Testing Asset Pricing Models With Coskewness
Giovanni BARONE AOES I Institute of Finance, University of Lugano, Lugano, Switzerland (giovanni,barone-adesi@/u.unisi.ch)
Patrick GAGLIAROI NI Institute of Finance, University of Lugano, Lugano, Switzerland (patrick.gagliardini@/u.unisi.ch)
Gi ov an ni URGA Faculty of Finance, Cass Business School, London, U,K. (g.urga@city ac.uk)
In this article we investigate portfoli o cos kcwnc ..s using a qua dnuic market model as a rerum -generati ng proce ss. We show that the portfolio, of small /lar ge ) firm ... have negative (positive ) coskcwncss with the market. We tcs\ an asset pricing model includ ing coskcwncss by checking the validity of the restriction s that it impo ...ex on the return-generating process. \1./1.' find evidence of an addi tional component in expected C:(CC'iS return .... which i ... nor explained by either covariance or cos kcwncss with the market. However, thi... unexplain ed compo nent i... homogeneous :lcrOSS portfolios in our sample and modest in magnitude, Finall y, we investigate the implication s of erroneously neglecting coskcwncss for testin g a...set pricing model... , with particular attention to the empirica lly detected exp lana tory power of linn ... ize.
KEY WO RDS : Asset pricing ; Asymp totic least squares : Co skewuc ss: Generali zed method of moments; ~ lonlt' Car lo simulation.
1. INTRODUCTION Alth ough the persistence of these anomalies ove r time is still subject to debate. the evidence suggests that the mean-variance Asset pn cmg models generally express expected returns CAPM is not a satisfactory description of market equilibrium. on financial asse ts as linear functions of co variances of re These pricing anomalies may be related to the possibility thai turns with some systema tic risk facto rs, Several formul ation s useless factors appea r to be priced. Of course. it is also possi of this genera l paradigm have been proposed in the litera hie that pricin g anomalies arc due to omitted factors. Alth ough ture (Sharpe 1964: Lint ner 1965: Black 1972: Merton 1973: statistical tests do not allow us to choose between these two Rubin stein 1973: Kraus and Litzenberger 1976: Ross 1976: possible explanations of pricin g anoma lies. Kan and Zhang Breeden 1979: Barone Adesi and Talwar 1983: Barone Adesi ( 1999a.h) sugges ted that perhaps a large increment in R2 and 1985: Jagannathan and Wang 1996 : Harvey and Siddique 2000: the persistence or sign and size of coe fficie nts over time are Dittmar 2002). However. most of the empirical tests sugges ted most likely to be associated with trul y priced factors. to date have produ ced negative or ambiguous results. These In the light of the forego ing. the aim of this article is to findings have spurred renewed interest in the statistical prop· consider market cos kew ncss and to investigate its role in test erties of the currently available testi ng methodologies. Among ing asse t pricing models. A dataset of monthly return s on rece nt studies. Shunken ( 1992) and Kan and Zhang 11999a.h) 10 stock port folios is used . Following Harvey and Siddique analyzed the common ly used statistical methodologies and (2000) . an asse t is defined as having "positive cos kewness' with highlighted the sources ofambig uity in their I1nd ings. the market when the residuals of the regression of its returns Although a full speci ficat ion of the return-generating process on a co nstant and the market returns are positively correlated is not needed for the formulation of most asse t pricing mod with squared market returns, Therefore. an asset with positive els. it appears that only its a priori knowledge may lead to the (negative) cos kewness reduces (inc reases) the risk of the port des ign of reliable tests, Because this condition is neve r met in folio to large absolut e market returns. and should comma nd practice. researchers arc forced to make unpalatable choices be a lower (hig her>expected return in equilibrium. tween two altern ative approac hes. On the one hand. powerful Rubin stein ( 1973). Kraus and Litzenberger (1976).Barone tests ca n be designed in the co ntext ofa (fully) specifie d return Adcs i ( 1985). and Harvey and Siddique (2000) studied non generating process. hut they arc misleadin g in the presence of norm al asse t pricing models related to cos kew ness . Kraus possible model misspecifications. On the other hand . more tol and Litzenb erger ( 1976) and Harvey and Siddique (2000) formul ated expected returns as a function of covariance and era nt tests may be co nsidere d. hut they may not be powerful. coskewness with the market port folio, In particular. Harvey as noted by Kan and Zhou (1999) and Jagannathan and Wang and Siddique (2000 ) assessed the importance of coskewness (200 1I. Note that the first choice may lead not only to the rejec in explaining expected returns by the increment