ISSN: 2038-7296 POLIS Workng Papers [Onlne] Dparmeno d Polche Pubblche e Scele Colleve POLIS Deparmen of Publc Polcy and Publc Choce POLIS POLIS Workng Papers n. 178 December 2010 Reransformaon bas n he adjacen ar prce ndex Andrew M. Jones and Robero Zanola UNIVERSITA DEL PIEMONTE ORIENTALE Amedeo Avogadro ALESSANDRIA Perodco mensle on-lne "POLIS Workng Papers" - Iscrzone n.591 del 12/05/2006 - Trbunale d Alessandra
Reransformaon bas n he adjacen ar prce ndex Andrew M. Jones and Robero Zanola + ABSTRACT The leraure on hedonc prce ndces, such as he adjacen ar prce ndex, ofen uses a logarhmc ransformaon of he prce daa o deal wh he non-normaly. However, hs creaes he problem of reransformng he predcons back o an economcally meanngful scale. Ths paper nvesgaes he mpac of dealng wh he reransformaon problem for esmaes of ar marke reurns. The emprcal resuls show how falure o allow for reransformaon may resul n a based prce ndex. Key words: Reransformaon; heeroskedascy; Pcasso; hedonc prces. JEL classfcaon: C6, D2, Z1. Deparmen of Economcs and Relaed Sudes, Unversy of York, UK. + Correspondng auhor. Deparmen of Publc Polcy and Publc Choce, Unversy of Easern Pedmon, Ialy, and Rmn Cenre for Economc Analyss, Rmn, Ialy, e-mal: robero.zanola@unpmn.
1. Inroducon In recen years many papers have dscussed he hedonc esmaon of prces for ar collecbles and measured her dynamcs usng hedonc prce ndces (see e.g., Gnsburgh and Throsby, 2006). To deal wh he non-normaly of ar prces many of hese sudes use a log-lnear specfcaon for he hedonc models and ndces. However, o our knowledge, he problem of reransformng predcons back o an economcally meanngful scale n order o compue he ar prce ndex has been gnored n hs leraure. To address hs problem, hs paper nvesgaes he effecs on esmaes of ar marke reurns of a modfed verson of Duan s (1983) smearng facor. Emprcal resuls show how falure o conrol for reransformaon ssues wll resul n based esmaes of he prce ndex. 2. Modellng framework A se of qualy characerscs =,...,K, s denfed for a regresson of he log prce of panng, wh dummy varable z, k 1 = 1,..., N, sold a me, wh = 1,..., T on s k-characerscs and a se of d, whch are equal o 1 n perod and zero oherwse, such ha: K T = α + βk, k δ τ + ε k= 1 = 2 (1) log( p ) z d whereα s he nercep; he β can be nerpreed as (mplc) prces of he varous characerscs descrbng he panng; he d can nerpreed as a measure of he componen of prces ha s no arbuable o he denfed characerscs, whch can be used as an esmae of he pure prce change; and ε s a random error erm. The error erm s assumed o have a zero condonal mean bu he oher momens of s dsrbuon may be funcons of he regressors z and d, for example here may be heeroskedascy on he log-scale. The qualy of a panng s defned n erms of s characerscs and a regresson of prces on hese characerscs, along wh he me dummes, holds qualy consan.ths enables he consrucon of consan-qualy prce ndces. In parcular, he adjacen qualy-adjused prce 2
