Semi Tobit

8/11/2019 Semi Tobit

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2 S . Ch en lJou rna l o f Economet r i cs 80 1997) 1 34

which is a Type-2 Tobit model. The labor supply model in Heckman (1974) isa variation of the Type-3 Tobit model. More recently, Udry (1994) proposed amore complicated sample selection model generalizing the st::ad~:~'d Type-3 Tobitmodel, in the context of informal credit transaction markets in determining theeffect of loans on risk sharing among households in Northern Nigeria.

Traditionally, estimation of sample selection models and other limited depen-dent variable models has been based on the maximum likelihood and otherlikelihood-based methods with the error distribution specified to be in a para-metric form known up to a finite-dimensional vector of parameters. As is wellknown, misspecification of the error distribution in sample selection models willin general render likelihood-based estimators inconsistent. Lee (1982) adopted arich parametric family to lessen the misspccifieation problem to a certain extent.Since a parametric form of error distribution cannot generally be justified byeconomic theory, much of recent econometrics literature has focused on semi-parametric estimation methods which only assume weak restrictions on the errordistribution to guard against possible misspeeifieation.

Formally, the Type-3 Tobit model is a two-equation model of the followingform:

l* = w,~o + ut, ( I )

y= : x f lo + t ,2 , (2)

where the first equation is the selection equation, the second equation is the mainequation, and the dependent variable y* can only be observed when the selec-tion variable I* is positive. This model reduces to the Type-2 Tobit model whenthe selection variable is only the sign of I*, in which case some exclusion re-striction on fl0 will be necessary ibr model identification I)nder the assumptionthat the error terms are independent of the regressors (see Chamberlain, 1986~Powell, 1989). Various semiparametric estimators for the Type-2 Tobit modelhave been proposed in Andrews (1991), Cosslett (1991), Gallant and Nychka(1987), Ichimura and Lee (1991), Newey (1988), and Powell (1989), among

others. It was shown by Lee (1994) that the information on the selection variableI* other than its sign makes the aforementioned exclusion restriction unnecessaryfor the model identification in the Type-3 Tobit model under the independencerestriction. Recently, Honor~ et al. (1992) and Lee (1994) proposed some semi-parametric estimators for the Type-3 Tobit model by exploiting this extra infor-mation on the selection equation. This article provides two new semiparametricestimators for fl0 under the independence condition. Our procedures differ fromtheirs in the way the selection bias is being corrected in the main equation.

Like those of Honor6 et al. and Lee, the estimators presented here are two-

step estimators where the selection equation is estimated first, and the estimatorobtained from that equation is used to eliminate the effect of the selection on thesecond equation. Our first estimator (henceforth called estimator I), constructed by


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C l i e n l d o u r n a l o f E c o n o m e tr ' s 8 0 ( 1 9 9 7 ) 1 - 3 4 3

first transforming the selection correction function into a constant term throughtrimming the original sample, is a simple least-squares-type estimator. For thesecond estimator (estimator lI), a Buckley-James-type approach is used to directlyestimate the selection correction function, with estimates of the parameters inthe selection equation given in the first step. in the second step, a weightedleast-squares-type procedure is performed by including the estimated selectioncorrection function as part of the regressors in the main equation. Our estimator IIis similar to Lee's (1994), except that in ours the estimated selection correctionfunction is not smooth and in Lee's it is. Hunor~ et al. (1992) proposed two-stepestimators by adopting a pairwise difference approach to eliminate the effect ofthe selection.

Semiparametric efficiency hounds can serve as a criterion for evaluating existingestimators and suggesting improved estimators. In this paper we also calculate thes~-miparametric efficiency bound for the Type-3 Tobit model under the indepen-dence restriction. For the Type-2 Tobit model under the independence restriction(Chamberlain, 1987), a nonsingular information matrix requires some exclusionrestriction in the main equation and the existence of a continuous regressor inthe selection equation. However, neither requirement is needed to ensure a non-singular information matrix for the Type-3 Tobit model.

The next section describes the model and discusses the e~timators and theirmotivation. Large sample properties of the estimators are investigated in Sec-tion 3. Section 4 contains a limited Monte Carlo study to investigate practicalperformance of the proposed estimators. In Section 5 we provide the semiparamet-ric efficiency bound for the Type-3 Tobit model under the independence condition.Section 6 concludes the paper with some discussions. The proofs of theorems arein the appendices.

2 . The model a n d e s t i m a t o r s

We consider estimation of the parameter vector//0 of the Type-3 Tobit model

defined by the latent variables I* and y* which are of the form

(3)

(4)

r =w~0 + Ul,

y= = x~ 0 + u2 .

Instead of I* and y*, we observe 1 and y which satisfy

I = m a x l * , 0 ) ,

y = y* I II>0t,

(5)

6)

where llat represents the indicator function of the event A, d= Iff>0t is thebinary selection indicator, y and 1 are the observable dependent variables, x and


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4 S . C h e n l J o u r n a l o f E c o n o m e z r ic s 8 0 1 9 9 7 ) 1 - 3 4

w a r e r o w v e c t o r s o f e x o g e n o u s v a r ia b l e s w i th d i m e n s i o n s k a n d p , r e s p e c ti v e l y,flo a n d 6 o a r e c o n f o r m a b l e c o l u m n v e c t o r s o f u n k n o w n p a r a m e t e rs , a n d u l a n du2 a re e r r o r t e rm s , x a n d w a r e a l lo w e d t o h a v e c o m m o n c o m p o n e n t s . L e t z b e av e c t o r c o n s is t i n g o f t h e d i s ti n c t c o m p o n e n t s i n ( w, x ) . H e r e y * i s o b s e r v a b l e o n l yw h e n d = 1 ; h e n c e , t h e s a m p l e i s s e le c t e d a c c o r d i n g t o t h e v a l u e s o f w a n d u ~.I n t h i s p a p e r w e p r o p o s e t w o e s t i m a t o r s o f flo u n d e r t h e r e s t r ic t i o n t h a t (u ~ , u 2 )i s i n d e p e n d e n t o f z . A s u s u a l , t h e i n d e p e n d e n c e r e s t ri c t io n c a n o n l y f a c i li t a tei d e n t i fi c a ti o n a n d e s O m a t i o n f o r t h e s l o p e p a r a m e t e r s . T h u s , i t i s a s s u m e d t h a t zd o e s n o t c o n t a i n a c o n s t an t t e rm . C o n d i t io n a l o n z a n d o n y * b e i n g o b s e r v a b l e ,t h e r e g r e s s i o n f u n c t i o n o f y i s

E ( y [ i > O, z ) --- xflo + E u21l> 0 , z ) . ( 7 )

T h e m a i n d i f f i cu l t y w i t h e s t i m a t i n g flo is t h e s e l e c t i o n c o r r e c t i o n t e r m E (u 2 1 ul >w ~ o , z ) b e i n g a n o n d e g e n e r a t e r a n d o m v a r i a b l e b e c a u s e u l i s r a n d o m l y c e n -

s o r e d b y - w O o . C o n s e q u e n t l y, t h e re w i13 b e n o n z e r o c o r r e l a ti o n b e t w e e n t h er e g re s s o r s a n d e r r o r t e r m i n th e m a i n e q u a t io n f o r th e s e l e c te d s u b s a m p l e w h e nt h e r e is d e p e n d e n c e b e t w e e n e r r o r t e r m s a c r o s s t h e t w o e q u a t io n s , w h i c h i ntu rn , causes t he l , , . a s t - squa res e s t ima toc fo r f l o app l i ed tot h e s e l e c t e d s a m p l et o b e i n c o n s i s t e n t . I n t h i s p a p e r w e p r o p o s e t w o w a y s t o d e a l w i t h t h i s i n -c o n s i s t e n c y p r o b l e m a s t h e r e s u lt o f t h e p r e s e n c e o f th e s e l e c t io n c o r r e c t i o nt e r m .

T h e i d e a b e h i n d o u r e s t i m a t o r 1 i s t o t ri m t h e o r i g i n a l s a m p l e s o t h a t th es e l e c t i o n c o r r e c t i o n t e r m i n t h e t r i m m e d s u b s a m p l e i s e q u a l t o a c o n s t a n t . C o n -s e q u e n t l y, t h e n e t e f f e ct o n t h e r e s u l t i n g s u b s a m p l e o f t h e s e l e c t i o n a n d t r i m m i n gp r o c e s s i s o n l y t o s h i f t t h e i n t e r c e p t t e r m i n t h e m a i n e q u a t i o n w i t h o u t a f f e c t i n gt h e s l o p e p a r a m e t e rs . T h e r e f o r e , th e l e a s t- s q u ar e s b a s e d o n t h e t r i m m e d s u b s a m -p i e y i e l d s c o n s i s t e n t e s t i m a t e s f o r t h e s l o p e p a r a m e t e r s i n t h e m a i n e q u a t i o n .S p e c i fi c a ll y, w e c a n c r e a te a c o n s t a n t c e n s o r i n g p o i n t, s a y z e r o , i n s te a d o f - w O of o r u j t h r o u g h t r i m m i n g t h e o r i g i n a l s a m p l e s o t h a t t h e c o n d i t i o n a l e x p e c t a t i o no f t h e e r r o r t e r m u2 i n t h e t r i m m e d s a m p l e i s a c o n s t a n t . A f t e r th e t r i m m i n g , t h e

r e s u lt i n g n e w s a m p l e c o n t a i n s o b s e r v a t io n s w i t h u j > 0 a n d w o o > 0 . I f ( u b u 2 ) i si n d e p e n d e n t o f z , t h e n t h e " n e w e r r o r te r m " u ~= u 2 g i v en [(U l > 0 , w O o > 0 ) a n dd = t , z ] w i l l h a v e a c o n s t a n t m e a n , c o n d i t i o n a l o n t h e r e g r e s s o r s z , i .e ., a o - -E u2lz , u l> 0 , w O o > 0 , d = 1)=E u21u+ > 0 ) i s a c o n s t a n t u n d e r t h e a s s u m p t i o nt h a t t h e r e g r e s s o r s a n d e r r o r t e r m s a r e i n d e p e n d e n t . T h e r e f o r e , t h e r e g r e s s i o nf u n c t i o n f o r t h e t ri m m e d s a m p l e i s

E ( y [z , u , > 0 , w O o > O , d = I ) = E ( y lz , u l > 0 , w O o > O , ul + w O o > O )

= E ( y l z , u l > 0 , w / i o > O )

= E ( y l u t > O , z ) = x f l o ao ( 8 )


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S. Chen/Journal of Econometrics 80 r997) 1-34 5

w h i c h s u g g e s t s a s i m p l e le a s t- s q u a re s p r o c e d u r e a p p l i e d t o t h e t r i m m e d s u b s a m p l et o e s t i m a t e / / o b y

~ , : ar g ra in 1 ~ {,~,,>o.w,~>0)(Yi - x i / / - u)2,/~.z rt i= l ( 9 )

where ~1~= l ~ - w ~ i s an es t im a te fo r ul~ , and 6 i s a roo t -n con s i s t en t e s t im a to r o f/~0 in a f i r st s t ep . No te tha t und er so m e m i ld con d i t ions (d i scu ssed b e low ) , x w i l lb e o f f u ll ra n k in th e s u b s a m p l e fo r w h i c h u l > 0 , w S 0 > 0 e v e ni f x = w ; t hus , theexc lus ion res t r i c t ion a l luded to above i s no t necessa ry fo r mode l iden t i f i ca t ion .Ins tead o f r e s tr i c ting u l to the in te rva l (0 ,oo ) , w e can a l so cons ide r K d i ffe ren ts u b s a m p l e s f o r w h i c hu l E ck - - l ,Ck) ,Wi60> - -C~- - I )h o l d s i n t h e t h s u b s a m p l e ,a n d c o < c t < . < c K , t h e n t h e c o n d i t i o n a l m e a n o f t h e e r ro r t e r m u 2 in t h e k t h

subsample i s a coi l s tant , i . e .~k=E(uelz, ek>U,>Ck-,, w6o> Ct bd= 1)=E u2}ck > ui > c k - i ) , a n d

E ( y l z , ck > u l> C , - 1 , W 6 0 > - - C k - - h d= 1 ) =X/~o + Z, . ( I 0 )

S im i la r ly, a new leas t - squares - type es tima to r /~p l fo r /~o can be de f ined by p oo l ingthe K sub samp les :

/~ p , = a r gm i n l~ - ]~ E d il{c> a :.~ ,.,,w ~ .~ _( t l )N o t e t h a t m o r e o b s e r v a t io n s w i ll b e i n c l u d e d i n d e f in i n g ~p , t h a n t h o s e u s e d i n

c o n s t r u c t in g ~ t i f w e s e t c 0 < 0 a n d c K m o d e r a t e l y la rg e . T h e r e f o r e , in g e n e r a l,~pl w ou ld b e mo re e ff i c ien t i f t he sam ple s i ze i s l a rge enou gh to o ff se t the losso f d e g r e e s o f f r e e d o m t a s a r e s u l t o f e s ti m a t i n g K i n te r c ep t t e r m s ( ~ k ) f o rk = 1 , 2 . . . . K .

