A Smoothed Least Squares Estimator For Threshold Regression Models

Transcription

1 A Smoothed Least Squares Estimator For Threshold Regressio Models M. Seo y ad O. Lito z Lodo School of Ecoomics July 0, 005 Abstract We propose a smoothed least squares estimator of the parameters of a threshold regressio model. Our model geeralizes that cosidered i Hase (000) to allow the thresholdig to deped o a liear idex of observed regressors, thus allowig discrete variables to eter. We also do ot assume that the threshold e ect is vaishigly small. Our estimator is show to be cosistet ad asymptotically ormal thus facilitatig stadard iferece techiques based o estimated stadard errors or stadard bootstrap for the threshold parameters themselves. We compare our co dece itervals with those of Hase (000) i a simulatio study ad show that our methods outperform his for large values of the threshold. We also iclude a applicatio to cross-coutry growth regressios. Some key words: Idex model; Sample Splittig; Segmeted Regressio; Smoothig; Threshold Estimatio. JEL Classi catio Number: C, C3, C4. y Departmet of Ecoomics, Lodo School of Ecoomics, Houghto Street, Lodo WCA AE, Uited Kigdom. address: m.seo@lse.ac.uk. z Departmet of Ecoomics, Lodo School of Ecoomics, Houghto Street, Lodo WCA AE, Uited Kigdom. address: litoo@lse.ac.uk. Sciece Research Coucil. This research was supported through a grat from the Ecoomic ad Social

2 Itroductio The threshold model (ofte called sample splittig or segmeted regressio) has wide applicatio i ecoomics. Hase (000) brought may of those applicatios to the attetio of ecoometricias. The literature divides accordig to autoregressio ad regressio, accordig to smooth, cotiuous, or discotiuous threshold, ad accordig to oparametric or parametric fuctioal form. It is di eret from the regime-switchig literature, see e.g. Kim ad Nelso (999) for a review, i that the switchig variable is observable. The smooth trasitio autoregressive models have bee widely used i macro ad acial applicatios, see the recet review paper of va Dijk, Terasvirta, ad Frases (000). The discotiuous threshold e ect has foud applicatios i macro ad i crosssectio growth regressios, see Hase (000) for discussio. There is also a oparametric literature i applied ecoomics associated with the cocept of regressio discotiuity desig, see for example Hah, Todd ad va der Klauw (00). I fact, a whole methodology has bee built aroud this, ad there are may applicatios. I that case the threshold poit is usually assumed kow. The paper of Delgado ad Hidalgo (000) work with the more geeral case of multiple ukow threshold poits i a oparametric regressio ad obtai a full set of results for estimatio ad iferece. This paper is about the parametric threshold regressio model. Ufortuately, this model does ot have a satisfactory basis for iferece eve i the case of least-squares estimatio. It has bee established that the threshold parameter estimate coverges faster tha the slope parameter estimates ad that its asymptotic distributio is ot Normal. O the other had, the slope parameter estimates coverges to a Normal distributio idepedetly of the threshold parameter estimate. I the cotext of threshold autoregressio, Cha (993) establishes that the threshold parameter estimate coverges to a fuctioal of a compoud Poisso process; the distributio is too complicated to be used i practice due to the depedece o the margial distributio of the covariates. Hase (000) develops a asymptotic distributio for the threshold parameter estimate based o the dimiishig threshold e ect assumptio, i which the threshold model becomes the liear model asymptotically. The limitig distributio is symmetric about zero ad has moderate tails but is ubouded at zero. Although the distributio is readily available through a simulatio, the validity of the asymptotic distributio may be limited to the small e ect case, as he calls it. It should be oted, however, that it provides a coservative co dece iterval for the threshold estimate for the case where the threshold e ect is held xed, uder the auxiliary assumptio of the ormality of ad the idepedece of the error from the regressors. These however are strog assumptios. Recetly, Gozalo ad Wolf (005) have proposed usig subsamplig to coduct iferece i threshold autoregressive models. They cosider the set-up of Tog (990) ad Cha (993) but also allow for the cotiuous threshold case of Cha ad Tsay (998). They allow for regime speci c het-

3 eroskedasticity as i Cha (993) (this was excluded i Hase (000)) but otherwise the iovatio process is i.i.d. They establish cosistecy of tests about ad co dece itervals for the threshold parameters based o the least squares estimator uder costat threshold assumptio. We cosider a threshold model that is more geeral tha the oe i Hase (000), which permits oly a pre-assiged cotiuous variable. I cotrast, we allow the threshold variable to be a liear combiatio of the regressors ad/or other variables, validatig the use of discotiuous variables for sample splittig i additio to cotiuous variables. It may be of iterest because it allows di eret threshold values for subsamples divided by a discrete variable like geder. Furthermore, we ca make decisio o the iclusio of a (some) variable(s) based o a test such as the t- or Wald test. This paper proposes the least squares estimatio of the threshold model after smoothig the objective fuctio i the spirit of the smoothed maximum score estimator of Horowitz (99). It is based o the replacemet of the idicator fuctio i the objective fuctio with a itegrated kerel. While the maximum score estimator by Maski (975) is asymptotically distributed as the radom variable that maximizes a certai Gaussia process, the smoothed maximum score estimator exhibits asymptotic ormality. The smoothig also brigs about a chage i the covergece rate. Uder smoothess coditios the smoothed maximum score estimator coverges faster tha the maximum score estimator. We develop a asymptotic theory for the smoothed least-squares estimatio of the threshold model i the regressio cotext. Ulike the previous literature, the threshold estimate is distributed as asymptotically ormal. Its covergece rate to esure the ormality is slower tha that obtaied i Cha (993) ad depeds o the choice of badwidth. Ulike i the maximum score case, smoothig reduces the rate of covergece. It is worth otig that Hase (000) also attais a maageable distributio at the expese of the covergece rate. The slope estimates are square root cosistet ad asymptotically ormally distributed, ad idepedet of the threshold estimate. Our developmet allows for time series data, a special case beig the threshold autoregressio of Tog (983; 990) : The cosistecy of the HAC estimatio i Adrews (99) is exteded to allow for the discotiuity i the threshold model. Our set-up is more geeral tha Gozalo ad Wolf (005) i that we allow both regime speci c heteroskedasticity ad covariate depedet heteroskedasticity as would be commo i cross-sectioal regressio applicatios. Also, our method has the usual advatage over subsamplig that we ca work with pivotal test statistics ad hece expect to obtai asymptotic re emets. We also ivestigate two slightly di eret implemetatios of the smoothig over approach. Although the two di eret methods result i the same asymptotics for the slope estimates, the limitig distributio of the threshold estimates are di eret, ad ot i geeral rakable. But we should iclude at least oe cotiuous variable ad the coe ciet is ormalized to.

