Composite Quantile Generalized Quasi-Likelihood Ratio Tests for Varying Coefficient Regression Models Jin-ju XU 1 and Zhong-hua LUO 2,*

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1 07 d Iteratioal Coferece o Iformatio Techology ad Maagemet Egieerig (ITME 07) ISBN: Comosite Quatile Geeralized Quasi-Likelihood Ratio Tests for Varyig Coefficiet Regressio Models Ji-u U ad Zhog-hua LUO, School of Medicie ecoomic ad Maagemet, Ahui Uiversity of Chiese Medicie, Hefei, Chia, 3003 School of Ecoomics ad Busiess Maagemet, Gasu Uiversity of Traditioal Chiese Medicie, Lazhou, Chia, Corresodig author Keywords: Comosite quatile regressio, Varyig coefficiet model, Geeralized quasi-likelihood ratio tests. Abstract. A ew test rocedure, called comosite quatile geeralized quasi-likelihood ratio (CQGQLR) test is roosed i this aer to test whether all or artial coefficiets are ideed costats or some secific fuctios for the varyig coefficiet regressio models. The test statistics are costructed based o the comariso of the comosite quatile quasi-likelihood fuctios uder ull ad alterative hyotheses. The roosed test methodologies are alied to aalyze the Bosto house rice data. The simulatio results ad the real examle illustrate the effectiveess ad ractical usefuless of the roosed test statistics. AMS subect classificatios. 6G056G060G4 Itroductio Sice the semial work of Koeker ad Basset (978), there has bee a abudace of literature o various alicatios ad theoretical extesios of quatile regressio. Regressio quatiles have the imortat advatage over coditioal mea regressio of beig able to directly estimate the effects of the covariates o quatiles other tha the ceter of the distributio. Quatile regressio has bee extesively alied i ecoomics, fiace, biology, medicie, ad may other discilies. May dimesio-reductio techiques have bee adoted for quatile regressio to deal with this roblem, such as additive model, sigle idex model ad varyig coefficiet quatile regressio models. Hoda (004) ad Cai ad u (008) cosidered the quatile varyig coefficiet model for time series data, Wu, Yu ad Yu (00) ivestigated the sigle idex model for quatile regressio, Zogwu Cai(0) discussed Semiarametric Partially Varyig Coefficiets Quatile Regressio Estimatio i Dyamic Models. A varyig coefficiet regressio model is a useful ad atural extesio of a classical liear regressio model. The varyig coefficiet models assume the followig coditioal mea structure: Y = = a (U ) + ε = A(U ) T + ε (.) a (U )deotes the ukow smooth fuctios,the ε is the radom errors. Comosite quatile regressio (CQR) has recetly gaied cosiderable attetio due to its ability to combie iformatio across differet quatile fuctios. CQR was recetly roosed by Zou ad Yua [008] for estimatig the regressio coefficiets i the classical liear regressio models. Kai et al. cosidered local CQR estimatio for semiarametric varyig-coefficiet artially liear models. However, to the best of our kowledge, the roblem i varyig coefficiet models of test is cosidered by few eole although it has broad otetial alicatios. This motivates us to cosider the roblem withi the framework of varyig coefficiet models. We roose a ew test rocedure, termed as comosite quatile geeralized quasi-likelihood ratio (CQGQLR) test, to test whether all or artial coefficiets are ideed costat or of some secific fuctios for the varyig coefficiet 00

