ISSN 440-77X Ausralia Deparmen of Economerics and Business Saisics hp:wwwbusecomonasheduaudepsebspubswpapers Mulivariae ess of asse pricing: Simulaion evidence from an emerging marke Javed Iqbal, Rober Brooks and Don UA Galagedera April 2008 Working Paper 208
Mulivariae ess of asse pricing: Simulaion evidence from an emerging marke Javed Iqbal, Rober Brooks* and Don UA Galagedera Deparmen of Economerics and Business Saisics, Monash Universiy ABSRAC he finie sample performance of he Wald, GMM and Likelihood Raio (LR) ess of mulivariae asse pricing ess have been invesigaed in several sudies on he US financial markes his paper exends his analysis in wo imporan ways Firsly, considering he fac ha he Wald es is no invarian o alernaive non-linear formulaion of he null hypohesis he paper invesigaes wheher alernaive forms of he Wald and GMM ess resul in considerable difference in size and power Secondly, he paper exends he analysis o he emerging marke daa Emerging markes provide an ineresing pracical laboraory o es asse pricing models he characerisics of emerging markes are differen from he well developed markes of US, Japan and Europe I is found ha he asympoic Wald and GMM ess based on Chi-Square criical values resul in considerable size disorions he boosrap ess yield he correc sizes Muliplicaive from of boosrap GMM es appears o ouperform he LR es when he reurns deviae from normaliy and when he deviaions from he asse pricing model are smaller Applicaion of he boosrap ess o he daa from he Karachi Sock Exchange srongly suppors he zero-bea CAPM However he low power of he mulivariae ess warrans a careful inerpreaion of he resuls Keywords: Zero-bea CAPM, Mulivariae es, Wald, LR, Emerging Markers *Correspondence o: Rober Brooks, Deparmen of Economerics and Business Saisics, Monash Universiy, PO Box 07, Narre Warren Vicoria 3805, Ausralia Phone: 6-3-99047224, Fax: 6-3- 99047225, Email: Roberbrooks@busecomonasheduau
I INRODUCION Asse pricing models and heir empirical ess consiue a major componen of he finance lieraure Univariae esing of he Capial Asse Pricing Model (CAPM) inroduced by Fama and MacBeh (973) employed a wo-sage es procedure his wo-sep procedure has been criicized on wo concerns Firsly he cross secion ess involve esimaed regressors and herefore are subjec o errors-in-variable bias Secondly asse pricing ess in paricular and economeric mehods in general ha involve esimaion or esing in sages are shown o lack efficiency and herefore are less powerful Affleck-Graves and Bradfield (993) conclude hrough simulaions ha frequen rejecion of CAPM ess or equivalenly non-rejecion of hypohesis ha here is no posiive linear relaionship beween bea and reurns is due o he low power of he univariae ess associaed wih smaller sample sizes According o Shanken (996) he saisical properies of muli-sage ess are difficul o assess Gibbons (982) developed a mulivariae es of he Black s (972) zero-bea CAPM In his es he zero-bea CAPM resricions are direcly imposed on he sysem of mulivariae marke model equaions wih each equaion corresponding o an asse he es resuls in a Likelihood Raio saisic which is asympoically Chi-Square disribued his es does no involve esimaed beas as he regressors so he errorsin-variable problem is no of any concern he es also makes beer use of available cross equaion informaion he mulivariae es of Gibbons, Ross and Shaken (989) is perhaps he mos widely used es of he Sharpe-Linner form of he CAPM his es provides an exac F-es in he mulivariae esing framework I is valid in small samples if an appropriae risk-free rae of is available hese mulivariae ess have been widely used in US and oher developed markes daa Boh of hese mulivariae 2
asse pricing ess assume ha he reurns and he residuals are normally disribued and are cross secionally dependen bu serially uncorrelaed and homoskedasic ypically a general asse pricing model should saisfacorily describe empirical daa under varied marke condiions Unforunaely he mulivariae asse pricing sudies have no been performed for emerging markes Emerging markes provide an ineresing pracical laboraory o es asse pricing models Several sudies have suggesed ha he characerisics of emerging markes are differen from he well developed markes of he US, Japan and Europe For example, Harvey (995) found ha (i) emerging markes have a higher level of volailiy and price changes han developed markes, (ii) a majoriy of he emerging markes had non-normal reurns and (iii) he reurns are more predicable han he developed markes Consequenly any mulivariae asse pricing es applied o emerging marke daa need o be robus o hese disribuional characerisics Greene (2003, p-0) poins ou ha amongs he hree asympoic ess namely he Wald, LR and LM, only he Wald es is asympoically valid under non-normaliy Is compuaion requires unconsrained parameers esimaes for which OLS (or SUR in sysem conex) can be readily applied he Wald es assumes he reurn o be idenically and independenly disribued (iid) he GMM based version of he es allows he es o be conduced wih weaker disribuional assumpions Applicaion of asse pricing ess in emerging markes possesses anoher difficuly Due o fricions in he money marke here are resricions in unlimied lending and borrowing and so an appropriae risk free-rae is difficul o secure for he emerging capial markes Forunaely, he Black-CAPM does no require specifying a risk-free rae his CAPM version is herefore a poenial financial model for hese markes Consequenly we focus on applicaion and 3