ndex of a panng beween perod and s, o (Trple, 2006):,s PI, for any gven se of characerscs, z, s equal PI s, exp = exp E p z, d = s E p z, d K ˆ ˆ ˆ α + βkz, k + δ E exp ( ε ), k 1 z d = K s ˆ ˆ ˆs E α β exp ( ) z, d kz ε +, k + δ k = 1 ( ˆ δ ) s ( δ ) ( ε ) s ( ε ) exp E exp z, d = exp ˆ E exp z, d (2) Whenever a change n qualy occurs s aken care of by he assocaed characerscs, and he qualy-adjused prce change wll be capured by he produc of he exponenal of he regresson coeffcen of he me dummy varable and he condonal expecaon of he exponenal of he unobserved error erm. Ths second erm, whch s he source of he reransformaon bas, has been negleced n he leraure on hedonc ar prce ndexes. In order o correc for hs reransformaon bas, Duan s (1983) nonparamerc smearng esmaor can be appled, hs esmaes he condonal expecaon of exp( ε ) by s sample mean. However, as he error on he log-scale of ar prces s expeced o be heeroskedasc, a varan of he sandard Duan esmaor has been adoped (see Mannng, 1998). Ths calculaes a separae smearng facor for each year, such ha: PI ( ˆ δ ˆ δ ) s, s = exp s where ϕ ϕ N K N 1 ˆ ˆ 1 ϕ = exp log( p ) ˆ kz, k exp N α β δ = ε = 1 k= 1 N = 1 ( ˆ ) (3) 3
3. Daa and Resuls The daase consss of 716 Pcasso panngs sold a aucon worldwde durng he perod 1988-2005. The daa se s colleced from he 2006 edon of he Ar Prce Index on CD-Rom. I conans records of panngs sold a he world s major aucons. Prces are gross of he buyers and sellers ransacon fees pad o aucon houses and are expressed n US dollars, deflaed usng US CPI prces (2000=100). Varables ncluded n he sudy are sze, meda, saleroom, syle perods (Czujack, 1997); and year of sale. Table 1 repors descrpve sascs for he varables used n he analyss. [TABLE 1 ABOUT HERE] Table 2 dsplays he resuls of he OLS esmae of hedonc log-prce equaon (1). Sandard errors of he coeffcens have been compued usng he Huber-Whe heeroskedascy-robus procedure. Accordng o he se of physcal characerscs n he regresson model greaer fnancal value s placed boh on larger sze, wh decreasng reurns o sze, and on ol works execued on canvas, whle prces decrease for mxed echnques. The se of explanaory varables relaed o he sale characerscs of he works show ha aucons a Chrse s and Soheby s ncrease prces over oher aucon houses. The fnal se of varables relaes o he dfferen syle perods. Works execued before 1954 command hgher prces. [TABLE 2 ABOUT HERE] Tme dummy varable coeffcens are hen used o compare he un-smeared prce ndex wh he smeared prce ndex. Calculang a separae smearng facor for each year (Table 3) as n eq.(3), he resuls for he wo ndces are repored n Fgure 1. Ths comparson cass doub on he capacy of he sandard ndex o esmae he correc reurn from an nvesmen n Pcasso panngs. Durng he perod he un-smeared esmae les boh above and below he smeared esmae whou any obvous regulary o he bas. [TABLE 3 ABOUT HERE] [FIGURE 1 ABOUT HERE] 4
In order o focus on he bas generaed by gnorng he reransformaon problem, ssue Fgure 2 shows he percenage dfferences beween he smeared and un-smeared esmaes over he me. These range from -22.3 up o +32.7 per cen, sgnallng ha here may be a subsanal bas n he emprcal leraure on ar prce ndces. Moreover, o he exen ha he ar marke s nfluenced by hese ndces, he ssue of reransformaon may lead o marke falure. [FIGURE 2 ABOUT HERE] 5. Concluson Ths paper nvesgaes he problem of reransformng predcons back o an economcally meanngful scale when ar prce ndces are calculaed from log-scale regressons. The emprcal resuls show how falure o deal wh reransformaon may resul n a based prce ndex, poenally creang msleadng nformaon upon whch choces are made n he fnancal ar markes. 5