4 We provide some simulatio evidece o the rate of covergece ad the ite sample distributio of our procedures. Co dece itervals based o our procedure perform better tha those of Hase (000) i his desig i the larger threshold case. The paper is orgaized as follows. Sectio itroduces the smoothed LS estimators ad their cosistecy ad asymptotic Normality is established i Sectio 3. Sectio 4 provides methods to costruct the asymptotic ad bootstrap co dece itervals. Sectio 5 discusses some extesios. Numerical results are preseted i Sectio 6 ad a applicatio to a cross-sectio growth model i Sectio 7. Sectio 8 cocludes. The proofs of Theorems are collected i a Appedix. The followig otatios are used. The itegral R is take over ( ; ) uless speci ed otherwise. Let jjgjj = R g (s) ds for ay fuctio g: For ay matrix A; let jjajj = tr(a > A) = : The Smoothed LS estimator. The Model Write the model y t = x > t + > ex t q > t > 0 + " t ; () where x t ; ex t ; ad q t may have commo variables. A leadig case is where ex t = x t but ex t ca also be a strict proper subset of x t : Let q t be the rst elemet of q t ; ad q t the other elemets of q t : Let X t whose rst elemet is q t deote all the regressors ad E (" t jx t ) = 0: Furthermore, assume the rst elemet of q t is the costat : Similarly, X t deotes q t ad X t the other elemets i X t. The rst elemet of is ormalized to ; ad the others are deoted as ; so that q t > = q t + q t > : This model icludes may cosidered i the literature as special cases, for example, the threshold autoregressio of Tog (983) as used by Potter (995). Hase (000) cosidered the special case where q t is oly a costat. It may be the case i practice where oly a few variables are employed to costruct the threshold idex.. Estimators The least squares (LS) estimator miimizes the objective fuctio S () = = X X y t x > t ~x > t q t + q > t > 0 y t x > t + X ~x > t ~x > t y t x > t o t ( ); () 3

5 where = > ; > ; > > R k ad t ( ) = q t + q > t > 0. The solutio is obtaied by pro led least squares, see Hase (000). Let LS deote the least squares estimator. De e a bouded fuctio K () satisfyig that lim K (s) = 0; lim K (s) = : s! s!+ It is worthwhile otig that this fuctio is aalogous to a cumulative distributio fuctio rather tha a desity fuctio. The, de e a smoothed objective fuctio S (; ) = X y t x > t X + ~x > t ~x > t y t x > t o qt + q t > K ad a smoothed least squares (SLS) estimator ; (3) = > ; > ; > > = arg mi S (; ) : (4) We assume that the parameter space is compact ad that the true parameter 0 = > 0 ; > 0 ; > > 0 is a iterior poit of : To distiguish the slope parameters, let s = ( > ; > ) > ad s 0 = ( > 0 ; > 0 ) > : I practice, oe solves the optimizatio problem by computig ( ); ( ) by a explicit least squares formula for give ; this is " # " P ( ) = x tx > t P ( ) ~x tx > t K t ( ) P x t~x > t K t ( ) P ~x t~x > t K t ( ) # " P x ty t P ~x ty t K t ( ) where K t ( ) = K(q t +q > t )= ); ad the optimizig the pro led criterio over : Practical di culty arises oly i the case of large dimesioal : There is a alterative approach, which is based o just replacig t ( ) i () by K t ( ); thus istead of (3) oe has S + (; ) = X y t x > t ~x > qt + q t > t K ad the smoothed least squares (SLS) estimator + = +> ; +> ; +> > = arg mi S+ (; ) : # (5) As before this optimizatio is doe i two stages with the pro led least squares estimators " # " + P ( ) = x P tx > t x # " P t~x > t K t ( ) + P ( ) ~x P x # ty t tx > t K t ( ) ~x P t~x > t Kt ( ) ~x ; ty t K t ( ) which are the plugged back ito (5) for optimizatio over : Note that although t ( ) = t ( ); K t ( ) 6= K t ( ) ad the estimators de ed by (3) ad (5) are di eret. I the case of the slope coe ciets this di erece vaishes asymptotically, but i the case of the threshold parameters it does ot. I the expositio we cocetrate maily o the estimator ; although similar commets apply to + : 4 ;

6 3 Asymptotic Properties 3. Cosistecy We assume the followig coditios to show the cosistecy of the SLS estimator. Assumptio (a) fx t ; " t g is a sequece of strictly statioary strog mixig radom variables with mixig umbers m ; m = ; ; : : : ; that satisfy m = o m 0=( 0 ) as m! for some 0 : (b) For some > 0 ; E Xt X t > < ad E kx t " t k < : (c) E (" t jx t ) = 0 a.s. (d) For almost every X t ; the probability distributio of X t coditioal o X t has everywhere positive desity with respect to Lebesgue measure. Coditio (a) correspods to Assumptio B of Adrews (987). Give a compact parameter space, the geeric uiform law of large umbers by Adrews (987) is applied for the followig developmet of the cosistecy proof, supported by the strog law of large umbers of de Jog (995, Theorem 4). For the asymptotic ormality, we eed to stregthe the mixig coditio. The followig theorem establishes the strog cosistecy of the SLS estimator. Theorem Let Assumptio hold. The,! 0 ad +! 0 almost surely. 3. Asymptotic Normality The asymptotic distributio is developed based o the stadard Taylor series expasio. Suppose S (; ) is twice di eretiable with respect to ; we de e T (; ) (; ) =@ Q (; ) S (; ) =@@ > : The superscript s ad to T ad Q ; whe applied, idicate the obvious partitios of T ad Q accordig to the slope parameter s ad the threshold parameter : We make a reparameterizatio to express the limitig distributios coveietly. Let z t = q t + q t > 0 : This ivolves decomposig ex t ito the part measurable with respect to z t ad the part that is ot so. There is a oe-to-oe relatio betwee z t ; X t > > ad Xt for ay 0 : Let T be the mappig such that z t ; X t > > = T Xt ad let S be the selectio matrix such that ~x t = SX t : Let _ = T > S > so that z t ; X t > _ = ~x > t : As above, we deote the rst elemet of _ as _ ad the others as _ : For example, if x t = ~x t = q t ; whose dimesio is k; the S = I k, T =!! 0 ad _ = : 0 I k + 5

