctios of mixig processes are mixigales, asymptotic theories for mixigales ca be applied (see Gallat ad White 988 for further discussios). Therefore, the 2

15 NED assumptio is geeral eough for the SCGMM estimator to apply to models with a variety of depedece structures. Assumptio P: The parameter space Θ is a compact subset of R k, ad the true parameter vector β 0 is a iterior poit of Θ. Assumptio ID: i) z t is a subset of w t such that z t =(w t, x 2t ), ad x t β 0 z t β 0, for all t. ii) The matrix N 0 = t= E[(z t β 0 >ε 0 ) w t w t ] is positive defiite uiformly i, wheever > 0, for some positive 0 ad ε 0. iii) [ E(z t β 0 y t >0)]>0, uiformly i, wheever > 0, for some positive 0. t = iv) The rak of the matrix t= E[w t x t ] is k. Assumptio ID is essetial for idetificatio. Coditio i) is the boud coditio for a subset of the istrumet discussed i the previous sectio, ad it essetially requires that z t β 0 fall ito the regio of u t* that is ot cesored by -x t β 0. Coditio ii) imposes a lower boud to z t β 0, ad requires z t β 0 to be o-egative for a substatial proportio of the sample. It also rules out the situatios that some elemets of w t are perfect colliear. Coditio iii) requires that the average value of z t β 0 for positive y t* to be positive, ad it esures the covergece of the compact subset Θ. Based o coditio ii), z t β 0 is positive for a substatial proportio of the sample. Sice the true depedet variable y t* is cesored to be above zero, x t β 0 is more likely to be positive if y t* itself is positive. Hece, z t β 0 is also more likely to be positive whe y t* >0 tha whe y t* <0. This assumptio is especially coveiet for implemetig the SCGMM estimator because the costrait becomes liear. Coditio iv) is the usual rak coditio for IV estimatio, ad requires w t ad x t be sufficietly liearly related. Assumptio W: The sequece of weightig matrices A is symmetric, positive defiite, 3

16 ad A p A *, where A * is ostochastic, symmetric, ad uiformly positive defiite. 8 The above assumptios permit applicatio of Gallat ad White s (988) versio of the uiform law of large umbers (Theorem 3.8) based o ear epoch depedece ad the Lipschitz coditio. The Lipschitz coditio is defied below. Defiitio 3.3: Let (Ω, F, P) be a probability space ad (Θ, ρ) be a Euclidea space. The sequece {q t : Ω Θ R} is defied to be almost surely Lipschitz-L o Θ if ad oly if for each θ i Θ, q t (,θ) is measurable-f, t =, 2,..., ad for ay θ, θ 0 i Θ there exist a fuctio L t 0 : Ω R + measurable-f/b(r + ) such that E[L t0 ] is O(), ad t = 0 q t (θ) q t (θ 0 ) L t θ θ 0, t =, 2,..., a.s. The followig Lemmas are eeded to establish the large sample properties. Lemma 3.: Give assumptios DM ad P, the elemets of f t (β) are almost surely Lipschitz-L o Θ. Lemma 3.2: Give assumptios DM, MX, NE, ad P, the elemets of f t (β) are ear epoch depedet o {w t, x t, v t } of size -/2 uiformly o (Θ, ρ), where ρ is Euclidea orm o R k. such that Lemma 3.3: Give assumptio E, P, ad ID, there exists a sequece of subset Θ cotais β 0, ad for every β i Θ, [ t = Θ of Θ, E(z t β y t >0)]>0 uiformly i ad uiformly i β, wheever > 0 for some positive 0, ad the momet coditio M(β)=0 is oly satisfied at β 0 over Θ. 8 A * is used here because, ulike that i Hase (982), A * is ot limited to coverge to a costat limit (see Bates ad White, 985). 4

