PROPERTIES OF TRANSFORMATION QUANTILE REGRESSION MODEL

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1 ACTA UNIVERSITATIS LODZIENSIS FOLIA OECONOMICA 285, 2013 Grażyna Trzpo * PROPERTIES OF TRANSFORMATION QUANTILE REGRESSION MODEL Absrac. We presen n hs paper a few mporan drecon on research usng quanle regresson. We sar from some movaon for hs mehod of regresson. Secondly we presen some man areas of applcaon hs mehod. Fnally we waned o pon ou ransformaon of he man model. Ths model, nroduced by Powell (1991) and furher analyzed by Chamberlan (1994) and Buchnsky (1995), specfes he condonal quanles of he Box-Cox ransformaon of he varable under apprasal as a lnear funcon of he covaraes. I provdes, whn a smple se-up, he needed flexbly, as boh he ransformaon parameer and he coeffcens of he lnear funcon are allowed o vary freely a each pon of he dsrbuon. The Box-Cox quanle regresson, whch has he lnear and log-lnear models as parcular cases, wll provde, herefore, a drec answer o he queson of he approprae ransformaon o be used. Key words: quanle regresson, quanle regresson model, Box-Cox ransformaon. I. QUANTILE REGRESSION MOTIVATION From sandard regresson o quanle regresson Regresson s used o quanfy he relaonshp beween a response varable and some covaraes. Sandard regresson has been one of he mos mporan sascal mehods for appled research for many decades. More complcaed models, such as polynomal regresson models, may also be used o model dfferen relaonshp. From condonal skew dsrbuons o quanle regresson Fg. 1(a) dsplays wegh agans age for a sample of 4011 US grls (Cole,1988). The nuvely reasonable noon of a relaonshp beween wegh and age s furher suppored by Fg. 1(b) whch presens several smoohed quanle regresson curves. These sugges ha he assocaed condonal dsrbuons are skew o he rgh. Two quesons of neres are: frs: wha s a ypcal wegh profle as a funcon of age second: wha s a ypcal wegh profle as a funcon of age for overwegh and underwegh people? * Prof., Unversy of Economcs n Kaowce. [125]

2 126 Grażyna Trzpo A sensble answer o he frs queson s no provded by sandard mean regresson, as he mean a any specfc year s pulled downwards. Hence, he medan curve s a more approprae curve o dsplay. Ths medan curve corresponds o he mddle quanle regresson curve dsplayed n Fg. 1(b). If s hough ha grls whose weghs le on or above he 97% curve for he populaon are overwegh, hen he approprae curve o dsplay s ha based on quanle regresson wh p = 0,97. Smlarly, he p = 0,03 quanle regresson curve dsplays he relaonshp of he wegh of underwegh grls wh age. II. APPLICATIONS OF QUANTILE REGRESSION In hs secon we presen some ypcal applcaons of quanle regresson o medcal reference chars, survval analyss, fnancal research, economcs research and he deecon of heeroscedascy 2.1. Applcaons o reference chars n medcne In medcne, reference (or cenle) chars provde a collecon of useful quanles. These are wdely used n prelmnary medcal dagnoss o denfy unusual subjecs n he sense ha he value of some parcular measuremen les n one or oher al of he approprae reference dsrbuon. The need for quanle curves raher han a smple reference range arses when he measuremen (and hence he reference range) s srongly dependen on a covarae such as age, as Cole and Green (1992) and Royson and Alman (1994) have dscussed. The chosen quanles are usually a symmerc subse of {0,03; 0,05;0,1;0, 25;0,5;0,75;0, 9;0, 95;0, 97}. An example of a reference char s shown n Fg. 1, Hahn (1995) wh he Y-varable beng wegh and he X-varable beng age. How can hese quanle regresson curves be obaned? Fgure 1. Wegh agans age for a sample of 4011 US grls Source: own work.