7 We the have > ex t q t > > 0 = (z t _ + X t > _ )(z t > 0) = z t _ (z t > 0) + X t > _ (z t > 0); where the rst term o the right had side is cotiuous i z t at z t = 0 with probability oe, while the secod term is ot. By Assumptio, the distributio of z t coditioal o X t has everywhere positive desity with respect to Lebesgue measure for almost every X t : Let f (jx ) deote this desity give X t = X ad f () the desity of z t : For each positive iteger i; de e f (i) (zjx ) i f (zjx ) =@z i wheever the derivative exists. De e E " t jx t = lim E " z!0 + t jz t = z; X t s>0 K 0 (s) ds + lim z!0 E " t jz t = z; X t s<0 K 0 (s) ds; (6) ad V s 4 P s= = Ex x > s " " s 4 P s= Ex! ~x > s " " s fz > 0; z s > 0g 4 P s= E~x x > s " " s fz > 0; z s > 0g 4 P s= E~x ~x > s " " s fz > 0; z s > 0g V = kk 0 k (X E t > _ ) X t > _ E " t jx t q t q tjz > t = 0 f (0) h V + = 4E (X t > _ i ) E " t jx t qt q tjz > t = 0 f (0)! Q s Ex t x > t Ex t ~x > t fz t > 0g = E~x t x > t fz t > 0g E~x t ~x > t fz t > 0g h Q = K 0 (0) E (X t > _ i ) q t q tjz > t = 0 f (0) Q + = kk 0 k h(x E t > _ i ) q t q tjz > t = 0 f (0) : If we impose a stroger assumptio that f" t g is a martigale di erece sequece, the all the autocovariaces drop out of V s : I cotrast, the threshold estimates do ot ivolve the log-ru variace as is the case i the dyamic biary choice model of de Jog ad Wouterse (004) ad i the threshold LAD model of Caer (00) : If K 0 is symmetric aroud zero, E " t jx t = kk 0 k lim E " z!0 + t jz t = z; X t + lim E " t jz t = z; X t =: z!0 If additioally, E (" t jz t ; X t ) is cotiuous at z t E (" t jz t = 0; X t ) : The assumptios we eed are collected i the followig. = 0, this expressio simpli es further to kk 0 k 6

8 Assumptio (a) For all vectors such that jj = ad r > 4; E X t > _ " t q t > r < ad E (X t > _ ) q t > r <, or (a + ) the coditio E (X t > _ ) q t > r < i (a) is replaced by E X t > _ q t > r < : (b) fx t ; " t g is a sequece of strictly statioary strog mixig radom variables with mixig umbers m ; m = ; ; : : : ; that satisfy m Cm (r )=(r ) for positive C ad ; as m! : (c) For some iteger h ad each iteger i such that i h ; all z i a eighborhood of 0, almost every X ; ad some M <, f (i) (zjx ) exists ad is a cotiuous fuctio of z satisfyig f (i) (zjx ) < M. I additio, f (zjx ) < M for all z ad almost every X. Furthermore, E (" 4 t jx t ) < M for almost every X t ; (d) ad the coditioal joit desity f (z t ; z t m jx t ; X t m ) < M; for all (z t ; z t m ) ad almost all (X t ; X t m ) ; ad the coditioal expectatio E (" t m jx t ; X t m ) < M for almost all (X t ; X t m ) : (e) 0 is a iterior poit of a compact parameter space : (f) Ad V s ; V ; Q s ; ad Q are ite ad positive de ite. I case of Hase s model where z t = q t +, V ad Q are de ed without q t q > t ad the coditio (a) is simpli ed to EjX > t _ " t j r < ad Ej(X > t _ ) j r <, or to EjX > t _ j r < i (a + ). The momet coditios are to esure the cosistecy of the variace covariace matrix estimators that are itroduced later. The coditio (a + ) is aalogous to de Jog ad Wouterse (004) ad is slightly stroger tha that of Cha (993) or Hase (000), which requires a ite fourth momet. The mixig coditio (b) is more geeral tha mixig i Hase (000), which icludes may oliear time series such as TAR processes as discussed there: The coditios (c) - (f) are commo i the smoothed estimatio as i Horowitz (99), oly (d) beig a aalogue of a iid sample to a depedet sample. The smoothess coditio here is stroger tha that of Cha (993) sice the boudedess of the rst derivative of the desity implies the uiform cotiuity. While (f) is stadard, the positivity of Q excludes a cotiuous threshold model, so does Assumptio.7 of Hase (000). The iteess of V s ca be implied by the -mixig coditio with a slightly stroger assumptio o the mixig coe ciet m plus a momet coditio. See Adrews (99; Lemma ). Ulike Hase (000), we do ot impose the cotiuity of E (" t jz t ) at z t = 0, thus allowig for a regime speci c heteroskedasticity. This type of heteroskedasticity is quite plausible i applicatios ad we would certaily wat to allow for it. I such a case, oe may wat to employ a weighted least squares although this requires further estimatio. It is expected that the asymptotics i Hase (000) ca be modi ed to allow such discotiuity, 7

9 but the the studetizig of the threshold estimate seems to become more cumbersome. We make the followigs assumptios regardig the smoothig fuctio K ad the badwidth parameter : Assumptio 3 (a) K is twice di eretiable everywhere, jk 0 ()j ad jk 00 ()j are uiformly bouded, R ad each of the followig itegrals is ite: jk 0 j 4 ; R jk 00 j ; R jv K 00 (v) dvj : (b) For some iteger h ad each iteger i ( i h) ; R jv i K 0 (v) dvj < ; ad s i sg (s) K 0 (s) ds = 0; ad s h sg (s) K 0 (s) ds 6= 0; ad K (x) K (0)? 0 if x? 0: (c) For each iteger i (0 i h) ; ad > 0; ad ay sequece f g covergig to 0; lim! i h s i K 0 (s) ds = 0; ad lim jk 00 (s)j ds = 0:! (d) lim sup h < ad! j sj> lim h jk 0 (s)j ds = 0:! j sj> (e) For some (0; ]; a positive costat C; ad all x; y R; (f) For some sequece m ; ad " > 0; jk 00 (x) K 00 (y)j C jx yj : j sj> log (m ) 6=r m! 0 3k 3=r+" m! 0: These coditios are similar to those i Horowitz (99). Coditio (b) is a aalogous coditio to that de ig the so-called h th order kerel, ad requires a kerel K 0 that permits egative values. A kerel that satis es these coditios is K (x) = (x) + x (x) ; where ad are the distributio fuctio ad the desity of the stadard ormal, respectively. For this kerel K 0 (0) = p = = 0:798 ad jjk 0 jj = 0:776: The limit distributio i Theorem of Hase (000) is expected to chage to arg max f! ( jrj = + W (r)) fr > 0g +! ( jrj = + W (r)) fr < 0gg ; r where! ad! are the right ad left limit i (6) : Thus,! ad! does ot average out as it does i our case. 8