comparison of he performance of several mulivariae ess of Black s zero-bea CAPM in he emerging marke conex he finie sample performance of Wald and LR ess of zero-bea CAPM have been invesigaed in several sudies on US markes he resuls usually favour he LR es he Wald es is no invarian o alernaive non-linear formulaion of he null hypohesis herefore i is of ineres o sudy wheher a given non-linear form of he Wald es resuls in considerably differen resuls in finie sample size and power performance of he Black-CAPM especially in comparison o he LR es Previous sudies have no considered his aspec o be more specific, he null hypohesis ha he zero-bea CAPM holds is expressed as H : α = γ ( β ), i,,n () 0 i i = Here γ represens he zero-bea rae, α and i β i are respecively he inercep and slope of ih asse in he sysem of marke model equaions We consider wo alernaive formulaions of his hypohesis: αi αi + g i = = 0, i =,,N β β i i+ (2) and g 2i = i i+ i+ βi α ( β ) α ( ) = 0, i =,,N (3) he firs formulaion which we referred o as raio ype is employed by Chou (2000) using he US daa and Chou and Lin (2002) for OECD counries daa for he Black- CAPM ess Bealieu e al (2004) argue ha such a formulaion suffers from an idenificaion problem due o disconinuiy as bea approaches one As many his was firs demonsraed by Gregory and Veall (985) via a simulaion analysis hey show ha he Wald es resuling from wo formulaions of he same hypohesis eg H β β 0 and H β β 0 are numerically no idenical in finie samples 0 : 2 = 02 : 2 = 4
porfolios beas end owards one [See Blume (975)], he sampling disribuion of he es saisic wih his form may behave poorly in he righ ail herefore he associaed es developed from such a formulaion converges poorly o he asympoic Chi-Square disribuion A formulaion similar o ha in (2) which we refer o as a muliplicaive formulaion is considered in Amsler and Schmid (985) his paper addresses he issue of invariance propery of he Wald es by considering he non-linear formulaions of he Wald and GMM ess and compares hem wih he LR es We show ha asympoic Wald and GMM ess resul in serious size disorions while he LR es gives more accurae sizes when he reurns are allowed o follow cerain parameric disribuion For he case when he residuals are nonparamerically resampled from he observed daa he performance of he LR es is equally poor he boosrap ess recify he size disorions and render he Wald and GMM ess a par wih he LR es Comparing he alernaive formulaions of he GMM es i is found ha when here are smaller deviaions from he asse pricing model he muliplicaive form of he GMM es ouperform he LR and oher ess As he deviaions from he asse pricing ess increase he abiliy of he LR ess o deec he difference increase rapidly compared o he oher ess he ess are applied o he monhly porfolio reurns from he Karachi Sock Exchange 2 which is he larges of he hree sock markes in Pakisan Khawaja and Mian (2005) remarks his marke has he ypical feaures of an emerging marke In addiion invesigaing his marke migh be ineresing for invesors as for 2002 he marke was declared he bes performing marke in he World in erms of he percen 2 he Karachi Sock Exchange is he larges of he hree sock markes in Pakisan In mid April, 2006 he marke capializaion was a US$ 57 billion which is 46 percen of Pakisan s GDP for he Fiscal Year 2005-06 (Ref: Pakisan Economic Survey 2005-06) 5
increase in he local marke index See Iqbal and Brooks (2007) for implicaions of porfolio allocaion in his marke he plan of he paper is as follows Secion II discusses he formulaion of he Wald, GMM and LR ess of he Black s CAPM his secion also describes he boosrap ess Secion III describes he daa used in he sudy and provides some specificaion ess on he marke model residuals Secion IV briefly discusses he resul of he empirical ess In secion V he empirical size and power of he ess are evaluaed using Mone Carlo simulaion experimens Secion VI provides conclusion II MULIVARIAE ESS OF HE ZERO-BEA CAPM A he Wald es We assume ha he reurn generaing process is he familiar marke model: R = α + β r + ε,,, (4) m = Here [ r r2 rn ] R = is he N vecor of raw reurns on N porfolios, ε is he N vecor of disurbances,α and β are N vecor of he inercep and slope parameers respecively he zero-bea CAPM specifies he following cross secional relaion: E( R ) γi = β( E( r ) γ ) (5) N m Here γ is he parameer represening reurns on he zero-bea porfolio Applying he expecaion on (4) yields E( R ) = γ ( β ) + βe( r ), i,,n (6) m = Comparing (5) and (6) he join resricions on he parameer imposed by he zero-bea CAPM are expressed in he following hypohesis 6
H : α = γ ( β ), i,,n (7) 0 i i = his is essenially a non-linear consrain on he sysem of marke model equaions and he ieraive esimaion and an LR es for he hypohesis is provided in Gibbons (982) Chou (2000) developed a Wald es ha permis he model o be esimaed enirely in erms of alpha and beas by expressing he null hypohesis as: α i H 0 : = γ, i =,,N β i (8) his is equivalen o N join hypoheses α α2 αn H0 : = = = β β β 2 N (9) α Le i α i+ gi =, i =,N β β i i+ (0) Denoe g( θ ) = [ g g N ] he hypohesis o be esed is N N ],where θ = [ α β α β H 0 : g( θ ) = 0 Noe ha he under normaliy and iid assumpion on he error erm he OLS esimae θˆ is asympoically normally disribued θˆ~ N(0, Σ ( X' X ) ) Here X is he 2 design marix wih a column of s and a column conaining reurn of he marke porfolio If he normaliy assumpion is violaed hen under he iid assumpion he limiing disribuion of g(θˆ ) can sill be approximaed by a normal disribuion hus he Wald es for he zero-bea CAPM can be formulaed as W ˆ g ˆ g ˆ g( θ ) ˆ Σ ( X' X ) ˆ g( ) = θ θ θ θ d 2 N = χ θ = () θ' θ' 7
