Nonparametric Quantile Estimations For Dynamic Smooth Coefficient Models

Transcription

1 Nnparametric Quantile Estimatins Fr Dynamic Smth Cefficient Mdels Zngwu Cai and Xiaping Xu This Versin: January 13, 2006 In this paper, quantile regressin methds are suggested fr a class f smth cefficient time series mdels. We emply a lcal linear fitting scheme t estimate the smth cefficients in the quantile framewrk. The prgramming invlved in the lcal linear quantile estimatin is relatively simple and it can be mdified with few effrts frm the existing prgrams fr the linear quantile mdel. We derive the lcal Bahadur representatin f the lcal linear estimatr fr α-mixing time series and establish the asympttic nrmality f the resulting estimatr. Als, a bandwidth selectr based n the nnparametric versin f the Akaike infrmatin criterin is prpsed, tgether with a cnsistent estimate f the asympttic cvariance matrix. The asympttic behavirs f the estimatr at the bundaries are examined. A cmparisn f the lcal linear quantile estimatr with the lcal cnstant estimatr is presented. A simulatin study is carried ut t illustrate the perfrmance f the estimates. An empirical applicatin f the mdel t the exchange rate time series data and the well-knwn Bstn huse price data further demnstrates the ptential f the prpsed mdeling prcedures. KEY WORDS: Bandwidth selectin; bundary effect; cvariance estimatin; kernel smthing methds; nnlinear time series; quantile regressin; value-at-risk; varying cefficients. Tentatively accepted by Jurnal f the American Statistical Assciatin Zngwu Cai is Prfessr, Department f Mathematics & Statistics and Department f Ecnmics, University f Nrth Carlina, Charltte, NC 28223, USA and Special-term Appinted Prfessr, Department f Ecnmics and Finance, Aetna Schl f Management, Shanghai Jiatng University, Shanghai, China ( zcai@uncc.edu). Xiaping Xu is Prfessr, Department f Statistics, China University f Gesciences, Wuhan, China ( jxiapxu@yah.cm). Cai s research was supprted, in part, by the Natinal Science Fundatin grant DMS and funds prvided by the University f Nrth Carlina at Charltte. The authrs thank thank the editr, the assciate editr and tw referees fr their cnstructive and detailed suggestins that led t significant imprvements in the presentatin f the article.

2 1 Intrductin Over the last three decades, quantile regressin, als called cnditinal quantile r regressin quantile, intrduced by Kenker and Bassett (1978), has been used widely in varius disciplines, such as finance, ecnmics, medicine, and bilgy. It is well-knwn that when the distributin f data is typically skewed r data cntains sme utliers, the median regressin, a special case f quantile regressin, is mre explicable and rbust than the mean regressin. Als, regressin quantiles can be used t test heterscedasticity frmally r graphically (Kenker and Bassett, 1982; Efrn, 1991; Kenker and Zha, 1996; Kenker and Xia, 2002). Althugh sme individual quantiles, such as the cnditinal median, are smetimes f interest in practice, mre ften ne wishes t btain a cllectin f cnditinal quantiles which can characterize the entire cnditinal distributin. Mre imprtantly, anther applicatin f cnditinal quantiles is the cnstructin f predictin intervals fr the next value given a small sectin f the recent past values in a statinary time series (Granger, White, and Kamstra, 1989; Kenker, 1994; Zhu and Prtny, 1996; Kenker and Zha, 1996; Taylr and Bunn, 1999). Als, Granger, White, and Kamstra (1989), Kenker and Zha (1996), and Taylr and Bunn (1999) cnsidered an interval frecasting fr parametric autregressive cnditinal heterscedastic (ARCH) type mdels. Fr mre details abut the histrical and recent develpments f quantile regressin with applicatins fr time series data, particularly in finance, see, fr example, the papers and bks by J.P. Mrgan (1995), Duffie and Pan (1997), Jrin (2000), Kenker (2000), Kenker and Hallck (2001), Tsay (2000, 2002), Khindanva and Rachev (2000), and Ba, Lee and Saltğlu (2001), and the references therein. Recently, the quantile regressin technique has been successfully applied t plitics. Fr example, in the 1992 presidential selectin, the Demcrats used the yearly Current Ppulatin Survey data t shw that between 1980 and 1992 there was an increase in the number f peple in the high-salary categry as well as an increase in the number f peple in the lw-salary categry. This phenmena culd be illustrated by using the quantile regressin methd as fllws: cmputing 90% and 10% quantile regressin functins f salary as a functin f time. An increasing 90% quantile regressin functin and a decreasing 10% quantile regressin functin crrespnded t the Demcrats claim that the rich gt richer and the pr gt prer during the Republican administratins; see Figure 6.4 in Fan and Gijbels (1996, p. 229). Mre imprtantly, by fllwing the regulatins f the Bank fr Internatinal Settlements, many f financial institutins have begun t use a unifrm measure f risk t measure the 1

3 market risks called Value-at-Risk (VaR), which can be defined as the maximum ptential lss f a specific prtfli fr a given hrizn in finance. In essence, the interest is t cmpute an estimate f the lwer tail quantile (with a small prbability) f future prtfli returns, cnditinal n current infrmatin. Therefre, the VaR can be regarded as a special applicatin f the quantile regressin. There is a vast amunt f literature in this area; see, t name just a few, J.P. Mrgan (1995), Duffie and Pan (1997), Engle and Manganelli (2004), Jrin (2000), Tsay (2000, 2002), Khindanva and Rachev (2000), and Ba, Lee and Saltğlu (2001), and references therein. In this paper, we assume that {X t, Y t } t= is a statinary sequence. Dente F (y x) the cnditinal distributin f Y given X = x, where X t = (X t1,..., X td ) with denting the transpse f a matrix r vectr, is the assciated cvariate vectr in R d with d 1, which might be a functin f exgenus (cvariate) variables r sme lagged (endgenus) variables r time t. The regressin (cnditinal) quantile functin q τ (x) is defined as, fr any 0 < τ < 1, q τ (x) = inf{y R 1 : F (y x) τ}, r q τ (x) = argmin a R 1E {ρ τ (Y t a) X t = x}, (1) where ρ τ (y) = y (τ I {y<0} ) with y R 1 is called the lss ( check ) functin, and I A is the indicatr functin f any set A. Clearly, the simplest frm f mdel (1) is q τ (x) = β τx, which is called the linear quantile regressin mdel well studied by many authrs. Fr details, see the papers by Duffie and Pan (1997), Kenker (2000), Tsay (2000), Kenker and Hallck (2001), Khindanva and Rachev (2000), and Ba, Lee and Saltğlu (2001), Engle and Manganelli (2004), and references therein. In many practical applicatins, hwever, the linear quantile regressin mdel might nt be rich enugh t capture the underlying relatinship between the quantile f respnse variable and its cvariates. Indeed, sme cmpnents may be highly nnlinear r sme cvariates may be interactive. T make the quantile regressin mdel mre flexible, there is a swiftly grwing literature n nnparametric quantile regressin. Varius smthing techniques, such as kernel methds, splines, and their variants, have been used t estimate the nnparametric quantile regressin fr bth the independent and time series data. Fr the recent develpments and the detailed discussins n thery, methdlgies, and applicatins, see, fr example, the papers by He, Ng, and Prtny (1998), Yu and Jnes (1998), He and Ng (1999), He and Prtny (2000), Hnda (2000, 2004), Tsay (2000, 2002), Lu, Hui and Zha (2000), Khindanva and Rachev (2000), Ba, Lee and Saltğlu (2001), Cai (2002a), De Gijer, and Gannun (2003), Hrwitz and Lee (2004), Yu and Lu (2004), and Li and Racine (2004), and references therein. In particular, fr the univariate case, recently, 2