On e exam ple o f an es t im a to r fo r 60 is a pa i rwis d i ffe renced es t ima to r inH o n o r 6 a n d P o w e i ( 1 9 9 4 ) :

= arg i.n in ~ (1 -- l{ i,< max{0,O~,-w,)6),i~


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6 S. ChenlJo urnal of Economelrics 80 1997) 1-34

F o r o u r e s t i m a t o r l I , i n s t e a d o f t r i m m i n g o b s e r v a t i o n s t o t r a n s f o r m t h e s e -l ec t ion cor rec t ion t e rm in to a cons tan t , we es t ima te the se lec t ion cor rec t ion t e rmE ( u 2 lu t > - w ~ o . z ) d i r e c t ly f o r e a c h o b s e r v at i o n ; t h e n w e p r o p o s e a l e a st -s q u a r es -t y p e p r o c e d u r e b y i n c l u d i n g t h e e s t i m a t e d s e l ec t io n c o r r e c ti o n t e r m a s p a r t o f th eregressor s in the m a in equa t ion . M ore spec i f i ca lly, w e con s t ruc t

nl n ^

E t z i , f l ) = - ~ Y ~-~ i D q ( 6 ) ( y j - - x j f l )n ~ (12 )~ i . O i A )--

t o e s ti m a t e E ( y - x f l l u l , > - w, ~ 0 , z ~ ) , w h e r e 6 is a fi rs t- st ep p r e l i m i n a r y e s ti m a t o ro f / i 0 ,

Dq(tS)---- I{t,>O,,,-w,)o>o} = I {6 -,v~ 6> -w ,& -w ,~-w ,6}

a n d t h e s a m p l e a v e r a g e i n ( 1 2 ) i s t a k e n o v e r t h e o b s e r v a t i o n s f o r w h i c h u t j >- w i r o , - w i i ~ o > - w j , ~ o i f Dq(c~) i s replaced by Di2(~0)- By the law o f la rgenumbers , I~ (z ; , f l ) w i l l be cons i s t en t fo r

E [ ( y - x f l ) l {,,, >-,,',~0,,,'60>,~ ,~ }] = E [(y - xB )iu l > - w i~ 0, Wro >w i ~ o , z i ]E [ I {,,, > _ ,,. ~.,,.~o > ,~,oo} ]

w hich wi l l be the i th se lec t ion cor rec t ion t e rm w hen f l = flo unde r the con d i -t i o n t h a t t h e e r r o r t e n n s a n d r e g r e s s o r s a r e i n d e p e n d e n t . O u r p r o p o s e d s e c o n des t im a t ion a te hod i s a we ig h ted sem iparam et r i c t~as t -squares approach

argm i , n 2 . , y, x , f l E z , , f l ) , d ~m T. ( 1 3 )p n t = l

w h e r e rn i = ( l / ( n - I ) ) ~ '~ j # i D i J (~ ) i s a w e i g h t f u n c t i o n . T h is w e i g h t i n g s c h e m eis s imi la r in sp i r i t t o the den s i ty we igh t ing in Po wel l e t a l . (1989 ) , to avo id the e r-ra ti c behav io r a s so c ia ted wi th sm al l va lues o f the r an do m den om ina to r o f l~ (z,.,f l )a n d t o s i m p l i f y t h e t e c h n i c a l a n a l y s i s . F u r t h e r d i s c u s s i o n s o n t h e m o t i v a t i o n a n diden t i f i ca t ion con d i t ions r e l a t ed to the w e igh ted sem iparamet r i e l eas t- squares e s -t i m a t i o n a p p r o a c h a r e a v a i la b l e i n L e e ( 1 9 9 4 ) . I n c o n t ra s t , b y i m p o s i n g s o m es t r o n g s m o o t h n e s s c o n d i t i o n s o n t h e d i s tr i b u t io n o f t h e r eg r e s so r s , L e e ( 1 9 9 4 )p r o p o s e d a s e m i p a r a m e t r i e l e a s t - s q u a r e s e s t i m a t o r b y s o l v i n g

m in _ i ~ l w ( w D ( y i - x i f l - E n i) 2d i, ( 1 4 )

wh ere E~,. i s a sm oo thed ve r s ion o f f :. (z , f l) , andl w ( ) i s a f ixed t r imming func-t io n . N o t i c e th a t w e u s e t h e w e i g h t f u n c t i o n ra,. t o d e a l w i t h t h e r a n d o m d e n o m -

i n a t o r o f E ( z ,, f l) , w h i l e L e e a d o p t e d a f i x e d t r i m m i n g a p p r o a c h t o e n s u r e t h a tt he r a n d o m d e n o m i n a t o r o f E~i is u n i f o r m l y b o u n d e d a w a y f ro m z e r o a s y m p t o t -i c a l l y. F o r L e e ' s m e t h o d , i t i s n e c e s s a r y t o c h o o s e s o m e s m o o t h i n g p a r a m e t e r s


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S. ChenlJournal of Econometrics 80 1997) 1-34 7

in cons t ruc ting E .~ . Also the f ixed t r imm ing approach cou ld cause som e loss o fe ff ic iency, espec ia l ly w hen the d im ens ion o f w i s h igh . Recent ly, Honor6 e t a l .( 99 2) propo sed pa irwise differenced es t imators /~hl and f lh2 such that

h l : a r g r n ~ n E ( l - - l{l ,0}(Yi - xi fl - ct)2,

T n : ~ i : lwhere ~ is a prel iminary es t imator of g0 in the f i rs t s tep.

We make the fo l lowing assumpt ions :

Assumpt ion 3 .1(Random sampl ing) . The vec tors( l~ ,w~,x~,y~)sat isfying (3)and (4) are independent and ident ical ly dis t r ibuted across i .

The assumpt ion tha t observa t ions a re independent o f each o ther i s made forconvenience ; i t might be poss ib le to a l low for dependent da ta . The iden t ica l

d i st ribu tion condi t ion , how ever, i s essen t ia l fo r /~z and th e es t imators o f Lee andHonor~ et a l . , s ince they e i ther involve cross observat ion comparisons ( for /~hland /~h2) or cross obse rvat ion es t imat ions ( in def ining.(zi, ff)and En,). For /~l


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8 s. Chen Journal of Econometrics 80 (1997) 1-34

t h e e s s e n t ia l r e q u i r e m e n t i s t h e c o n d i t i o n t h a tE uill)i>O)i s a c o n s t a n t a c r o s so b s e r v a t i o n s .

A s s u m p t i o n 3 . 2 .T h e e r r o r t e r m ( u t , u 2 ) is i n d e p e n d e n t o f t h e r eg r e s s o r s( x , w ) ,a n d ( u l , u 2 ) is c o n t i n u o u s l y d i s t r ib u t e d w i t h i ts d e n si U : h a v i n g u n i f o r m l y b o u n d e dp a r t i a l d e r i v a t i v e s .

W h e n x i n c l u d e s e n d o g e n o u s v a r i a b l e s , a ll t h e e s t im a t o r s e x c e p t ~ hl c a n b ee a s i l y m o d i f i e d t o a l l o w i n s t n nn e n ~ a l v a r i a b l e a p p r o a c h e s . H o w e v e r , a s i m i l a rm o d i f i c a t i o n i s l e s s o b v i o u s f o r / ~ h ~ -

L et ~ = ( I , x ) an d Z(/)) = Es~(u2~ - ~o)d~ 1 ( t, >w,a>o}-

A s s u m p t i o n 3 . 3E a c h e l e m e n t o fZ ( b ) i:~ dif fer en tia ble at 6o.

A s s u m p t i o n 3 . 3 T h e v e c t o r o f r e g r e s s o r s i n t h e s e l e c ti o n e q u a t i o n s a t is f i esP(wdi0 = 0 ) = O.

A s s u m p t i o n 3 . 4( M o m e n t c o n d i t i o n )E[llzl]43 a n d E[ll.vll 4]a r e f i n i t e , w h e r eI1" IId e n o t e s t h e u s u a l E u c l i d e a n n o r m .

A s s u m p t i o n 3 . 5 ( P r e l i m i n a r y e s t i m a t o r ) T h e p r e l i m i n a r y e s t i m a t o r t~ o f 60 isW - c o n s i s t e n t , a n d i t h a s a n a s y m p t o t i c l i n ea r re p r e s e n t at i o n

v:~(~ - ~o) = ~ ~ ~, + oo(l ), (15 )V tt i=l

w h e r e q~ i : -O( l i , w i ) ,E t ~ i = 0 , a n d EIt~II 2 is f ini te.

A s s u m p t i o n 3 . 6( I d e n t i f i c a t io n ) T h e m a t r i xE [s sil {t ,>,,~0>0}] is no ns in gu la r.

A s s u m p t i o n 3 .3 i s m a d e t o e n s u r e a v al i d Ta y l o r s e ri e s e x p a n s i o n w h e n e x p e c -t a t io n i s f ir st t a k e n. A m o r e p r i m i t i v e c o n d i t i o n i s A s s u m p t i o n 3 . 3 '. F o r a g i v e nf i r st ; st e p e s t i m a t o r ~ f o r 6 0, t h e o b s e r v a t i o n s i n t h e t r i m m e d s u b s a m p l e s a t i s fy( w i 6 > 0 ) . A s s u m p t i o n 3 . 3 ' is m a d e t o e n s u r e t h at th e t r i m m i n g c r i te r io n(w~6 > O)

i s a s y m p t o t i c a l l y e q u i v a l e n t t o t h e c o n d i t i o n (w ~(~0 > 0 ) . I n f a c t, A s s u m p t i o n s3 .2 , 3 . 3 ' a n d t h e M o m e n t C o n d i t i o ~ 3 .4 a r e s u f fi c ie n t f o r A s s u m p t i o n 3 .3 . 2A s s u m p t i o n 3 . 3 ' h o l d s w h e n w t~ o h a s d e n s i t y w i t h r e s p e c t to L e b e s g u e m e a s u r e .