10 Coditio (e) serves to determie the rate for : Whe the data are i.i.d. ad the regressors possess a momet geeratig fuctio, the coditios ca be weakeed to log ()! 0; (7) sice m = 0 ad we ca set m = i this case. Cotrary to the smoothed maximum score estimatio, we choose the badwidth that coverges to zero as fast as permissible. Although coditio (e) i Assumptio 3 provides permissible rates for the badwidth selectio, it may ot be sharp. I fact, Delgado ad Hidalgo (000) study the oparametric estimatio of the locatios ad sizes of the discotiuities i coditioal expectatio. They establish asymptotic p ormality at rate ; where p is the umber of covariates i the oparametric regressio, uder p the restrictios that p+! ad lim sup p+5 < : If oe could take p = 0 (oe caot i their theory); which would correspod to parametric regressio, i their results, this would suggest asymptotic ormality holds at rates arbitrarily close to : Theorem Let Assumptios - 3 hold with Assumptio (a) : The p ( s s 0) =) N p ( 0 ) =) N ad they are asymptotically idepedet. Similarly, we have 0; Q s V s Q s ; 0; Q V Q ; Corollary 3 Let Assumptios - 3 hold with Assumptio (a + ) : The ad they are asymptotically idepedet. Remarks. p s+ s 0 =) N 0; Q s V s Q s ; p + 0 =) N 0; Q + V + Q + ;. The covergece rate of is p, which meas that faster covergece of to zero accelerates the covergece of : This is i cotrast to the smoothed maximum score estimator for which the faster covergece of the badwidth reduces the covergece rate of the estimator. I the i.i.d. case, the badwidth = log = p satis es the coditio (7) ad lim sup h < for ay! h : I this case we obtai that is (apart from a logarithmic factor) 3=4 cosistet. However, 9

11 the badwidth restrictios are su ciet ad ot ecessary ad it is quite plausible that oe obtais p covergece but perhaps ot asymptotic ormality for smaller badwidths.. As i the least squares estimatio of the threshold model, the slope estimate s is ot a ected asymptotically by the estimatio of the threshold parameter i either case. 3. The assumptio that h is bouded is imposed to esure the asymptotic idepedece of from s : With a badwidth covergig slower, we may obtai the covariaces betwee them, which may prove bee cial for ite sample iferece o the slope parameters sice s deped o regardless of the choice of i ite samples. It is also likely, however, that it may itroduce a asymptotic bias for as it is the case i the smoothed maximum score estimator. The covergece rate of s is ot a ected by this chage i the rate of covergece of the badwidth. 4. Our coditios are stroger tha those of Hase (000) ad Cha (993) with regard to smoothess. Speci cally, they do ot require the distributio of z t jx t to be smooth. Whe the smoothess coditios do ot hold, our estimator coverges at a slower rate due to the presece of a bias term of large order. This is as foud i Pollard (993) regardig the smoothed maximum score estimator of Horowitz (99). 5. Although we do ot explicitly treat it, the small threshold case of Hase (000) ca be aalyzed withi the same framework. Speci cally, whe 7! =! 0 oe still obtais asymptotic ormality, provided > 0 ad is ot too large, but at a slower rate of covergece re ectig the presece of i the score ad Hessia fuctios. Notice that the asymptotic variace of the score fuctio (of ) is somewhat simpler i this case because the term E[(X > t _ ) 4 q t q > tjz t = 0] is of smaller order relative to E[4(X > t _ ) E (" t jx t ) q t q > tjz t = 0]. Compare with Hase (000). 6. If q t cosists of the costat oly, the is the threshold estimate i the usual sese. If a dummy such as geder or regio is icluded i additio to the costat, the the coe ciet estimate for the dummy meas the di erece i the threshold values betwee two subsamples. Therefore, the t -test o the coe ciet examies whether the threshold poits are the same across two subsamples or ot. 7. The case where the thresholdig variable is time ca also be hadled i this framework. The results obtaied above apply to the estimate of the break fractio (0; ) with some modi catios. The terms costitutig the asymptotic variaces are de ed with f (0) =, q t = ; ad the coditioal expectatios replaced with the ucoditioal oes. 8. The asymptotic distributios of ad + do ot deped o the error autocorrelatio fuctio, whereas the asymptotic distributios of the slope parameter estimates does. 9. The two estimators ad + have di eret asymptotic variaces. The rakig could go either way, as the followig example illustrates, ad so there is othig a priori to favour oe approach over 0

12 the other. Cosider the desig of Hase (000) y t = > x t + > x t (q t ) + " t ; where x t = (; x t ); q t N (; ) ; " t N (0; ) ; = ( ; ) > ; = 0; = 0; ad = : I case I, x t = q t ad i case II, x t N(0; ): The theoretical asymptotic variace of the two smoothed estimators ad + i these desigs is give below. This shows that as! 0 the asymptotic variace icreases for both estimators. For small ; has slightly lower asymptotic variace but for bigger tha about 0:5; + has smaller variace i cases I ad II. Case I p 64 Case II p 64 4 Iferece Methods avar( ) avar( + ) avar( )=avar( + ) + p p The costructio of the asymptotic co dece set is straightforward by ivertig the t or Wald statistic give the asymptotic ormality. Ways to estimate the asymptotic variaces are described below. We also discuss the likelihood ratio statistics. We also discuss the bootstrap co dece itervals. 4. Asymptotic Variace Estimatio, t ad Wald Statistics We ow discuss various estimators of the asymptotic variace of our estimators. As usual there are may alterative estimators of the asymptotic variace depedig o which iformatio is imposed. I the simulatio experimets below we ivestigate some of the proposals made here. Let ;t ( ) = + ;t + where e + t = y t x > t + ~x > t + qt +q t K > + are de ed, respectively: ~x > o t ~x > t y t x > q t t K 0 qt + q t > = e + t ~x > q t t K 0 qt + q t > ; : The, the variace estimators for the threshold parameter ^V = X ;t ( ) ;t ( ) > ad ^V + = X + ;t + + ;t + > : (8)