of R2 in cross tion of the co rrect model, but also to the acceptance of irrclc sectional regress ions. More recently. Dittm ar (2002) presented vant factors as sources of systematic risk. as noted by Kan and a framework in which age nts are also adve rse to kurtosis. irn Zhang ( 1999a .h). plying that asse t retu rns are influenced by both coskcwness and Tn complicate the pict ure. a number or empirical regularities cok urtos is with the return on aggrega te wea lth, The author tests have been detected that arc not consistent with standa rd asse t pricing mod els. such as the mean-variance capi tal asse t pricing © 2004 Am erican Stat istical Association model (CA PM I. Amo ng other studies, Bunz ( 198 1) related ex Journal of Business & Econ om ic Stati sti cs peered returns to firm size. and Fama and French (1995) linked Octo be r 2004 , Vol. 22, No. 4 expected returns also to the rurio of book value to mar ket value, 00110.11 98/07350010 400 000 0244
474 Barone Adesi, Gagliardini, and Urga: Coskewness in Asset Pricing 475 an ex tended asset pricing model within a ge neralized method 2.1 The Quadratic Market Model of mom ent s (G!\l M ) framework (see Han sen 1982). Most of the foregoin g formulations are very ge neral. becau se Fac tor models arc among the mosl widely used return the specification of an underl ying return-ge nerating process is ge nerating pro cesses in financial econome trics, Th ey explain not requ ired . However, we arc conce rned about their possible comoverne nts in asset return s as arising fro m the co mmo n ef lack of pow er. made worse in this context by the fac t that co fect of a (s mall) number of underl ying vari ables, called factors varia nce and coskcwness with the market arc almo st perfect ly (sec . e.g.. Ca mpbe ll. 1.0. and MacK inlay 1987: Gourieroux and collinear across port foli os. Of course. in the ex treme case. Jasiak 2001). In thi s article we use a linear two-factor mod el. in whic h market cov ariance is proportion al to market cos kcw the qu ad ratic market mod el. as a return-generating process. ness. it will be imp ossibl e to identify covariance and coskew Market returns and the square of the mark et returns are the two ness premia separately. Th erefore. to identify and acc urately factor s. Specifica lly. we denote by R, the N x I vector of returns measu re the co ntribution of cuskcwncss. in thi s article we pro in pe riod t of N portfolios and by R,H.I the return of the market. pose an approa ch (sec also Baron e Ade si 19X5) based o n the If R F.t is the return in pe riod t of a (conditionally) rivk -free prior specification of an appro pria te return -generat ing process. ass et. then port fol io and market excess ret urns are de fined by the qu ad rat ic market mod el. Th e quadr atic market mo de l i ~ an r . = Rt - R,...,l and r,H.' = RM., - R,.." . whe re l is a N x I vecto r extension of the tradition al mark et mo del (Sha rpe 1964; Lintne r of ls. Similar ly. the excess squared market return is defined by 1965). including thc ' quare of the market returns as an add i liM,' = R ,~J , ' - RF.I ' Th e quadratic market mod el is specified by tional factor. The coefficien ts of the qu adr atic factor measure r/ =a +{3rM., +YQM.I + E,. t= 1• •••• 1'. the margin al contribution of co skew ness to expec ted excess re ( I ) turns . Becau se mark et returns and the square of the mark et H I' : y E]RN ret urns are almost orthogo nal regressor s. we obtain a precise where a is a N x I vector o f intercepts. fJ and y are N x I vec test of the significance of quadratic coefficients. In add ition. tors ofsensitivities. and e, is an N x I vector of errors satisfying thi s fra mework allows us to test an as set pricing mod el with coskewness by checking the restricti on s that it imposes on the E[E/IRM.,. Rr ./ ] = II. coefficie nts of the quad ratic market mod el. Th e spec ification of a ret urn -generating process provides more powerful tests. as with R,'l,' " and R"'.t denoting all presen t and past values of co nfir med in a series of Mon te Ca rlo simulations (see Sec. 5). RM., and R,.../. In addi tion to evaluating asset pri cing mod els that include Th e quad ratic mark et model is a direct extension o f the well coskewncss. it is also important to investigate the consequences known rnarket model tSha rpc 1964 : Lintner 1965 ). which cor on asset pricin g tests when coskew ness is erroneo usly omit responds to restriction y = 0 in ( I). ted . \Ve consider the possib ility that port folio characteristics, r, = a + fJrM" + e., I = I. . .. . 