2To show this, letO(uhu2) be the density function of 0q,u2) with respect to Lebesguemeasure. Let E.- denote the expectation operator with respect to z. Define Z'(~l,&e)=Es~(u2 --~)dl(l>,,.a~.,,s=>o}. Then from Assumption 3.2 and the definitions o f ~ and Zffi), we haveZ (b )- -Z *( 6, b) and ~*(/io,/~)----0. Consequently, by Assumptions 3.2, 3.3 J and 3.4

i.(6o + zlg) - z(~o) = x'( 6o + zl~,6o + zl&) - Z'(~o ,~o + d~i)o

= E: | | , ,~6o+~: i )>O}sJwf f (u 2 - - ~to)g(tq,u2)du du2w~O


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S. Chen.iJournal o f Econometrics 80 1997) 1-34 9

F u r t h e r m o r e , A s s u m p t i o n 3 . 3 ' d o e s n o t r u l e o u t t h e c a s e t h a t a l l t h e c o m p o n e n t so f w a r e d i s c re t e, i n t h i s c a s e i f w i s o n e - d i m e n s i o n a l , t h e n A s s u m p t i o n 3 . 3 ~ho lds i f P (w = 0 ) - - 0 . Ass um pt ion 3 .5 a s su m es a f i r s t- s t ep e s t im a to r ~ i s ava i l ab l e .S o m e e x a m p l e s o f ~ s a t is f y i n g A s s u m p t i o n 3 . 5 a r e li s te d i n t h e p r e c e d i n g s e c t i o n .S i n c e P (u t > 0 ) > 0 h o l d s g e n e r a ll y, t h e n t h e I d e n t if i ca t io n C o n d i t i o n 3 .6 re q u i r e st h at ( l , x ) b e o f f u ll r a n k i n t h e h al f -s p a c e { w , w ~ 0 > 0 } , w h i c h i n t u rn , h o l d sg e n e r a l l y w i t h o u t e x c l u s i o n r e s t r i c t i o n i m p o s e d o n t h e r e g r e s s o r s . I n t h e c a s eo f a s ing le r eg re s so r an d x = w, t hen the fu ll r ank con d i t i on i s s a t is f i ed i f t hec o n d i ti o n a l v a r ia n c e o f x g i v e n x > 0 i s n o n z e r o . L e t ~ - - ( ~ , / ~ ) ' . N o t e t h at t h ee s ti m a to r ~ o f ~ r o - - ( ~ , ~ ) ' c a n b e e x p re s s ed in a c l o se d fo r m .

T h e o r e m 1. U n d e r A s s u m p t i o n s3 . - 3 . 6 , ~ i s a cons i s t en t e s t im a to r o f ~ .o , an d

x/~(~ - no) N(O ,X, ) ,w h e r e

Xt - - A , - ' (C ;.a. + C ) . , f~ + [2 , C~ . + ~ 2 1 C ~ )A~ -1 ,

w i t h

~ t = ~ 3 / .( ~ o )

At = Fa~s~di1U, >w, ao>0~

a n d

i = s [ ( u 2 i- - ~ o ) d i I i f , > w, a 0 > o } ,

C~,~. = E 2 2 ',w i t h C ~ , C ~ , C ~ . a n a l o g o u s l y d e fi ne d .

P roof . S e e A p p e n d i x A .

L i k e t h e O L S e s t i m a t o r f o r t h e l i n e a r r e g r e s s i o n m o d e l , r~ c a n b e e x p r e s s e d i na c l o s e d f o r m , t h e r e f o r e i ts a s y m p t o t i c n o r m a l i t y c a n b e p r o v e d d i r e c t l y w i t h o u t

showing i t s cons i s t ency f i r s t .O n c e t h e a s y m p t o t i c p r o p e r t i e s o f ~ a r e d e t e rm i n e d , i t is e a s y t o in _ 'et : h o s e o f^

]~ l, s ince lff i s a sub ve c to r o f ~ . L e t J = (0 ,1kk) , w he re 0 i s a nu l l v ec to r and Ikk

i s t h e k - d i m e n s i o n a l i d e n t it y m a t r ix . F r o m T h e o r e m I , w e h a v e ~ ( ~ I /0 ) dN ( 0 , J Z I J ' ) u n d e r A s s u m p ti o ns 3 .1 - 3 .6 .

-~ IE~ I {~,,~o> o} s'w ]f (ut - ~o)~(O,u~)du2,1~ + o(,~a)H e n c e X ( / i )is different/able ~ o -


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10 S . C h e n l J o u r n a l o f E c o n o m e tr ic s 8 0 1 9 9 7 ) 1 - 3 4

F o r t h e p o o l i n g e s t i m a t o r ~pl ' l e t d d k ( f ) = l { c ~ > t _ ) ~ 6 > ~ ; _ , . , ~ . 6 > _ ~ , _ , }f o rS ' e6 '~ x - ~Kk = I . . . . . K . L e ts x ( 6 ) = ( d d l ( 6 ) . . . . . d dK (6 ),x ~- ]~ K = td d k ( 6 ) ) , ) .K = K~ 0] 2_~t,=)

d d ~ . ( 6 o ) ( u 2 - o i l ) ,and Z K (6 )= Es~(6)Y'j~kx=id d k ( 6 ) ( u 2 - ~ k ) .We d e f i n e

~zt~(60)O x - 06 '

A K = Es~(6o)SK (30 ) .

Def ine - -vx s imi l a r t o ,S t by r ep lac ing l ] l , A i and ), byO K , A Kand 2x - , r e spec -t i ve ly. Le t nx = (~ t . . . . . 0 t~ ,/~o)'; t hen un de r s imi l a r r egu la r i t y co nd i t i on s i t i ss t r a igh t fo rward to show the e s t ima to r ~ t c fo r nK i s a sympto t i c no rma l w i th Z 'Ka s th e a s y m p t o t ' c c o v a r i a , c e m a t r i x b y t h e s a m e s e t o f a rg u m e n t s . A g a i n i t i s

e a s y t o d e r i v e t h e a s y m p t o t i c p r o p e r ti e s o f /~ p l s i n c e / ~ i s a s u b v e c t o r o fn K .

3 .2 . E s t i m a t o r 1 I

In th i s subsec t ion , we de r ive the l a rge sam ple p rope r t i e s o f e s t im a to r I I , f12,of [ io . Le t

zl ( 6 ) = E d j D i j ( 6 ) D i k ( 6 ) ( x i - x k ) ( x i - x ) )

a n d

D ( 6 ) = E d i D i j ( 6 ) D i ~ ( 6 ) ( u 2 i - u 2y ) ( x i - x k )

f o r i # j # k . T h e f o l l o w i n g a d d i ti o n a l a s s u m p t i o n s a re m a d e .

A s s u m p t i o n3. Z Eac h e le m en t o f z2(6) i s d i ffe ren t iab le a t 6 i0-

A s s u m p t i o n 3 . 7 .F o r o b s e r v a t i o n s i n d e x e d b y i a n d j w i t h i # j u n d e r r a n d o msam plin g, P( w iro = u5-3o, wi # wj) = 0.

A s s u m p t i o n 3 . 8 .T h e m a t r i x r t ( 6 o ) =E [ d i D i y ( 6 o ) D i k ( ~ o ) ( x i - x k ) ( x i - x j ) ]f o ri # j # k i s nons ingu la r.

S i m i l a r t o A s s u m p t i o n 3 . 3 , A s s u m p t i o n 3 . 7 i s a s m o o t h n e s s c o n d i t i o n . Wec a n s h o w a s i n lc ,~ t n ot e 2 t h a t A s s u m p t i o n s 3 ,2 , 3 .4 a n d 3 . 7 ' i m p l y A s s u m p -t ion 3 .7 . N o te t ha t t he o bse rv a t ions used to con s t ruc t I~ .(z i,/~ ) i n (12 ) s a t i s fy0 v j ~ > w i ~ ) . G i v e n A s s u m p t i o ~ 3 . 7 ' t h i s r e q u i r e m e n t w i l l b e a s y m p t o t i c a l ly e q u i v -a l en t t o t h e c o n d i t i o n ( ~ 6 o >w i $o ). A s s u m p t i o n 3 . 7 ' d o e s n o t r u l e o u t t h e e a s etha t a l l t he componen t s i n w a re d i sc re t e . When w i s one -d imens iona l , t henA s s u m p t i o n 3 . 7 ' h o l d s t r i v i a l l y. L i k e A s s u m p t i o n 3 . 6 , A s s u m p t i o n 3 , 8 i s a ni d e n t i f i c a t i o n c o n d i t i o n , z l ( 6 o ) c a n b e r e w r i t t e n a s E { d i [ x i -E ( x l w r o > w i 6 o ) ]Ix ; - E (x lwr0 > wi ro ) ] [ P (m > - w,60 , wra > wi ro )]2 } ; t he re fo re o u r i den t i fi ca t ion


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s. ChenlJourn al o f Econometrics 80 (1997) -3 4 I I

c o n d i t i o n i s e ss e n t ia l ly e q u i v a l e n t t o t h a t o f L e e 3 ( A s s u m p t i o n 5 i n L e e , 1 9 9 4 ) .I n g e n e r a l , A s s u m p t i o n 3 . 8 d o e s n o t r e q u i r e a n y e x c l u s i o n r e s t r i c t i o n o n t h ereg res so r s . See Lee (19 94 ) f o r fu r the r d i scuss ions . In t he case tha t t he e r ro rt e r m u~ h a s p o s i t i v e d e n s i ty e v e r y w h e r e , th e i d e n t if i c at i o n c o n d i t i o n s r e q u i r ex~ - E (x dw ~6 0> 0 ) t o be o f fu ll r ank fo r f it andxi - E(xtlw~5o>w ~5o)t o b eo f f u l l r a n k f o r / ~ 2 a n d L e e ' s e s t i m a t o r, c o m p a r e d w i t h a s l i g h t l y w e a k e r c o n d i -t i on tha t x i - Ex~ be o f fu l l r ank fo r i den t i f ica t ion fo r t he e s t im a to r s i n H ono r6e t a l . F i n a l l y, o n l y L e e ' s e s t i m a t o r i n v o l v e s s m o o t h i n g , w h i c h , i n t u r n , r e q u i r e ss t ro n g s m o o t h n e s s c o n d i t io n s ; i n c o n s t r a s t, b o t h o f o u r e s ti m a t o r s a n d t h o s e i nH o n o r ~ e t a l. r e q u i r e m i n i m a l s m o o t h n e s s c o n d i t i o n s .

T h e o r e m 2 . f f A s s u m p t i o n s3.1 , 3 .2 , 3 .4 , 3 .5a n d 3.7 , 3 .8 hold, then f12 is ac o n s i s t en t e s t i m a t o r o f flo , a n d

~ (/ i2 - f lo) N(0, Z2) ,

w h e r et I

Z2 = A z1(C ,m + C,l$~r2 +Q2C0,1 + Q2 C0 $Q 2)A 2

w i t h

f~2 := 03 ' '

A2 = Ed iDo(6o)D~ k(~o)(x i - x~ ) ' (x t- :9 )an d ~h i s de f ined by( 5 1 ) i n t he p roo f . C )p l = E~hPl~, w i th C ~ an d C ~ ana logo us lydefined.

P r o o f . S e e A p p e n d i x A .

T h e c o n s i s t en c y a n d a s y m p t o t i c n o r m a l i ty o f /~ 2 is p r o v e d b y m a k i n g u s e o ft h e r e c e n t d e v e l o p m e n t s in U p r o c e s s e s ( se e , f o r e x a m p l e , A r c o n e s a n dGif ie1993) .

S im i l a r ly, t he pa i rwi se d i f f e r enced e s t ima to r ] ?h2 in H ono r6 e t a l . ( 199 2) cana l s o b e e x p r e s s e d i n t e r m s o f U - st a ti s ti c s. I ts c o n s i s t e n c y a n d a s y m p t o t i c n o r m a l -i t y c a n b e s h o w n b y f o l l o w i n g t h e s a m e l in e o f r e a s o n i n g u n d e r s i m i l a r re g u l a r i tycond i t i ons .