8 Here he parial derivaives g θ are evaluaed a he OLS esimaes from he unresriced sysem Keeping in view he concern of Bealieu e al (2004) regarding his form of he Wald es we consider an alernaive formulaion of he zero-bea CAPM hypohesis wih a muliplicaive form for he non-linear resricion,n, i ), ( ) ( g i i i i i = = + + β α β α (2) In his case he marix of he parial derivaives is as follows: = N N N N 2 2 3 3 2 2 ) ( ) ( 0 0 0 0 0 0 ) ( ) ( 0 0 0 0 0 0 ) ( ) ( g α β α β α β α β α β α β θ (3) he Wald es can be formulaed similar o previous case and is given by () wih i g and g θ replaced by (2) and (3) respecively he es saisic is disribued asympoically as a Chi-Square disribuion wih N- degrees of freedom B he GMM es Alhough he Wald ess are jusified under non-normaliy hey sill require he assumpion of iid disurbances I is widely repored especially for emerging markes ha he residuals may be serially correlaed For example, Harvey (995) repors such evidence for a group of emerging markes ha he reurns show greaer predicabiliy in hese markes han he developed markes Evidence of serial
correlaion is also repored in ables and 2 for our porfolio reurns calculaed from he Karachi Sock Exchange daa One approach o deal wih he non-spherical residuals is o employ an esimaed robus covariance marix in he Wald saisics and proceed wih he es W ˆ g g = g( θ ) Vˆ g( ˆ ˆ ˆ ) θ= θ θ= θ (4) θ θ θ Here V is he HAC covariance marix of he parameer esimaes his es is asympoically disribued as Chi-Square, for deails see Ray e al (998) We can show ha he same Wald saisics can be derived using he Hansen s (982) Generalized Mehod of Momens he GMM ess do no require srong disribuional assumpion regarding normaliy, heeroskedasiciy and serial independence of he residuals Wih N asses and ime series observaion on each asse he momen condiions vecor can be defined as: ε ( α, β ) ε ( α, β )r m ε ( α, β ) 2 2 2 ε ( α, β )r 2 2 2 m f ( θ ) = ε ( α, β ) N N N ε ( α, β )r N N N m ε and h ( θ ) = ε ( θ ) x (5) = ( θ ) x = Now we have 2N momen condiions and 2N parameers o be esimaed herefore he mulivariae sysem of 2N equaions is exacly idenified Here [ rm ] x = ( θ ) [ ε ε 2 ε N ] ε = and ε i= Ri α i βirm 9
he GMM 3 esimae of he parameer minimizes he quadraic form of he sample momen resricion vecor ˆ θ = arg min h ( θ ) W h ( θ ) (6) GMM Here W is a posiive definie weighing marix whose elemens can be funcions of parameers and daa Hansen (982) shows ha he opimal weighing marix is W S = { Asy Var [ h ( )]} = θ (7) he asympoic covariance marix of he GMM esimaor is V = [ D S D ] (8) Where D = P lim [ h ( θ )] In pracice S and D are unknown bu he θ' asympoic resuls are valid for some consisen esimaor S and D For he exacly idenified case Mackinlay and Richardson (99) show ha he porfolio efficiency can be esed by firs esimaing he unresriced sysem and hen compuing he es saisics of he efficiency hypohesis which involve hese unresriced esimaes Moreover in his case he GMM esimaor is independen of he weighing marix and is he same as he OLS esimaor; however he covariance marix mus be adjused o allow for heeroskedasiciy and serial correlaion he GMM esimaes are asympoically normally disribued ( θ ˆ θ ) ~ N(0,V ) Here V is as defined above Any non-linear funcion g( ˆ θ ) of he parameer is also asympoically normal [ g( θ ) g( ˆ θ )] ~ N [0,( g θ ) V ( g θ ) ] 3 he jus idenified sysem herefore leads o a simple mehod of momen esimaor raher han a generalized mehod of momen esimaor We coninue o use he erm GMM following use of he erm in lieraure in his case 0
herefore he GMM based Wald es for he can be formulaed as W ˆ g g = g( θ ) Vˆ g( ˆ ˆ ˆ ) (9) θ = θ θ = θ θ θ 2 θ Noe ha GMM procedure arrives a he same Wald es ha was developed wihou resoring o he GMM framework In his case V = [ D S D ] (20) We esimae hese marices as follows D = I N = x x = I N X X (2) S is esimaed by Newey-Wes (987) HAC covariance marix, for deails see Ray e al (998) S p v = ˆ η ˆ η + ( ) [ ˆ η ˆ η v + ˆ ηv ˆ η ] (22) + p = v= = Here η = e x so ha η η = ε ε x x ( ) η η v and v ( ) η η are auo covariance marices of lag v Here p is he lag lengh beyond which we are willing o assume ha he correlaions beween η and η v are 2 9 essenially zero We use he Newey-Wes fixed bandwidh p = in[ 4( ) ], 00 where in[ ] denoes he ineger par of he number Mackinlay and Richardson (99) and Chou (2000) employed he Whie (980) covariance marix as S which corresponds o p=0 in our case hus hese auhors assume ha he disurbances are heeroscedasic bu serially independen he reurn predicabiliy evidence from he emerging markes calls for a robus covariance marix such as he Newey-Wes (987) covariance marix
C he LR es Gibbons (982) employed he LR es LR d 2 = (log ˆ Σ * log ˆ Σ ) χ N (23) Where ˆΣ * and Σˆ are he resriced and unresriced covariance marices of he esimaes respecively he es is derived under he assumpion ha he reurns follow a mulivariae normal disribuion Following Gibbons, his es has been widey applied in he mulivariae asse pricing sudies including Jobson and Korkie (982), Chou (2000) and Amsler and Schmid (985) among ohers D he Boosrap ess of he Zero-Bea CAPM As discussed in he subsequen analysis, he asympoic ess especially he Wald and GMM ess of he Black-CAPM have serious size disorions which impede heir validiy in empirical applicaions In his case he residual boosrap provides an alernaive mean of obaining more reliable p-values of he ess I is well esablished ha if he es saisic is asympoically pivoal ie he null disribuion does no rely on unknown parameers, hen he error in size of he boosrap es is only of he order O( n j 2 ) compared o he error of he asympoic es which is of he order O[ n ( j+ ) 2 ] for some ineger j 4 See for example Davidson and MacKinnon (999) he boosrap p-values are obained as follows: he unresriced sysem of marke model is esimaed by he seemingly unrelaed regression he residuals { ε } are obained and he five es saisics are compued he sysem is also esimaed subjec o he zero-bea CAPM resricions and he 4 he ess considered in our invesigaion are all asympoically pivoal and are asympoically Chi- Square (N-) 2