3 Properes of Transformaon Quanle Regresson Model 127 An obvous approach s o use a known condonal dsrbuon F(y x) o f he underlyng condonal dsrbuon. The 100 % quanle curve corresponds o q (x) = F 1 ( x). Now, f he dsrbuon s normal, hen esmang he 100 % quanle curve s sraghforward. If, however, he dsrbuon s skew, as s more usual, hen ofen a ransformaon o normaly s appled. A ypcal ransformaon s he Box Cox ransformaon o whch we shall reurn, see Cole (1988), Alman (1990) and Royson and Wrgh (2000) Applcaons o survval analyss Applcaons o survval analyss nclude sudyng he effec of a specfc covarae on he survval me of an ndvdual. A gven covarae may have a dfferen effec on low, medum and hgh rsk ndvduals. These effecs can be undersood by consderng several quanle funcons of survval me; see Koenker and Gelng (2001) for deals. Fg. 2 presens hree quanle regresson curves wh p = 0,1; 0,5; 0,9 based on he 184 survval mes of paens wh covarae age beween 12 and 64 years from he Sanford hear ransplan survey (Crowley and Hu,1977); see Yang (1999) for furher deals abou censored medan regresson. Fgure 2. Survval mes of paens wh covarae age beween 12 and 64 years Source: own work. Cox s proporonal hazard model s ofen used for survval analyss. Alernavely, he acceleraed falure me approach ha models he logarhm of he survval me as a funcon of covaraes can be employed. The basc model poss survval mes T, =1,..., n, ha may be censored and ha depend on covaraes x. In he absence of censorng, s naural o consder he pars {T, x } n = 1 as a mulvarae ndependenly and dencally dsrb-

4 128 Grażyna Trzpo ued sample. If he h observaon has been censored, hen we observe Y for T. The log -ransformaon of T provdes he usual acceleraed falure me model, whch regresses he logarhm of T lnearly on x,.e. log(t ) = x T β +, where, =1,..., n, are ndependenly and dencally dsrbued wh an unknown dsrbuon funcon. The mean of s no assumed o be zero because we observe Y nsead of T n he case of censorng and so he nercep erm s no ncluded n he vecor β. Because of hs, mean regresson analyss s no a good esmaon echnque for he acceleraed falure me approach. However, he quanle regresson echnque ha models he quanles of he survval me or a monoone ransform hereof, as a funcon of he covaraes and he nercep s approprae (see Yang (1999) Applcaons n fnancal research Fnancal regulaons usually requre banks o repor her daly rsk measures called value a rsk (VaR). VaR models are he mos commonly used measure of marke rsk n he fnancal ndusry (Laurdsen,2000). Le Y be he fnancal reurn, so ha he y sasfyng P(Y y) = p for a gven low value of p s he VaR. The varable Y may depend on covaraes x such as exchange raes. Clearly, VaR esmaon relaes o exreme quanle esmaon hrough esmang he al of fnancal reurn. The dsrbuon of fnancal reurn could also be llusraed by several quanles. For example, he common approach o esmang he dsrbuon of oneperod reurn n fnancal models s o forecas he volaly and hen o make a Gaussan assumpon (see Hull and Whe (1998)). Marke reurns, however, are frequenly found o have more kuross han a normal dsrbuon. A general dscusson of usng quanle regresson for reurn-based analyss was gven by Basse and Chen (2001) Applcaons n economcs research Quanle regresson s useful n he sudy of consumpve markes as he nfluence of a covarae may be very dfferen for ndvduals who belong o hgh, medum and low consumpon groups. Smlarly, changes n neres raes may have a dfferen nference on he share prces of companes whch belong o hgh, medum and low profs groups. In parcular, quanle regresson s now regarded as a sandard analyss ool for wage and ncome sudes n labour economcs; see, for example, Buchnsky