4 Hnda (2000) and Lu, Hui and Zha (2000) derived the asympttic prperties f the lcal linear estimatr f the quantile regressin functin under α-mixing cnditin. Fr the high dimensinal case, hwever, the afrementined methds encunter sme difficulties such as the s-called curse f dimensinality and their implementatin in practice is nt easy as well as the visual display is nt s useful fr the explratry purpses. T attenuate the abve prblems, De Gijer and Zerm (2003), Hrwitz and Lee (2004), and Yu and Lu (2004) cnsidered an additive quantile regressin mdel q τ (X t ) = dk=1 g k (X tk ). T estimate each cmpnent, fr the time series case, De Gijer and Zerm (2003) first estimated a high dimensinal quantile functin by inverting the cnditinal distributin functin estimated by using a weighted Nadaraya-Watsn apprach, prpsed by Cai (2002a), and then used a prjectin methd t estimate each cmpnent, as discussed in Cai and Masry (2000), while Yu and Lu (2004) fcused n the independent data and used a back-fitting algrithm methd t estimate each cmpnent. On the ther hand, t estimate each additive cmpnent fr the independent data, Hrwitz and Lee (2004) used a tw-stage apprach cnsisting f the series estimatin at the first step and a lcal plynmial fitting at the secnd step. Fr the independent data, the abve mdel was extended by He, Ng and Prtny (1998), He and Ng (1999), and He and Prtny (2000) t include interactin terms by using spline methds. In this paper, we adapt anther dimensin reductin mdelling methd t analyze dynamic time series data, termed as the smth (functinal r varying) cefficient mdelling apprach. This apprach allws appreciable flexibility n the structure f fitted mdels. It allws fr linearity in sme cntinuus r discrete variables which can be exgenus r lagged and nnlinear in ther variables in the cefficients. In such a way, the mdel has the ability f capturing the individual variatins. Mre imprtantly, it can ease the s-called curse f dimensinality and cmbines bth additivity and interactivity. quantile regressin mdel fr time series data takes the fllwing frm A smth cefficient d q τ (U t, X t ) = a k (U t ) X tk = X t a τ (U t ), (2) k=0 where U t is called the smthing variable, which might be ne part f X t1,..., X td r just time r ther exgenus variables r the lagged variables, X t = (X t0, X t1,..., X td ) with X t0 1, {a k ( )} are smth cefficient functins, and a τ ( ) = (a 0,τ ( ),..., a d,τ ( )). Here, sme f {a k,τ ( )} are allwed t depend n τ. Fr simplicity, we drp τ frm {a k,τ ( )} in what fllws. It is ur interest here t estimate the cefficient functins a( ) rather than the quantile regressin surface q τ (, ) itself. Nte that mdel (2) was studied by Hnda (2004) 3

5 fr the independent sample, but ur fcus here is n the dynamic mdel fr nnlinear time series, which is mre apprpriate fr ecnmic and financial applicatins. The general setting in (2) cvers many familiar quantile regressin mdels, including the quantile autregressive mdel (QAR) prpsed by Kenker and Xia (2004) wh applied the QAR mdel fr the unit rt inference. In particular, it includes a specific class f ARCH mdels, such as heterscedastic linear mdels cnsidered by Kenker and Zha (1996). Als, if there is n X t in the mdel (d = 0), q τ (U t, X t ) becmes q τ (U t ) s that mdel (2) reduces t the rdinary nnparametric quantile regressin mdel which has been studied extensively. Fr the recent develpments, refer t the papers by He, Ng and Prtny (1998), Yu and Jnes (1998), He and Ng (1999), He and Prtny (2000), Hnda (2000), Lu, Hui and Zha (2000), Cai (2002a), De Gijer and Zerm (2003), Hrwitz and Lee (2004), Yu and Lu (2004), and Li and Racine (2004). If U t is just time, then the mdel is called the time-varying cefficient quantile regressin mdel, which is ptentially useful t see whether the quantile regressin changes ver time and in a case with a practical interest is, fr example, the afrementined illustrative example fr the 1992 presidential electin and the analysis f the reference grwth data by Cle (1994), Wei, Pere, Kenker and He (2003), and Wei and He (2005), and the references therein. Hwever, if U t is time, the bserved time series might nt be statinary. Therefre, the treatment fr nn-statinary case wuld require a different apprach s that it is beynd the scpe f this paper and deserves a further investigatin. Fr mre applicatins, see the wrk in BLIND2 (2005). Finally, nte that the smth cefficient mean regressin mdel is ne f the mst ppular nnlinear time series mdels in mean regressin and has varius applicatins. Fr mre discussins, refer t the papers by Chen and Tsay (1993), Cai, Fan, and Ya (2000), Cai and Tiwari (2000), BLIND1 (2003), Hng and Lee (2003), and Wang (2003), and the bk by Tsay (2002), and references therein. Althugh ur interest in cnditinal quantile estimatin is mtivated by the statistical inferences such as the frecasting and ecnmetric and financial applicatins fr time series data, we intrduce ur methds in a mre general setting (α-mixing) which includes time series mdeling as a special case. Our theretical results are derived under α-mixing assumptin. Fr the reference cnvenience, we first intrduce the mixing cefficient. Let F b a be the σ-algebra generated by the statinary prcess {(U t, X t, Y t )} b t=a. Define α(t) = sup{ P (A B) P (A) P (B) : A F 0, B F t }. It is called the strng mixing cefficient f the statinary prcess {(U t, X t, Y t )}. α(t) 0 as t, the prcess is called strngly mixing r α-mixing. If 4