1'. suc h as size. arc e mp irically found to explain ex pec ted ex ( 2) cess ret urn s because or the o missio n of a trul y priced factor. Hi .: y = II in (1). nam ely coskcwness . To explain this prob lem. le t us assume that The motivation for including the square of the mar ket retu rns coskcwness is trul y priced but is omitted in an asset prici ng is to fully acco unt for coskewness with the mark et port folio . mod el. Th en. if market cos kcwncs s is correlated with a variable In fact. dev iations from the linear relation betwee n asset return s suc h as size. this var iable will have spurious explanatory power and market returns implied by (2) arc e mp irica lly observed. for the cross-section of ex pected returns. because it pro xies for More specifica lly, for some classes of assets. residu als fro m the omitted coskcw ness. In our empirica l applica tion (sec Sec . 4 ). regression of returns on a constant a nd market returns ten d to be we ac tually tind tha t co skcwness a nd firm size are correlated . positively (neg atively ) corre lated with squared mark et returns. This finding sugg es ts that the e mp irically observed relation be These assets therefore show a tenden cy to have relativel y high er tween size and assets excess returns may he explained by the (lowe r) retu rns whe n the mark et expe riences high absolute re omissio n of a systematic ris k factor. namel y mark et co skew nes s turn s. a nd arc sa id to have positive (negative ) coskewncss with (see also Harvey and Sidd iqu c 2000. p. 1281). the mark et. Thi s findi ng is supported by OUT empirical inves Th e article is orga nized as follows. Sectio n 2 introdu ces tigation s in Section 4 . whe re. in accorda nce with the results the qu adr ati c mark et mod el. An asset pricin g model includ ing of Harvey and S iddiquc (2000 ). we lind that portfolios formed cos kcwness is der ived usin g arbitrage pricing. and the testing by asse ts of sma ll fir ms te nd to have negat ive cos kewncss with of various related statistical hypotheses is discussed . Sect ion .3 the market. wherea s port folios formed by assets o f large firm s reports estima tor s and test statistics used in the e mp irical part have positive market coskcwness. In addition to class ical beta. of the article. Secti on 4 descri bes the data and rep ort s em pirica l market cos kewncs s is therefore a nother importan t risk churuc results. Secti on 5 provides Monte Car lo simulations for inves tcristic : all asset that has positive cos kcwn css with the mar ket tigating the tinite sample pro perties of our test statistics . and diminishes the sensitivity of a portfolio to large ab solute ma r Sec tion 6 concludes . kct returns . Th erefore. everything else hein g eq ua l. investor s should prefer assets with positi ve market cos kewness to those 2. ASSET PRICING MODELS WITH COSKEWNESS wi th negat ive co skewness. Th e qu adrat ic market model ( 1) is In this section we introdu ce the econo metric specifications a spec ification that pro vides us with a very simple way to take co ns idered in the article. \Ve describe the return-gen erating into account mark et coskcwncss. Indeed. we have process. deri ve the corres ponding restricted equ ilibri um mod 1 , y = ~[E I covlc.. R,il.t l. (3 ) els. and finall y compare OUT approac h with a G MM fra mework. ".1 476 Journal of Business & Economic Statistics . Octobe r 2004
where I I (rcsp, ( " .1) arc the rcsidualv fro m a theoret ical re us wit h informati on about the sensitivi ties of our port foli os to gression of portfolio returns R, (market square returns Hi l ,l' thi s factor. Furthe rmore. variab les represent ing portfolio cha r resp.) on a constant and market return HM". Because coef ac teristics. which are correlated with ii across portfolios. will ficien ts y arc pro portional to co vle.. R ~ I.tI . we ca n lise the have sp ur ious ex plana tory power for expected excess returns. es tima te of y in model ( I ) to investigate the coskcw ness because they are a proxy for the se nsitivit ies to the omi tted pro pert ies of the N portfolios in the sa mple. Moreover. al facto r. A case of particular interest is whe n a is hom ogen eou s though y docs not correspo nd exactly 10 the us ual probabilistic across assets. ii = Ao t. where Ao is a scalar. that is. definition of market co skcwncss. coe fficien t y is a very good E( r,) = l AO + 13 )., + y A ~ . (1) proxy for cnvt r.. R~ f.t ) / V (R .~ f . t ) ' as po inted out by Kra us and correspondi ng (0 the specification Litzen be rger ( 1976). With in our samp le. the approxima tion er ror is sma ller than Ifh (sec App. A) , Finall y. the statistical r, = 13 1".11 ., + Y'I.II ., + y ll + AO I +' " I = I. .... T . (joint) sign ifica nce ofco skcwncss coe fficie nt y call be assessed (8) H~ ; 3 1J. AO; 0' = IIy + AOI in II). hy testin g the null hypothesis H;. against the a lternative Hr. Specifica tion (8) co rres po nds to the case where the fac to r omit 2.2 Restricted Equilibrium Models ted in model (4) has hom ogeneou s se nsitivit ies across portfo lios. Fro m (7). Ao may be interpreted as the expected excess From the standpoi nt of financial econo mics. a linear Iac returns of a port foli o wit