5 Properes of Transformaon Quanle Regresson Model 129 (1995). I s also mporan o sudy how ncomes are dsrbued among he members of a populaon, e.g. o deermne ax sraeges or for mplemenng socal polces. Oher applcaons nclude modelng household elecrcy demand over me n erms of weaher characerscs. The low quanle curves correspond o background use, where as possbly he hgh quanle curves reflec hgh use durng acve perods of he day parcularly due o ar condonng; see Hendrcks and Koenker (1992) Applcaons o deecng heeroscedascy Recognzng heeroscedascy s an mporan ask for he daa analys. Quanle plos can provde a useful descrpve ool. These plos no only help o deec heeroscedascy bu also provde an mpresson of he locaon,spr ead and shape of he condonal dsrbuon of Y gven X = x. Quanle regresson can be used o assess deparures from he assumpons of he model Y = x T β +. If he dsrbuon of does no depend on he value of he covarae X, all regresson quanles wll be parallel. For example, he seven quanle curves for he US grls daa n Fg. 1 are clearly no parallel, ndcang heeroscedascy III. ESTIMATION METHODS AND ALGORITHMS We wll now presen esmaon mehods and algorhms for quanle regresson The paramerc quanle regresson model To quanfy he relaonshp beween a response varable Y and covaraes x, we ofen assume ha E[Y X=x] can be modeled by a smple lnear combnaon x T β. Smlarly, he basc quanle regresson model specfes he lnear dependence of he condonal quanles of Y on x. Consder he followng regresson model (Trzpo, 2009b) y g( x ) e (1) where he dependen varable y = (y 1, y 2,... y n ) and ndependen x= (x 1, x 2,... x n ) where yr and xr p, g() s real valued and unknown. We are neresed n esmang he regresson funcon g() gven x. In he paramerc

6 130 Grażyna Trzpo framework of he lnear regresson model when g(x ) = ()x he quanle regresson was proposed as a soluon of 1 n mn ( y x ) (2) R p n 1 where (z) = I(z < 0) z, I s he ndcaor funcon 1. The condonal quanle of y gven x, by monooncy of quanle funcon, Q( x) g( x) D 1 ( x) g ( x) (3) where D 1 (x) s condonal h quanle of error erm and Q( x) nf{ : P( y x) }. In equaon (3) g(x) and D 1 ( x) are no denfed separaely. However g (x), he condonal h quanle can be denfed, hen he equaon(1) can be rewren as y g ( x ) v (4) where v = D 1 ( x) and v s a new error erm whch has a zero condonal quanle. Gven (y ; x ), he quanle model can be esmaed by regresson quanles, whch are defned by he mnmzaon problem, *( ) mn w ( y xb ) w 1 y xb (5) br y x b y x b where he weghs w are nroduced o accoun for dfferen varably of x and he dfferen number of observaons a each x. There s no explc soluon for he regresson coeffcens under hs paramerc quanle regresson model snce he check funcon s no dfferenable a he orgn. However, usng recen advances n neror pon mehods for solvng lnear programmng problems dscussed by Pornoy and Koenker (1997),hs 1 I[A] = 1 f A s rue, I[A] = 0 oherwse.

7 Properes of Transformaon Quanle Regresson Model 131 mnmzaon can be performed by usng he algorhm ha was provded by Koenker and D Orey (1987) The Box Cox ransformaon quanle model Le y denoe response varable and x a vecor of k covaraes represenng ndusry arbues. For n (0,1), he h quanle of he condonal dsrbuon of y gven x, s defned as Q ( y x) nf{ y F( y x) } where F(x) denoes he condonal dsrbuon funcon. The sascal model used n hs paper specfes he h condonal quanle of y gven x as he nverse of he Box-Cox power ransformaon (Box and Cox, 1964) of an affne funcon of he covaraes, where Q ( y x) g( x( ( ), ( )) (6) (1 ) g(, ) e 1/ for 0 for 0 (7) Model (6) s que flexble snce no only he coeffcens bu also he whole ransformaon may change from quanle o quanle. Of course, he case where = 1 yelds he lnear model for he condonal quanles. By analogy wh he lnear model, he populaon quanle regresson parameers may be defned as (, x) Q ( y x) / x g1( x ( ), ( )) ( ), j = 1,., k j j j where x denoes he vecor of he regressors' sample means and g1(, ) g(, ) / The esmaon of hese regresson quanles for values of y n (0,1) consues he man am of hs sudy as hey descrbe he relevancy of covaraes a dfferen pons of response varable dsrbuon.