In o rde r fo r l a rge sample in fe rence on f l0 t o be ca r r i ed ou t u s ing the e s t ima to r s~1, ]~2, ]~pl,a n d f ib 2 , c o n s i s t e n t e s t im a t o r s o f t h e a s y m p t o t i c c o v a r i a n c e m a t r ic e so f t h e s e e s ti m a t o r s n e e d t o b e p r o v i d e d . C o n s i d e r, f o r e x a m p l e , e s t im a t i n g X t .

3 I n fa ct, t h e f ix e d t r im m i n g s e t W i n L e e i s c o n s h o c te d s uc h t h a t P ( u t > - - w , ~ o , w & > w v ~ o )is bounded away from zero for 1)~(EW. Therefore, our identification cond ition is imp lied by tha.)eFLee.


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12 S. Chen lJournal of Econometrics 80 1997) 1-34

A cons i s t en t e s t ima to r o f Z 'l can be eas i ly ob ta ined i f we can f ind cons i s ten tes t imators of ~21,C ; ; , C ~ , , C ~ a n d A l . O b v i o u s l y,

/ / l 1 ~ ,= - s i s i i ~ l , > w, ~ > o } d

il

i s a cons i s t en t e s t ima to r o f Al . I t fo l lows d i rec t ly f rom Pakes and Po l l a rd (1989 ,pp . 104 3-10 44) tha t one cons i s t en t e s t ima to r o f I21 i s a numer ica l de r iva t ivees t ima to r ~ t

~ = ~ ~'~ (h ,( ~ . oi. ~ +e , n q u ) - h,(,, ~. ~ --e , . q , y ) )2neln i=l

f o r t h e j t h c o l u m n o f ~ t , w h e r e

q l j i s the un i t vec to r wi th 1 in i ts j t h p lace ande ln c o n v e rg e s t o z e r o a s s a m p l es ize increases wi thn - ~ / ~ e ~ ~= Op( ).

Cons i s t en t e s t ima t ion o f the ma t r i ces C6~ . ,C~6 and C;.;, requi res su i table es-t ima tes o f the com pon en t s {2 i} , {4'i} in the a s ym pto t i ca l ly linea r r ep resen ta t ionof ~ g iven in (48) . Speci f ica l ly, i t i s useful to assume tha t sequences {q~i} ,{) . /}ex i s t such tha t

_l ~ lid , - ~/112 = op(I ),r / i= l

1 ~ I1~.~ ~.,I]2 : o p ( l ) .F l i l

O ne ex am ple o f {q~i} i s sugges ted in Honor~ and Pow el l (1994) . An ana log oussequence o f { ) . i } can be cons t ruc ted as

, .~ = s i 1{t, >,,.,~>o} (Yi --s i fz )d i .

W ith the sequ ences {q~ i} and {2 ,} g iven , the r emain ing com pon en t s o f the a sym p-to t ic cov ar iance m at r ix o f Z l , nam ely, C~.s.,C~_ and C , can be cons i s t en t lyes t ima ted as in Powel l (1989) . Def ine

1 _~n -.i

w ith an alo go us defin i t ions o f C~.~. and C ~. i t is ea sy to sh ow ihat ~.~ , C'~.~.an d C'~ are con~ ~tent for C~.~, C~.~. an d C~4, , respe ctively . B y fol lo w ing th e sam eapp roach, i t is not d i fficul t to cons t ruct cons is tent es t imators for the asym ptot iccovariance matrices of/~pL,i~2 and /~h2-


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S . C h e n I J o u r n a l o f E o l to m e t r ic s 8 0 1 9 9 7 ) 1 - 3 4 [3

4 . F i n i te s a m p l e b e h a v i or

In th i s sec t ion w e p resen t a sm al l M onte Car lo s tud y o f tbe p roposed es t i-ma to r s . W hi le i t cann o t com ple te ly cha rac te r ize the sam pl ing d i s tr ibu t ion o f thees t imators , i t does reveal cer ta in aspects of the i r behavior.

In a ll o f the d es igns

I = m ax {w161, + w2612 + u l ,0 } ,

y = (W lP, i + wz[Jt2 u2)lU ~>0 },

I 6 )

( 1 7 )

w h e r e t he t ru e p a ra m e t er s a re ( 11 , 6 t 2 ) = ( 1 , 1 ) a n d ( ~ l h ~ 1 2 ) = ( | , 2 ) . T h e r e-g ressor s wl and w2 a re d raw n f rom a no rmal N(0 , i ) d i s t r ibu t ion and a un i fo rm

U ( -2 ,2 ) d i s t ribu t ion , r e spec t ive ly, Di ffe ren t des igns a re cons t ruc ted by va ry in gthe d i s t ribu t ions o f the e r ro r t e rms . Da ta on u l a re genera ted f rom th ree d i ffe r-en t d i s t ribu t ions , nam ely, the s t andard no rmal d i s t r ibu t ion N(0 , 1 ) (No rma l ) ; am i x e d g a m m a a n d n o r m a l d is tr ib u t io n ( G a m m a * N o r m a l ) : v / '~ ' ~G a m m a ( 0 , i ) +v~/~ .2N(0 , 1 ); and a m ixed nega t ive g am m a an d normal d i s t ribu t ion ( -Ga m m a (0 ,1 ),N or m al) : - [v / '0- ' .8Gam ma(0, I)-t-v /0--~ .2N(0, 1) ] , wh ere G am m a(0 ,1) i s a s tan-d a r d i z e d g a m m a r a n d o m v a r ia t e w i t h z e r o m e a n a n d u n i t v a ri a n c e o f w h i c h t h ed e n s i t y f u n c t io n i s f a ( t ) = 8 / 3 ( t -t- 2 ) 3 e x p [ - 2 ( t + 2 ) ] , t > - 2 , w i t h i t s m o d e a t_ The d is turba nce u2 i s ob ta ine d f rom u2 = v / '0~ut + V~- .Su~, wh ere u~ is a2"

N ( 0 , | ) r a n d o m v a r ia b l e i n d e p e n d e n t o f u t . T h e s e d e s i g n s a r e s im i l a r t o t h o s e i nLee (1994) , and the censor ing l eve l i s abou t 50%.Here we cons id e r the r e su lt s o f e s t ima t ing (/lmm,~12) by d i ffe ren t e s t ima t ion

approaches . Lee (1994) compared the f in i t e sample pe r fo rmance be tween h i ssemiparamet r i c l eas t - squares e s t ima to r fo r the Type-3 Tob i t mode l and somesemiparamet r i c e s t ima to r s p roposed fo r the Type-2 Tob i t mode l , i n the con tex to f the Type-3 Tob i t m ode l w i th ce r ta in exc lus ion res tr i ct ion on ~o a lluded toear l ie r imp osed ; the resul t s favored Le e ' s es timat~,r. W e wi l l focu s on co m par in gv a r io u s s e m i p a ra m e t r ic e s ti m a t o rs d e s i g n e d f o r t he T ~ e - 3 To b i t m o d e l . M o r espec if ical ly, th e seco nd -step est im ators are/~pl , /12, ~hl , ~t,2, ~ffl, an d f ill , wh ere thecensor ing in tervals for ~pl are chosen to be equispaced wi th K--- -20 wi th co= - 4and CK = 4 . W e a l so inves t iga te :he pe r fo rmanc e o f ~pl w i th d i ffe ren t c e n t r in gintervals . ~1 and ~H are the es t imators by Lee wi th the f ixed t r imming se ts b~ingIwl[ < 1.9, Iw2l < 1.8 a n dI w z l< 1.7, [w-z[ < 1.6, re sp ec tiv el y;also ]~2 is our est ima-tor I I , and/~h, and ~h2 are the pa i rwise d i fferenced es t imators wi th the absoluteand square loss func t ions , r e spec t ive ly. Note tha t ~s l i s the on ly one wi thou t ac losed -form solut ion g iven a f i r s t-s tep es t imator. The p ai rw ise d i fferenced es t i -m a t o r w i t h th e a b s o lu t e l o ss f u n c ti o n p r o p o s ed b y H o n o ~ a n d P o w e l l i s u s e das the f ir st -s tep es t ima to r due to i ts go od f in i te sample p e r fo rma nce (H onor~ an dPowel l , 1994) . Because es t imatb I , f l , , i s a specia l caseo f Jffpl w ith on ly on e


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14 S C/wn t Journal of Econometrics 80 1997) 1-34

c e n s o r i n g i n t e r v al , n a m e l y, ( 0 , + o o ) , t h e r e c o u l d b e s e r i o u s lo s s o f o b s e r v a t i o n sd u e t o h e a v y t r i m m i n g , h e n c e i t s p e r f o r m a n c e i s n o t r e p o r t e d h e r e .

T h e r e s u lt s f r o m 3 0 0 r e p l i c a ti o n s f r o m e a c h o f th e t h r e e m o d e l s a r e p r e s e n t e dw i t h s a m p l e s i z es o f 5 0 , 1 0 0, a n d 2 0 0 . F o r e a c h o f t h e s ix e s t i m a t o r s u n d e rc o n s i d e r a t i o n , w e r e p o r t t h e m e a n v a l u e ( M e a n ) , t h e S t a n d a r d D e v i a t i o n ( S . D . ) ,a n d t h e r o o t m e a n s q u a r e e r r o r ( R M S E ) . F o r bo t h /11 a n d / } t l , a l l t h e s m o o t h i n gp a r a m e t e r s a n d k e r n e l s m o o t h i n g f u n c t i o n s a r e c h o s e n a c c o r d i n g t o L e e ( 1 9 9 4 )( s e e L e e f o r f u r t h e r d e t a il s ). 3 a e b i a se s o f t h e e st i m a t e s c a n b e d e r i v e d b yc o m p a r i n g t h e i r m e a n v a l u e s w i t h t h e t r u e p a r a m e t e r s .

Tab le I r epo r t s s imu la t ion re su l t s o f t he s ix sgm ipa rame t r i c e s t ima t ion ap -p r o a c h e s w i th v a r i o u s s a m p l e s i z e s w h e n t h e e r r o r t e r m u l i s n o r m a l ( N o r m a l ) .N o t e t h a t t h e r e a r e o n l y s m a l l s a m p l e b i a s e s e v e n w h e n t h e s a m p l e s i z e i s 5 0 ,w h i c h in t u r n i m p l ie s t h a t t h e e f f e ct iv e s a m p l e s i ze is o n l y a b o u t 2 5 f o r t h e m a i n

e q u a t i o n a s a r e s u l t o f t h e s e l e c t io n s c h e m e . T h e b i a se s a n d v a r i a n c e s d e c r e a s eas t he sam ple s i ze s i nc rease . A l so n o t i ce t ha t a s t he sam ple s i ze i nc reases f rom50 to 100 , t he r e l a t i ve pe r fo rm anc e o f ti p1 im prov es s ign i f i can t ly, wh ich i s t o bee x p e c t e d s i n ce t h e l o ss o f d e g re e s o f f r e e d o m c a n b e v e r y i m p o r t a n t w h e n t h es a m p l e s i ze i s s m a l l d u e t o t h e e s t im a t i o n o f se v e r a l i n t e rc e p t t e rm s c o r r e s p o n d -i n g to t h e s a m e n u m b c r o f ce n s o r in g i n te rv a ls . T h e p r o b l e m b e c o m e s l es s s e v c r cas t he samp le s i ze i nc reases wh i l e t he num ber o f t he in t e rcep t t e rms i s he ld f ixed .