resriced parameers are esimaed he es saisic for he zero-bea CAPM say W n is calculaed 2 he following seps are repeaed 5000 imes a A block boosrap sample { ε * } is drawn from { ε } he block lengh chosen is same as he lag lengh p in he HAC covariance marix hen he resampled reurns are obained as R * = ˆ( γ ˆ β ) + ˆ β rm + ε * b he es saisics say W n * is compued 3 he boosrap p-values are compued as he percenage of imes W n * is greaer han W n III DAA AND HE DIAGNOSIS OF HE MARKE MODEL RESIDUALS A he Daa he daa for his sudy comprise porfolios formed from a sample of socks lised on he Karachi Sock Exchange (KSE) and are obained from he DaaSream daabase he sample period spans nearly 3 ½ years from Ocober 992 o March 2006 he daa consis of monhly closing prices of 0 socks and he Karachi Sock Exchange 00 index (KSE-00) he crieria for socks selecion was based on he availabiliy of ime series daa on coninuously lised socks for which he prices have been adjused for dividend, sock spli, merger and oher corporae acions he KSE-00 is a marke capializaion weighed index I comprises op companies from each secor of KSE in erms of heir marke capializaion he res of he companies are picked on he basis of marke capializaion wihou considering heir secors We consider he KSE-00 as a proxy for he marke porfolio he 0 socks in he sample accoun for 3
approximaely eighy per cen of he marke in erms of capializaion Marke capializaion daa is no rouinely available for all firms in he daabase However he financial daily, he Business Recorder 5 repor informaion on firms over he recen pas 6 he marke capializaion of all seleced socks is collec a he beginning of July 999 which roughly corresponds o he middle of he sample period considered in he sudy We use monhly daa and compue he raw reurns assuming coninuous compounding o invesigae robusness of he empirical resuls we consider porfolios based on hree differen formaion schemes namely size, bea and indusry 7 Forming porfolios serves wo imporan purposes Firsly hey provide an effecive way of handling he curse of dimensionaliy of he mulivariae sysems Wih large number of parameers working wih individual socks may resul in highly imprecise esimaes Secondly forming porfolios wih respec o size, bea and indusry provides a means of conrolling he confounding effecs of hese characerisics and hus enables unambiguous inerpreaion of resuls We consruc seveneen equally weighed size and bea porfolios his number was considered keeping in view he desire o include a leas five socks in each porfolio and o avoid over aggregaion by forming oo few porfolios Firs he socks are ranked on marke capializaion in ascending order he firs porfolio consiss of he firs five socks while he res comprise of six socks each he bea porfolios are based on ranking of he socks on he bea esimaed via he marke model Porfolio reurn is calculaed as he equally weighed average reurn of he socks in he porfolio For indusry porfolios he socks are classified ino sixeen major indusrial 5 wwwbusinessrecordercompk 6 Due o he lack of sufficien daa on capializaion and relaively shor sample period he porfolio rebalancing is no performed 7 Some sudies, such as Groenewold and Fraser (200), repor ha he conclusion of an analysis may be differen and even conflicing when differen porfolios are employed 4
secors he secor sizes range from wo socks in he ranspor secor and hireen socks in he communicaion secor 8 hese secors serve as naural porfolios B Residual diagnosic ess All residual diagnosics and he asse pricing ess are performed for wo disinc subperiods: Ocober 992 o June 999, July 999 o March 2006 and he whole period- Ocober 992 o March 2006 he objecive here is o examine he sabiliy of he risk reurn relaionship in he wo sub-periods his is imporan as he volaile poliical and macroeconomic scenario in emerging markes migh make he reurn disribuion non-saionary and unsable Each sub-period consiss of 8 monhly observaions ha correspond o 6 ¾ year of monhly daa able repors he Mardia (970) es of mulivariae normaliy of he residuals of he unresriced marke model for he size, bea and indusry porfolios his es is based on mulivariae equivalens of skewness and kurosis measures he resuls are repored for he es based on skewness and kurosis measures separaely Boh skewness and kurosis based saisics are significan indicaing and overwhelming rejecion of mulivariae normaliy of he residuals he ess are significan for all cases wih size, indusry and bea porfolios able 2 repors he Hosking (980) mulivariae pormaneau es of no auocorrelaion for up o lag 3 in he marke model residuals his es is a mulivariae generalizaion of he univariae es 9 of Box and 8 he indusry secors employed are Auo and allied, Chemicals, Commercial Banks, Food producs, Indusrial Engineering, Insurance, Oil and Gas, Invesmen banks and oher financial companies, Paper and board, Pharmacy, Power and uiliy, Synheic and Rayon, exile, exile Spinning and Weaving, ranspor and communicaion and Oher Miscellaneous firms ha include obacco, meal and building maerial companies 9 he univariae JB ess for normaliy and he LB es of auocorrelaion are also performed which indicae ha normaliy and serial independence is rejeced for many individual porfolios regressions he resuls are no repored o save space 5