h covaria nce and cos kcwncss with the tor mo de l is only a return-generating process. which is not mark et both equal to O. Such a portfolio may correspond to necessarily consistent with notion s of economic eq uilib rium, the ana logous of the zero-be ta portfoli o in the Black version Const raints on its coefficients arc imposed for exa mple by arbi of the ca pita l asset prici ng model t flluck 1972 ). Alterna tively. trage pr ic ing IRoss 1976; Chamberlai n and Rothschil d 1983). AO > 0 (Ao < 0) may he du e to the lise of a risk-free rate lower The arbitrage pricin g theory implies that e-x pec ted excess re (higher) than the act ua l rate faced by inves tors. With reference turns ofassets fo llow ing the fac tor model ( I) satisfy the restric to the observed empirical regul arities and mod el miss pec itica tion (Barone Ade si 1985 ) tions mentioned in Section I. the imp ort ance of model (8) is that if hypothesis H :! is not rejected agains t Hr, then we ex pec t El r,) = 13 ).1+ y A ~ . (4) po rtfolio c haracte ristics suc h as size to not have add itio na l ex where i'l and A:! arc expected excess ret urn s o n portfo lios planatory power for expec ted excess ret urns. once coskcw ncss whose excess returns arc pe rfectly co rrelated wit h factors is taken into account. In additio n. a more powerful evaluation of the validity of the asset pricing mode l (4 ) shou ld he provided 1)1.t and «.\1.t . Eq uat ion (4) is in the for m of a typical linear by a test ofHI agai nst the alternative 'H:! . asset prici ng mode l. which re lates expected excess returns to covarianccs and coskew nes xes w ith the market. In this article 2.3 The Generalized Method of Moments Framework we tes t the asscr pricing model with coskewncss (4) throu gh the restr iction s that it im pose- on the coefficients of the return Asset pricin g mod els of the type (4) have been conside red generating process ( I). We derive these restr ictions. Becau se by Krau s and Litze nberge r ( 1976) and Har vey and Siddique (2000). Har vey and Sidd ique (2000) int rodu ced the ir specifi the e xcess market return ru, sutistiex (4). it must he that ca tion as a model in whic h the stoc has tic disco unt fac tor is A, = £1,-.11 .,) . (5) quadratic in mark et returns . Specifica lly. in our nota tion. the asset pricing mo de l with coskewness (4) is equivalent to the or A similar restriction does not ho ld for the second fac tor lIM .t thogonal ity co ndition because it is not a traded asse t. However, we expect i.:! < O. because assets wi th positive coskcwncxs decrease the risk of a /;'[ r,III,l o)J = u, (9) portfo lio wit h res pect to large absolute mark et return s and thu s where the stoc has tic di scount fac tor mdtS ) is given hy m,( e5) = sho uld comma nd a lower risk premi um in an arbitrage equ ilih I- I'M .181 - tJJ/ .,tJ2 a nd e5 = (8]. (5 :!) is a two-dimension al para rium, By taking expectations on ho th sides of ( I ) and substi meter. A quadratic stochas tic di scount fact or 111 ,(e5) ca n he j usti tuting (4 ) and (5) . we deduce that the asset pricing mod el (4) fied as a (formal) second-order Taylor ex pansion ofa stochas tic implies the cross-e quation restriction a = (Jy . whe re I? is the discount factor, whic h is no nlinear in the market returns . Thus scalar parameter () = ). :! - £'((/'\1./). Th us ar bitrage pricin g is in the GMM a pproach. the derivation and tes ting of the orthog co nsiste nt wit h the restricted model onali ty condition (9) do l10t require a prior speci fica tio n o f a data-gen eratin g pro cess. I = I. .... T. (6) More recently. in a con dit ional G M t\. 1 framework. Dit tmar H, ; 3 ,1: 0' = lI y in t l ). (2002) used a stoc hast ic di scount fa, tor mode l embodying hot h quadratic and cubic terms. Th e validity of the mode l is tested Th erefore. the asset pricin g model with coskewness (4) is tested by a G ~H\ t statistic using the we ighti ng matr ix proposed by by test ing 'HI aga inst Hr· Jagannathan and Wan g (1996) and Han sen and Jagarmathan When model (4) is not supported by data. there exists an ( 1997). As ex plained earlier. the main contribution of our ar additional co mponent a (a N x I vect or) in ex pected excess ticle. beyond the result s obtained hy Har vey and Siddique returns that ca nnot he fully related to mark et risk and coskew (2000) and Dittmar (2002). is that we foc us 0 11 testi ng the as ness risk. E( r ,) = f3 AI + y i. ~ + a . In this case. intercepts a of set prici ng model w ith coskcwness (4) through the res trict ions model ( 1) sat isfy a = ()y +a. It is crucial to investigate how that it imposes 0 11 the return-gen eratin g process ( I ). instead o f the additional co mpo nent a varies across assets. lndccd , if this ado pt ing a meth odology using an uns peci fied a lternative (e.g.. compone nt arise s from an omitted fac tor. the n it w ill provide a G MM test).