8 132 Grażyna Trzpo 3.3. Inference procedures for Box-Cox quanle regresson model The esmaon of model (6) s based on an equvarance propery of he quanle regresson o monoonc ransformaons of he dependen varable and follows Chamberlan (1994). Specfcally, makng z() =g 1 (y, ) where g 1 (,) s he Box-Cox ransformaon, he specfcaon (1) mples ha he quanles of z are lnear,.e. Q ( z x) x ( ) Therefore, for gven, () can be esmaed by mnmzng n (Koenker and Basse, 1978), n 1 ( z x ) (8) n 1 wh u ( u) ( 1) u for u 0 for u 0 Hence, for any gven, model (1) can be esmaed exacly n he same way as a sandard lnear quanle regresson. Of course, he usual mean regresson does no have hs propery unless = 1. Denoe by ˆ (, ) a soluon of model (8). Chamberlan (1994) suggesed esmang () by mnmzng n : n 1 ( y g( x ( ˆ(, ), )) (9) n 1 Fnally, () n model (1) s esmaed by () = (, ()). We proceeded by solvng model (8) for a grd of values of l and hen choosng he par (,()) ha yelds he smalles value for model (9). Under regulary condons, can be shown ha he jon dsrbuon of ˆ ( ) ( ˆ( ), ˆ( )) for m values of n (0,1), n( ˆ( ) 1 ( 1),..., ˆ( ) ( ) ) m m

9 Properes of Transformaon Quanle Regresson Model 133 wll converge o a m (k + 1)-varae normal dsrbuon, wh 0 mean and covarance marx whose j h block s gven by wh 1 1 V, ) H ( ) L(, ) H ( ) (10) ( j l j j l l H ( ) A( ) E[ fu ) (0) d( x, ( )) d2( x, ( )) ] ( (11) L, ) (mn{, } ) A( ) E[ d( x, ( )) d ( x, ( )) ] A( ) (12) ( j l j l j l j j 2 l l ' where f u ( ) denoes he densy of u( ) y g( x ( ), ( )) gven ( ) x ' ' ' x, d( x, ( ))' ( xg1 xg 2 ) ( xd2( x, ( ))), g ( ' 1 g1 x ( ), ( )), g2 g x ( ), ( )) 2 ( ' g1(, ) g(, ) / and g2(, ) g(, ) / wh I k A ( ) ' 0k ' 0 0 k k 0 '(, ) / k 1 a (k + 1) (2k + 1) marx where (, ) [ Ef u (0) g ' x x ] [ Ef (0) g x ] 1 1 u 2 A rgorous reamen of hs dervaon may be found n Powell (1991). Buchnsky (1995) develops he heory of he Box-Cox quanle regresson for he case of dscree regressors where he esmaon of ˆ ( ) can be accomplshed by mnmum dsance mehods. Inerval nferences for he quanle regresson parameers requre he conssen esmaon of he asympoc covarance marces (10). The crcal feaure of hs mehod s he nonparamerc esmaon of fu( ) ( x) n (11) based on he hsogram mehod of Sddqu (1960). Alernavely o hs ype of esmaor, one could have consdered he boosrap esmaon of he asympoc covarance marx V() as dd Chamberlan (1994), for he lnear model wh ndependen errors, and Buchnsky (1994), also for he lnear model bu wh general errors.