Pierce (970) he resuls provide evidence of predicabiliy in he residuals for bea and indusry porfolios and for he whole sample period for he size porfolios he residual correlaion is however no found for boh sample periods wih he size porfolios IV HE RESULS OF EMPIRICAL ANALYSIS able 3 presens he resuls of esing he zero-bea CAPM via he five ess he ess are repored for he hree ses of porfolios and for he wo sub-periods: Ocober 992-June 999, July 999-March 2006 and for he whole period As he asympoic ess resuls in considerable size disorions only he boosrap p-value are repored All five ess provide srong evidence in suppor of he Black s zero-bea CAPM Excep for one case of he GMM es wih muliplicaive formulaion wih indusry porfolios in he firs sample period all he boosrap es resuls in p-values above 09 0 he ess are also robus across differen ses of porfolios he Wald ess wih he wo non-linear formulaions resuls in numerically smaller values of he es saisics compared o he LR es which in urn is smaller in value o he GMM ess Comparison of numerical values of he ess wih he wo alernaive non-linear formulaions resuls in a mixed conclusion bu he alernaive formulaions do no appear o aler he decision o srongly suppor he financial model hus i appears ha when he daa provide srong suppor for he asse pricing model as in he presen case, forming he es hypohesis in alernaive ways is unlikely o change he conclusion drawn from empirical daa regarding he es oucome 0 Chou and Lin (2002) also repor he p-values of GMM and Wald ess in excess of 090 for he zerobea es on he OECD daa 6
V SIMULAION EXPERIMEN o invesigae how well he LR es and he Wald and GMM ess wih he wo formulaions of he zero-bea CAPM perform under various disribuional specificaion and o examine heir finie sample behaviour we invesigae heir size and power for he case of size sored porfolios Assuming ha he null hypohesis is rue we evaluae he rejecion probabiliies ha esimae he percen of ime of he null hypohesis is rejeced in he simulaion experimen and compare hem wih he nominal significance levels he larger differences beween he nominal and empirical rejecion raes would indicae ha he ess have larger size disorions hereby making he ess saisically unreliable For power comparison we have chosen he alernaive form similar o ha employed by Gibbons (982) ie H : α = γ ( β ) + N( c,) (24) We chose c = o 3 wih an incremen of 05 ha is a normally disribued componen is added o inercep vecor According o Gibbons (982) his ype of alernaive is compaible wih a variey of asse pricing models ha are compeiors of he CAPM such as he Meron (973) iner-emporal model wih one sae variable his alernaive will es he sensiiviy of he zero-bea CAPM ess if he average reurns are sysemaically over esimaed relaive o ha prediced by he zero-bea CAPM We firs consider he case when he reurns are generaed by boosrapping he error erms from he residuals of he marke model As able 2 indicaes ha he residuals may be serially correlaed we performed block boosrap wih block lengh ha is equal o he lag lengh of he HAC covariance marix his choice of block lengh is consisen wih Inoue and Shinani (2006) In his way he finie sample performance A sensiiviy analysis indicaes ha increasing he block lengh does no aler he conclusion significanly 7
of he ess are invesigaed when he ess encouner real daa Nex we evaluae he size and power of he ess under he assumpion ha he residuals and reurns follow an iid normal disribuion o examine he behaviour of he ess when he reurns have higher kurosis and heavier ails relaive o normal disribuion we also considered he case of errors following a -disribuion wih 5 degrees of freedom o invesigae he performance of he ess when he reurns daa have skewness we considered a mixure normal disribuion as an alernaive model of he residuals Finally we examine he non-iid disribued daa by specifying an auoregressive model of order one for he errors he sample sizes considered are = 60 and 62 he firs sample size of five years monhly daa is considered in mos US sudies of mulivariae asse pricing ess he second sample size corresponds o enire available sample period which correspond o 35 years monhly daa o generae residual vecor from a mulivariae normal disribuion wih zero mean and covariance marix Σ we se ε = L z, (25) Here L is he Cholesky facor of Σ (ie Σ = L' L ) and z is an N vecor of sandard normal random numbers he residuals from he -disribuion were generaed by seing ε = L z, (26) ( χ v ) 2 v 2 Where χ v denoes a Chi-Square random variae wih v degrees of freedom We se v = 5 o inroduce lepokurosis relaive o he normal disribuion We simulaed he residual from a mixure normal disribuion by seing ε = pn(0, Σ ) + ( p )N( η, τσ ) (27) 8