L Barone Aoest. Gagliardini, and Urga: Coskewness in Asset Pricing 477
3. ESTIMATORS AND TEST STATISTICS \\ here vcchtI ) is a (N~l )N x I vec tor representation of I co n raining only cleme nt... on and above the mai n d iagonal. Th e In this sectio n we der ive the es tima tors and test statistic :"> Pxll. est imator of (J based on the normal fami ly is ob ta ined by used in our empirical ap plicatio ns. Foll owing an approach maximizing widely adopted in the literature (sec. c.g.. Campbell et al. 1987: Gouricroux and Javiuk 200 I ). we conside r the ge neral frame work of p..cudo-rnaximum likel ihood ( P ~ 1 L) methods, \Ve de rive the stativtical properties of the estimators and test statistics within the different co ... kcwnes... a ...... ct pri ci ng models pre ...cntcd where in Section 2.1 and 2.2. For com plete ness. we provide full r = J.... . T. derivation in the Appendixes. As is well known. the Pi\I L estimator for la'. ff . y ' )' is equiv alent to the geucralized lea ... t squares (GLS ) c ... tirnator on the 3.1 The Pseudo-Maximum Likelihood Estimator SU R ...ystem and also to the ordinary least ...quare... t (O LS) esti mato r performed equation hy equation in model (I ). LeI n de \Vc a".. urne that the error term £1 in model (I) with 1 = note the N x matrix defined bv II = Y T he P~l L I . .... T is a homo..ccda... tic martingale difference -cqucnce 3 la Ii I. c ... timator ii = Iii 91is consisten t when T - oc, and its ... ati'ifying: P asymptotic di... tribution is given hy
1101 I:'[E/£; I£/_I. R,H ". RF.I ] = I . where F, = ( I. rv» q.H ,t )'. where I i... a po ... hive-definite N x N matrix. T he fac tor Let u ... now conside r the test of the (joint) ... tati stica! sig f, = (",H." C/ ," ',) ' is -uppo...cd 10 he exogenous in the sen-e niticunce of the cos kewncss coefficie nts y . that is the test of o f Eng le. Hendry. and Rich ard (1988). Th e expectatio n and hyp oth e sis 'H;'.: y = Il. again... t N·F. Th is test ca n he easily the var iance-covariance matri x of fac tor r, arc den ot ed by performed co mputing a \Vald statis tic . which is given by Ji and I f. Suui-ticul inference in the ass et pr icing models pre ,t. _ ·I·_I_ _ , ~ - I c;r - -." y L. y. 1\ 4 1 se nted in Sect ion 2 is conve niently cast in the ge ne ral frame ~- work of Pt\lL methods (While 1981 ; Gouri croux , M onfor t, and t Trogn on 1984 ; Boller-l ev and Wooldridge 1992). If (I denotes (Upper indi ces in a matrix denote clements of the inversc.)Th e the parameter of intere st in the model und er con- ideration. the n statis tic ~ r* is asymptotically X2("l-distrihuled. with" = N. the PML es tima tor b defined by the ma ximizat ion whe n T - 00.
II = arg ma x L r( Hl. I I I) 3.3 Restricted Equilibrium Models " where the c rite rio n l.r UIl is a (condit ional) pse udo -log-likeli We now consider th e- co ns trained models (6) a nd (8) deri ved hood. More specifically. Lr IO) is the (co ndit iunal} log by arbitrage eq uili brium. The est imation of these models is lcs ... likeli hood of the model when we adopt a given conditional simple. bccau ...c they e ntai l cross-eq uation revtrictions. We let d istribution for error E, that satisfies ( 10) a nd is such that "u = ("Q'.y'". ". AU. 'ee. h("L. )' )' • the rc-ulting p... cudo-truc density of the model is exponential quadratic. Under ... tandurd regularity a ... surnptions. the P~1L c ... denote the vector of parameters of model (S) . The P~1L. est ima timator (i i..consistent for any chosen conditional di ... tribution of tor of iJ ba-ed on a normal pseudo-conditional log -likelihood is error e, ...ati ... fvin g the foregoing condition... ( ...cc the aforemen defined by minimization of tioned references}. Estimator 1i is efficient when the P...cudo conditional distribution of e, coincides with the true one. being then the P~1 L e ... tirnaror identical with the maximum likelihood ( ~ 1 L) e ... timator, Fina lly. because the Pi\1L estimator is ha ...ed on the maximization of a ... t ~lti ... tica' critcrion, hypothesi te ... tin g clln he l·onJu<.:ted hy the u,ual general asymphHk tesl. . E,(O) = r,- {J r.\f,' - Yc/'\I., - y lJ - i.ol. r=J. .... T. In what follows. we ,y ... te mat ica lly analy/c in Ihc Pi\11. T he P~lL estimator (j i ... givcn hy the followi ng ... y... tem of im t'rmnework thl' altemative spccificmion... introducl'd in Sec plici t equations (sec App. B): tion 2.