Ta b l e 2 r e p o rt s s im u l a t io n r e s u l ts i n t h e c a s e o f a G a m m a * N o r m a l e r r o r t e r mi n t h e s e l e ct i o n e q u a t io n . T h e p a t t e rn o f t h e p e r f o r m a n c e o f t h e e s t i m a t o r s i s

c l o s e t o t h a t r e p o r t e d i n Ta b l e I . M o s t e s t i m a t i o n p r o c e d u r e s p e r f o r m w o r s e t h a ni n t h e f ir st d e s i g n . A s p o i n t e d o u t b y L e e , t h e G a m m a * N o r m a l e r r o r t e r m h a s ahea v ie r up pe r ta i l, and the th in ne r l e f t t a il i s cens o red du e to t he sam ple se l ec t iona n d t h e p o s i t i v e c o r r e l at i o n b e t w e e n t h e e r r o r te r m s a c r o s s th e t w o e q u a ti o n s .

Ta b l e 3 c o n t a i n s s i m u l a t i o n r e su l ts w i t h a n e g a t i v e G a m m a * N o r m a l e r r o r te r mi n t h e s e l e c ti o n eq u a t io n . C o m p a r e d w i th t h e G a m m a * N o r m a l c a se , h e re t h eth inne r t a i l i s censo red wh i l e t he heav ie r f i gh t t a i l r ema ins ; consequen t ly, a l l t hee s t i m a t o r s h a v e t h e s m a l l e s t R M S E s c o m p a r e d w i t h t h e o t h e r t w o d e s i g n s .

O v e r a l l , w e o b s e r v e t h a t fip l i s q u i t e c o m p e t i t i v e a m o n g t h e s i x e s t im a t o r s

exar.'Aned w hen the sam p e s i ze i s m od era t e ly l a rge , and /12 i s s l i gh t ly w orseth an lffhl, /~h2 an d /~t-

Tab le 4 r epo r t s t he s im u la t ion r e su l t s for/~p~ wi th d i f f e ren t cens o r ing in t e rva l s .T h e s a m p l e s i z e i s 1 00 . A l l t h e c e n s o r i n g i n t e r v al s a r e c h o s e n t o b e e q u i s p a c e dw i t h c o = - 4 . T h e i n d i v i d u al i n te r v a l le n g t h f o r v a r i o u s c e n s o r i n g s c h e m e s is0 .2 , 0 ,4 , 0 .6 , 0 .8 and 1 .0 wi th the cor responding c~c ' s equal to 4 , 4 , 3 .8 , 4a n d 4 , r e s p e c t i v e ly. 4 T h i s r a n g e s e e m s t o b e q u i t e w i d e . T h e r e is a t r a d e - o f fb e t w e e n l o s s o f d e g r e e s o f f r e e d o m f o r c e n s o r i n g w i t h s m a l l i n t e r v al s w i th l o s s

4 S i m u l a t i o n r e s u l ts a r c n o t v e r y s e n s i t iv e t o t h e e q u i s p a c i n g s ch e m e . T h e i n t e r v a l ( t o . o K ) i schosen to be wide enoug h ~o ensure the loss o f observations to b minimal.


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S. Chen /Journal o f Econometrics 80 (1997) 1-34 15

TableNormal design

n Estima',ors Tru e M ean S.D. RM SE

50 /~pl 1.000 1.000 0.31 9 0.3182.000 2.025 0.279 0.279

f12 1.000 0.98 5 0.297 0.2972.000 2.002 0.289 0.288

/~hl 1.000 0.988 0.289 0.2 892.000 2.006 0.270 0.270

/~1,2 1.000 0.984 0.2 76 0.2 762.000 2.006 0.260 0.260

1~1 1.000 0.982 0.319 0.3192.000 1.999 0.265 0.264

[ ltt 1.000 0.980 0.369 0.3692.000 1.998 0.285 0.285

100 /~p~ 1.000 1.010 0.1 97 0.1 97Z000 1.984 0.189 0.189

/~2 l .O00 1.016 0.21 0 0.21 02.000 1.984 0.203 0.203

tim 1.000 1.012 0.194 0.1942.000 1.988 0.183 0. i 83

]~h2 1.000 1.007 0.192 O. 19 I2.000 1.987 0.183 0. |83

/~I 1.000 .005 0.214 0.2132.00 0 1.984 O. 188 O. 188/$ll 1.000 i.000 0.229 0.229

2.000 1.989 O. 195 O. 195

2 /~p~ 1.000 1.005 0.128 0.1282.000 2.003 0.121 0.121

f12 1.000 1.0OO 0.14 5 0.14 52.000 2.002 0.130 0.129

flJ, 1.000 l.OO[ 0.133 0. 1332.00 0 1.999 O. 123 O. 23

fib2 1.000 1.000 O. 132 O. 1322.000 2.000 0.121 0.121

/~t 1.000 0.997 0.137 0.1372.000 1.994 0.121 0.121

/~u 1.000 0.997 0.154 0. 532.600 1.998 0.130 0.130

o f o b s e r v a t io n s a s s o c i a te d w i t h c e n s o r i n g w i t h l a rg e i n t e r v a ls . T h e v a l u e s o f / ~ p ta s s o c i a t e d w i t h d i f fe r e n t c e n s o r i n g i n t e rv a l s d o n o t a p p e a r t o b e v e r y s e n s i t i v et o d i f fe r e n t i n t e rv a l c e n s o r i n g s c h e m e s . O n e p r a c t i c a l a p p r o a c h i s t o t a k e a v e r a g ev a l u e s o f t h e e s t im a t e s . T h e r o w m a r k e d ' Av e " r e p o r ts th e p e r f o r m a n c e o f t h e


16/34

16

Table 2G a m m a * N o r m a l d e s i g n

S . C h e n l J o u r n a l o f E c o n o m e tr ic s 8 0 1 9 9 7 ) 1 - 3 4

n Es t ima lo r s True M ean S .D . RM SE

50 /~r~ 1.000 0.953 0.354 0.35 72.00 0 1.981 0.351 0.35 I

/~ 1 .000 0 .950 0 .349 0 .35 I2 ,000 i .970 0 ,298 0 .299

/3~,t ,000 0.94 5 0.32 7 0.3312.000 1 .972 0 .288 0 .289

fiJ~2 1.000 0.9 46 0.3 17 0.32 I2.000 1.971 0.284 0.285

li t 1.000 0.934 0+347 0.3522.0 00 1.981 0,2 99 O. 299

/l t t 1.0O0 0.922 0,390 0.39 72.000 .977 0,33 0.33 I

100 l ivt 1.000 0.996 0.196 0. i 962.000 2.005 O. 98 0. t 98

~2 1 .000 0 .995 0 .220 0 .2192.000 1.997 0.221 0.220

f l / ,1 1.000 0.991 0.197 0.1972 . 0 0 0 1 . 9 9 3 0 . 2 1 1 0 . 2 1 1

['t,2 1.0 00 0. 98 9 O. 19 5 O. 1952.000 i .992 0.211 0.21 0

/~j 1.000 0.993 0.2080.2082.000 1 .987 0 .216 0 .216

/~lt 1.000 0.992 0.245 0.2452.000 1 .993 0 .229 0 .229

20 0 ~pt 1.000 1.005 O. 128 0.1282.000 1 ,999 0 .116 0 .116

/ J2 1.000 1.005 O. 149 0.1 492.000 1 .992 0 .143 0 .143

~J~1 1.000 1.003 0. 4 0. 14 02.000 1.995 O. 34 0,134

/~t 2 1.000 | .000 O. 38 0. 372.000 1.993 O. 32 0,132

~, , 1.000 1.004 0.154 0. i 542.000 1.992 0.135 0.135

~ t 1 .000 1 .008 0 .174 0. 732.000 1.991 0.145 0.145

a v e r a g e d e s t i m a t e s . T h e r e s u l t s a r e q u i t e e n c o u r a g i n g w i t h t h e a v e r a g e e s t i m a t o r

o u t p e r f o r m i n g i n d i v i d u a l i n t e r v a l c e n s o r i n g e s t i m a t o r s . O n e p o s s i b l e e x p l a n a t i o n

f o r t h e g o o d p e r f o r m a n c e o f t h e a v e r a g e e s t i m a t o r i s i ts f u l l e r u t i l i z a t i o n o f t h ed a t a .


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X Chenldourna l o f Econometr ics 80 1997) 1 -34

Table 3Negative Gam ma Normal design

17

n Es t ima tors "[ rue Me an S .D, RM SE

50 ~p, 1.000 1.023 0.293 0.2942.000 2 .021 0 .282 0 .282

f l , 1 .000 0 .997 0 .283 0 .2822.000 1 .999 0 .269 0 .268

[~h 1.000 0.99 6 0.277 0.27 72.000 1 .997 0 .267 0 .266

[~b2 1.000 0.9 94 0.266 0 .2652.000 . '397 0.252 0.251

~ : 1.000 0 .988 0 .294 0 .2942 .000 1.995 0.261 0.260

/~u 1 0 0 0 0 . 9 9 6 0.328 0 .3282.000 i .997 0 .278 0 .278

100 / tpl i .000 0.992 0.177 O.1772.000 1 .998 0 .160 0 .160

,f~2 i .000 0.984 0.185 O. 1852.0 00 1.983 O. 173 O. 173

[~1 1 .000 0 .985 0 .179 0 .1802.000 1 .984 0 ,167 0 .168

[il,2 1.000 0.985 O,177 0.1772.000 i .983 0.160 0.161

~1 1.000 0.981 0.197 0.1982.000 1 ,989 0 .162 0 .162/~1 1.000 0.97 8 0,214 0.2 5

2.000 1.993 0.171 0.171

2 0 0 ~pl 1.000 1 .004 0.118 0 . 11 82.000 2 .004 0 .112 0 .112

~2 1 .000 0 .996 0 . 32 0 .1322,000 .999 0.114 0. [ 14

fl~l 1.000 0.9 96 0.124 0.12 42.000 1 .998 0 .11I 0.111

~ .000 0 .997 0 .124 0 . I242.000 1 .997 0 .110 0 .109

~t 1 .000 0 .998 0 .130 0 .1302 0 0 0 1 9 9 5 G. i06 0 .106

~11 1.000 0.997 0.146 O. 1462.000 1 .993 0 .111 0 .112

In li g ht o f t h e c l o s e r e la t io n s h ip b e t w e e n o u r es t im a t o r II ( /~ 2 ) a n d L e e s ( 1 9 9 4 )e s ti m a t o r, s o m e c o m p a r i s o n s c a n b e m a d e . T h e w e i g h t i n g s c h e m e i n d e f i n i n g / ~ 2i s a d o p te d l a rg e ly f o r c o n v e n i e n c e r ath er th a n f o r e ff i c ie n c y c o n c e r n s . H o w e v e r ,it i s n e c e s s a r y t o c h o o s e s o m e s m o o t h i n g p a r a m e t er s to i m p l e m e n t L e e s