Following Chou (2000) we se he parameer as p=07 and τ =5 he negaive skewness 2 in he reurns was inroduced by seing η as he vecor of he sandard deviaion of he observed marke model residuals o ensure zero mean of he disurbances we subraced ( p ) η from he generaed residuals o inroduce a non-iid disribuion for he reurns we generaed he residuals from a AR () model by seing ε A + e (28) = ε he parameers in he diagonal marix A and he covariance marix of he residual vecor were esimaed from he observed daa on he marke model residuals As discussed subsequenly he Wald and GMM ess wih asympoic Chi-Square criical values resul in quie erraic es sizes he correc sizes were herefore obained using a compuaionally inensive boosrap procedure he size simulaion was carried ou as follows: () he null hypohesis is incorporaed in he marke model and reurns are generaed from he following equaion: R * = ˆ( γ ˆ β ) + ˆ β rm + ε * (29) he parameers γ, β and Σ are esimaed from he observed daa in he respecive sample he daa on he residuals are drawn from one of he alernaive disribuion he observed marke porfolio reurns is employed in he simulaions he parameers of resriced and unresriced sysem of marke model are esimaed and he es saisic say W n is compued (2) We obain B = 200 boosrap runs by resampling he reurns from equaion (29) again bu his ime using he parameer esimaes from sep () For each boosrap 2 Nine ou of 7 observed residuals from he marke model regressions have negaive skewness herefore we have chosen o inroduce negaive skewness in he reurns 9
he sysems is esimaed and he es saisics say W n * is compued Boosrap criical value correspond o a nominal level of a as (-a)h quanile of he boosrap disribuion formed from he 200 W n * values are obained (3) he above wo seps are repeaed 5000 imes he size of asympoic ess is compued as he empirical rejecion probabiliy ha W n exceeds he Chi-Square (N-) criical values in hese 5000 simulaions We compue he size of boosrap ess as empirical rejecion probabiliy ha W n exceeds quanile of he boosrap disribuion For power simulaions he following wo sep procedure is employed () he reurns from he following equaion is generaed R * = ˆ( γ ˆ β ) + N( c,) + ˆ β rm + ε * (30) We esimae γ, β and Σ from he observed daa in he respecive sample he daa on he residuals are drawn from one of he alernaive disribuion he parameers of resriced and unresriced sysem of marke model are esimaed and he es saisic say W n is compued (2) he sep () is repeaed 5000 imes for each c from o 3 in he incremen of 05 he power of he ess is compued as he empirical rejecion probabiliy ha he compued es exceed he boosrap criical values obained in he size simulaions As here are some size disorions even a boosrap criical values we evaluae only he size correced power Our simulaions design for size and power evaluaion resembles Hall and Horowiz (996) alhough we employ a larger number of simulaions (5000 insead of 000) and number of boosraps (200 insead of 00) able 4 presens he rejecion probabiliies of he five ess of he zero-bea CAPM a he criical values obained from asympoic Chi-Square disribuion Excep for he case of boosrap residuals he sizes of he LR es are closer o he nominal values compared o he Wald and GMM ess Similarly excep for he firs case of boosrap 20
disribuion boh non-linear formulaions of he Wald ess resuls in severe underrejecion In general no form of he Wald es can be preferred over he oher One he oher hand boh formulaions of he GMM ess over-rejec Increase in sample size from 60 o 62 make he size disorion nearly half bu o eliminae he size disorions compleely would require much larger sample sizes which are difficul o secure especially for emerging markes Overall i can be concluded ha he LR es resuls in smaller size disorions for iid daa his resul is also found o be quie robus o disribuional deviaion from normaliy eiher in he form of higher skewness or excess kurosis I appears ha when he asympoic ess face real daa (he firs case in panel ) no one of hem performs saisfacorily As he asympoic es give quie severe size disorions i is worhwhile o consider he ess wih boosrap criical values he rejecion probabiliies of he boosrap ess are presened in able 5 he firs and very obvious observaion from he boosrap ess is heir closer approximaion o he nominal sizes Excep for a few cases he error in approximaion of he es sizes o he nominal sizes is wihin % for all he five ess For example he average percen approximaion error for he Wald es wih raio and muliplicaive formulaion is 033, 05, 03 percen and 025, 04, 042 percen respecively wih nominal size of %, 5% and 0 % Generally drawing any conclusion regarding he relaive meris of he size of he wo formulaions is difficul he wo formulaions of boh he Wald and he GMM ess compare quie favourably wih he LR es when he size is evaluaed a boosrap criical values I is concluded ha he asympoic LR es dominae he oher ess especially wih he iid daa bu his generalizaion o boosrap ess is no exended In fac boosrapping he Wald and GMM ess have rendered hese ess a par wih he LR es despie heir poor asympoic performance 2
As he asympoic ess resul in quie erraic es sizes we invesigae he power only a he boosrap criical values Figure presen he empirical rejecion probabiliies of he five ess of he zero-bea CAPM a boosrap criical values obained from he size simulaions wih sample size of 60 he power is compued using he firs five years marke reurn daa and using he parameer esimaes over his period he LR es clearly dominaes he oher in he cases of Normal, Mixure Normal and Auoregressive errors For oher wo cases GMM es wih muliplicaive formulaion perform beer when he es is subjec o a smaller deviaions from he null hypohesis A relaively larger deviaions LR es dominaes he oher ones his resul appears o be robus o various ype of non-normaliies inroduced in he simulaion experimen as far as he reurn daa remain idenically disribued Also as he disance from he null hypohesis increases he gain in power is more rapid for he LR es he power of he Wald and GMM ess remain low a he convenional sample size consising of 5 years monhly daa he power of he Wald and GMM ess do no exceed 040 in any of he cases Figure 2 presens he power when sample size increases o 62 which corresponds o all available sample daa he LR es generally dominaes he oher ess especially when he reurns are subjec o larger deviaions from he asse pricing model A relaively smaller deviaions he GMM es wih muliplicaive formulaion performs beer han he LR ess in cases ha represen deviaions from non-normaliy such as he excess kurosis capured by Suden errors, higher deviaion from skewness represened by a Mixure Normal disribuion and he case when errors are generaed from he real daa by boosrapping For he normal case he muliplicaive form of Wald and GMM es appear o perform beer han he raio formulaions Figure and 2 reveal ha power of he Wald and GMM ess appear o increase less rapidly compared o he LR ess 22