T )( r Cp'·Y')' = Dr,- i:',
Th e 4uad ratie ma rket model ( I) land the mark et mode l 12 11 117) are scc mi ng ly unrelat ed regressio ns (SUR) sys te ms (Zellne r 196:!1. w ith the sa me reg n:ss nr ... in each eq uat io n. Let (I dcnot c and the parameter ... orinter est in model ( I) . 118) o= (a '"'. Ii . y . vee hl 1:) j' . 478 Journal of Business & Economic Statistics, October 2004
where By applying these general results. we derive the ALS statistic for testing the hypotheses HI and H2agai nst the a lternative HF = r,- y q.\f.l - ~ L Et prM ,'- yO- (see App. C for a full derivation), Th e hypothesis H I against H, = ( I"M., . qu .. + iJ) '. HF is tested by the stat istic
Z= (y.I). I (a- ii y)' i -'(a- ii y) d , ~ 1' = T _,_ ,_ ~ X-It» . (2 1) d f 1 ,,1' - 1 ,,1' d - 1,,1' an r = f L.r=1 r ., r AJ = T L...t= I '"A f.,. an qM = T L...I= I li M ." I + l. 1:/ l. An estimator for A.. = p..\. }.2)' is given by with /' = N - I. where 'i = Ii + (0 . ii)' and
-l. =I'_+ (iJ0) . ( 19 ) ii = argmin(a - iJ y )'i - l(a - iJ y) " Estimator (P'. y'f is obtained by (time series) OLS regressions of r,- fa l on H, in a SUR sys tem. performed equat ion by equatio n. whereas (jj . fa) ' is obta ined by (c ross-sectional) G LS The ALS statistic for testing hypothesis 112 against 111' is given regression of r - PrM- YtiMon Z. A ste p of a feas ible algo by ri thm consists of ( 1) start ing from old estimates: (2) comput -- -.... - , ~ - I __ -_ - 2 (a - iJ y -AO') 1: (a - iJ y -AOI) " 1 ing (P'. y')' from ( 16); (3) co mputing (iJ . fa) ' fro m ( 17) using ~1'=T __ 1- ~x (/'), (22) new estima tes for P.Y. and Z;and (4) co mputing i from ( 18). I + l.'1:/ l. usin g new estimates. The procedure is iterated until a conver gence criterion is met. Th e starti ng values for p. y . and 1: are with P = N - 2. whe re 'i = Ii + (0. ii)' and pro vid ed by the unrestricted es tima tes on model ( I). whereas (iij:o)' = argmin( a - iJ y -Aol) ' i -I(a - iJ y - AO I) for the pam meters l.o and iJ they are provid ed by ( 17). where tt,).o est imates from ( I) arc used . The asymptotic distributi ons of the PML estimator are reported in Appendix B. In particular, it is shown that the as Finally. a test of hypothesis 'H I against 'H2 is simply per ym ptotic variance of the est imator of (P'. y ', iJ.i.O. AI. A2 ) is formed as a r-test for the para meter 1..0. independentof the tru e distribut ion of the erro rterm E,. as long as it satisfies the conditions for PMLestimation, The results for the co nstra ined PML es tima tion of model (6) foll ow by setting 4. EMPIRICAL RESULTS AO = 0 and Z= Y and delet ing the vector I. In this section we report the results of our empirical appli \Ve now consider the problem of testing hypotheses 11I cation. performed on monthly returns of stock portfolios. We and H2, corres po nd ing to mod el s (6) and (8), against the a lter fir st estimate the quadratic market model. then test the different native HF. If e denotes the parameter of mod el ( I). then each associated asset prici ng models with cos kewness. Finally. we of these two hypotheses can be written in mixed form. investigate the consequences of erroneously neglecting coskcw Ie ;3 a E A C IRq ogle. a) = 01 . (20) ness when testing asset pricing models. The section hegins with a brief description of the da ta. for an appropriate vector function g with values in ;R' and suit able dimensions q and r. Let us assume thatthe rankconditions 4 .1 Data Description g g il ) and rank -il ) = q Our dataset comprises 450 (percentage) monthly return s of rank( ao' = r ( aa' the 10 stock port folios formed according to size by French. are satisfied at the true values eO, aO Th e test of hyp oth esis (20) for the peri od Jul y I 963-Dccembe r 2000. Data are ava ilable at based on asy mptotic lea stJquares (A LS) co nsists of ve~fyi ng hltp://H'eb.mil.edl//kjrellch/uww/dalt1\ _lihrary.html. in the tile whether the constra ints g(O. a) = Uare satisfied. where 0 is an "Port folios formed on size." The port folios are constructed at unconstrained estimator of O. the PML estimator in our case the end of Jun e each year. using Ju ne market equity data and (Gourieroux et aJ. 1985). More speci fica lly. the test is based on NYSE breakpoin ts . The portfoli os from Jul y of year I to Juoe the statistic of t + I include all NYSE. AM EX. and NAS DAQ stocks for which we have market equity data for June of year 1. Portfolios arc ra nked by firm size. with portfolio I be ing the smallest and portfo lio 10 the largest. whereS is a consistentestimator for The market return is the value-weighted return on all NYSE. AM EX . and NAS DAQ stocks. Th e risk -free rule is the I-month ag il g' ) - I SO= ( ae' Qo ae Treasury Bill rate fro m Ibbotson Associ ates. Market returns and risk-free return s areavailable at hltp:/h.:eb.mil.edu/ kjrl'11cltl eva luated at the tru e values eOand aO. where Qo = V"' Jt x ln n r/dalll\_lib rary.hlml, in the files "Fa ma-Frcnch benchmark (e- eOII. Unde r regulari ty co nditions, ~T is asy mptotica lly factors"and "Fama-French factors."We use the T-Bill rate. be x l(r - q)-di stributed and is asy mptotica lly equiva lent to the cause othermoney-market series arenot available forthe whole other asymptoti c tests. period of our tests. l Barone Adesi. Gagliardini, and Urga: coseewnese in Asset Pricing 479