10 134 Grażyna Trzpo The heorecal bass for boosrappng quanle regresson esmaors are provded n Hahn (1995) and Fzenberger (1998). Mone Carlo comparsons n Koenker (1994) sugges ha n..d. suaons he sparsy esmaor fares beer han does he boosrap. IV. QUANTILE REGRESSION FOR TIME SERIES Mos research n quanle regresson has assumed ha he observaons of he response varable Y are condonally ndependen. Recenly, several researchers have dscussed dfferen mehods for me seres quanle regresson modellng. For example,a mehod based on esmang he condonal dsrbuon s gven by Ca (2002),w hereas a mehod based on he check funcon s gven by Gannoun e al. (2003). In he mehod of Ca (2002),he me seres Y s assumed o be relaed o he me seres X hrough he model Y ( X ) ( X ) where μ(x ) s he regresson funcon and s he model error. The dependence of σ(x ) on X means ha he model s heeroscedasc. The mehod frs esmaes he condonal dsrbuon of Y gven X and hen esmaes he condon quanle by he nverse of he condonal dsrbuon funcon. In he mehod of Gannoun e al. (2003) for he esmaon of he condonal quanle of a srcly saonary real-valued process Z gven he presen and pas records, he quanle of Z s characerzed as q ( x) arg mn{ E[ ( Z ) X x]} R 4.1. Quanle regresson as a rsk measure We should solve a problem of fndng an mnmum of coheren rsk measures, whch s equvalen o fnd a maxmum of Choque expeced value usng lnear form of he uly funcon and a concave dsoron funcon v. When we wre quanle regresson problem n general case we have a problem of esmaons a vecor of unknowns parameers b, for a sample of ndependen observaons form a random varables Y 1,Y 2,...,Y T accordng o rule: P(Y < y) = F(y x b), =1, T (12)

11 Properes of Transformaon Quanle Regresson Model 135 where {x, = 1, T} s a row n know marces of observaons (sze T K) and dsrbuon of F s unknown (Trzpo 2007). Gven (y ; x ), for = 1, T, he quanle model can be esmaed by regresson quanles, whch are defned by he mnmzaon problem: mn R y 1 y. (13) { : { : y } } y Wrng as {x, = 1, T} sequence of K vecors (rows) of observaon marces, we assume, ha {y, = 1, T} s a random sample of regresson process: u = y x b havng dsrbuon F. Then regresson quanle, for 0 < < 1 s done as a soluon of a problem: mn R y x 1 y x. (14) { : y x { : y x } } If K = 1 and x = 1for all, a problem (14) can reduce o problem (13). The smalles absolue error s hen equales o medan. The problem (14) always hale a soluon, for a connuous dsrbuon hs soluon s unque. The problem of fndng mnmum can be reformulaed as equvalen lnear programmng problem: where mn{α1 r (1 α) 1r } (15) y Xb r r K 2T ( b, r, r ) R R where 1 s a uny vecor of sze T. V. FINAL REMARKS Quanle regresson s emergng as a comprehensve approach o he sascal analyss of lnear and non-lnear response models, parly because classcal lnear heory s essenally a heory jus for models of condonal expecaons. We have llusraed ha quanle regresson has srong lnks o hree very useful sascal conceps: regresson, robusness and exreme value heory. We ry o