as he deviaions from he null hypohesis increase In pracice i is difficul o deermine he exen of deparure from he null hypohesis bu i is clear ha when he disance beween he null and alernaive models is smaller i will be exremely difficul for he boh asympoic and he boosrap ess o deec he difference Consequenly he accepance of he asse pricing model pose a quesion of wheher he daa acually suppor he model or he resuls merely reflec he low power of he ess under consideraions his commen applies o he resuls of empirical ess repored in able 3 for he daa from he Karachi Sock marke I is neverheless expeced ha boosrap LR and GMM ess wih larger sample sizes will deec he economically and saisically significan differences beween he null and alernaive models hus even he boosrap based version of hese ess requires a careful consideraion in pracical applicaions of mulivariae ass pricing wih finie samples especially in emerging markes VI CONCLUSION he paper examines he finie sample performance of five mulivariae ess of he zero-bea CAPM he empirical performance of he ess is examined on an emerging marke daa I is well esablished ha reurn characerisics of he emerging markes differ from ha of he developed markes Moreover we accoun for he fac ha money markes in he emerging markes are no perfec so ha a reliable risk-free rae is difficul o obain he ess considered are Gibbons (982) LR es and wo nonlinear formulaions of he Wald and an associaed GMM es he formulaion of he Wald saisic of he Black-CAPM resricion employed in earlier research [for example, Chou (2000) and Lee e al (997)] migh be associaed wih idenificaion problems when he parameers of he model (bea of he porfolios) approach one he 23
paper herefore invesigaed an alernaive formulaion of he Wald es and he associaed GMM es of he asse pricing model A Mone Carlo simulaion experimen demonsraes ha he size disorions of asysmpic ess are considerably higher especially for he Wald and GMM ess However boosrap ess recify he size disorions and render he Wald and GMM ess a par wih he LR es Comparing he alernaive formulaions of he GMM es i is found ha when here are smaller deviaions from he asse pricing model he muliplicaive form of he GMM es ouperform he LR and oher ess As he deviaions from he asse pricing ess increase he abiliy of he LR ess o deec he difference increase rapidly compared o he oher ess While he asympoic LR es for iid daa resuls in correc sizes he Wald and GMM es require compuaional inensive resampling procedures o recover he correc rejecion probabiliies In larger samples he GMM es wih he muliplicaive formulaion generally resuls in higher power compared o he es wih a raio ype formulaion he ess resuls are based on monhly porfolio daa from an emerging marke-he Karachi Sock Exchange he ess srongly suppor he zero-bea CAPM However he finie sample properies of he mulivariae ess found in he sudy indicaes ha non-rejecion migh be caused by he low power of he ess Acknowledgmen: he auhors wish o hank, Gael Marin, Bruce Hansen, Mervyn Silvapule, Don Poski, Muhammad Akram, he paricipans in Economeric Sociey Ausralasian Meeings 2006 and he seminar paricipans a Monash Universiy for heir helpful commens on earlier versions 24
able : es of mulivariae normaliy of Marke Model residuals Sample Period Size Porfolios Bea Porfolios Indusry Porfolios Skewness Kurosis Skewness Kurosis Skewness Kurosis Oc 92 Jun 99 9504 (0000) Jul 97 Mar 06 8989 (0000) Oc 92 Mar 06 5530 (0000) 3505 (0000) 3447 (0000) 37247 (0000) 879 (0002) 8406 (0000) 507 (0000) 34546 (0000) 34628 (0000) 37744 (0000) 7862 (0000) 7573 (0000) 4767 (0000) 30780 (0000) 3089 (0000) 33732 (0000) his able repors he ess of mulivariae normaliy of he residuals of he unresriced marke model P-values are given in he parenhesis he Mardia (970) es of mulivariae normaliy is based on he mulivariae skewness and kurosis 3 2 measures D = r 2 s and D2 = r where rs = ( X X ) S ( X s X ), = s= X and S are he sample mean and sample covariance marix of he residuals 2 respecively D 6 ~ χ ( f ) where f = N( N + )( N + 2) 6, and N( N + 2) D2 ~ N(0,) 8N( N + 2) able 2: es of serial independence of unresriced Marke Model residuals Sample Period Size Porfolios Bea Porfolios Indusry Porfolios Lag Lag2 Lag3 Lag Lag2 Lag3 Lag Lag2 Lag3 Oc 92 Jun 99 3492 (04) 62620 (0080) 89947 (025) 32250 (0085) 62834 (0072) 92050 (00) 32284 (0002) 59457 (0006) 87834 (0003) Jul 97 Mar 06 3227 (065) 60384 (022) 9288 (0073) 33995 (0020) 63704 (0044) 93506 (0053) 28630 (0093) 56000 (0069) 8588 (02) Oc 92 Mar 06 34626 (00) 6332 (0055) 92288 (009) 36382 (000) 6562 (003) 9432 (0036) 33335 (0000) 58095 (008) 8550 (005) his able repors he ess of serial independence of he residuals of he unresriced marke model P-values are given in he parenhesis he Hosking (980) mulivariae pormaneau es is a mulivariae generalizaion of he univariae pormaneau es of Box and Pierce (970) he es saisic a lag lengh s is s 2 2 Q( s) = r( C 0 jc00 CojC00 ) ~ χ 2 ( N ) j j= = rs U ru s, i wherec U is he N residual marix lagged i periods he es is performed for s = 2, 3 he iniial missing values are filled wih zero = 25