References Czujack C., 1997, Pcasso Panngs a Aucon, (1963-1994). Journal of Culural Economcs 21, 229-247. Duan, N., 1983, Smearng esmae: a nonparamerc reransformaon mehod. Journal of he Amercan Sascal Assocaon, 78, 605-610. Gnsburgh, V. Throsby, D. (eds.), 2006, Handbook of he Economcs of Ar and Culure, Norh-Holland, Amserdam. Mannng, W. (1998). 'The logged dependen varable, heeroscedascy, and he reransformaon problem.' Journal of Healh Economcs, 17, 283-95. Trple, J., 2006, Handbook on Hedonc Indexes and Qualy Adjusmens n Prce Indexes, OECD. 6
TABLE 1. Descrpve sascs Mean Sd. Dev. Descrpon prce 2,122,799 5,984,123 Prce of panng sze.5171.5275 Area sze2.5452 1.1117 Squared area canvas.7251.4467 Ol on canvas panel.0580.2340 Ol on panel mxed.0557.2294 Mxed echnque oher_ech.1896.3922 Oher echnques (omed caegory) chrlon.1505.3578 Sold a Chrse s London chrny.2666.4424 Sold a Chrse s New York sohlon.1458.3530 Sold a Soheby s London sohny.2868.4525 Sold a Soheby s New York ohauc.1256.3316 Sold a oher aucon houses (omed caegory) syle1.0564.2309 Chldhood and Youh (1881-1901) syle2.0184.1344 Blue and Rose Perod (1902-1906) syle3.0589.2356 Analycal and Synhec Cubsm (1907-1915) syle4.1055.3074 Camera and Classcsm (1916-1924) syle5.1043.3058 Juggler of he Form (1925-1936) syle6.1595.3664 Guernca and he Syle Pcasso (1937-1943) syle7.1472.3546 Polcs and Ar (1944-1953) syle8.3497.4772 The Old Pcasso (1954-1973) (omed caegory) d88.0140.1174 1988 dummy d89.0475.2128 1989 dummy d90.0670.2503 1990 dummy d91.0182.1336 1991 dummy d92.0335.1801 1992 dummy d93.0517.2215 1993 dummy d94.0461.2098 1994 dummy d95.0642.2454 1995 dummy d96.0517.2215 1996 dummy d97.0768.2665 1997 dummy d98.1271.3333 1998 dummy d99.0824.2752 1999 dummy d00.0475.2128 2000 dummy d01.0503.2187 2001 dummy d02.0531.2243 2002 dummy d03.0349.1837 2003 dummy d04.0601.2378 2004 dummy d05.0377.1906 2005 dummy 7
TABLE 2. Hedonc regresson resuls Coef. Robus Sd. Err. Physcal characerscs area 0.0004*** 0.0000 area2-1.13e-08*** 1.02e-09 canvas 0.3853*** 0.1161 panel -0.2831 0.2208 mxed -1.0101*** 0.2855 Sale characerscs chrlon 0.3823*** 0.1406 chrny 0.4037*** 0.1401 sohlon 0.3231** 0.1455 sohny 0.5995*** 0.1371 Syle characerscs syle1 1.6199*** 0.2042 syle2 2.2112*** 0.3544 syle3 1.8137*** 0.1816 syle4 0.9535*** 0.1180 syle5 1.2625*** 0.1207 syle6 0.9340*** 0.1232 syle7 0.2699*** 0.1090 Year dummes [ncl.] cons 10.1389*** 0.2154 R-squared 0.64 Noe: ***, **, * sgnfcance a.01,.05, and.10 respecvely 8
TABLE 3. Esmaes of smearng facors Smearng facor d88 1.2216 d89 1.1883 d90 1.4520 d91 1.1282 d92 1.3874 d93 1.3047 d94 1.7316 d95 1.6501 d96 1.5572 d97 1.5777 d98 1.8095 d99 1.6935 d00 1.6123 d01 1.4645 d02 1.5459 d03 1.2256 d04 1.3234 d05 1.3188 9
FIGURE 1. Comparson beween un-smeared and smeared prces ndces 200 160 unsmeared prce ndex smeared prce ndex 120 80 40 0 d88 d89 d90 d91 d92 d93 d94 d95 d96 d97 d98 d99 d00 d01 d02 d03 d04 d05
FIGURE 2. Percenage dfference beween smeared and unsmeared prce ndces 40.0 30.0 32.7 22.2 23.0 20.0 14.7 10.0 5.6 8.0 1.3 0.0-10.0 d89 d90 d91 d92 d93 d94 d95 d96 d97 d98 d99 d00 d01 d02 d03 d04 d05-2.7-4.7-6.0-5.6-4.8-6.4-9.2-0.3-20.0-22.3-20.7-30.0 11
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