18/34

18 S . C h e n t J o u r n a l a f E c o n o m e tr ic s 8 0 1 9 9 7 ) 1 - 3 4

Table 4Resul ts with vario us interval censoring .schemes. Sample size: 100

Design Interval width True Mean S.D. RMSE

Normal 0.2 1.000 1.016 0.211 0.21 I2.000 1.989 0.198 0.198

0.4 1.000 1.010 0.197 0.1972.000 1.984 0.189 0.189

0.6 1.0(30 1.010 0.192 0.1 922.000 1.985 0.184 0.184

0.8 1.000 1.019 0,198 0,1992.000 1.993 O. 182 O. 18 I

1.0 .000 1.018 0.211 0.2 122.000 1.991 0.179 0. 79

Ave 1,000 1,015 0.191 0.1912.000 1.989 0.174 0. i 71

Gam ma* Normal 0.2 1.000 0.994 0.208 0.2082.000 2.004 0.214 0.214

0.4 i .000 0.996 0. i 96 0.1962.000 2.005 0.198 0.198

0.6 .000 1.007 0.196 0.1962.000 2.008 O. 190 O. 190

0,8 1.000 1.002 0.200 0.2002.000 2,007 0.205 0.205

1.0 1,000 1,012 0.207 0.207

2,000 2.014 0.199 0.199Ave 1.000 1.002 0.190 0.189

2.000 2.007 0.189 0.189

Negative 0.2 1.000 0.993 0.190 0.190Ga mm a* Norm al 2.000 1.999 0.162 0.161

0.4 1.000 0.992 0.177 0.1772.000 1.998 0.160 0.160

0.6 1.000 0.986 0,177 0.1772.000 1.996 O, 166 O. 166

0.8 1.000 0.990 O 1 8 4 O I g 42.000 2.000 0.171 0,171

.0 1.000 0.988 0.191 0,1912.000 .995 0.179 0.179

Ave i .000 0.990 0.173 0.1732.000 i .998 0.156 0.155

e s t im a t o r. L e e ( 1 9 9 4 ) r e p o r t e d t h a t h i s e s ti tn a t o es a r e . t , t v e r y s e n s i t iv et o t h e s m o o t h i n g p a r a m e t e r s i n h i s M o n t e C a r l o s t u d y. S e c o n d , L e e ' s e s t i m a -t o r i n v o l v e s a f i x e d t r i m m i n g s e t w h o s e c h o i c e c o u l d a f f e c t i t s p e . r f o r m a n e e .T h i s p o i n t i s i l l u s t r a t e d f r o m t h e r e l a t i v e l y p o o r p e r f o r m a n c e o fl u d u e t o


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X ChenlJournal of Econometrics 80 1997) 1-34 19

heav ie r t r imming (27% t r imming fo r /~n vs . 15% fo r /~ t ) , and th i s p rob lem i s ex -pec ted to become more severe when more reg ressor s a re inc luded in the se lec t ionequat ion .

5 . S e m i p a r a m e t r i c e f fi c ie n c y b o u n d f o r t h e Ty p e - 3 T o b i t m o d e l

Sem iparamet r i c e ff i c iency bou nds can .serve as a c r i t e rion fo r com par ing ex i s t -ing es t ima to r s and sugges t ing improved es t ima to r s . In th i s sec t ion we ca lcu la tet h e s e m i p a r a m e t r i c b o u n d f o r t h e Ty p e - 3 To b i t m o d e l u n d e r t h e c o n d i t i o n t h a tthe e r ro r t e rms and regressor s a re independ en t . For genera l tr ea tmen t o f e ff ic i encyb o u n d s , s e e B e g u n e t a l . ( 1 9 8 3 ) a n d N e w e y ( 1 9 9 0 ) .

Here we fo l low the approach o f Coss le t t (1987 ) ( a l so B egu n e t a l ., 1983). The

d i s c u s s io n i s i n f o rm a l , w h i l e t h e a rg u m e n t s i n t h e f o l lo w i n g c a n b e m a d e r i g o ro u su n d e r a s et o f a s s u m p t i o n s in A p p e n d i x B , S e c t io n B . I , b y f o l l o w i n g C o s s l e tt(19871 . L ike C oss le t t, we adop t He l l inger d i ff e ren t ia l approach ( see B egu n e t a l .,1983 fo r de ta i l s ) . Cons ide r a more genera l sample se lec t ion mode l

I = m a x { - v l ( z , O ) + u l , 0 } , ( 1 8 )

y = i{l>O}{- -V2 z,O) + u2} . ( 1 9 )

Note tha t under the independence res t r i c t ion the l ike l ihood i s

ca'~ y , I , z , O , # ,h ) = h ( z ) { G o ( v l ) l { l = o } + g ( l + v b y + ~ 2 ) l ( t > o i } , ( 2 0 )

w he re v~ = vl (z, 0),v2 =- '~(~'~ 0) , 0 (u l, u= ), h ( z )a n d . f ( y , L z , O, a , h ) a ret he den-s i t i e s o f ( u l , u z ) , z a n d( y , L z ) wi th respec t to s igma- f in it e m easures ~ , oJ and / . t ,r e spec t ive ly, on some measurab le spaces . Here ( i s Lebesgue measure , whi l e ~ )i s the p roduc t o f measures co r respon d ing to each exp lan a to ry va r iab le : Lebesg uemeasure fo r con t inuous va r i ab les , coun t ing measure fo r d i sc re te va r i ab les , andL e b e s g u e m e a s u r e p l u s c o u n t i n g m e a s u r e f o r a n y m i x e d ( d i sc r e te a n d c o n t i n u o u s )va r i ab les . Also #0 and Go a re the marg ina l dens i ty and cumula t ive d i s t r ibu t ion

func t ions o f u i . Le t L2( /~ ) ,L2(~) and L2(~) ) be the usua l L2-spaces o f square -in tegrab le func t ions , inne r p roduc t s and norms in these spaces a re deno ted by( - , . ) a n d I 1 I I ,wi th subsc rip ts # ,w and ~ to d i s t ingu i sh the d i ffe ren t L2-spaces .In our app l i ca tion the m easure ~( i s the p roduc t o f co wi th coun t ing m easure on{1 = 0 , y = 0} p lus Leb esgue m easure on { I > 0} , so tha t

I1 11 = I I ( 0 ,0 , -) l l 2 , + f f dy d l l l ( y, I , IE l,=o f l , - - o o

fo r any func t ion ~b(y, I , z ) in L2(p) .W e then p roceed to ca lcu la te the ' de r iva t ives ' o f the square - roo t l ike l ihood

wi th re spec t to 0 ,g and h , r e spec t ive ly. S imi la r to C oss le tt , we can ob ta in the


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20 S. Chen/Journal o f Econometrics 80 (199 7) 1-34

fo l lowin g ' s co res" ( t imes f~12) fo r 0 and the non param et r i c com pon en t s g andh :

p o ( y , l , z ) = 2 h l / 2 ( z ) { ~ v ' - [ , ( I + v , , y + v 2 ) ~ Of f ~ o~ r j ) G o ~ 2 v l 1 { i = o ~ +

~ v z ] g _ l / z ( I } (22)+ ff2 (l + v l , y + v 2 ) - f f f f ] + v t , y + v 2 ) l u > o I ,

~ j f l ( l , y, z ) = h t / 2 ( z ) { G o 1 /2 (v l ) d ( v t ,fl) l {/=0} + fl( l + vl, y + v2)l {t>o} },

( 2 3 )

where .q l ( ' , ' ) and g2 ( ' , ' ) deno te , r e spec t ive ly, the pa r ti a l de r iva t ives o f g wi threspec t to i t s f i r s t and second componen t s ,

jo~

= - f f al 2(ul,u2)fl(u,.u2)dul du2 (24)F --OO

for/~ ~ .~;, = {/~ ~ ~ 2 ( ( ) I


21/34

X Chen lJournal o f Econometrics 80 (1997) 1-34 21

- O- l /2 (u l , ux ) [Ol (u t , 2 )k l (u t ) + O2( t , u2 )k2 (u t)] 1 {t> o}

o ' / 2 ( u l ,us)go u1)kKul 1{ t o } }+ G o ( u 1

(27

w h e r e u l = 1+e l a n d u2 = y + v2 , kL(u )= E(OVL/~OI v l < u ) andk2(u ) = E(~v2/aO Iv t < u ) . H enc e w e a r r ive a t t he fo l low ing re su l t.

P r o p o s i ti o n 3 . U n d e r th e c o n d it io n s in A p p e n d i x B , S e c t i o n B . I , t h e s e m i p a r a -m e t r i c e f f ic i e n cy b o u n d f o r 0 in t h e n o n l in e a r s a m p l e s e l e c ti o n m o d e l( 1 8 ) - ( 1 9 )i s g iven by I , I , whe re

I . = 4{ (Po .p~}v - -(=* ,o r* r ) , }

]- f H v, ( u ) V a r [ ~ - v l < u d~ k Go u)+ . f 0 - ' ~ . . , , . , )O ~ : . , . . 2 ) V, , r[ ' l- = LaO I vt < ul

+ a ' 2 ~ t2 ( u 2 c [ av'O0 ~v200, v, u ,

+ o , < . , . . . ) o a ( u , . . , . ) c o vL o o' ~ - I v , < . ,

w h e r e

Ov; c~v < u , ] = E [a O 0 0 ,o y [ 00 " S0 ' I vl vl

f o r L j = 1 ,2

a n d H e , ( u ) = P ( v t < u ) .

( 2 9 )

W hen v l = v2 andu i = u 2 t h e m o d e l r e d u c e s t o t h e c e n s o r e d n o n l i n e a r r e g r e s -s i o n m o d e l i n C o s s l e t t ( 1 9 8 7 ) , t h e n th e r e s u lt s w il l b e e x a c t l y th e s a m e a s t h o s ein Coss l~ t t .

W e n o w s p e c i a l iz e t o t h e Ty p e - 3 To b i t m o d e l w i t h a tw o e q u a t i o n l i n e a rr eg res s ion m ode l a s t he u nd e r ly in g l a t en t mo de l , w he re v i = -W 6 o , v2------x flo .


22/34

22 S. C/ten Journal o f Econometrics 80 (1997) 1-34

L e t O ( u l ) = { - w O o < Ul } ,

o)( u n ) = 0

a n d (: o)( m ) = r ~

w h e r e

(0B(,~l ) = 0

a n d

~ , . = E ( w ' w l D ( u ) ) - E ( w ' I D ( u I ) )E ( w I D ( U l ) ),

F,~x = E(W'x ID(Ul ) ) -- E ( w ' lD (u l ) )E (x ID (u t ) )

l'~,~ = E (x 'x[ D (u l) ) - E ( x '[D(ut ) ) E ( x l D ( u l ) ) ;

t h e n t h e l o w e r b o u n d f o r 0 = ( 6 ,1 ~ ) w o u l d b e V - t , w h e r e

V = - - o ~ H v , ( u ) A ( u ) d \ ~ ]

+ f ( J - I (u l ,U 2)(g2 1(U I ,U2 M (U l) + ffI (UI ,,U2),q2(Ul ,U2)B(Ul )- o o

+ a l ( t , t t , u 2 ) g 2 ( U l , l Q ) n t ( U l ) +,~2(Ul ,u 2 )C ( U l ) d u 2 ]d u l H r , ( u ) . ( 3 0 )J

L e t

M t = a t ( u , , u 2 ) - aoO 'q ) e ( u l , u 2 ) / G o ( u l ) ( wf f l / 2 ( u t ,u 2 ) - - E ( w [ D ( u ) D ( u l )

= O2(u_. l.u ...3 .2 ) . E ( x I D ( u , ) ) ) D ( u , ) .N I o l / 2 U l , u 2 ) x - -

L e t ~ * a n d ~ d e n o t e t h e m e a s u r e s h a v i n g d e n s i t ie sO(uE,u2)and ffo (u l ) , r e s pe c-t i v e l y, w i t h r e s p e c t t o L e b e s g u e m e a s u r e s o n R 2 a n d R . T h e f o l l o w i n g p r o p o s i t i o np r o v i d e s a s et o f su f fi c ie n t c o n d i t i o n s f o r t h e i n f o r m a t i o n m a t r i x i n ( 3 0 ) t o b en o n s i n g u l a r.