2 regressio models. The test statistics are costructed based o the comariso of the comosite quatile quasi-likelihood fuctios uder ull ad alterative hyotheses. I also aly the roosed test methodologies to test if the existig models i the literature used to aalyze the Bosto house rice data are aroriate or ot. The simulatio results ad the real examle illustrate the effectiveess ad ractical usefuless of the roosed test statistics. Estimatio of the Regressio Coefficiets The varyig coefficiet quatile regressio model takes the form qτ(ut, t) = = 0 a,τ (U t ) t = A τ (U t )T t (.) U t R d is called the smoothig variable ad t = ( t0, t,, t ) with t0 = are i.i.d observatios, A(U t ) = A τ (U t ) = (a 0,τ, a,τ,, a,τ )T are smooth coefficiet fuctios which might be some fuctio of t0,..., t or time or some other exogeous variables. Without loss of geerality, I cosider oly the case i which U t i (.) is oe dimesioal (d = ). For simlicity, we dro τ from a,τ (.) i what follows. To estimate the coefficiet fuctios A( ), I aly the local fittig techique as follows. Assume A(U ) has a cotiuous first derivative. For a give oit u, oe ca aly Taylor exasio to aroximate A(U i ) as A(U i ) = β 0 + β (U i u), (.) β 0 = A(u) ad β = A (u) is the first derivative of A(u). Let c τk deote the 00 τk % quatile of ε. The For a give q, let ρ τk (r) = r(τ k I (r<0) ), τ k = k/(q + ) for k =,,, q. Thus, followig the local CQR techique, β 0 (u) ad β (u) ca be estimated via miimizig the locally weighted CQR loss: q k = i= ρ τk {Y i i T (β 0 β (U i u))}k h (U i u), (.3) K( ) is a kerel fuctio, K h (x) = K h (U i u), ad h = h is a sequece of ositive umbers h tedig to zero, which cotrols the amout of smoothig used i estimatios. we ca get the local liear estimate of A(u), deoted by A = β 0. Test Statistics Test of Fuctioal Form of Varyig Coefficiets Sectio is devoted to fittig a varyig coefficiet quatile regressio model. Now, it turs to oe geeral ad iterestig testig roblem to check whether the varyig coefficiet are of some secific fuctioal form. This is equivalet to the followig hyothesis: H 0 : A τ (u) = A 0,τ (u) versus H : A τ (u) A 0,τ (u) (3.) A 0,τ (u) is a vector of kow fuctioals. The likelihood ratio tye test was roosed by Cai, Fa ad Yao (000) for the hyothesis testig roblems formulated i (3.) for the coditioal mea regressio models i (.).The geeralized likelihood ratio is defied as follows: RSS0 RSS0 RSS λ = l( H) l ( H0)= log (3.) RSS RSS l(h ) is the log-likelihood uder H with ukow regressio fuctio relaced by a reasoable oarametric regressio estimator, 0 ˆ i i ad i= = R SS = ( Y a )

3 0 = ˆ i 0 i i= = a0 R SS ( Y a ) ˆ ( u ) is the true or estimated value of coefficiets uder H 0. Motivated by Cai, Fa ad Yao (000), for the varyig coefficiets quatile regressio models, by takig the loss fuctio as the check fuctio istead of the sum of squared errors, I roose the similar test statistic for the testig roblems i (3.). As elaborated i Komuer (005), ρ τ i i i= = l ( H )= { Y a } ca be regarded as the egative logarithm of quasi-likelihood. So the corresodig comosite quatile geeralized quasi-likelihood ratio (CQGQLR) test statistic is defied as follows: q T = l( H ) l( H ) = ρ Y aˆ 0 τ k i i k = i= = q ρ Y a τ k i 0 i k = i= = q ρ τ ˆ k i i k = i= = (3.3) l ( H )= { Y a } ad a ˆ (u) is the oarametric estimate of a (u) by usig local liear estimatio techique uder the alterative hyothesis, ad q 0 ρ τ k i 0 i k = i= = l ( H )= { Y a } with a 0 (u) is the true fuctio uder the ull hyothesis. Test of Costacy of Varyig Coefficiet Oe secial case of the hyothesis i (3.) is to check is that A 0,τ (u) is a vector of costats. The, the test hyothesis becomes to checkig whether the varyig coefficiets are ideed varyig. That is equivalet to : ( ) versus : A A0 (3.4) H0 Aτ u =A0 τ H τ τ,, With a kow costat vector, by the discussio above, the CQGQLR test statistic is defied as follows (3.5) ad a ˆ (u)is the oarametric estimate of a (u) by usig local liear estimatio techique uder the alterative hyothesis, ad, 0