12 136 Grażyna Trzpo demonsrae ha quanle regresson s wdely used n many mporan applcaon areas, such as medcne and survval analyss, fnancal and economc sascs and envronmenal modelng. REFERENCES Alman, N. S. (1990) Kernel smoohng of daa wh correlaed errors. J. Am. Sas. Ass., 85, Basse G.W., Chen H. (2001) Porfolo syle: reurn-based arbuon usng quanle regresson. Emp. Econ., 26: Box G., Cox D An analyss of ransformaons revsed. Journal of he Royal Sascal Socey, Seres B 26: Buchnsky M. (1995) Quanle regresson, Box-Cox ransformaon model and he U.S. wage srucure, J. Economer., 65: Ca Z. (2002) Regresson quanles for me seres. Economer. Theory, 18, Chamberlan G. (1994). Quanle regresson, censorng and he srucure of wages. In Advances n Economercs, Sms C (eds), Cambrdge Unversy Press: New York; Cole T. J. (1988). Fng smoohed cenle curves o reference daa (wh dscusson). J. R. Sas. Soc. A, 151: Cole T. J., Green P. J. (1992) Smoohng reference cenle curves: he LMS mehod and penalzed lkelhood. Sas. Med., 11: Crowley J., Hu M. (1977) Covarance analyss of hear ransplan daa. J. Am. Sas. Ass., 72: Fzenberger B. (1998). The movng blocks boosrap and robus nference for lnear leas squares and quanle regressons. Journal of Economercs 82: Gannoun A., Saracco J., Yu K. (2003) Nonparamerc predcon by condonal medan and quanles. J. Sas. Planng Inf., o be publshed. Hahn J. (1995). Boosrappng quanle regresson esmaors. Economerc Theory 11: Hendrcks,W., Koenker,R. (1992) Herarchcal splne models for condonal quanles and he demand for elecrcy. J. Am. Sas. Ass., 93: Hull J., Whe A. (1998) Value a rsk when daly changes n marke varables are no normally dsrbued. J. Derv., 5: Koenker R Confdence nervals for regresson quanles. In Asympoc Sascs: Proceedngs of he 5 h Prague Symposum, Mandl P, Huskova M (eds), Physca-Verlag: Berln. Koenker R. and Gelng R. (2001) Reapprasng med fly longevy: a quanle regresson survval analyss. J. Am. Sas. Ass., 96: Koenker R., Machado J. (1999) Goodness of f and relaed nference processes for quanle regresson. J. Am. Sas. Ass., 94: Koenker R., Park B.J. (1996). An neror pon algorhm for nonlnear quanle regresson. J. Economer.,71: Koenker R. W., D Orey V. (1987) Algorhm AS 229: Compung regresson quanles. Appl. Sas., 36: Koenker R., Pornoy S., Ng P. (1992) Nonparamerc Esmaon of Condonal Quanle Funcons:L1 Sascal Analyss and Relaed Mehods (ed. Y. Dodge),pp Amserdam: Elsever. Koas A., Gelfand A. E. (2001) Bayesan semparamerc medan regresson model. J. Am. Sas. Ass., 96: Laurdsen S. (2000) Esmaon of value of rsk by exreme value mehods. Exremes, 3: Pornoy S. and Koenker R. (1997) TheGaussan hare and he Laplacan orose: compuably of squared-error versus absolue-error esmaors (wh dscusson). Sas. Sc., 12:

13 Properes of Transformaon Quanle Regresson Model 137 Powell J. (1991). Esmaon of monoonc regresson models under quanle resrcons. In Nonparamerc and Semparamerc Mehods n Economercs and Sascs: Proceedngs of he Ffh Inernaonal Symposum on Economc Theory and Economercs, Barne W., Powell J., Tauchen G (eds), Cambrdge Unversy Press: New York; Royson P., Alman D.G. (1994) Regresson usng fraconal polynomals of connuous covaraes: parsmonous paramerc modellng (wh dscusson). Appl. Sas., 43: Royson P., Wrgh E. M. (2000) Goodness-of-f sascs for age-specfc reference nervals. Sas. Med., 19: Taylor J. (1999) A quanle regresson approach o esmang he dsrbuon of mulperod reurns. J. Derv., 24: Trzpo G. (2007) Regresja kwanylowa a esymacja VaR, Prace Naukowe AE Wrocław, 1176, Wrocław. Trzpo G. (2009). Applcaon weghed VaR n capal allocaon, Polsh Journal of Envronmenal Sudes, Vol 18, No. 5B: Trzpo G. (2009). Esmaon mehods for quanle regresson, Suda Ekonomczne 53: 81 90, Zeszyy Naukowe AE Kaowce. Yang S. (1999) Censored medan regresson usng weghed emprcal survval and hazard funcons. J. Am. Sas. Ass., 94: Grażyna Trzpo WŁASNOŚCI TRANSFORMACJI MODELU REGRESJI KWANTYLOWEJ Przedsawamy arykuł, w kórym omawamy modele regresj kwanylowej. Omawamy moywacje dla sosowana klasycznego modelu, jak równeż główne kerunk zasosowań regresj kwanylowej. Nasępne przechodzmy do ransformacj podsawowego modelu. Ten model jes wprowadzony przez Powell a (1991) a kolejno analzowany przez Chamberlan a (1994) Buchnsky ego (1995), wprowadzono specyfczne warunkowe kwanyle znane jako ransformacja Box Cox a. Omawamy esymację model oraz esy sonośc.