6 Amng varius mixing cnditins used in literature, α-mixing is reasnably weak and knwn t be fulfilled fr many time series mdels. Grdetskii (1977) and Withers (1981) derived the cnditins under which a linear prcess is α-mixing. In fact, under very mild assumptins linear autregressive and mre generally bilinear time series mdels are α-mixing with mixing cefficients decaying expnentially. Auestad and Tjøstheim (1990) prvided illuminating discussins n the rle f α-mixing (including gemetric ergdicity) fr mdel identificatin in nnlinear time series analysis. Chen and Tsay (1993) shwed that the functinal autregressive prcess is gemetrically ergdic under certain cnditins. Furthermre, Masry and Tjøstheim (1995, 1997) and Lu (1998) demnstrated that under sme mild cnditins, bth ARCH prcesses and nnlinear additive autregressive mdels with exgenus variables, particularly ppular in finance and ecnmetrics, are statinary and α-mixing. T the best f ur knwledge, an pen questin remains t derive the general cnditins under which the time series generated by a dynamic smth cefficient quantile regressin mdel is statinary. It is well-knwn that an ergdic Markv prcess initiated frm its invariant distributin is (strictly) statinary. Nte that any autregressive mdel can be expressed as a vectr-valued Markv mdel. Thus, it is the cmmn practice t establish the statinarity by prving ergdicity. Recent results in this directin include thse f An and Chen (1997) and An and Huang (1996), which surveyed varius sufficient cnditins fr the ergdicity f nnlinear autregressive mdels, including sme special cases f dynamic smth cefficient quantile regressin mdels. The mtivatin f this study cmes frm an analysis f the well knwn Bstn husing price data, cnsisting f several variables cllected n each f 506 different huses frm a variety f lcatins. The interest is t identify the factrs affecting the huse price in Bstn area. As argued by Şentürk and Müller (2003), the crrelatin between the huse price and the crime rate can be adjusted by the cnfunding variable which is the prprtin f ppulatin f lwer educatinal status thrugh a varying cefficient mdel and the expected effect f increasing crime rate n declining huse prices seems t be nly bserved fr lwer educatinal status neighbrhds in Bstn. The interesting features f this dataset are that the respnse variable is the median price f a hme in a given area and the distributins f the price and the majr cvariate (the cnfunding variable) are left skewed. Therefre, quantile methds are suitable fr the analysis f this dataset. Therefre, such a prblem can be tackled by using mdel (2). In anther example, ne is interested in explring the pssible nnlinearity feature, heterscedasticity, and predictability f the exchange rates such as the Japanese Yen per US dllar. The detailed analysis f these data sets is reprted in Sectin 3. 5

7 The plan f this paper is as fllws. In Sectin 2, we define the lcal linear quantile estimatins f the smth cefficients and large sample results, including the lcal Bahadur representatin, cnsistency and asympttic nrmality f the estimatrs, cupled with a new data-driven fashined bandwidth selectr based n the nnparametric versin f the Akaike infrmatin criterin, as well as the cnsistent estimatr f the asympttic cvariance matrix, are given in the same sectin with sme discussins. Als, we discuss the lcal cnstant quantile estimatin and its asympttic prperties. Mrever, the asympttic behavirs f bth estimatrs at bundaries are examined. A cmparisn f tw estimatrs is presented. In Sectin 3, we illustrate the finite sample perfrmance f the estimatrs with a Mnte Carl experiment and als we give an applicatin t the exchange rate series and the Bstn huse price data. Finally, the derivatins f the therems are given in Sectin 4 with sme lemmas. The Appendix cntains the prfs f certain lemmas needed in the prfs f the therems in Sectin 4. 2 Mdeling Prcedures 2.1 Lcal Linear Quantile Estimate Nw, we apply the lcal plynmial methd t the smth cefficient quantile regressin mdel as fllws. Fr the sake f brevity, we nly cnsider the case where U t in (2) is ne-dimensinal, dented by U t in what fllws. Extensin t multivariate U t invlves fundamentally n new ideas althugh the thery and prcedure cntinue t hld. Nte that the mdels with high dimensin might nt be practically useful due t the curse f dimensinality. A lcal plynmial fitting has several nice prperties such as high statistical efficiency in an asympttic minimax sense, design-adaptatin, and autmatic edge crrectin (see, e.g., Fan and Gijbels, 1996). We estimate the functins {a k ( )} using the lcal plynmial regressin methd frm bservatins {(U t, X t, Y t )} n. We assume thrughut the paper that the cefficient functins a( )} have the (q + 1)th derivative, s that fr any given gird pint u 0, a k ( ) can be apprximated by a plynmial functin in a neighbrhd f the given grid pint u 0 as a(u t ) a(u 0 ) + a (u 0 ) (U t u 0 ) + + a (q) (u 0 ) (U t u 0 ) q /q! and q q τ (U t, X t ) X t β j (U t u 0 ) j, j=0 where β j = a (j) (u 0 )/j!. Then, the lcally weighted lss functin is n q ρ τ Y t X t β j (U t u 0 ) j K h (U t u 0 ), (3) j=0 6

8 where K( ) is a kernel functin, K h (x) = K(x/h)/h, and h = h n is a sequence f psitive numbers tending t zer, which cntrls the amunt f smthing used in estimatin. Slving the minimizatin prblem in (3) gives â(u 0 ) = β 0, the lcal plynmial estimate f a(u 0 ), and â (j) (u 0 ) = j! β j (j 1), the lcal plynmial estimate f the jth derivative a (j) (u 0 ) f a(u 0 ). By mving u 0 alng with the real line, ne btains the estimate fr the entire curve. Fr varius practical applicatins, Fan and Gijbels (1996) recmmended using the lcal linear fit (q = 1). Therefre, fr the expsitinal purpse, in what fllws, we nly cnsider the case q = 1 (lcal linear fitting). The prgramming invlved in the lcal (plynmial) linear quantile estimatin is relatively simple and can be mdified with few effrts frm the existing prgrams fr a linear quantile mdel. Fr example, fr each grid pint u 0, the lcal linear quantile estimatin can be implemented in the R package quantreg, f Kenker (2004) by setting cvariates as X t and X t (U t u 0 ) and the weight as K h (U t u 0 ). Althugh sme mdificatins are needed, the methd develped here fr the lcal linear quantile estimatin is applicable t a general lcal plynmial quantile estimatin. In particular, we nte that the lcal cnstant (Nadaraya-Watsn type) quantile estimatin f a(u 0 ), dented by ã(u 0 ), is β minimizing the fllwing subjective functin n ρ τ (Y t X t β) K h (U t u 0 ), (4) which is a special case f (3) with q = 0. We cmpare â(u 0 ) and ã(u 0 ) theretically at the end f Sectin 2.2 and empirically in Sectin 3.1 and the cmparisn leads t suggest that ne shuld use the lcal linear apprach in practice. 2.2 Asympttic Results We first give sme regularity cnditins that are sufficient fr the cnsistency and asympttic nrmality f the prpsed estimatrs, althugh they might nt be the weakest pssible. We intrduce the fllwing ntatins. Dente Ω(u 0 ) E[X t X t U t = u 0 ] and Ω (u 0 ) E[X t X t f y u,x (q τ (u 0, X t )) U t = u 0 ], where f y u,x (y) is the cnditinal density f Y given U and X. Let f u (u) present the marginal density f U. Assumptins: (C1) a(u) is twice cntinuusly differentiable in a neighbrhd f u 0 fr any u 0. 7