able 3: he es of Black-CAPM his able presens he values of he five es saisics resuling from he es of he Black CAPM Boosrap p-values based on 5000 simulaions are given in parenhesis Noe: he Wald and GMM ess are adjused by muliplying (-N-) and LR es saisics is adjused by (-5-N2) o improve heir small sample performances See Gibbons, Ross and Shanken (989) and Jobson and Korkie (982) for deail Porfolio Mehod Wald Wald 2 GMM GMM 2 LR Panel : Oc 92 Jun 99 Size 638 (0978) Bea 4885 (0989) Indusry 075 (0944) 6504 (098) 5029 (0989) 03 (0973) 2394 (0983) 5897 (0997) 5687 (0992) 5689 (094) 585 (0998) 902 ( 0896) 7445 (0996) 60238 (0994) 2295 (0983) Panel 2: Jul 97 Mar 06 Size 9500 (0944) Bea 0642 ( 0985) Indusry 2583 (0978) 889 (0974) 0648 (0976) 8757 (0992) 334 (0983) 7085 (098) 24350 (0996) 033 (0994) 9638 (0943) 663 (0995) 0482 (0995) 455 (0987) 4403 (0994) Panel 3: Oc 92 Mar 06 Size 8593 ( 0982) Bea 9775 (0959) Indusry 39 (0960) 9748 (0976) 99759 ( 0956) 8762 (0985) 527 (0962) 547 (0959) 7768 (0999) 3205 ( 0982) 209 ( 0984) 0833 ( 0998) 709 (0977) 893 (0966) 3927 ( 0974) 26
able 4: Size of he Black-CAPM ess wih Asympoic Chi Square Criical Values his able provides he empirical rejecion probabiliies of five ess of Black-CAPM model each evaluaed wih five alernaive disribuion specificaions of he residuals he resul for a nominal size of a % correspond o number of imes he es saisic exceeds he (-a) % quanile of Chi Square Disribuion wih N- degrees of freedom divided by he number of simulaion ie5000 he marke porfolio from observed daa is employed in hese experimens Wald Wald 2 GMM GMM 2 LR % 5% 0% % 5% 0% % 5% 0% % 5% 0% % 5% 0% Panel : Boosrap Disribuion =60 0023 0057 0099 000 0042 0074 0500 0648 0730 0475 0649 0728 0259 046 0522 =62 0045 038 0230 003 035 0220 0448 0648 075 0408 064 0726 059 0364 0507 Panel 2: Normal Disribuion =60 000 000 0002 0000 000 0004 043 0249 0326 047 0260 0347 002 0052 009 =62 0002 002 0032 0002 005 0036 0056 047 0220 0056 046 023 0008 0046 0096 Panel 3 Suden Disribuion wih 5 degrees of freedom =60 000 000 000 0000 0000 000 020 026 0293 06 028 0293 000 0055 0099 =62 0002 0005 006 0000 0005 003 0043 05 08 0038 06 086 0009 0049 0097 Panel 4 Mixure Normal Disribuion =60 0002 0002 0002 0000 0000 0000 0093 083 0248 006 02 0286 000 0056 006 =62 000 0004 002 000 0007 004 003 0096 06 0035 00 060 002 0048 0098 Panel 5 : AR() =60 0000 000 0003 0000 0000 000 04 0254 0333 044 0265 0349 00 0048 0096 =62 0003 002 0027 000 00 0030 0054 030 0206 0052 04 023 000 0042 0083 27
able 5: Size of he Black-CAPM ess wih Boosrap Criical Values his able provides he empirical rejecion probabiliies of five ess of Black-CAPM model each evaluaed wih five alernaive disribuion specificaions of he residuals he resul for a nominal size of a % correspond o number of imes he es saisic exceeds he (-a) % quanile of he boosrap disribuion consruced from boosrapping he es saisics 200 imes he rejecion probabiliies are based on 5000 simulaions he marke porfolio from observed daa is employed in hese experimens Wald Wald 2 GMM GMM 2 LR % 5% 0% % 5% 0% % 5% 0% % 5% 0% % 5% 0% Panel : Boosrap Disribuion =60 0020 0063 09 00 005 0096 004 0052 0099 004 0053 006 006 0059 00 =62 003 0048 0096 003 0049 003 004 0059 00 007 0059 009 002 005 002 Panel 2: Normal Disribuion =60 000 0046 0097 002 0046 0090 007 0055 0097 0029 0076 024 003 0054 004 =62 005 0054 0099 008 0057 006 002 0055 006 002 0050 0096 00 0049 0098 Panel 3: Suden Disribuion wih 0 degrees of freedom =60 0008 0038 0080 009 0037 0078 006 0053 0099 002 0050 0094 006 0058 003 =62 004 0050 0098 003 005 000 005 0058 005 004 0046 0088 003 0053 00 Panel 4: Mixure Normal Disribuion =60 000 0036 0078 0008 0042 0085 00 0045 0087 000 0042 0087 003 0057 006 =62 006 0052 008 006 0053 006 003 0050 004 006 0056 003 005 0054 009 Panel 5: AR() =60 002 0049 00 00 005 000 003 0053 0099 003 0045 0085 006 0055 005 =62 004 0049 0093 005 0049 0096 003 0054 002 003 005 0099 008 0055 002 28
Boosrap Errors Normal Errors 0 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 00 5 2 25 3 c c Suden (5) Errors Mixure Normal Errors 0 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 00 5 2 25 3 c c AR () Errors 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 c Fig : he size correced power of he boosrap ess of he zero-bea CAPM under errors generaed from alernaive disribuion a =60 (a 5% nominal level) Noe: he size correcions are done by employing criical values from size simulaions he rejecion probabiliies are compued from 5000 simulaions Here c measures he deviaion from he null hypohesis he sample size is =60 his corresponds o 5 years monhly daa 29
Boosrap Errors Normal Errors 0 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 00 5 2 25 3 c c Suden (5) Errors Mixure Normal Errors 0 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 00 5 2 25 3 c c AR () Errors 0 Empirical Rejecion Probabiliy 09 08 07 06 05 04 03 02 0 Wald Wald 2 GMM GMM 2 LR 00 5 2 25 3 c Fig 2: he size correced power of he boosrap ess of he zero-bea -CAPM under errors generaed from alernaive disribuion a =62 (a 5% nominal level) Noe: he size correcions are done by employing criical values from size simulaions he rejecion probabiliies are compued from 5000 simulaions Here c measures he deviaion from he null hypohesis he sample size is =62 his corresponds o 35 years monhly daa 30
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