13 These impose the absece of ay theoretical autocorrelatio but allow for heteroskesdasticity. We may also make some degrees of freedom adjustmet replacig by k; where k is the total umber of estimated regressio parameters. Oe may wish to impose homoskedasticity, which ca be achieved by separatig out the residuals, for example replace ^V + H = X (e + t ) 4 X (~x > t ) K 0 qt + q > t q tq > t: Regardig the estimatio of Q ad Q + there are several possibilities. First, just take Q ( ; ) ad Q + + ; : Secod, as with oliear least squares oe ca drop some terms that are asymptotically zero. For example, the Hessia is Q + + ; = + > + e+ e > ; (9) ad the secod term is asymptotically zero. Istead therefore, compute the OPG (outer product of the gradiet) estimate ^Q + = + > : (0) Ulike Hase (000) we do ot eed to explicitly do oparametric estimatio of desity ad coditioal expectatio. We ow tur to V s ; which requires HAC estimatio because the e ect of error autocorrelatio does ot die out. Let e t = y t x > t ~x > qt + q t > t K s ;t ( ) = ^j = x > t ; ~x > qt + q > > t t K ( P t=j+ s ;t ( ) s ;t j ( ) > e t e t j for j 0; P t= j+ s ;t+j ( ) s ;t ( ) > e t+j e t for j < 0: Let w () : R! [ ; ] be a cotiuous fuctio such that w (0) = ; w (x) = w ( x) ; ad kwk < : The, de e ^V s = X j= + j w ^j; l where l is a lag trucatio parameter that is o (). Similarly we ca de e ^V s+ : For more discussio regardig the choice of the kerel ad lag trucatio parameter, see Adrews (99) : It should be oted, however, that his cosistecy results regardig the HAC estimator do ot hold for the

14 threshold models due to the lack of smoothess. Fially Q s ad Q s+ ca be estimated by 0 P ^Q s x P tx > t x t~x > qt +q t t K > P ~x tx > qt +q t t K > P A ~x t~x > qt +q t t K > ^Q s+ = X s+ ;t + s+ ;t + > : The above stadard errors have imposed the block diagoal structure betwee the estimates of ; s foud i the asymptotics. I small samples it may be preferable to ot impose this restrictio; ideed, Hase (000) proposed to use Boferoi-type bads to take accout of the small sample e ect of estimatio error i o the the estimatio of s. We have a much more atural ad simple way of doig this. Istead, compute the diagoal elemets of the matrix ^Q NB b V NB ^Q NB ; where: bv NB = 0 ^V s = ^Q s ^V s >! p ^Qs bv s s ^V ^Q s ; ^QNB = ^V ^Q s > ^Q P x t~x > q t t > K 0 qt +q t > P ~x t ~x > t + ~x t x > q > t ~x t y t t K 0 qt +q t >! ; () A : Similarly, we may de e ^Q + with 0 P ^Q s + x t~x > t + q t > K 0 qt +q t > + P ~x t ~x > qt +q t t K > + + ~x t x > t + q > ~x t y t t K 0 qt +q t > + The followig theorem establishes the cosistecy of the proposed stadard errors. Theorem 4 Uder Assumptio -3, ^V s ; ^V ; ^Q s ad Q ( ) coverge i probability to V s ; V ; Q s ad Q ; respectively. It follows that t ad Wald statistics based o ay of the above estimates are asymptotically correctly sized. A : 4. Likelihood Ratio Dufour (997) argues that the t or Wald statistic behaves poorly whe the parameter space cotais a regio where ideti catio fails. Therefore, Hase (000) ; i which the threshold parameter is ot ideti ed asymptotically, proposes the co dece iterval for the threshold parameter iverted 3

15 from the LR statistic that is costructed uder the auxiliary assumptio that the error is i.i.d. ormal. We may de e LR ( ) = S ( ) S ( ) ; () S ( ) ad similarly LR ( ) + usig S + : If is oe-dimesioal, the statistics are distributed as s X asymptotically where the scalig factors are s = V =Q or s = V + =Q +, where = var(" t ): Uder homoskedasticity, the scalig factor of LR ( ) + is equal to oe. Apart form this special case, oe must adjust the critical values or repivot the test statistics by dividig through by a estimate of s obtaied i the previous sectio. The resultig co dece regio is the set C = f : LR ( ) =bs X ()g; where X () is the upper -critical value of the X distributio ad bs is a cosistet estimate of s. Note that i ite samples C is ot ecessarily a iterval ad may be a uio of disjoit itervals, as happes quite ofte i practice, see Hase (000, Figure ). I this case, oe may prefer the iterval C it = [ mi ; max ]; where mi = if C ad max = sup C : Asymptotically, C it C. Whe is multidimesioal, the adjustmet ad C are the same, but i ite samples C it for heteroskedasticity is more complicated ad this reduces the attractiveess of the likelihood ratio. 4.3 Bootstrap A alterative approach to iferece here is based o the bootstrap. I the i.i.d. case this is particularly simple. Let fw t g be the dataset, where W t = (y t ; X t ). The let fwt g be a radom sample draw with replacemet from fw t g : Compute from fwt g i the same way as was computed from fw t g : Suppose that oe wats a two-sided symmetric level co dece iterval for the scalar quatity (): The rst method is to just obtai the empirical quatiles x ; of the distributio of ( ) coditioal o fw t g ; ad the let the iterval be [( ) x ;= ; ( ) + x ; = ]: This would be called the percetile method. A perhaps more desirable approach is based o the statistic T = (( ) ())=s ; where s is a estimate of the asymptotic stadard deviatio of ( ): I the evet that is di eretiable we would have s = r( ) > b Q b V b Q r( ); where b V ad b Q are the matrices with sub-blocks ^V s ad ^V ad b Q s ad b Q described above (i the i.i.d. case oe does ot compute the covariaces). By the bootstrap simulatio oe obtais the critical values z ;= of T = (( ) ( ))=s ad the the iterval [( ) z ;= s ; ( ) + z ; = s ]: This is usually called the bootstrap-t method. This co dece iterval is asymptotically correct, refer to Theorem. of Horowitz (00) : Sice the asymptotic distributio of T does ot deped o uisace parameters, we ca expect the bootstrap to achieve asymptotic re emets, see Shao 4