9 (C2) f u (u) is cntinuus and f u (u 0 ) > 0. (C3) f y u,x (y) is bunded and satisfies the Lipschitz cnditin. (C4) The kernel functin K( ) is symmetric and has a cmpact supprt, say [ 1, 1]. (C5) {(X t, Y t, U t )} is a strictly α-mixing statinary prcess with mixing cefficient α(t) satisfies t 1 t l α (δ 2)/δ (t) < fr sme psitive real number δ > 2 and l > (δ 2)/δ. (C6) E X t 2δ < with δ > δ. (C7) Ω(u 0 ) is psitive-definite and cntinuus in a neighbrhd f u 0 (C8) Ω (u 0 ) is cntinuus and psitive-definite in a neighbrhd f u 0. (C9) The bandwidth h satisfies h 0 and n h. (C10) f(u, v x 0, x s ; s) M < fr s 1, where f(u, v x 0, x s ; s) is the cnditinal density f (U 0, U s ) given (X 0 = x 0, X s = x s ). (C11) n 1/2 δ/4 h δ/δ 1/2 δ/4 = O(1). Remark 1: (Discussin f Cnditins) Assumptins (C1) - (C3) include sme smthness cnditins n functinals invlved. The requirement in (C4) that K( ) be cmpactly supprted is impsed fr the sake f brevity f prfs, and can be remved at the cst f lengthier arguments. In particular, the Gaussian kernel is allwed. The α-mixing is ne f the weakest mixing cnditins fr weakly dependent stchastic prcesses. Statinary time series r Markv chains fulfilling certain (mild) cnditins are α-mixing with expnentially decaying cefficients; see the discussins in Sectin 1 and Cai (2002a) fr mre examples. On the ther hand, the assumptin n the cnvergence rate f α( ) in (C5) might nt be the weakest pssible and is impsed t simplify the prf. Further, (C10) is just a technical assumptin, which is als impsed by Cai (2002a). (C6) - (C8) require sme standard mments. Clearly, (C11) allws the chice f a wide range f smthing parameter values and is slightly strnger than the usual cnditin f n h. Hwever, fr the bandwidths f ptimal size (i.e., h = O(n 1/5 )), (C11) is autmatically satisfied fr δ 3 and it is still fulfilled fr 2 < δ < 3 if δ satisfies δ < δ 1 + 1/(3 δ), s that we d nt cncern urselves with such refinements. Indeed, this assumptin is als impsed by Cai, Fan and Ya (2000) fr the mean regressin. Finally, if there is n X t in mdel (2), (C5) can be replaced by (C5) : α(t) = O(t δ ) fr sme δ > 2 and (C11) can be substituted by (C11) : n h δ/(δ 2) ; see Cai (2002a) fr details. 8

10 Remark 2: (Identificatin) It is clear frm (2) that Ω(u 0 ) a(u 0 ) = E[q τ (u 0, X t ) X t U t = u 0 ]. Then, a(u 0 ) is identified (uniquely determined) if and nly if Ω(u 0 ) is psitive definite fr any u 0. Therefre, Assumptin (C7) is the necessary and sufficient cnditin fr the mdel identificatin. T establish the asympttic nrmality f the prpsed estimatr, similar t Chaudhuri (1991), we first derive the lcal Bahadur representatin fr the lcal linear estimatr. T this end, ur analysis fllws the apprach f Kenker and Zha (1996), which can simplify the theretical prfs. Define, µ j = u j K(u) du and ν j = u j K 2 (u) du. Als, set ψ τ (x) = ( ) τ I {x<0}, U th = (U t u 0 )/h, X t = Xt U th X, Yt = Y t X t t[a(u 0 ) + a (u 0 ) (U t u 0 )], and θ = ( ) β0 a(u n h H 0 ) β 1 a with H = diag{i, h I}. (u 0 ) Therem 1: (Lcal Bahadur Representatin) Under assumptins (C1)- (C9),we have θ = [Ω 1(u 0 )] 1 n h fu (u 0 ) where Ω 1(u 0 ) = diag{ω (u 0 ), µ 2 Ω (u 0 )}. n ψ τ (Y t ) X t K(U th ) + p (1), (5) Remark 3: Frm Therem 1 and Lemma 5 (in Sectin 4), it is easy t see that the lcal linear estimatr â(u 0 ) is cnsistent with the ptimal nnparametric cnvergence rate n h. Therem 2: (Asympttic Nrmality) Under assumptins (C1)- (C11), we have the fllwing asympttic nrmality [ n h H ( ) â(u0 ) a(u 0 ) â (u 0 ) a h2 (u 0 ) 2 ( a ) ] (u ) µ p (h 2 ) where Σ(u 0 ) = diag{τ(1 τ) ν 0 Σ a (u 0 ), τ(1 τ) ν 2 Σ a (u 0 )} with N {0, Σ(u 0 )}, Σ a (u 0 ) = [Ω (u 0 )] 1 Ω(u 0 ) [Ω (u 0 )] 1 /f u (u 0 ). (6) In particular, n h [â(u 0 ) a(u 0 ) h2 µ 2 2 a (u 0 ) + p (h 2 ) ] N {0, τ(1 τ) ν 0 Σ a (u 0 )}. 9

11 Remark 4: Frm Therem 2, the asympttic mean squares errr (AMSE) f â(u 0 ) is given by AMSE = h4 µ a (u 0 ) 2 + τ(1 τ) ν 0 n h f u (u 0 ) tr(σ a(u 0 )), which gives the ptimal bandwidth h pt by minimizing the AMSE and the ptimal AMSE is h pt = ( ) 1/5 τ(1 τ) ν0 tr(σ a (u 0 )) n 1/5 f u (u 0 ) a (u 0 ) 2 ( ) 4/5 τ(1 τ) ν0 tr(σ a (u 0 )) a (u 0 ) 2/5 n 4/5. AMSE pt = 5 4 f u (u 0 ) Further, ntice that the similar results in Therem 2 were btained by Hnda (2004) fr the independent data. Finally, it is interesting t nte that the asympttic bias in Therem 2 is the same as that fr the mean regressin case but the tw asympttic variances are different; see, fr example, Cai, Fan and Ya (2000). If mdel (2) des nt have X (d = 0), it becmes the nnparametric quantile regressin mdel q τ ( ). Then, we have the fllwing asympttic nrmality fr the lcal linear estimatr f the nnparametric quantile regressin functin q τ ( ), which cvers the results in Yu and Jnes (1998), Hnda (2000), Lu, Hui and Zha (2000), and Cai (2002a) fr bth the independent and time series data. Crllary 1: If there is n X t in (2), then, n h [ q τ (u 0 ) q τ (u 0 ) h2 µ 2 2 q τ (u 0 ) + p (h 2 ) ] N { 0, σ 2 τ(u 0 ) }, where στ(u 2 0 ) = τ(1 τ) ν 0 fu 1 (u 0 ) f 2 y u (q τ(u 0 )). Nw we cnsider the cmparisn f the perfrmance f the lcal linear estimatin â(u 0 ) btained in (3) with that f the lcal cnstant estimatin ã(u 0 ) given in (4). T this effect, first, we derive the asympttic results fr the lcal cnstant estimatr but the prf is mitted since it is alng the same line with the prf f Therems 1 and 2; see BLIND2 (2005) fr details. Under sme regularity cnditins, it can be shwn that where n h [ ã(u 0 ) a(u 0 ) b + p (h 2 ) ] N {0, τ(1 τ) ν 0 Σ a (u 0 )}, b = h2 µ 2 2 [ a (u 0 ) + 2 a (u 0 ) f u(u 0 )/f u (u 0 ) + 2 {Ω (u 0 )} 1 Ω (u 0 ) a (u 0 ) ], 10

