Retransformation bias in the adjacent art price index

Transcription

1 ISSN: 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/ Trbunale d Alessandra

2 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.

3 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

4 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

5 3. Daa and Resuls The daase consss of 716 Pcasso panngs sold a aucon worldwde durng he perod 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

6 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 up o 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

7 References Czujack C., 1997, Pcasso Panngs a Aucon, ( ). Journal of Culural Economcs 21, Duan, N., 1983, Smearng esmae: a nonparamerc reransformaon mehod. Journal of he Amercan Sascal Assocaon, 78, 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, Trple, J., 2006, Handbook on Hedonc Indexes and Qualy Adjusmens n Prce Indexes, OECD. 6

8 TABLE 1. Descrpve sascs Mean Sd. Dev. Descrpon prce 2,122,799 5,984,123 Prce of panng sze Area sze Squared area canvas Ol on canvas panel Ol on panel mxed Mxed echnque oher_ech Oher echnques (omed caegory) chrlon Sold a Chrse s London chrny Sold a Chrse s New York sohlon Sold a Soheby s London sohny Sold a Soheby s New York ohauc Sold a oher aucon houses (omed caegory) syle Chldhood and Youh ( ) syle Blue and Rose Perod ( ) syle Analycal and Synhec Cubsm ( ) syle Camera and Classcsm ( ) syle Juggler of he Form ( ) syle Guernca and he Syle Pcasso ( ) syle Polcs and Ar ( ) syle The Old Pcasso ( ) (omed caegory) d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy d dummy 7

9 TABLE 2. Hedonc regresson resuls Coef. Robus Sd. Err. Physcal characerscs area *** area2-1.13e-08*** 1.02e-09 canvas *** panel mxed *** Sale characerscs chrlon *** chrny *** sohlon ** sohny *** Syle characerscs syle *** syle *** syle *** syle *** syle *** syle *** syle *** Year dummes [ncl.] cons *** R-squared 0.64 Noe: ***, **, * sgnfcance a.01,.05, and.10 respecvely 8

10 TABLE 3. Esmaes of smearng facors Smearng facor d d d d d d d d d d d d d d d d d d

11 FIGURE 1. Comparson beween un-smeared and smeared prces ndces unsmeared prce ndex smeared prce ndex d88 d89 d90 d91 d92 d93 d94 d95 d96 d97 d98 d99 d00 d01 d02 d03 d04 d05

12 FIGURE 2. Percenage dfference beween smeared and unsmeared prce ndces d89 d90 d91 d92 d93 d94 d95 d96 d97 d98 d99 d00 d01 d02 d03 d04 d

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