4 .2 Results Table 2. Variance Estimates of Model ( 1)
4.2.I Quadratic Market Model. We beg in with the es 2 3 4 5 6 7 8 9 10 tima tion of the quadratic market model ( I ). PML- SUR csti 1 17.94 13.42 10.69 9.4 1 6.93 5.20 4.02 2.64 .51 - 3.11 2 11.50 9.02 8.27 6.35 4.81 3.69 2.61 .58 - 2.72 mates of the coefficients 0'. fj . and y and of the variance E 3 8.24 7.18 5.65 4.51 3.34 2.39 .68 - 2.40 in model (I) arc reported in Tables I and 2. As explained in 4 7.39 5.56 4.37 3.40 2.41 .78 - 2.33 Section 3.2. these estimates are obtained by OLS regressions. 5 5.07 3.71 2.82 2.21 .77 - 1.93 6 3.67 2.42 1.85 .78 - 1.59 performed equation by equation on system (\ ). As expected. 7 2.56 1.68 .75 - 1.29 the bela coefficients are strongly significant for all portfolios. 8 1.93 .85 - 1.05 with smaller portfolios having higher betas in general. From 9 1.04 -.SO 10 .96 the estimates of the y parameter. we sec that small portfolios have significantly negative coefficients of market co skewness NOTE ThIStable reports the eslrmale of the varl8nce l. = EI-,-; ~ , q ..,, 1 of the error _, I'l the quadrallCmarkel model (e.g .. y = -.0 17 for the smallest portfolio). Co skcwness cocf ficierus are significantly positive for the two largest portfolios rr = fI + ~r"" . "q.., + _ " 1,.1._ .. T. (y = .003 for the largest portfolio). In parti cular. we observe wher. r," R, - R. rl, r.. , = R"'r - R. r- and q.. , '" R;..- R" R, ISthe N·vectOfof portfolios returns, R.." (R, ,) IS!he ma,ke' retum Ithe flsk·lr" relum), and l ISa N·veclOfof l 's tha t the fJ and y coefficients arc strongly correlated aero" port folios. \Ve ca n test for joint sign ificance ofthe co skewne...... para ~r ' ~r' meter y by using the Wald statistic in ( 14). The statistic ass ume s the value ~r · = 35.34. which is strongly significant at the 5'k level. because its cri tical value is X.~5 ( 10 ) = 18.31. Table 1. Coefficient Estimates at Model ( 1) Finally. from Tab le 2. we also see that smaller portfol ios are characterized by larger varia nce s of the residual error term s. Portfolio i Ct ; \Ve performed several spec ification tests of the functional .418 1.101 - .017 for m of the mean portfolios return in ( 1). First, we es tima ted (1.84) (24.23) ( - 3.32) (1.70) [20.24] 1- 2.94[ a factor SU R model inclu d ing also a cubic power o f market re 2 .299 1.188 - .0 13 turn s, RXt .l - R,...,. as a factor in add ition to the constant. market (1.65) (32.62) ( - 3.05) excess returns. and market squared excess returns. The cubic [1.56J [27.07J [- 2.65J factor was found to be not significant for all port folios. Further 3 .288 1.182 -.010 (1.88) (38.37) (- 2.84) more, ( 0 test for more ge neral [arms of rnisspecifications in the [1.86J [29.18] ( - 2.45J mea n. we performed the Ram sey ( 1969) reset test on eac h port 4 .283 1.166 - .0 10 folio. inclu ding quadr atic and cubic filled values of (I) among (1.96) (39.99) (- 3.00) the regresso rs. In this case. too . the null of co rrect speci fication [1.83] (30.98J [- 2.82] of the quadratic market model was accepted for all portfolios in 5 .328 1.135 - .009 (2.73) (46.94) ( - 3.34) our tests. [2.51J [34. 16] ( - 2.68) From the sta ndpoint of our ana lysis, one central res ult from 6 .162 1.110 -.006 Table I is that the coskewncss coefficien ts are (significantly) (1.59) (54.02) ( - 2.58) (1.53] (37.85J [- 2.28] different from 0 for all portfolios in our sample, except for two 7 .110 1.105 -.002 of moderate size. Furthermore. coskewncss coefficients tend (1.29) (64. 37) (-.88) to be correlated with size. with small portfolios having nega [1.24) [SO.66] (-.84) tive coskewness with the marke t and the largest portfolios hav 8 .076 1.083 -.000 ing posit ive market co skcwness . Thi s resuh is consistent with (1.02) (72.59) (-.18) (.90] (56.61J [-.23] the finding, of Harvey and Siddiquc (2000). It i, worth noting 9 -.016 1.017 .003 that the dependence between portfolios returns and market re (-.30) 192.76) (2 .06) (Urns deviates from tha t of a linear specification (as assumed (- .28) [98.4 3] (2.26J in the market model). generating smaller (larger) returns for 10 -.057 .933 .003 (- 1.10) (88.77) (2.64) small (large) portfolios when the market has a large absolute (-.99] (66.71) [2.73] return. This finding has important con sequences for the assess
NOTE ' ThIS lable fflpof1S toreach portfolio i, ' ''''' 1.. .• 10. lhe PML-SUR eSlrmalltS of the ment of risk in various portfolio classes: sma ll finn portfolios. coetflcoenlS " ., /I.. and y. of the quadra!lcmarl<.et model having negative market coskcwncss, arc exposed to a sou rce r.. _ .. + f!.r..",+y.q.."+f,,. 1=1 . . . ,T,I= l •... .N. of risk additional to market risk . related to the occurrence of
...... ,.,. r.I " R 1 - R", 'Ill ,. R..,.. - Rf , ,. and q"" = Rtr- RIO ... R.r IS lhe return 01portfolIO I large absolute ma rket returns, In ad dition. as we have already ....rnonltl I. and R..,r (R", ) oerctes the market return (the flsk·lree relurn ) In paran lheses we see n. market model (2). whe n te .... ted against quadratic ma rket repo rt I·slallstlCS compu ted under the assumphon model ( I). is rejected with a largely significant Wald statistic. In the light of our find ings . we con clude that the ex tension of the return-generating process to include the squared market re '"d turn is valuab le.