12 which implies that the asympttic bias fr ã(u 0 ) is different frm that fr â(u 0 ) but bth have the same asympttic variance. Therefre, the lcal cnstant quantile estimatr des nt adapt t nnunifrm designs: the bias can be large when f u(u 0 )/f u (u 0 ) r {Ω (u 0 )} 1 Ω (u 0 ) is large even when the true cefficient functins are linear. It is surprising that t the best f ur knwledge, this finding seems t be new fr the nnparametric quantile regressin setting althugh it is well dcumented in literature fr the rdinary regressin case; see Fan and Gijbels (1996) fr details. Finally, t examine the asympttic behavirs f the lcal linear and lcal cnstant quantile estimatrs at the bundaries, we ffer Therem 3 belw but its prfs are mitted due t their similarity t thse fr Therem 2 with sme mdificatins and fr the rdinary regressin setting (Fan and Gijbels, 1996); see BLIND2 (2005) fr the detailed prfs. Withut lss f generality, we cnsider nly the left bundary pint u 0 = c h, 0 < c < 1, if U t takes values nly frm [0, 1]. A similar result in Therem 3 hlds fr the right bundary pint u 0 = 1 c h. Define µ j,c = 1 c uj K(u)du and ν j,c = 1 c uj K 2 (u)du. Therem 3: (Asympttic Nrmality) Under assumptins f Therem 2, we have the fllwing asympttic nrmality f the lcal linear quantile estimatr at the left bundary pint, ] n h [â(c h) a(c h) h2 b c 2 a (0+) + p (h 2 ) N {0, τ(1 τ) v c Σ a (0+)}, where b c = µ2 2,c µ 1,c µ 3,c µ 2,c µ 0,c µ 2 1,c and v c = µ2 2,cν 0,c 2 µ 1,c µ 2,c ν 1,c + µ 2 1,c ν 2,c [ µ2,c µ 0,c µ 2 1,c] 2. Further, we have the fllwing asympttic nrmality f the lcal cnstant quantile estimatr at the left bundary pint u 0 = c h fr 0 < c < 1, n h [ ã(c h) a(c h) b c + p (h 2 ) ] N { 0, τ(1 τ) ν 0,c Σ a (0+)/µ 2 0,c}. where [ b c = hµ 1,c a (0+) + h2 µ 2,c 2 { a (0+) + 2a (0+)f u(0+) f u (0+) Similar results hld fr the right bundary pint u 0 = 1 c h. + 2Ω 1 (0+)Ω (0+)a (0+) }] /µ 0,c. Remark 5: We remark that if the pint 0 were an interir pint, then, Therem 3 wuld hld with c = 1, which becmes Therem 2. Als, as c 1, b c µ 2, and v c ν 0 and these limits are exactly the cnstant factrs appearing respectively in the asympttic bias and variance fr an interir pint. Therefre, Therem 3 shws that the lcal linear estimatin 11

13 has the autmatic gd behavir at bundaries withut the need f bundary crrectin. Further, ne can see frm Therem 3 that at the bundaries, the asympttic bias term fr the lcal cnstant quantile estimate is f the rder h by cmparing t the rder h 2 fr the lcal linear quantile estimate. This shws that the lcal linear quantile estimate des nt suffer frm bundary effects but the lcal cnstant quantile estimate des, which is anther advantage f the lcal linear quantile estimatr ver the lcal cnstant quantile estimatr. This suggests that ne shuld use the lcal linear apprach in practice. As a special case, Therem 3 includes the asympttic prperties fr the lcal cnstant quantile estimatr f the nnparametric quantile functin q τ ( ) at bth the interir and bundary pints, stated as fllws. Crllary 2: If there is n X t in (2), then, the asympttic nrmality f the lcal cnstant quantile estimatr is given by [ ] n h q τ (u 0 ) q τ (u 0 ) h2 µ 2 {q τ (u 0 ) + 2 q 2 τ(u 0 )f u(u 0 )/f u (u 0 )} + p (h 2 ) N { 0, στ(u 2 0 ) }. Further, at the left bundary pint, we have n h [ q τ (c h) q τ (c h) b c + p (h 2 ) ] N { 0, σ 2 c }, where b c = [ hµ 1,c q τ(0+) + h2 µ 2,c 2 {q τ (0+) + 2 q τ(0+)f u(0+)/f u (0+)} ] /µ 0,c and σc 2 = τ(1 τ) ν 0,c fu 1 (0+) f 2 y u (q τ(0+))/µ 2 0,c. 2.3 Bandwidth Selectin It is well knwn that the bandwidth plays an essential rle in the trade-ff between reducing bias and variance. T the best f ur knwledge, there has been almst nthing dne abut selecting the bandwidth in the cntext f estimating the cefficient functins in the quantile regressin even thugh there is a rich amunt f literature n this issue in the mean regressin setting; see, fr example, Cai, Fan and Ya (2000). In practice, it is desirable t have a quick and easily implemented data-driven fashined methd. Based n this spirit, Yu and Jnes (1998) r Yu and Lu (2004) prpsed a simple and cnvenient methd fr the nnparametric quantile estimatin. Their apprach assumes that the secnd derivatives f the quantile functin are parallel. Hwever, this assumptin might nt be valid fr many applicatins in ecnmics and finance due t (nnlinear) heterscedasticity. Further, the mean regressin apprach can nt directly estimate the variance functin. T attenuate 12