16 ad Tu (995) ad Horowitz (00). Similar commets apply to the likelihood ratio statistics or the repivoted likelihood ratio statistics. I the time series case, oe geerally has to use a more complicated resamplig method like the block bootstrap to capture the e ect of the depedece structure o the limitig distributio. However, i the special case of the threshold parameter or fuctios thereof, oe ca obtai cosistet co dece itervals from the i.i.d. resamplig because the limitig distributio of the estimator is ot a ected by the depedece structure. O the other had, oe does ot obtai asymptotic re emets by this method. I order to obtai asymptotic re emets for the threshold parameters or to compute cosistet itervals for the slopes we may use the o-overlappig (viz., Carlstei (986)) ad overlappig (viz., Küsch (989)) block bootstrap procedures. The observatios to be bootstrapped are the vectors fw t : t = ; : : : ; g as before. Let L deote the legth of the blocks satisfyig L _ for some 0 < <. With o-overlappig blocks, block is observatios fw j : j = ; : : : ; Lg; block is observatios fw L+j : j = ; : : : ; Lg; ad so forth. There are B di eret blocks, where BL = : With overlappig blocks, block is observatios fw j : j = ; : : : ; Lg; block is observatios fw +j : j = ; : : : ; Lg; ad so forth. There are T fw t L + di eret blocks. The bootstrap sample : t = ; : : : ; g are obtaied by samplig B blocks radomly with replacemet from either the B o-overlappig blocks or the order sampled. L + overlappig blocks ad layig them ed-to-ed i the 5 Some Extesios 5. The Cotiuous Case Suppose that _ = 0; (3) where _ was de ed i sectio 3.. The, the model () becomes cotiuous, sice ~x > t = z t ; X > t _: I this case, the formula Q + V + Q + we gave for the asymptotic variace of the threshold parameter estimate is ot well-de ed, sice V ad Q are zero; however, lower order terms ca be foud that are o-zero i both quatities. Let V = 4 _ E E " t jx t qt q tjz > t = 0 f (0) Q = _ s sg (s) K 00 (s) ds E q q > jz t = 0 f (0) A = A _ (h ) (= (h )!) f zjx (0jX ) q df X (X ); 5

17 where E (" t jx t ) = R s>0 s K 0 (s) ds lim z!0 + E (" jz; X ) + R s<0 s K 0 (s) ds lim z!0 A = R s h sg (s) K 0 (s) ds: The, Theorem ca be modi ed as follows. E (" jz; X ) ad Corollary 5 Let Assumptios - 3 hold with V ad Q replaced by V ad Q respectively. Furthermore, assume (3) ad p h has a ite limit : The, p ( s s 0) =) N 0; Q s V s Q s ; p ( 0 ) =) N Q A; Q V Q ; ad they are asymptotically idepedet. Note that the covergece rate of the threshold estimate is chaged from p to p : This rate is slower tha that of the usmoothed LSE of a TAR model i Gozalo ad Wolf (005) ; where both the slope ad threshold estimates are joitly asymptotically ormally distributed with the p rate ad they are correlated. The bias correctio is straightforward sice p T ( ) is a cosistet estimator of A ad the studetizig ca be doe as described i Sectio 4. Whe the co dece iterval is costructed as i Sectio 4 with the badwidth satisfyig (7) ; it will be a asymptotically correct oe eve whe the true model is cotiuous sice = 0 i that case. We ca also costruct a test for the cotiuity of the model. Sice _ = T delta-method ad Theorem 5. > S > ; we ca test the hypothesis (3) by the X test, utilizig the 5. Multiple Threshold Case Suppose that there are multiple thresholds determied by variables q > tj j that eter i a additively separable fashio y t = x > t + px ~x > t j q tj > j > 0 + " t : j= The, the estimatio strategy ad theoretical results are essetially as before. Speci cally, let! q jt + q jt > j K tj = K j ; j = ; : : : ; p j ad de e for give = ( ; : : : ; p ); b s ( ) = (W ( ) > W ( )) W ( ) > y; where W ( ) is the k( + p) matrix with rows w t = (x > t ; ~x > t K t ; : : : ; ~x > t K tp ) > : The de e to miimize S + ( ) = X y t w > t ( ) b s ( ) : 6

18 I this case, we expect the rate of covergece of to be the same as before, although the asymptotic variace will be di eret. Bai (997) has show, i the structural chage cotext, that a sequetial strategy ca work: estimate a sigle threshold model ad the a secod threshold coditioig o the rst oe ad so o. This is very coveiet computatioally. Simulatios show that this approach also works i this case: the domiat threshold is ideti ed i the rst roud etc. O the other had if oe has thresholds of the type y t = x > t + ~x > t q t > > 0; : : : ; q tp > p > 0 + " t ; the the smoothig based method will su er severely from curse of dimesioality because the smoothig operatio is of dimesio p: 5.3 Alterative Estimatio Criteria The least squares method ca sometimes be strogly i ueced by outliers ad oe may wish to use a more robust method for estimatig parameters like the LAD. Our secod method ca easily be adapted to this case. Thus for example cosider the criterio S LAD+ (; ) = X y t x > t ~x > qt + q t > t K ad let LAD+ = ( LAD+> ; LAD+> ; LAD+> ) > = arg mi S LAD+ (; ) : Although there is ot a explicit formula for the pro led slope estimators i this case, the pro led slopes are regular LAD regressio estimators ad ca be computed e cietly by liear programmig, Koeker (997). The threshold estimate ca easily be computed i the scalar case by grid search but otherwise it requires some care. It ca be show that LAD+ is cosistet ad asymptoptically ormal with the same rates as the least squares estimators, uder some coditios. 6 Numerical Results 6. Mote Carlo We ivestigated agai the desig of Hase (000). I this case, y t = > x t + > x t (q t ) + " t ; where x t = (; x t ); q t N (; ) ; " t N (0; ) ; = ( ; ) > ; = 0; = 0; ad = : I case I, x t = q t ad i case II, x t N(0; ): We compute ; + usig the kerel K(x) = (x) + x(x); where ad are the stadard Gaussia c.d.f. ad desity fuctios respectively. The estimators 7

19 are computed by grid search over the sample of observed threshold values. We cosider parameter values f0:5; 0:5; :0; :5; :0g ad sample sizes f50; 00; 50; 500; 000g ad do s = 000 replicatios for each experimet. I other work we have examied larger sample sizes, ad we commet o these results. 6.. Performace of the Estimator I this sectio we describe the performace of the usmoothed ad smoothed threshold estimators. We take badwidth parameter = (log ) = : I Table a we report results for the estimates of ; while i Table b we preset the results for the estimates of : We preset the iterquartile rage divided by.35, which is a robust estimate of the stadard deviatio of the estimates. The biases are very small i all cases ad are ot reported. There are several mai results:. Results improve with sample size ad with the value of. The small sample variability of all estimates is much higher tha predicted by the asymptotic theory, but this overpredictio reduces cosiderably with sample size ad with : This overpredictio is also implicitly true for the usmoothed least squares estimator. 3. The estimator + is early always better tha We have also examied the case with very large sample sizes ad d that with = 0; 000 the mea squared errors are withi 5% of the asymptotic predictios. Also i this case q-q plots reveal that ormality is a good approximatio. 6.. Performace of the Co dece Itervals We ext compare our co dece itervals with those of Hase (000). We compute the estimators by the two di eret smoothig methods ad we ivestigated three di eret t-statistic co dece itervals: those based o estimates of the asymptotic variace, those based o the percetile bootstrap, ad those based o the pivotal bootstrap usig the asymptotic stadard errors to studetize. Hase (000) used the likelihood ratio, which ca be expected to work particularly well i this desig as it assumes ormality ad homoskedasticity. We report results for the parameter ad for the fty di eret combiatios of sample sizes ( f50; 00; 50; 500; 000g) ad parameter values ( f0:5; 0:5; :0; :5; :0g) for case I ad II. We implemeted the two methods as i the previous sectio. The results of the simulatios are show i the tables. I Table abc we give the coverage rate for itervals based o percetile bootstrap, pivotal bootstrap, ad asymptotic method. I Table 8