17 Sice the radom set Θ will coverge to the fixed set Θ, based o Lemma 3.3, the limitig SCGMM criterio fuctio Q 0 (β) has idetifiably uique miimizer β 0 o these coditios, the followig result ca be established: Θ. With THEOREM 3.4: For the cesored regressio model (2.), uder assumptios E, MX, NE, DM, P, ID ad W, there exists a measurable fuctio β : Ω Θ, such that Q ( β )= if Θ Q ( β ) ; ad the SCGMM estimator β β is weakly cosistet, β p β 0. 9 IV Asymptotic Normality of the SCGMM Estimator The asymptotic distributio of the SCGMM estimator is established through the empirical process method via stochastic equicotiuity. There are primitive coditios available to show stochastic equicotiuity. If the empirical momet fuctio f t (β) is idepedet or m- depedet, the correspodig empirical process ca be show to be stochastically equicotiuous via symmetrizatio because f t (β) belogs to the type I class of fuctios which satisfy Pollard s etropy coditio(see Adrews 994a). If f t (β) is mixig, stochastic equicotiuity ca be show via bracketig coditios (Adrews, 993). However, as f t (β) is ear epoch depedet here, the above primitive coditios are ot applicable. Nevertheless, sice f t (β) is Lipschitz at β 0 ad differetiable at the true parameter β 0 with probability oe, a weaker stochastic equicotiuity coditio ca be established followig Newey ad McFadde (994). 0 The followig assumptios are eeded for the asymptotic distributio of the SCGMM estimator. 9 It could be strogly cosistet if the weightig matrix A could coverge almost surely. However, as discussed i sectio V, the optimal weightig matrix oly coverges i probability. 0 As discussed o the Appedix, the stochastic equicotiuity coditio for Lipschitz momet fuctios defied i Theorem 7.2 ad 7.3 i Newey ad McFadde (994) ca be exteded to the NED case. 5

18 Assumptio NE : The disturbace process {u t* } is ear epoch depedet with respect to {x t, v t } of size -2(r-)/(r-2) for some r>2. Assumptio MX : The process {x t, w t, v t } is a mixig sequece such that either φ m is of size -r/(r-), r 2 or α m is of size -2r/(r-2) with r>2. Assumptios NE ad MX stregthe assumptios NE ad MX by icreasig their sizes. Defie B 0 = Var ( F (β 0 )), ad G 0 = E[( z t β 0 < u t* < z t β 0 ) ( w t x t )]. t = Assumptio PD: The sequece {B 0 } is uiformly positively defiite. I order to establish the asymptotic distributio, the followig cetral limit theorem (Wooldridge ad White 988) for NED fuctios of a mixig process is used. Cetral Limit Theorem: Let {Z t } be a double array such that Z t r < for some r>2, E(Z t )=0,, t =,2,..., ad {Z t } is ear epoch depedet o {V t } of size, where {V t } is a mixig process with φ m is of size -r/(r-2) or α m is of size -2r/(r-2). Defie v 2 Var t = Zt, -2 ad suppose that v is O( - - ). The v t = Zt A N(0,). The followig lemmas give the limitig distributio of F (β 0 ) ad the stochastic equicotiuity coditio. The stochastic equicotiuity i Lemma 4.2 is a weaker result tha that i Adrews (994a) because of the deomiator term. Lemma 4.: Give assumptios E, DM, P, ID, MX, NE, ad PD, the (B 0 ) -/2 F (β 0 ) A N(0, I). Lemma 4.2: Give assumptios E, DM, P, ID, ad MX, the for ay δ 0, sup F( β) ( β ) ( β) / [ + β β ] β β δ F 0 M p 6

19 THEOREM 4.3: Give assumptios E, DM, P, ID, MX, NE, ad PD, if β is a cosistet estimator of β 0, the (G 0 A * B 0 A * G 0 ) /2 (G 0 A * G 0 ) ( β β0 ) A N(0, I). If G 0, A *, ad B 0 have limits G 0, A *, ad B 0, respectively, the result ca be writte i the followig form: ( β β 0 ) A N[0, (G 0 A * G 0 ) G 0 A * B 0 A * G 0 (G 0 A * G 0 ) ]. V. Cosistet Estimator of the Asymptotic Covariace Matrices I order to use the asymptotic ormality of β to coduct hypothesis tests for the parameter vector β 0, cosistet estimators of the asymptotic covariace matrices must be derived. I additio, a cosistet estimator of the asymptotic covariace matrix also provides the optimal weightig matrix to get a more efficiet SCGMM estimator. 0 Based o Theorem 4.3, to estimate the asymptotic covariace matrices, both G 0 ad B must be estimated. A atural estimator of G 0 is G ( β ) = ( z t β t = < ut <zt β ) ( wt x t ), where ut = yt x t β. It ca be show that G ( β ) is a cosistet estimator of G0. I the presece of arbitrary forms of heteroskedasticity ad autocorrelatio, B 0 ca be 0 writte as B Γ( j), where j= + Γ(j) = t= j+ t= j+ E[ ft( β 0) f t E[ f t + j ( β 0)] j( β 0) f t( β 0)] for for j 0 j < 0. I the simplest case, such for a idepedet sequece, B 0 equals Γ(0), where 7