14 these prblems, we prpse a methd f selecting bandwidth fr the freging estimatin prcedure, based n the nnparametric versin f the Akaike infrmatin criterin (AIC), which can attend t the structure f time series data and the ver-fitting r under-fitting tendency. This idea is mtivated by its analgue f Cai and Tiwari (2000) and Cai (2002b) fr nnlinear time series mdels. The basic idea is described belw. By recalling the classical AIC fr linear mdels under the likelihd setting 2 (maximized lg likelihd) + 2 (number f estimated parameters), we prpse the fllwing nnparametric versin f the bias-crrected AIC, due t Hurvich and Tsai (1989) fr parametric mdels and Hurvich, Simnff and Tsai (1998) fr nnparametric regressin mdels, t select h by minimizing AIC(h) = lg { σ } τ (ph + 1)/[n (p h + 2)], (7) where σ τ 2 and p h are defined later. This criterin may be interpreted as the AIC fr the lcal quantile smthing prblem and seems t perfrm well in sme limited applicatins. Nte that similar t (7), Kenker, Ng and Prtny (1994) cnsidered the Schwarz infrmatin criterin (SIC) f Schwarz (1978) with the secnd term n the right-hand side f (7) replayed by 2 n 1 p h lg n, where p h is the number f active knts fr the smthing spline quantile setting, and Machad (1993) studied similar criteria fr parametric quantile regressin mdels and mre general M-estimatrs f regressin. Nw the questin is hw t define σ τ 2 and p h in this setting. In the mean regressin setting, σ τ 2 is just the estimate f the variance σ 2. In the quantile regressin, we define σ τ 2 as n 1 t ρ τ (Y t X t â(u t)), which may be interpreted as the mean square errr in the least square setting and was als used by Kenker, Ng and Prtny (1994). In nnparametric mdels, p h is the nnparametric versin f degrees f freedm, called the effective number f parameters, and it is usually based n the trace f varius quasi-prjectin (hat) matrices in the least square thery (linear estimatrs); see, fr example, Hastie and Tibshirani (1990), Cai and Tiwari (2000), and Cai (2002b) fr a cgent discussin fr nnparametric regressin mdels and nnlinear time series mdels. Fr the quantile smthing setting, the explicit expressin fr the quasi-prjectin matrix des nt exist due t its nnlinearity. Hwever, we can use the first rder apprximatin (the lcal Bahadur representatin) given in (5) t derive an explicit expressin, which may be interpreted as the quasi-prjectin matrix in this setting. T this end, define S n = S n (u 0 ) = a n n ξ t X t X t K(U th ), 13

15 where ξ t = I(Y t X t a(u 0 ) + a n ) I(Y t X t a(u 0 )) and a n = (n h) 1/2. It is shwn in the Appendix that Frm (5), it is easy t verify that θ a n S 1 n q τ (U t, X t ) q τ (U t, X t ) 1 n S n (u 0 ) = f u (u 0 ) Ω 1(u 0 ) + p (1). (8) n s=1 n ψ τ (Y t ) X t K(U th ). Then, we have ψ τ (Ys (U t )) K h ((U s U t )/h) X 0 t S 1 n (U t ) X s ( ) where X 0 t = Xt. The cefficient f ψ 0 τ (Ys (U s )) n the right-hand side f the abve expressin is γ s = a 2 n K(0) X 0 s S 1 n (U s ) X 0 s. Nw, we have that p h = n s=1 γ s, which can be regarded as an apprximatin t the trace f the quasi-prjectin (hat) matrix fr linear estimatrs. In the practical implementatin, we need t estimate a(u 0 ) first since S n (u 0 ) invlves a(u 0 ). We recmmend using a pilt bandwidth which can be chsen as the ne prpsed by Yu and Jnes (1998). Similar t the least square thery, as expected, the criterin prpsed in (7) cunteracts the ver-fitting tendency f the generalized crss-validatin due t its relatively weak penalty and the under-fitting f the SIC f Schwarz (1978) studied by Kenker, Ng and Prtny (1994) because f the heavy penalty. 2.4 Cvariance Estimate Fr the purpse f statistical inference, we next cnsider the estimatin f the asympttic cvariance matrix t cnstruct the pintwise cnfidence intervals. In practice, a quick and simple way t estimate the asympttic cvariance matrix is desirable. In view f (6), the explicit expressin f the asympttic cvariance prvides a direct estimatr. Therefre, we can use the s-called sandwich methd. In ther wrds, we need t btain a cnsistent estimate fr bth Ω(u 0 ) and Ω (u 0 ). T this effect, define, Ω n,0 = 1 n n X t X t K h (U t u 0 ) and Ω n,1 = 1 n n w t X t X t K h (U t u 0 ), where w t = I(X t â(u 0) δ n < Y t X t â(u 0) + δ n )/(2 δ n ) fr any δ n 0 as n. It is shwn in the Appendix that Ω n,0 = f u (u 0 ) Ω(u 0 ) + p (1) and Ω n,1 = f u (u 0 ) Ω (u 0 ) + p (1). (9) Therefre, the cnsistent estimate f Σ a (u 0 ) is given by Σ a (u 0 ) = [ Ω n,1 (u 0 ) ] 1 Ω n,0 (u 0 ) [ Ω n,1 (u 0 ) ] 1. Nte that Ω n,1 (u 0 ) might be clse t singular fr sme sparse regins. T avid this cmputatinal difficulty, there are tw alternative ways t cnstruct a cnsistent estimate f 14

16 f u (u 0 ) Ω (u 0 ) thrugh estimating the cnditinal density f Y, f y u,x (q τ (u, x)). The first methd is the Nadaraya-Watsn type (r lcal linear) duble kernel methd f Fan, Ya and Tng (1996) defined as, n n f y u,x (q τ (u, x)) = K h2 (U t u, X t x) L h1 (Y t q τ (u, x))/ K h2 (U t u, X t x), where L( ) is a kernel functin, and the secnd ne is the difference qutients methd f Kenker and Xia (2004) such as f y u,x (q τ (u, x)) = (τ j τ j 1 )/[q τj (u, x) q τj 1 (u, x)], fr sme apprpriately chsen sequence f {τ j }; see Kenker and Xia (2004) fr mre discussins. Then, in view f the definitin f f u (u 0 )Ω (u 0 ), the estimatr Ω n,1 can be cnstructed as, Ω n,1 = 1 n f y u,x ( q τ (U t, X t )) X t X t K h (U t u 0 ). n By an analgue f (9), ne can shw that under sme regularity cnditins, bth estimatrs are cnsistent. 3 Empirical Examples In this sectin we reprt a Mnte Carl simulatin t examine the finite sample prperty f the prpsed estimatr and t further explre the pssible nnlinearity feature, heterscedasticity, and predictability f the exchange rate f the Japanese Yen per US dllar and t identify the factrs affecting the huse price in Bstn. In ur cmputatin, we use the Epanechnikv kernel K(u) = 0.75 (1 u 2 ) I( u 1) and cnstruct the pintwise cnfidence intervals based n the cnsistent estimate f the asympttic cvariance described in Sectin 2.4 withut the bias crrectin. Fr a predetermined sequence f h s frm a wide range, say frm h a t h b with an increment h δ, based n the AIC bandwidth selectr described in Sectin 2.3, we cmpute AIC(h) fr each h and chse h pt t minimize AIC(h). 3.1 A Simulated Example Example 1: We cnsider the fllwing data generating prcess Y t = a 1 (U t ) Y t 1 + a 2 (U t ) Y t 2 + σ(u t ) e t, t = 1,..., n, (10) where a 1 (U t ) = sin( 2 π U t ), a 2 (U t ) = cs( 2 π U t ), and σ(u t ) = 3 exp( 4 (U t 1) 2 ) + 2 exp( 5 (U t 2) 2 ). U t is generated frm unifrm (0, 3) independently and e t N(0, 1). 15