20 3abc we give the same for itervals based o + : These tables correspod to Table II of Hase (000). Apart from the smallest value of ; the bootstrap coverage rates are close to the omial rate ad because of the small umber of replicatios are geerally withi stadard errors of the target value (0.0) except for the small case. The coverage rates of the asymptotic itervals are less satisfactory for smaller samples sizes, but improve steadily with sample size ad are competitive for = 000. There does ot seem to be much di erece betwee the itervals based o ad the itervals based o +. I Table 4ab ad 5ab we give the bootstrap itervals for based o the two estimators. These correspod to Table III of Hase (000). 3 The coverage rates of the bootstrap itervals are close to the omial throughout. We also ivestigated the bootstrap for the usmoothed estimator. The coverage rates were very low (ad ot reported here) eve i the largest sample sizes ad we take this as evidece of icosistecy. The results suggest that the small threshold case, = 0:5; is problematic. Ideed the asymptotic itervals are cosiderably udercovered for this case, although the bootstrap itervals are overcovered. This suggests that a combiatio of the two itervals may be useful i practice. Figure shows a typical sample from this process - the threshold e ect is ideed very small i this case. We ivestigated some di eret asymptotic co dece itervals for the special case z = q; = 0:5; = :06 =5 : The results are reported i Tables 6 ad Tables 7. We cosider badwidths = :06 =5 ad = (log ) = : The results suggest that larger badwidth gives better coverage. It also suggests that the Likelihood ratio itervals have the most accurate coverage, followed by the o-block diagoal co dece itervals. We also report the media legth of the co dece itervals; the smaller badwidth procedures gives smaller legth. 6. Applicatio 6.. Growth with multiple equilibria We illustrate our methodology by examiig the hypothesis that iitial coditios may determie cross-sectio growth behavior usig the Summers-Hesto data set. Durlauf ad Johso (995) studied it by a regressio tree method due to Breima et al. (984) ad Hase (000) by a threshold regressio usig the same data set. We specify the model similarly to the previous studies. Let y i;t be real GDP per member of the populatio aged 5-64 i year t; i be ivestmet to GDP ratio, i be growth rate of the workig-age populatio, ad S i be the fractio of workig-age populatio erolled i secodary school. The variables other tha y i;t are the aual averages over the period I Table III of Hase, the critical level of the table is 95%. Ad the co dece iterval is costructed as uio of co dece itervals based o a give set of threshold values that is a co dece iterval for the threshold estimate. 9

21 The, the log real GDP growth, l y i;985 l S i : l y i;960, is explaied by l y i;960 ; l i ; l ( i + 0:05) ; ad Durlauf-Johso proposed the iitial output y i;960 ad the literacy rate (lr i ) at the year 960 as the possible threshold variables. Hase examied each variable separately by the Lagrage multiplier test of Hase (996) ad foud some evidece for the presece of a threshold e ect based o the iitial output. Multiple threshold variables are ot allowed i Hase (960) or i Hase (000) : Thus, he took a sequetial approach i which he estimates the threshold with the iitial output ad test for the further threshold withi the subsamples splitted by the threshold estimate. This procedure is repeated util we caot d further evidece of threshold. He reports the rst sample split at the output level of $863 ad the secod at the iitial literacy rate of 45% withi the subsample whose iitial output is larger tha $863. We rst estimate the model where the output is the threshold variable. The SLS estimate + is $78 with the stadard error of $36. There are a couple of di eret methods to compute the stadard error as explaied i Sectio 4.. The oes reported here are the most coservative ad are robust to heteroskedasticity 4. The 95% bootstrap co dece iterval is [0; 6675] ; much wider tha the asymptotic iterval. Figure displays the smoothed sum of squares residuals as a fuctio of the threshold i output. There are 47 of the 96 coutries below the threshold. The secod split is also based o the iitial output of $777 with the stadard error of $33. See Figure 3 for the sum of squared residuals.ulike i Hase, we could ot d evidece for the threshold i the literacy rate i the subsamples geerated by the iitial output of $78 usig the LM test of Hase (996) : We also estimate the model with the threshold i the liear combiatio of the iitial output ad the iitial literacy rate. The coe ciet of the output is ormalized to, ad the estimates are obtaied by bivariate grid search. The estimated splittig lie is y 960 = 46 lr 94; which reduces the sum of squared residuals by approximately 0% compared to that of oe threshold i output above. The coe ciet of lr appears sigi cat as its stadard error is 0 ad its 95% bootstrap co dece iterval is [4:7; 390] : O the cotrary to the sequetial approach, this idicates that both the iitial output ad the iitial literacy rate may be related to the determiatio of the growth path. The estimates for slope parameters are reported i Table 8. We observe that the iitial output ad the populatio growth have egative e ect o the growth rate. I the subsample where the output is above $78, 54% additioal growth rate is expected while the average is 44%. Ad i the 4 The heteroskedasticity appears clear. For example, the sample variace i the regime i which the output is below the threshold $78 is 0.0 while that of the other regime is

22 subsample where y 960 > 46 lr 94; the 57% icrease is expected. 7 Coclusios We have show that the smoothed threshold estimator is asymptotically ormal albeit at a slower rate tha the correspodig usmoothed estimator. This is bor out by simulatios. O the other had, our simulatios show that our co dece itervals ca be more accurate tha the co dece itervals of Hase (000) especially for larger thresholds. It may be possible to show that the rate at which the estimator (or correspodig test statistics) approaches its limit is quite fast, see Hall (99) for correspodig results for desity ad regressio estimators ad Horowitz (998) for results for smoothed LAD (SLAD) estimators, ad perhaps faster tha is the case for the usmoothed estimator. Furthermore, we expect the smoothed estimatio will eable the higher-order correctio by the pivotal bootstrap, as is the case i the SLAD estimatio i Horowitz (998): He shows that the SLAD estimator has much simpler higher-order asymptotics tha the LAD estimator ad thus the bootstrap ca correct the secod-order term. Sice the smoothig also makes the objective fuctio of the threshold estimatio di eretiable, which is ecessary for the Taylor-series expasio, we ca expect a simpler expasio ad the higher-order correctibility of the bootstrap. This would provide a theoretical ratioale for the simulatio results ad give oe motivatio for preferrig our estimator/test statistic over the usmoothed oe. I practice, it is importat to have some strategy for choosig the smoothig parameter : The aswer is likely to deped o the purpose to which the estimatio is put. For estimatio itself, a small of the order (log ) = seems to perform well. For testig problems badwidth is likely to a ect size ad power i di eret ways so small is ot ecessarily best.