4 with a 0 is the true fuctio uder the ull hyothesis Test of Costacy of Varyig Coefficiet with Ukow Value I some alicatios, it may be more iterestig i checkig the costacy of the varyig coefficiet with the true value A 0τ ukow. Therefore, we cosider the test statistic for the hyothesis i (3.4) with a ukow costat vector. Uder the ull hyothesis, oe ca estimate the coefficiet aˆ0k for the liear quatile regressio ad costruct the quasi-likelihood as follows The, the comosite quatile geeralized quasi-likelihood ratio (CQGQLR) test statistic for hyothesis testig roblem i (3.4) with ukow A 0,τ is defied by T = l( H ) l( H ) 0 q q = ρ ˆ τ k Yi a i ρτ k Yi a0 i k = i= = k = i= = q q + ρ ˆ τ k Yi a0k i ρτ k Yi a0 i k = i= = k = i= = T + T reect H 0 for large value of T. (3.6) A Real Examle I this sectio, I cosider the alicatio of these methodologies to a real examle. Here I aalyze a subset of the Bosto house rice data (htt://lib.stat.cum.edu/datasets/bosto) of Harriso ad Rubifeld (978) which is used to study the effect of air ollutio o real estate rice i the greater Bosto area i 970s.The data set cosist of 506 observatios o 4 variables. As idicated i Cai ad u (008) which aalyzed this data set by usig a varyig coefficiet quatile regressio model, we focus o exlorig the ossible (liear, oarametric or semiarametric) relatioshis betwee the deedet variable ad some maor factors which might factors the house rice. Here I adot the same otatio as i Cai ad u (008) i order to do a comariso. Y will be used to deote the deedet variable, the media value of ower-occuied homes i $,000 s (house rice).u is roortio of oulatio of lower educatioal status. is the average umber of rooms er house i the area. deotes the er caital crime rate by tow. 3 is the full roerty tax rate er $, is the ulil/teacher ratio by tow school district. Note that there are may aers ivestigatig this data set i the literature, ad the reader is referred the aer by Cai ad u (008) for details. I this sectio, I will focus o two models. First, we cosider the model from Cai ad u (008) which is the followig quatile smooth coefficiet model q ( U, ) = a ( U ) + a ( U ) + a ( U ) (4.) τ t t 0τ t τ t t τ t t t = 0 = log( ). We ow check whether the fuctioal coefficiets i A t (u) t ( a, a, a ) T τ τ τ i model(4.) are ideed varyig with u, that is,we test the ull hyothesis H 0 :A T (u)=a 0, A 0T is a vector of ukow arameters. For this testig roblem, I calculate the test statistic by usig the roosed test rocedure. The corresodig -value are 03

5 reorted i Table. Therefore, oe ca see that all the -values are less tha sigificat level 0.05 from Table, which imlies that the varyig coefficiets are ideed varyig. Table. The -values for testig costacy i model (4.). τ value It is clear that model (4.) does ot iclude two variables 3 ad 4. The reaso as claimed by Cai ad u (008) is that the fuctioal coefficiets for variables 3 ad 4 may be costat. Therefore, I use the roosed test rocedure to test whether the coefficiets of 3 ad 4 are costat or ot. To this effect, we cosider the followig model q ( U, ) = a ( U ) + a ( U ) + a ( U ) (4.) τ t t 0τ t 3τ t t3 4τ t t4 ad the cosider the testig roblem formulated as the ull hyothesis Aτ u a u a 0, 3, u a4, u τ τ τ H : A 0 τ = A 0, ( ) = ( ( ), ( ), ( )) T ad A 0,T is a vector of ukow arameters. By usig the test statistic by followig the test rocedure as i Sectio 3, I calculate the quasi-likelihood usig liear arametric comosite quatile regressio uder the ull hyothesis. The corresodig -values are reorted i Table, from which, oe ca see that all the -values are greater tha sigificat level 0.05.This imlies that the varyig coefficiets are ideed costat. Table. The -values for testig costacy i model (4.). τ value Refereces [] Koeker, R. ad Bassett, G, Regressio quatiles. Ecoometrica, 46, 33-50, 978. [] Cai, Z. ad u,, Noarametric Quatile Estimatios for Dyamic Smooth Coefficiet Models. Joural of the America Statistical Associatio, 03, , 008. [3] Hoda, T, Quatile Regressio i Varyig Coefficiet Models.Joural of Statistical Plaig ad Iferece,, 3-5, 004. [4] Wu, T., Yu, K. ad Yu, Y., Semiarametric Quatile Regressio Estimatio i Dyamic Models with Partially Varyig Coefficiets.Joural of Ecoometrics, 67, 43-45, 0. [5] Cai, Z., Fa, J. ad Yao, Q., Fuctioal-Coefficiet Regressio Models for Noliear Time Series. Joural of the America Statistical Associatio, 95, , 000. [6] Cai, Z. ad iao, Z., Sigle-idex Quatile Regressio. Joural of Multivariate Aalysis, 0, 607-6, 00. [7] B. Kai, R. Li, H. Zou. New efficiet estimatio ad variable selectio methods for semiarametric varyig-coefficiet artially liear models, Aals of Statistics, 39(0),

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