17 The quantile regressin is q τ (U t, Y t 1, Y t 2 ) = a 0 (U t ) + a 1 (U t ) Y t 1 + a 2 (U t ) Y t 2, where a 0 (U t ) = Φ 1 (τ) σ(u t ) and Φ 1 (τ) is the τ-th quantile f the standard nrmal. Therefre, nly a 0 ( ) is a functin f τ. Nte that a 0 ( ) = 0 when τ = 0.5. T assess the perfrmance f finite samples, we cmpute the mean abslute deviatin errrs (MADE) fr â j ( ), which is defined as MADE j = n 1 0 n 0 k=1 â j (u k ) a j (u k ), where â j ( ) is either the lcal linear r lcal cnstant quantile estimate f a j ( ) and {z k = 0.1(k 1) : 1 k n 0 = 27} are the grid pints. The Mnte Carl simulatin is repeated 500 times fr each sample size n = 200, 500, and 1000 and fr each τ = 0.05, 0.50 and We cmpute the ptimal bandwidth fr each replicatin, sample size, and τ. We cmpute the median and standard deviatin (in parentheses) f 500 MADE values fr each scenari and summarize the results in Table 1. Table 1: The Median and Standard Deviatin f 500 MADE Values The Lcal Linear Estimatr τ = 0.05 τ = 0.5 τ = 0.95 n MADE 0 MADE 1 MADE 2 MADE 0 MADE 1 MADE 2 MADE 0 MADE 1 MADE (0.520) (0.041) (0.041) (0.091) (0.032) (0.032) (0.517) (0.042) (0.039) (0.414) (0.023) (0.02) (0.056) (0.019) (0.018) (0.390) (0.023) (0.023) (0.071) (0.018) (0.017) (0.051) (0.014) (0.014) (0.072) (0.017) (0.017) The Lcal Cnstant Estimatr τ = 0.05 τ = 0.5 τ = 0.95 n MADE 0 MADE 1 MADE 2 MADE 0 MADE 1 MADE 2 MADE 0 MADE 1 MADE (2.937) (0.050) (0.051) (0.115) (0.027) (0.028) (3.188) (0.052) (0.051) (3.025) (0.024) (0.025) (0.062) (0.016) (0.015) (3.320) (0.025) (0.025) (0.462) (0.015) (0.014) (0.054) (0.012) (0.011) (0.427) (0.015) (0.015) Frm Table 1, we can bserve that the MADE values fr bth the lcal linear and lcal cnstant quantile estimates decrease when n increases fr all three values f τ and the lcal 16

18 linear estimate utperfrms the lcal cnstant estimate. This is anther example t shw that the lcal linear methd is superir ver the lcal cnstant even in the quantile setting. Als, the perfrmance fr the median quantile estimate is slightly better than that fr tw tails (τ = 0.05 and 0.95). This bservatin is nt surprising because f the sparsity f data in the tailed regins. Mrever, anther benefit f using the quantile methd is that we can btain the estimate f a 0 ( ) (cnditinal standard deviatin) simultaneusly with the estimatin f a 1 ( ) and a 2 ( ) (functins in the cnditinal mean), which, in cntrast, avids a tw-stage apprach needed t estimate the variance functin in the mean regressin; see Fan and Ya (1998) fr details. Hwever, it is interesting t see that due t the larger variatin, the perfrmance fr a 0 ( ), althugh it is reasnably gd, is nt as gd as that f a 1 ( ) and a 2 ( ). This can be further evidenced frm Figure 1. The results in this simulated experiment shw that the prpsed prcedure is reliable and they are alng the line f ur asympttic thery. Finally, Figure 1 plts the lcal linear estimates fr all three cefficient functins with their true values (slid line): σ( ) in Figure 1(a), a 1 ( ) in Figure 1(b), and a 2 ( ) in Figure 1(c), fr three quantiles τ = 0.05 (dashed line), 0.50 (dtted line) and 0.95 (dtted-dashed line), fr n = 500 based n a typical sample which is chsen based n its MADE value equal t the median f the 500 MADE values. The selected ptimal bandwidths are h pt = 0.10 fr τ = 0.05, fr τ = 0.50, and 0.10 fr τ = Nte that the estimate f σ( ) fr τ = 0.50 can nt be recvered frm the estimate f a 0 ( ) = 0 and it is nt presented in Figure 1(a). The 95% pint-wise cnfidence intervals withut the bias crrectin are depicted in Figure 1 in thick lines fr the τ = 0.05 quantile estimate. By the same tken, we can cmpute the pint-wise cnfidence intervals (nt shwn here) fr the rest. Basically, all cnfidence intervals cver the true values. Als, we can see that the cnfidence interval fr â 0 ( ) is wider than that fr â 1 ( ) and â 2 ( ) due t the larger variatin. Similar plts are btained (nt shwn here) fr the lcal cnstant estimates due t the space limitatins. Overall, the prpsed mdeling prcedure perfrms fairly well. 3.2 Real Data Examples Example 2: (Bstn Huse Price Data) We analyze a subset f the Bstn huse price data (available at f Harrisn and Rubinfeld (1978). This dataset cnsists f 14 variables cllected n each f 506 different huses frm a variety f lcatins. The dependent variable is Y, the median value f wner-ccupied hmes in $1, 000 s (huse price); sme majr factrs affecting the huse prices used are: prprtin f ppulatin f lwer educatinal status (i.e. prprtin f adults with high schl educatin and prprtin f male wrkers classified as labrs), dented by U, the average number f 17

19 rms per huse in the area, dented by X 1, the per capita crime rate by twn, dented by X 2, the full prperty tax rate per $10,000, dented by X 3, and the pupil/teacher rati by twn schl district, dented by X 4. Fr the cmplete descriptin f all 14 variables, see Harrisn and Rubinfeld (1978). Gilley and Pace (1996) prvided crrectins and examined censring. Recently, there have been several papers devted t the analysis f this dataset. Fr example, Breiman and Friedman (1985), Chaudhuri, Dksum and Samarv (1997), and Opsmer and Ruppert (1998) used fur cvariates: X 1, X 3, X 4 and U r their transfrmatins t fit the data thrugh a mean additive regressin mdel whereas Yu and Lu (2004) emplyed the additive quantile technique t analyze the data. Further, Pace and Gilley (1997) added the gereferencing factr t imprve estimatin by a spatial apprach. Recently, Şentürk and Müller (2003) studied the crrelatin between the huse price Y and the crime rate X 2 adjusted by the cnfunding variable U thrugh a varying cefficient mdel and they cncluded that the expected effect f increasing crime rate n declining huse prices seems t be nly bserved fr lwer educatinal status neighbrhds in Bstn. Sme existing analyses (e.g., Breiman and Friedman, 1985; Yu and Lu, 2004) in bth mean and quantile regressins cncluded that mst f the variatin seen in husing prices in the restricted data set can be explained by tw majr variables: X 1 and U. Indeed, the crrelatin cefficients between Y and U and X 1 are and respectively. The scatter plts f Y versus U and X 1 are displayed in Figures 2(a) and 2(b) respectively. The interesting features f this data set are that the respnse variable is the median price f a hme in a given area and the distributins f Y and the majr cvariate U are left skewed (the density estimates are nt presented). Therefre, quantile methds are particularly well suited t the analysis f this dataset. Finally, it is surprising that all the existing nnparametric mdels afrementined abve did nt include the crime rate X 2, which may be an imprtant factr affecting the husing price, and did nt cnsider the interactin terms such as U and X 2. Based n the abve discussins, it cncludes that the mdel studied in this paper might be well suitable t the analysis f this dataset. Therefre, we analyze this dataset by the fllwing quantile smth cefficient mdel q τ (U t, X t ) = a 0,τ (U t ) + a 1,τ (U t ) X t1 + a 2,τ (U t ) Xt2, 1 t n = 506, (11) where X t2 = lg(x t2 ). The reasn fr using the lgarithm f X t2 in (11), instead f X t2 itself, is that the crrelatin between Y t and X t2 (the crrelatin cefficient is ) is slightly strnger than that fr Y t and X t2 ( ), which can be witnessed as well frm We d nt include the ther variables such as X 3 and X 4 in mdel (11), since we fund that the cefficient functins fr these variables seem t be cnstant. Therefre, a semiparametric mdel wuld be apprpriate if the mdel includes these variables. But it is beynd the scpe f this paper and deserves a further investigatin. 18