20 Γ(0) = E [(z t β 0 > 0) (mi{(u t* ) 2, (z t β 0 ) 2 }) w t w t ]. = t A cosistet estimator for Γ(0) is: Γ (0) = (z t β > 0) (mi{( ut ) 2, (z t β ) 2 }) w t w t, where ut = t = yt x t β. If observatios are m-depedet ad m is a kow fiite iteger, a cosistet estimator for B 0 is: m B = Γ (0) + m j= [ Γ (j) + Γ (j) ], where we have used the fact that Γ ( j) = Γ (j), ad Γ (j) = [ t= j+ ft( β ) f t j ( β)], for j 0. I a geeral case, the autocorrelatio amog elemets of f t (β) may ot equal zero after a fiite umber of lags. Nevertheless, it is still possible to obtai useful estimators by usig the ear epoch depedece ad mixig coditios. I this case, we ca defie the heteroskedasticity ad autocorrelatio cosistet (HAC) covariace matrix estimator for B 0 as: B (p ) = Γ (0) + p j= ( j p + ) [ Γ (j) + Γ (j) ], where the weight ( j ) is suggested by Newey ad West (987) ad guaratees the p + positive semi-defiiteess of B. It is clear that as j icreases, the weights decrease. For the choice of the lag trucatio parameter p, it is ecessary to let p go to ifiity at some suitable rate. The rate give i the followig assumptio also follows Newey ad West (987). Assumptio TL: {p } is a sequece of itegers such that p as ad p =O( /4 ). Assumptio DM : The radom variables u t*, x t, ad w t are 4r-itegrable uiformly i t, for r>2, t =,2,... 8

21 Assumptio NE : The disturbace process {u t* } is ear epoch depedet with respect to {x t, v t } of size -4(r-) 2 /(r-2) 2 for some r>2. THEOREM 5.: Give assumptios E, P, ID, MX, TL, DM, ad NE, if β is a cosistet estimator of β 0, the the estimators G cosistet, G ( β ) p G 0 ad B (p ) ( β ) ad B (p ) defied above are weakly p B 0. Furthermore, if the weightig matrix is chose as [ B (p )] -, the SCGMM estimator will be relatively more efficiet. Based o Hase (982), the weightig matrix [ B (p )] - is called the optimal weightig matrix. It ca be show that the asymptotic covariace matrix based o the optimal weightig matrix is smaller tha that based o other weightig matrix. Corollary 5.2: If the weightig matrix is [ B (p )] -, the the SCGMM estimator β has a smaller variace tha that based o other weightig matrices with differet limits, ad the asymptotical distributio becomes [G 0 (B 0 ) - G 0 ] /2 ( β β 0 ) A N (0, I). I practice, two steps are eeded i order to obtai the optimal weightig matrix because it is ecessary to have a prelimiary cosistet estimate of the parameter before B (p ) ca be calculated. The prelimiary cosistet estimate ca be obtaied by usig ay arbitrary weightig matrix such as the idetity matrix. If the model is just idetified, the choice of weightig matrices is irrelevat, ad the asymptotic distributio has the same simple expressio as that with the optimal weightig matrix. With the SCGMM estimator, several specificatio tests are possible. A test of over- idetifyig restrictios ca follow Hase (982). It ca be show that F ( β ) B ( p) F ( β ) coverges i distributio to a chi-square distributed radom variable with l k degrees of freedom. A related test o the validity of the istrumets is also possible based o Newey (985) 9