23 A Proofs of Theorems Lemma. Suppose that Assumptio holds. The followig covergeces hold almost surely uiformly over the parameter space: (i) (ii) (iii) X qt + q t > <! Pr qt + q t > < for ay > 0; X x t x > t q t + q t > > 0! Ex t x > t q t + q t > > 0 X x t " t q t + q t > > 0! Ex t " t q t + q t > > 0 Proof of Lemma. We apply the geeric uiform law of large umbers by Adrews (987, Corollary ). Assumptio A ad B of that paper are also assumed here. Assumptio B is trivially satis ed i (i) sice the idicator fuctio is bouded, ad i (ii) ad (iii) ; sice: E sup xt x > t q t + q t > > 0! E xt x > t < E sup xt " t q t + q > t > 0! E jx t " t j < : Next, q t + q t > < ad q t + q t > > 0 satisfy Assumptio A3 as show i de Jog ad Wooterse (004, Lemma 4). The, by Cauchy-Schwarz iequality, (ii) ad (iii) satisfy Assumptio A3, which completes the proof of Lemma. Proof of Theorem. First, we show that js (; ) S (; )j! 0 almost surely uiformly over : To do that, ote that = js () S (; )j X ~x > t ~x > t y t x > t o q t + q > qt + q t > t > 0 K v v u t X ~x > t ~x > t y t x > t o u t X q t + q t > qt + q t > > 0 K ; the rst term of which almost surely coverges to a ite umber uiformly over > ; > > by Lemma. For the covergece of the secod term, ote that the same reasoig as i Lemma 4 of Horowitz (99) applies. The, it is su ciet to show that, for ay > 0; (A4) i that paper, i.e., X q t + q t > <

24 coverges to Pr q t + q t > < ; almost surely uiformly over ; which follows from Lemma. Next, we show that = arg mi S () is cosistet, which is su ciet for the cosistecy of : For a xed ; the least squares estimator of ad are the OLS estimators, which are deoted as ( ) ad ( ) respectively. Let S ( ) = S ( ( ) ; ( ) ; ) : Let " t ( ) = y t x > t ( ) ~x > t ( ) q t + q t > > 0 such that E (x t " t ( )) = 0 ad E x t q t + q t > > 0 " t ( ) = 0: Let X be the matrix stackig x > t ad ~x > t q t + q t > > 0 ad " with " t ( ) : The, S ( ) = "> " "> X X> X X> "! E" t ( ) ; almost surely uiformly over ; by Lemma. Note that = ( 0 ) ad = ( 0 ) ; ad that E" t ( ) is uiquely miimized at = 0 ; sice 0 de es the coditioal expectatio, which miimizes MSE, ad the threshold idex z t ( ) icludes at least oe cotiuous radom variable. By the latter, E" t ( ) is cotiuous o : Therefore,, which also miimizes S ( ) ; coverges to 0 almost surely. Furthermore, it i tur implies that ad coverge to 0 ad 0 almost surely by Lemma. Proof of Theorem. The asymptotic distributio developed here is based o the Taylor series expasio of T (; ) : T ( ; ) = T ( 0 ; ) + Q ( ~ ; ) ( 0 ) = 0; where ~ = ( ~ > ; ~ > ; ~ > ) > lies betwee ad 0 : Let the dimesio of s be k s ad de e a k- dimesioal diagoal matrix D whose rst k s elemets are ad the others are p ad ote that p D ( 0 ) = Q s p Q s > ~; ~; The followig is useful for the developmet below ~x > t p Q s ~; A Q ~; p T s ( 0 ; ) p T ( 0 ; ) ~x > t y t x > t o = ~x > t 0 ~x > t 0 y t x > t 0 + Rt () = ~x > t 0 ~x > t 0 ~x > t 0 fz t > 0g + " t + Rt () = ~x > t 0 ( fzt > 0g) ~x > t 0 " t + R t () = ~x > t 0 sg (zt ) + ~x > t 0 " t + R t () ; (4)! : where R t () = > ~x t x > t ( 0 ) + ( + 0 ) > ~x t ~x >t ~x >t fz t > 0g ~x >t " t ( 0 ) ; 3

25 ad sg (s) = if s is positive, ad otherwise. The, 0 P T s ( 0 ; ) y P t x > t 0 ( xt ) + ~x> t 0 x t K ~x > t 0 ~xt ~x t y t x > t 0 K ad Q s Q s~ Q ~ = T ( 0 ; ) = ~; ~; ~; = 0 0 P z t z t P x P t" t x t~x > z t 0 fz t > 0g K t P ~x z t" t K t P + ~x t~x > z t 0 fz t 0g K t X ~x > o t 0 ~x > t 0 y t x > q t t 0 K 0 zt X P x tx > t ~x > t 0 sg (zt ) + ~x > qt t 0 " t K 0 zt ; P x t~x > ~ t q> t P P = ~x > t P = P x t~x > t K P ~x t~x > t K K 0 qt +q t > ~ qt +q > t ~ qt +q t > ~ A ~x t ~x > t ~ + ~x t x > t ~ ~x t y t q > t K 0 qt +q > t ~ ~x > t (y t x > t ~ )o qt q > t K 00 qt +q > t ~ ~x > t 0 sg (zt ) + ~x > t 0 " t + R t ( ~ )o qt q > t A A K 00 A ; z t+q t( > ~ 0 ) ; where the last equality follows from (4) : We show the covergeces of T s ad Q s ad the others i the followig sequece of Lemma s. Lemma Suppose (~ 0 ) = o () : The, p T s ( 0 ; ) =) d N (0; V s ) ; ~;! p Q s : Q s Proof of Lemma. Assumptio 3 (d) implies that s fs > 0g K = o (5) for all ozero s R. Therefore, it follows from the domiated covergece theorem that the P followigs are o p () : p x tx > z t ( fz t > 0g K t P ), p x tx > z t fz t 0gK t, ad P p x z t" t (fz t > 0g K t ): The, P p p T s ( 0 ; ) = x! t" t P p ~x t" t fz t > 0g + o p () =) d N (0; V s ) : 4