20 Figures 2(c) and 2(d). In the mdel fitting, cvariates X 1 and X 2 are centralized. Fr the purpse f cmparisn, we als cnsider the fllwing functinal cefficient mdel in the mean regressin Y t = a 0 (U t ) + a 1 (U t ) X t1 + a 2 (U t ) Xt2 + e t (12) and we emply the lcal linear fitting technique t estimate the cefficient functins {a j ( )}, dented by {â j ( )}; see Cai, Fan and Ya (2000) fr details. The cefficient functins are estimated thrugh the lcal linear quantile apprach by using the bandwidth selectr described in Sectin 2.3. The selected ptimal bandwidths are h pt = 2.0 fr τ = 0.05, 1.5 fr τ = 0.50, and 3.5 fr τ = Figures 2(e), 2(f) and 2(g) present the estimated cefficient functins â 0,τ ( ), â 1,τ ( ), and â 2,τ ( ) respectively, fr three quantiles τ = 0.05 (slid line), 0.50 (dashed line) and 0.95 (dtted line), tgether with the estimates {â j ( )} frm the mean regressin mdel (dt-dashed line). Als, the 95% pint-wise cnfidence intervals fr the median estimate are displayed by the thick dashed lines withut the bias crrectin. First, frm these three figures, ne can see that the median estimates are quite clse t the mean estimates and the estimates based n the mean regressin are always within the 95% cnfidence interval f the median estimates. It can be cncluded that the distributin f the measurement errr e t in (12) might be symmetric and â j,0.5 ( ) in (11) is almst same as â j ( ) in (12). Als, ne can bserve frm Figure 2(e) that three quantile curves are parallel, which implies that the intercept in â 0,τ ( ) depends n τ, and they decrease expnentially, which can supprt that the lgarithm transfrmatin may be needed as argued in Yu and Lu (2004). Mre imprtantly, ne can bserve frm Figures 2(f) and 2(g) that three quantile estimated cefficient curves are intersect. This reveals that the structure f quantiles is cmplex and the lwer and upper quantiles have different behavirs and the heterscedasticity might exist. But unfrtunately, this phenmenn was nt bserved in any previus analyses in the afrementined papers. Frm Figure 2(f), first, we can bserve that â 1,0.50 ( ) and â 1,0.95 ( ) are almst same but â 1,0.05 ( ) is different. Secndly, we can see that the crrelatin between the huse price and the number f rms per huse is almst psitive except fr huses with the median price and/r higher than (τ = 0.50 and 0.95) in very lw educatinal status neighbrhds (U > 23). Thirdly, fr the lw price huses (τ = 0.05), the crrelatin is always psitive and it deceases when U is between 0 and 14 and then keeps almst cnstant afterwards. This implies that the expected effect f increasing the number f rms can make the huse price slightly higher in any lw educatinal status neighbrhds but much higher in relatively high educatinal status neighbrhds. Finally, fr the median and/r higher price huses, the crrelatin deceases when U is between 0 and 14 and then keeps almst cnstant until U 19

21 up t 20 and finally deceases again afterwards, and it becmes negative fr U larger than 23. This means that the number f rm has a psitive effect n the median and/r higher price huses in relatively high and lw educatinal status neighbrhds but increasing the number f rms might nt increase the huse price in very lw educatinal status neighbrhds. In ther wrds, it is very difficult t sell high price huses with high number f rms at a reasnable price in very lw educatinal status neighbrhds. Frm Figure 2(g), first, ne can cnclude that the verall trend fr all curves is decreasing with â 3,0.95 ( ) deceasing faster than the thers, and that â 3,0.05 ( ) and â 3,0.50 ( ) tend t be cnstant fr U larger than 16. Secndly, the crrelatin between the husing prices (τ = 0.50 and 0.95) and the crime rate seems t be psitive fr smaller U values (abut U 13) and becmes negative afterwards. This psitive crrelatin between the husing prices (τ = 0.50 and 0.95) and the crime rate fr relatively high educatinal status neighbrhds seems against intuitive. Hwever, the reasn fr this psitive crrelatin is the existence f high educatinal status neighbrhds clse t central Bstn where high huse prices and crime rate ccur simultaneusly. Therefre, the expected effect f increasing crime rate n declining huse prices fr τ = 0.50 and 0.95 seems t be bserved nly fr lwer educatinal status neighbrhds in Bstn. Finally, it can be seen that the crrelatin between the husing prices fr τ = 0.05 and the crime rate is almst negative althugh the degree depends n the value f U. This implies that increasing crime rate slightly decreases relatively the huse prices fr the cheap huses (τ = 0.05). In summary, it cncludes that there is a nnlinear relatinship between the cnditinal quantiles f the husing price and the affecting factrs. It seems that the factrs U, X 1 and X 2 d have different effects n the different quantiles f the cnditinal distributin f the husing price. Overall, the husing price and the prprtin f ppulatin f lwer educatinal status have a strng negative crrelatin, and the number f rms has a mstly psitive effect n the husing price whereas the crime rate has the mst negative effect n the husing price. In particular, by using the prprtin f ppulatin f lwer educatinal status U as the cnfunding variable, we demnstrate the substantial benefits btained by characterizing the affecting factrs X 1 and X 2 n the husing price based n the neighbrhds. Example 3: (Exchange Rate Data) This example cncerns the clsing bid prices f the Japanese Yen (JPY) in terms f US dllar. There is a vast amunt f literature devted t the study f the exchange rate time series; see Sercu and Uppal (2000) and the references therein fr details. Here we use the prpsed mdel and its mdeling appraches t explre 20