22 by comparig the estimates usig all istrumets ad usig a subset of valid istrumets to costruct a Hausma type test statistic. VI. Simulatio Results I this sectio, a small scale simulatio study of a simple liear regressio model is coducted. The depedet variable y is geerated from the equatio y * t =α+βx t +u t ad y t =max(0, y * t ). I order to focus o the performace of the trimmig procedure coducted by the istrumets, the simulatio model is geerated by i.i.d. processes. The parameter vector has a itercept α ad a slope β. I order to create a correlatio betwee them, x t ad u t are geerated as u t =ε t +e t ad x t =π+λe t +v t, where ε t is geerated from i.i.d. stadard ormal distributio N(0,), ad v t assumes evely-spaced values i the iterval [-b, b] so that the variace Var(v t )= (i.e., b.7), ad e t takes o the repeated sequece {-,, -,, } ad Var(e t )=. I this case, E(e t )=0 ad E(v t )=0, ad the correlatio betwee v t, e t ad ε t is zero. These specificatios also imply that E(u t )=0 ad the regressio error u t is symmetrically distributed about zero. The istrumet z t is geerated as z t =δ+v t. The coditio that x t β z t β for all t requires that π δ-λe t. Give the values of e t, this coditio requires π δ+λ. I this case, the λ correlatio betwee x ad u is ρxu = 2 / 2. The correlatio betwee x ad z is 2 / 2 ( 2( λ + )) ρxz =. ( λ + ) As i ay IV estimatio, the istrumet should be correlated with the edogeous regressor, ad this correlatio affects the asymptotic variace of the SCGMM estimator. Both correlatio coefficiets are determied by parameter λ. Clearly, the regressio error u is ot ormally distributed, ad thus the Tobit procedure caot be used. Moreover, sice the regressor x is correlated with the error u, the SCLS is ot cosistet either. The implemetatio of the SCGMM estimatio requires miimizatio of the o-smooth criterio fuctio Q (β) over the parameter space Θ. Usual algorithms based o ucostraied parameter spaces geerally do ot work ad almost always result i solutios where z t β 0 for all 20

23 t. Therefore, based o the costrait o the parameter space, a mathematical programmig method is eeded for the computatio. The costrait requires that the average predicted values of z t β for observatios with positive y t are positive, ad thus it becomes essetially liear. This coditio is especially coveiet for computatio. 2 Additioally, the criterio fuctio itself is ot cotiuously differetiable, which causes problems for fidig the optimal value i the mathematical programmig process. Some approximatios are used to smooth the criterio fuctio. 3 I the simulatio, the weightig matrix used is the idetity matrix. Because the model is just-idetified, the choice of weightig matrix is immaterial. I the basic sceario, the parameters are defied as α=, β=, λ=, δ=0. I this case, the correlatio betwee x ad u is 0.5, ad betwee x ad z is 0.7, ad the cesorig proportio of y * t is 25%. I additio, about 78% of the values of (α+βz t ) are positive. As idicated i the sectios above, values of egative (α+βz t ) are dropped i the estimatio. For each simulatio desig, the true parameter values are reported alog with the sample mea, stadard deviatio (SD), root mea squared error (RMSE), media, ad media absolute errors (MAE). For compariso, the results based o Powell s SCLS estimator are also reported. 4 For each desig, the simulatio is repeated for 400 times. 5 The simulatio results are summarized i Table. I the simulatio, the software package GAMS (the Geeral Algebraic Modelig System) is used for computatio via its mathematical programmig solvers MINOS. OLS estimates based o the cesored value y t are used as startig values. Other software programs such as GAUSS also have mathematical programmig packages. 2 I the implemetatio, a small positive costat is used as the lower boud for the costrait istead of usig 0, because the mathematical programmig solver allows the costrait to reach the boud. The results are ot sesitive to the choice of this costat For example, the followig approximatios are used: max( x,0) = 0.5 ( x + ξ + x), 2 ( x > 0) = ( x + x), where ξ is a small costat. (2x +ξ ) 4 The SCLS estimator is calculated usig (2.3) as a recursio formula (Powell, 986), also with OLS estimates as startig values. 5 Sometimes, the solver fails to fid the iterior optimal solutio (for example, the costrait reaches the lower boud). I this case the correspodig estimates are discarded. Such cases do ot occur ofte. For example, i Desig, amog 400 repetitios, oly oe fails to fid optimal solutio; i Desig 2, every repetitio fids the optimal solutio. Whe the degree of cesorig icreases, such cases seem to icrease. 2