Propo s i t ion 4 . Under cond i t ions tha t( i ) ( d . q o ( u l ) / d u t -g~(ut )~Go(u ) ) (w - E(w[D(Ul ) ) )D(u l ) i s no t l i nea r ly depen den t a lm os t su re ly w. r.t , ra ~ , an d( i i )


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S. C hen l Journal o f Econometrics 80 199 7) i~34 23

N I i s n o t l i n e a r l ) , d e p e n d e n t a l m o s t s u r e l y w. r . t , o~ e ~ * ; t h e i n f o r m a t i o n m a t r i xV in ( 3 0 ) is nonsinoular,

Proof . T h e i n f o r m a t i o n m a t r i x F i n ( 3 0 ) c a n b e r e w r i tt e n a s

J ' f f f N1 N ,

w h i c h i s si n g u l a r i f a n d o n l y i f ) ~ M i + 2 [ N i = 0 a l m o s t s u r e l y w i t h r e s p e c tt o c o x r e* , f o r so m e n o n z e r o v e c t o r 2 = ( 2 ~ , 2 ' ~ ) ' . T h e r e a r e t h re e e a s es : ( 1 )~.l = 0 , & ~ 0 ; ( 2 ) 2 t ~ 0 , ~ ~ 0 ; a n d ( 3 ) ,:.l ~ 0 , 2 2 = 0 . F o r t h e f ir st c a s e , w e h a v e(02( I / I ,r 4 2 ) / g t /2 U l , U 2 ) ) ) .t 2 X -E(x lD(u t) ) ) D ( u , ) = 0 a l m o s t s u r e ly w i t h r e s p e c t toco x if * , c o n t ra d i c t i n g a s s u m p t i o n ( i i ) . F o r t h e s e c o n d c a s e , ).~ M ~ q - ~ N I : 0 a l -

m o s t s u r e l y w.r.t co ~ . . T h e n w e h a v e

f (2~Mt + . ~ N l gl /2 (u l ,u2)du2 ~-0

a l m o s t s u r e ly w. r.t co ~ . H e n c e ,

( d g 0 ( u l ) / d u l - f l~(ul ) /Go (ut ) ) (w - E(w lD (ul) ) ) D ( u l ) = 0

a lm os t su r e ly w, r.t , co ff~ ', wh ich con t r ad i c t s a s su m pt io n ( i ) . S im i l a r l y, t he t h i rdease w i l l a l so l e ad t o a con t r ad i c t i on .

I f b o t h x a n d w a r e o f f u ll r a n k i n t h e h a l f - s p a c e { - w ~ o < c o } ,(dgo(co) /du -g~ (co ) /Go(co ) )~Oa n d g 2(c o, c l ) ~ O f o r s o m e (co, cl) , t h e n t h e a s s u m p t i o n s o fP r o p o s i t io n 4 w i l l b e s a ti sf ie d s i n c e b o t h Va r ( x [- w ~ 5 0 < u l ) a n d V a r ( w l - w f i 0< u l ) a r e l ef t c o n t i n u o u s i n u l , a n d ( d g 0 ( u l ) / d u l - g 2 0 ( u t) /Go(Ul))a n d 02 (ul , u2)a r e c o n t i n u o u s i n(u l ,u2) . W h e n x a n d w b o t h h a v e f u ll r a n k f o r th e s e l e c t e ds u b s a m p l e , i .e . b o t h E [ V a r ( x l l = 1 ) ] a n d E [ V a r ( w l l = I ) ] a r e n o n s i n g u l a r, t h e a s -s u m p t i o n s in P r o p o s i ti o n 4 h o l d g e n e r a l l y e x c e p t f o r s p e c i a l f o r m s 5 o f g . T h i sf u ll r a n k c o n d i t i o n , i n g e n e r a l , d o e s n o t r e q u ir e t h e e x i s t e n c e o f a c o n t i n u o u sr e g r e s s o r o r a n y e x c l u s i o n r e s tr ic t io n o n t h e r e g r e s s o r s . I n t h e c a s e o f a s in g l er e g r e ss o r a n d w = x , w e w i ll h a v e p o s i ti v e i n f o rm a t i o n i f x h a s n o n z e r o c o n -

d i t iona l var iance g l ,v e n { - x ~ 0 < co } , a n dd g ( c o ) / d u l< 0 and O2(co,c t ) ~ O f o rs o m e c o a n d c t .

F o l l o w i n g C h a m b e r l a in ( 1 9 8 6 ) , s in g u l ar it y o f i n f o r m a t io n m a t r ix w i l l i m p l yt h a t th e r e i s n o V ff f- ro o t c o n s i s t e n t e s t i m a t o r f o r 0 . W e c a n a rg u e a s C h a m b e d a i n( 1 9 8 6 ) t h a t t o e n s u r e a n o n s i n g u l a r i n f o r m a t i o n m a t r i x i n t h e Ty p e - 3 To b i t m o d e lno i n t e r cep t i s a l l ow ed ; t h i s i s na tu r a l s i nce no r e s t r i c t ion on t he l oe a t i oa o f t hee r ro r d i s t ri bu t i ons i s impo sed , an d t he i n t e r cep t t e rm s a r e no t i den t if i ed . I n co n -f ro s t t o t h e Ty p e - 2 To b i t m o d e l ( C h a m b e r l a i n , 1 9 8 6 ) , h o w e v e r, g e n e r a l e x c l u s io n

S No te t ha tIhe f a m i l y o f u n i f o r m d i s tr i b u ti o n s a r e n o t r e g u l a r i n th e s e n s e o f B e g u n e t a l, ( 1 9 8 3 ) ,a n d r u l e d o u t b y t h e r e g u l a r i t y c o n d i t i o n s .


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24 S. ChenlJournal of Econometrics 80 1997) 1-34

r e s t r i c t i on on f l i s no t needed ; a l so v l i s no t r equ i r ed to have a con t inuous d i s -t r i bu t ion .

6 . C o n c l u d i n g r e m a r k s

S e m i p a r a m e t r i c e s t i m a t i o n o f t h e Ty p e - 3 T o b i t m o d e l i s c o n s i d e r e d in t h isp a p e r u n d e r t h e r e s tr i ct i o n t h a t t h e e r r o r t e r m s a n d t h e r e g r e s s o r s a r e i n d: p e n -den t . Th e f ir s t m e tho d i s a s im p le l ea s t - squa res - type e s t ima t ion appro ach app l i edt o a n a d j u s t e d s a m p l e , a n d t h e s e c o n d o n e i s a w e i g h t e d l e a s t - s q u a r e s - t y p e a p -p roa ch th rou gh e s t ima t ing the se l ec t ion co rr ec tio ,1 func t ion . Bo th e s t ima to r s a r ec o n s i s t e n t a n d a s y m p t o t i c a l l y n o r m a l . E s t i m a t o r s f o r t h e a s y m p t o t i c c o v a r i a n c em a t r i c e s a r e p r o v i d e d t o c a r r y o u t l a rg e s a m p l e i n f e r e n c e . A l i m i t e d M o n t e C a r l o

s i m u l a t i o n i s u s e d t o s t u d y t h e p r a c ti c a l p e r f o r m a n c e o f t h e e s ti m a t o r s . W e a l s oc a l c u l a te t h e s c m i p a r a m e t r i c e f f i c ie n c y b o u n d f o r t h e Ty p e - 3 To b i t m o d e l u n d e rthe independence r e s t r i c t i on .

S i n c e t h e c o v a r i a n c e m a t r i c e s o f a ll t h e a v a i l a b l e e s t im a t o r s d e s i g n e d f o r t h eTy p e - 3 To b i t m o d e l a r e c o m p l i c a t e d a n d d e p e n d e n t o n a p a r t i c u l a r f i r s t - s t e pe s t i m a t o r, e x a c t a n a l y l i c r e l a t i o n s h i p b e t w e e n t h e s e m a t r i c e s a n d t h e s e m i p a r a -m e t r i c b o u n d w i l l b e d i f f ic u l t t o d e r i v e . A l s o a ll o f th e s e e s t i m a t o r s a r e b a s e do n s o m e c o n d i t i o n a l m o m e n t s w i t h o u t c o n s i d e r i n g t h e a s s o c i a t e d o p t i m a l i n s t r u -m e n t s ; t h e r e f o r e , n o n e o f t h e m a c h i e v e s t h e b o u n d . I t m a y b e p o s s i b l e t o ex te ~ ,dR i t o v ' s ( 1 9 9 0 ) m e t h o d f o r t h e Ty p e - I To b i t m o d e l t o c o n s t r u c t a n e f fi c ie n t e s t i-m a t o r f o r t h e Ty p e - 3 To b i t m o d e l , b u t t h i s i s a t o p i c f o r f u t u r e r e s e a r c h .

A p p e n d i x A

P roo f o f Theorem 1T o s im pl i fy the no ta t ion , de f ine s r =s~ l{ i ,> , , ,~>0 } . W i thth i s conve n t ion , r~ can be ex pres s ed a s

) -= d ~ st ~ disi y~i=1i t " x I n, ~ , , . , , ,=,E: ; ( ~ : o + u 2 , - ~ o

- - | . '1

= , ~ o \ N ' ' 7 -

h e n c e ,

/ 1 ~ \ t~( ,~- ~ o ) = ( ~ E d : ~ ~ ;~\ i= I / " ~

a : ; ( u 2 , - c ~o ) .i =

( A . l )


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S . C h e n l J o u r n a l o f E c o n o m e t r ic s 8 0 1 9 9 7 ) 1 - 3 4 2 S

To get the des i red resul t , we need to showt h a t ( l / n ) ~ / n _ l . t ~s s~ c o n v e rg e sin probabi l i ty to a nons ingular matr ix and ( I /vrn) ) - '~=ld ~ s ~ ' ( u 2 i - o t o )i s a symp-tot ical ly normal.

First . note that

1 x~ n--ds . , s . I nn-~= l i i i n~~di$i$i{>w.Jo>o}=l

l R I+ - ~ dts~s~(Itit >~.,~>0} - 1 { >w,a0 >o} )

f f i = 1

By the s t rong l aw o f l a rge numbers ,

I d i s ~ s i l l l , > w, ~ . > o ~ -~ E d i s i s ~ I f l ~ > w, ~ . > o I ( A . 2 )n i = I

w h e r e c o n v e rg e n c e m e a n s e l e m e n t b y e le m e n t c o n v e rg e n c e. A l s o b y th e C a u c h y -Schwar tz inequal i ty,

l ~ -- s s ~ s i ( 1 ' / g } - I ' ' ~ } )[]

n I I d : ~ s d l 2 n ( 1 | l '> w : > } - I f f ' > w ' > } ) 2 ( A . 3 )

By Lemma 12 in Pakes and Po l l a rd (1989) i t i s easy to ve r i fy tha t the c l a s so f func t ions o f{ I { i > w a > o } , 6 E R P }i s un i fo rmly bou nded and Euc l idean . (Fo rthe concept Eucl idean and o ther empir ica l processes re la ted te rms the reader i sr e fe r red to Pakes and Po l la rd . ) Then by the un ifo rm law o f l a rge num bers inLemma 8 in Pakes and Po l l a rd we can p rove tha t

1

n ~ l{/'>w"~>) ~p Elll ,>w,~o>o},

l{/,>,,;3> 0}l{tj> w:~ o>o } P-~ El(t~>w,~o>o}/1 i----I

a n d

2 i = l


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2 6 S . C h e n l d o u r n a l o f E c o n o m e t r ic s 8 0 1 9 9 7 ) 1 - 3 4

w h e r e - p - d e n o t e s c o n v e rg e n c e i n p r o b a b i l i t y. C o n s e q u e n t l y,

i ) _ ~. i i f, w, , ~ > o 1 { t, , , , , ~ o > O } 2h i =

__ _ l{ t ,>wf i>o t - - 2 1 Iu ,>w,~>o t I{ l,>w, ~o>on i= l n i=1

+ - i {1, > ,,; ,~> 0}r / i= l

---- Op ( 1 ). ( A . 4 )

T h e r e f o r e , f o r t h e tw o t e r m s o n t h e r i g h t- h a n d s i d e o f ( A . 4 ) , t h e f ir st o n e i sb o u n d e d i n p r o b a b i l i t y b y A s s u m p t i o n 3 . 4 a n d t h e s e c o n d t e r m c o n v e rg e s t o 0in p robab i l i t y ; hence

di s i s i - -- , tE d i s : i l u , > , . ~ o > o } i n p robab i l i t y./ r / i = I