24 Oe feature of the tabulated results is the dramatic differece betwee the SCGMM estimates ad the SCLS estimates. As we have show, whe regressors are correlated with the error terms, the SCLS estimatio is icosistet. This is evidet i the table, ad the SCGMM performs much better tha does the SCLS. The results also show the fiite sample bias of the SCGMM estimator, but it is small. The sample mea ad media of the SCGMM estimator are very close to each other. Throughout the table, the performace of the SCGMM improves as the sample size icreases, measured by both SD ad RMSE. Whe the sample size decreases by 50 percet (Desig 2), the resultig stadard deviatio icreases by about 2, which is i accord with the asymptotic theory. I Desig 3 ad 4, the itercept α is chaged to zero. I this case, the cesorig proportio of y * t icreases to about 38%, ad the proportio of positive values of (α+βz t ) reduce to 50%. The icreased cesorig appears to have a strog effect o the performace of the SCGMM estimator. The mea bias icreases, ad so does the RMSE. I the Desig 5 ad 6, λ is chaged to 0.6. This chage icreases the correlatio betwee x ad z to about 86% ad reduces the correlatio betwee x ad u to about 37%. As expected, such a chage improves the performace of the SCGMM estimator. The results show a reduced mea bias, ad especially reduced stadard deviatio ad RMSE for the itercept estimate. Sice the regressio error cosists of a stadard ormal radom variable ad a biary radom variable (with the values - ad ), its distributio depeds o these two radom variables. I Desig 7 ad 8, the distributio of ε is chaged to N(0, 3) to chage the distributio of the regressio error. The resultig ew distributio has a larger variace of 4 (double the previous variace). As expected, the resultig SCGMM estimates show a relatively larger SD ad RMSE. It is impossible to completely characterize the fiite sample behavior of the SCGMM estimator uder the coditios imposed i the sectios above. However, the simulatio still 22

25 shows the relative performace for a few differet cases characterized by differet sample sizes, cesorig proportios, correlatios betwee the regressor, the regressio error, ad the istrumets, ad differet distributios of the error term. I ay case, the SCGMM estimator performs very well ad much better tha does the SCLS estimator. VII. Coclusio This paper proposes a semi-parametric SCGMM estimator for cesored models. The estimator is robust to o-ormality ad heteroskedasticity of ukow forms for the distributio of the regressio error, to the presece of edogeous explaatory variables, ad to models with depedet observatios. The SCGMM estimator is show to be cosistet ad asymptotically ormal. A HAC estimator of the covariace matrices is also provided. The SCGMM estimator also adopts a differet approach to the idetificatio of the true parameter by eforcig a costrait for the parameter space. The simulatio results demostrate that the SCGMM works very well, much better tha does the SCLS estimator i the presece of edogeous regressors. The SCGMM estimator eeds a stroger requiremet o the istrumets tha the usual GMM or IV estimatio. However, as discussed, the requiremet is ot as restrictive as it appears i practice. I fact, the requiremet that z t β 0 x t β 0 for all t ca be relaxed. I particular, suppose there is oly oe edogeous regressor x with the istrumet is z, ad if x z for a substatial proportio i the populatio, the the coditio x t β 0 z t β 0 holds for those realizatios (suppose the true value β 0 is o-egative). Other realizatios of x t ad z t such that x t β 0 <z t β 0 caot help to idetify the true value of β 0 ad thus ca be deleted. Therefore, i such a sample, the subsample with x t z t ca be used for estimatio. As the sample size goes to ifiity, the size of the subsample will also go to ifiity, ad the the true value β 0 ca be idetified. This is extremely useful whe it is difficult to fid istrumets that satisfyig coditio (2.3) for all t. 23

26 The SCGMM estimator still requires that the error distributio be symmetric. A test o the symmetry assumptio ca be doe based o Newey (987), usig some odd fuctio of f t (β) to costruct additioal momet coditios. I future research, it will be useful to relax the assumptio of symmetry ad to exted this approach to geeral sample selectio where the cesorig (selectig) idex may be differet from the depedet variable of the curret regressio equatio. 6 6 Hooré ad Powell (994) propose a pairwise differece estimator that does ot require the symmetry assumptio for the error distributio, but the error terms must be a i.i.d. process ad all regressors must be exogeous. 24

27 REFERENCES Amemiya, T. (985): "Advaced Ecoometrics," Harvard Uiversity Press, Cambridge, Massachusetts. Adrews, D.W.K. (984): No-strog Mixig Autoregressive Processes, Joural of Applied Probability, 2, Adrews, D.W.K. (993): A Itroductio to Ecoometric Applicatios of Empirical Process Theory for Depedet Radom Variables, Ecoometric Reviews, 2(2), Adrews, D.W.K. (994a): Empirical Process Methods i Ecoometrics, Hadbook of Ecoometrics, Volume IV. Adrews, D.W.K. (994b): Asymptotics for Semiparametric Ecoometric Models via Stochastic Equicotiuity, Ecoometrica, 62:, Bates, C. ad H. White, (985): A Uited Theory of Cosistet Estimatio for Parametric Models, Ecoometric Theory,, Butcher, Kristi F. ad Ae Case (994), The Effects of Siblig Compositio o Wome s Educatio ad Earigs, Quarterly Joural of Ecoomics, 09: Card, David (995), Usig Geographic Variatio i College Proximity to Estimate the Retur to Schoolig, i: Louis N. Christofides, E. Keeth Grat ad Robert Swidisky, eds., Aspects of Labour Market Behavior: Essays i Hoor of Joh Vaderkamp (Uiversity of Toroto, Caada), Davidso, R. ad J.G. MacKio (993): Estimatio ad Iferece i Ecoometrics, Oxford Uiversity Press, Chapter 7, Davidso, J. (994): Stochastic Limit Theory--A Itroductio for Ecoometricias, Oxford Uiversity Press. Gallat, R. ad H. White (988): A Uified Theory of Estimatio & Iferece for Noliear Dyamic Models, Basil Blackwell Ltd., 988. Goldberger, A.S. (983): Abormal Selectio Bias, Studies i Ecoometrics, Time Series ad Multivariate Statistics, ed. By S. Karli, et al. New York: Academic Press. Hase, L. P.(982): "Large Sample Properties of Geeralized Method of Momets Estimators," Ecoometrica, 50, Hog, Ha ad Elie Tamer (2003): Iferece i Cesored Models with Edogeous Regressors, Ecoometrica, Vol. 7, No. 3, Hooré, B. E. ad J.L. Powell (994): Pairwise Differece Estimators of Cesored ad Trucated Regressio Models, Joural of Ecoometrics, 64,

28 Horowitz, J. L. (986): A Distributio-Free Least Squares Estimator for Cesored Liear Regressio Models, Joural of Ecoometrics, 32, Lee, Myoug-Jae (992): Wisorized Mea Estimator for Cesored Regressio, Ecoometric Theory, 8, Levitt, Steve D. (997): Usig Electoral Cycles i Police Hirig to Estimate the effect of Police o Crime, America Ecoomic Review, 87, Lewbel, A. (998): Semiparametric Latet Variable Model Estimatio with Edogeous or Mismeasured Regressors, Ecoometrica 66, Newey, W.K. (985a): Semiparametric Estimatio of Limited Depedet Variable Models with Edogeous Explaatory Variables, Aales de L INSEE, 59/60, Newey, W.K. (985): Geeralized Method of Momets Specificatio Testig, Joural of Ecoometrics, 29, Newey, W. K. (987): Specificatio Tests for Distributio Assumptios i the Tobit Model, Joural of Ecoometrics, 34, Newey, W.K. ad K.D. West (987). A Simple, Positive Semi-defiite, heteroskedasticity ad Autocorrelatio Cosistet Covariace Matrix, Ecoometrica, 55, Newey, W.K. ad D. McFadde (994): Large Sample Estimatio ad Hypothesis Testig, Hadbook of Ecoometrics, Vol. IV, Chapter 36. Powell, J.L. (984): "Least Absolute Deviatios Estimatio for the Cesored Regressio Model," Joural of Ecoometrics, 25, Powell, J.L. (986): "Symmetrically Trimmed Least Squares Estimatio for Tobit Models", Ecoometrica, 54, Robiso, P.M. (982): O the Asymptotic Properties of Estimators of Models Cotaiig Limit Depedet Variables, Ecoometrica, 50 (), Ruppert, D. ad J.R. Carroll (980): Trimmed Least Squares Estimatio i the Liear Model, Joural of the America Statistical Associatio, 75:372, Wooldridge, J.M. ad H. White (988): Some Ivariace Priciples ad Cetral Limit Theorems for Depedet Heterogeeous Processes, Ecoometric Theory, 4, Wooldridge, Jeffrey M. (994): Estimatio ad Iferece for Depedet Process, Hadbook of Ecoometrics, Volume IV. Chapter 45,

29 Table Simulatio Results True Mea SD RMSE Media MAE Desig : T=400, λ=, cesorig=25%, positive (α+βz)=78%, ρ xz =0.7, ρ xu =0.5 SCGMM α β SCLS α β Desig 2: T=200, λ=, cesorig=25%, positive (α+βz)=78%, ρ xz =0.7, ρ xu =0.5 SCGMM α β SCLS α β Desig 3: T=400, λ=, cesorig=38%, positive (α+βz)=50%, ρ xz =0.7, ρ xu =0.5 SCGMM α β SCLS α β Desig 4: T=200, λ=, cesorig=38%, positive (α+βz)=50%, ρ xz =0.7, ρ xu =0.5 SCGMM α β SCLS α β