N ext , w e inves t iga t e t he a sym pto t i c d i s t r ibu t ion o f (I /v~ )) --~ .in= ld~s~. (u2~-xo) .To s i m p l i f y t h e n o t a ti o n , w e d e f in er i = ( u 2 . d i , w i , I . s i ) a n d h ( r i, tS )= s~ l{ l,> w,~ > o}( u 2 i - c t o ) d i .S i n ce t h e c la ss o f f u nc ti o ns { I { I > ~ > o } , 6 E R p } is E u c l id e an , a n ds~ (u 2i - a o ) d i i s a f i x e d f u n c t i o n ( w h i c h i s n e c e s s a r i l y E u c l i d e a n ) , t h e n t h e c l a s s

o f fu n c t i o n s { h ( - , 6 ) ,6 E R p }i s a E u c l i d e a n c l as s w i t h a n e n v e l o p e F f o r w h i c hJ F 2 d p < c o b y L e m m a 1 4 i n P a k e s a n d P o l l a r d a n d A s s u m p t i o n 3 . 4 , a n d t h ep a r a m e t e r i z a t i o n i s L 2 ( p ) c o n t i n u o u s a t g o b y A s s u m p t i o n 3 . 3 . L e t

I ~ d i s ~ u 2 i O t o ) 1 { I , > , v , 6 > o ~ - E d i s ~ u 2 i C ~ o ) I { I , > , , , ,s 0 ~ ;

t h en b y L e m m a2 . 1 7 o f P a k e s a n d P o l l ar d1 9 8 9 ) , f o r e v e r y s e q u e n c e o f p o s i ti v en u m b e r s { tt n} c o n v e rg i n g to z e r o ,

s u p [vnh( ,6 ) - vnh( , 6o ) l= Op(1).LI~;-6oil t2n )---~0, a s n - -* oo . H ence ,

P I v. h - .g ) -v , , h . ,6o)1 > a )

~

\11 6- 6oll~< ~,

+ P ( ~ - ,~ol > t=.) ~ o

( A . 6 )


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S. Chen l Journa l o f Econometr i cs 80 (1997) 1 -34

as n- -- , oo , fo r a ny ~ > 0 . T herefore ,

1_~ ~ ai s[ u2 i -g o )l {1, >,~, $>o }V ~ i = l

___ 1 ~ ~_ d is ~(u 2 i - O ~o ) l i l> w , 6o>o}V / H i = I

+ v/-nE dis: u2i - go )1 {t. > w,6>0} I~=d + op ( l )

s ince

27

( A . 7 )

E : .~ = E d : ~ u 2~ - ~ o ) 1 {u,, >o.~,60 > o

= Es~(E(d~u2~ 1 0 ', > o.w,~o> 0~ I ) - ZoEd~ {~,,,> o,~.a>o} ) 1 w,~o>o}

~ 0 .

i f w e s h o w t h atE d ~ s ~ u 2 i -go)1 { , >,v,6>o)[6 =d is asy m pto tica l ly l inea r, the n thea s y m p t o t i c d i s tr ib u t io n o f ( I / V ~ ) ~ = ts~ u2i- go)di I {1, >w$>o} fo llo w s ea sily .

To d e r i v e t h e a s y m p t o t i c d i s t r i b u t i o n o fv ~ E d i s ~ u 2 i -~o )liI, >~,~>o}[~=~, w e

expand the expec ta t ion in Tay lo r se r i e s . By Assumpt ion 3 .3

vGEs ~(u2~ - ~ o) iv , >,v,6>ofl~=. ~

= vG Es ~tu 2~ - ~o ) V, > w, ~o >o~ + vG f~ l (~ - ,~o) + % ( l )

= V/-nfll(~ -- ~0 ) -I- Op(i ). (A .8 )

B y t he a sympto t i c l inea r r ep resen ta t ion o f~ i n A s s u m p t i o n 3 . 5

, 1 ~ ~l~bi + op(1) (A .9)/ ~ E s A u 2 i - go)t~ / , >w.~>o~l~=~ = - ~ t=

w i t h E ~ i = 0 . Consequen t ly,

1 ~-~At_I(A~+ ~ | ~ i ) + o p ( l ) . ( A . I O )

Hence ,

v ~ ( n - - ~ o ) ~ N (O , Z I )

as asser ted . [ ]


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28 S. ChentJournal of Econometrics 80 (1997) 1-34

P r o o f o f T h e o r e m 2 .Ins tead o f ana lyz ing the min imiza t ion p rob lem (13) d i -r ec t ly, we ob ta in a c losed fo rm fo r /~2 by some a r i thmet ic ,

I q

/~z = |(n)~- ~-]~ ~ d,D~y(~)D,i(6)(xi - xt,) (xi -L j i k i

x ( n ) ~ ~ - ~ ~ d i D ~ ] ( 5 ) D i i ( ~ ) ( y ~ - y j ) ( x ~ - x } ) ,j i k i

wh ere (n)~ =n ( n - l ) ( n - 2 ); t he n

]~ ( 2 -- [1o) ---- (n) ; ~ ~ d i D o ( ~ ) D i k ( 5 ) ( x ,-- x k ) ' ( x -- x j v/n n)3-1~ ] ~ d i D i j ~ ) D i k 5 ) u 2 i - u 2 j ) x i - xt )

= s i ~ s 2 .

(A.1 I )

(A.12)

We wi l l show tha t $1 converges to a nons ingu ta r ma t r ix in p robab i l i ty, andtha t $2 i s a sympto t i ca l ly normal ly d i s t r ibu ted , r e spec t ive ly. Le t

~ k ( 6 ) = d ~ D i j (~ ) D i ~ ( 5 ) ( x : - x~, ) ' ( x i - - x y ) ,

~ i~ kO S, f l ) = d i D i j ( 5 ) D i k ( 5 ) ( y i - y j - ( x i - x . i ) f l) ( x i - x k ) ' ,

+@A5, ~) + j~;(5, ~)

a n d

then i t i s easy to show tha t

= n ) ; -~ ~ ~ g ) + o p I )i


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X ChenlJournal o f Econometrics 80 (1997) i-3 4 29

and

i


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3 0 S. Chen I Journal of Econometrics 80 (1997) 1-34

- ( n ) 3 - 1 ,< j < ~ ( ~ b v k ( 6 o ) - E ~ ij,~ ( 6 o ) )]

= % ( I ) . ( A . 1 5 )

Simi lar to the pr oo f in Sect ion 3 .1 , by Assumpt ion 3 .7 , we expand the expecta t ionterm in Taylor ser ies

S ~ = ~ n ) f I ~ ~ k i i , ~ o ) - -E~qk ,~o)) + 122x/-~ 5 - - ~o ) + o p l ) A .1 6 )i < j < t ~

b e c a u s e

E~b~x.(6o = - E d i D i j ( 5 o ) D i k ( 6 0 ) ( u 2 i - u 2 y ) (x ~ - x k )= - E [ D i ~ ( 6 o ) ( x i -xt ,) ' 1 {,v,~o>,~,, >~ }

x (E [( u2 ~ l {,, ,, > _, ,. ,~0 } - u2 j I {,, , ,> . . . . ~o r ) l z . z j] )]

~ - 0 , ( A . 1 7 )

where 02 = ~E~b i~(60) /~ ' . Hence ,

$2 = v ~ (n )~ l ~ (@iyk(5o)- E~btjk(ao)) + O2 V~ (~ - 60) + op( t) . (A .18 )

i


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S. Ch enlJournal o f Econometrics 80 1997) 1-34 31

Appeml ix B

B. 1. Reou larity conditions Jb r S ectio n 5

The fo l low ing assumpt ions are mad e for ca lcula t ing the semiparam etr ic e ffi -c iency bound for the Type-3 Tobi t under the independvnce res t r ic t ion . The as-sumpt ions are s imi lar to those in Coss le t t (Sect ion 4) .

Assumption B.1.Both vt(z,O) and v2(z,O ) aredifferent iable with respect to 0for a lmost a l l z , and mea surable in z for each 0 E O ( the param eter space for0) ; a l so the square- root l ike l ihood i s Hel l inger d i fferent iable wi th respect to 0 ,i.e.

and

f d ~ z ) h z ) { G ~ :2 v ~ z,O+ 3 ) ) - G ~ / Z v~ z ,O ))

~ T g O U I Z , O ) G o I / 2 D I Z , O ) )~Vl(-~~1)= o(Izl 2)

f do~(z)h(z)f f d ld y g~/z(I+ vt(z,O + z ), y + v2(z,O + 3))0

l [ + v l , y + v 2 )~ O + [ t 2( 1+ v t , y +v 2 ) ~ ]

x [ / - t / z ( / + t h , y + v 2 ) ~

w h e r e vt = vt(z,0) , v2 = v2(z, 0) .

= o I T I 2 ) ,

Assump ion B .2.The funct iong(ul ,u2) i s cont inuously d i fferent iable wi thO(ut,u2) , gl(ul ,u2) and O2(ut,u2) un i fo rmly bounded . Bo th~(uhu2) /O(ul ,u2) and~3~(UhU2)/O(Ub u2 )are integrable.

Assumption B.3 . Thecondi t ional expected va luesE[(~vt/OO)lv~< u ] a n d E [ ( ~ v 2 /00) Iv I


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32 S Chen lJournal o f Econome+rics 80 (1997) 1-3 4

B.2. D er iva t ions fo r the e ff ic ien t score

Fo r ,cZ~ in ( 23) , its adjoint ~.0" is foun d to be

( '~"Tr)(ul ' u2)= gt /2(U U2) [ , , ,f~'l

( a . 2 )

Le t "~/.,i loft ~'; the n it ca n be ver ified that

@(Ul,U2) A/2 u u"~=d h r, ( v ) d ( v, ~ ' ).q t l , 2 J J v ~ - - , (B .3)f l ( u , , u 2 ) = H v, u l ) ., (H, , , (v) ) Go(v)

where hv, (v)and H,, , (v)are probabil i ty density and cumulative distr ibution func-t ions o f v, , r espec t ive ly, so lves (B .2) , i . e . f l (u l ,u2)=( .~ ; ,~q) - t~(uL,u2) ,

: / /2,l} = 0, and

( . ~ ' * . ~ , , ) ( .~ -1 % ) - I , ~ ( u i , u 2 ) = q ' ( ' , t , u 2 ) . B . 4 )

Hence,

o o

( . % * . ~ , , ) - ; . < , * p 0 ( - , . - 2 ) = - ou 2 ( ' . , , ,~ ) f ~ f v ) d o( ______2). , G o v )

- I - ~ g - 1 : 2 ( R I , U 2 ) [ g I ( U h U 2 ) k I ( U l )

+ 0 2 ( u , , u2 )k2 u l ) ]

g l / 2 ( u ; ,u 2 ) g O ( U t)k l (u l )2 G o ( u t )

(B.5)


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S. Chenl Jour nal of Econometri cs 80 1997) 1-34

F i n a l l y , w h a v e

= - h I / 2 z )G ol /2 v ,)Lf l c= tp )c lG - -- -o --~ l{ t= o

,,, \ G o v ) ] l { i o }

- -=I2- ~ 1 ( u i , u 2 ) [ g = ( u l , u 2 ) k t ( u = ) + g 2( u = , u 2 )k 2 (u l) ]1 { l > o )

g=/2(u~ u2 ~ u t )k l (u = ) 1 I+ G o ( u = ) i t > o } ,

w h e r e u l = 1 + V l a n d u 2 = y + v 2 .

33

8 . 6 )

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