_, = (fl '. . .. f .... ) . 4.2.2 Restricted Equilibrium Models. \Ve now investigate
Whareas ' ·slalishes, calculaled wllh Newey and west (1969) he' eroscedasIIClly ' and market cos kcwness in the contex t of model s that arc co nsis aul OCOrralal lOfl-consislanl esumeto r Wilh live lags, are in square brad Table 3. PMLEstimates of Model (6) Table 4. PML Estima tes of Model (8) Portfolio i ii, }/; Portfo lio i ii, Yi 1.106 - .017 1.100 - .0 17 (24.50) (- 3.25) (24.38) (-3.32) 2 1.191 -.0 12 2 1.187 - .0 12 (32.97) ( -2.99) (32.84) ( -3.05) 3 1.186 - .009 3 1.183 - .0 10 (38.79) ( - 2.74) (38.70) ( - 2.91) 4 1.170 - .009 4 1.167 - .010 (40.41) ( - 2.90) (40.31) ( - 3.07) 5 1.140 - .009 5 1.137 - .009 (47 .38) ( - 3.14) (47.35) ( - 3.52) 6 1.112 -.006 6 1.110 - .006 (54.56) ( - 250) (54.45) (- 2.62) 7 1.107 - .001 7 1.107 - .002 (65.07) (-.76) (65.07) ( - 1.06) 8 1.085 - .00 1 8 1.085 - .001 (73.37) (-.05) (73.40) (-.38) 9 1.017 .002 9 1.018 .002 (9366) (2.14) (93.72) (1.90) 10 .933 .003 to .934 .003 (89.53) (2.63) (89.60) (2.57) iI =-14.955 ;:; = 4.850 ii = - 27.244 G= - 7.439 f,i =.032 ( - 2.23) (.70) ( -3.73) (1.01) (3.27 ) NOTE: This table reports PML estimates 01the coetticients 01the restncted model (6), NOTE: This table reports PML estima tes 01the coefficients 01the restricted mode l (8), T/ = /Jr,,,, + r qu , + r ,1+ AUr + t ,, t =I ,. ,. T. where r' is a scalar parame ter, denved from the quadratICmar ket model (1) by Imposrng the where /1 and An are scalar parameters. cerwed Irom lhe quadrenc marke t model (1) by Imposing restrcnon given by the asse t prICing mode l With ccskewness. E( r, ) ;: A, fJ + A.1Y, The scalar rj the resmcncn E( f r) = Ani + Al fl +- }.1r . Under thiSresvcuoo. asset expected excess relurns con and the premium lor coske wness Io. ~ are related by II = ;,2 - E(qM,j , The restricted model (6) tam a component AO that IS nOI explamed by either covariance or cossewness wllh the marke t corresponds to hypothesis 'H, : 31"1: C/ "" l' r In {t]. r-steneucs are reported in patenmeses. The restricted model (8) corresoooos 10hypo thesis 112 :3 IJ, Ao :a - , ~ r + Aor In (1). r-sreustce are reported In paren these s P~'1L cstirnntion of models (0) and (8) . Specification (6) is oh rained from the quadratic market mod el after imposin g restr ic compone nt acros s asset s. we test hypothesis 'Hz aga inst 'Hr . .. ," ~i, tion s from the asset pricin g mod el (4). Speci ficatio n (8 ) instead TI1(' rest stansnc r;;, j In (22) assume s the value = 5.32. we ll allows for a homogen eou s additional constant in ex pec ted ex belo w the cri tical va lue X.Z)s (X) = 15.51 . A more powerful test cess returns. Th e co rrespond ing PML estimators arc obtained of asset pricing mod el (4) should he provided by testing hy from the a lgorithm based on ( 16)--(18). as reported in Sec pothesis 'HI againsl the alt ern ati ve. 'H ~ . T his test is performed tion 3.3 . Th e result s for mod el (6) arc report ed in Tabl e 3: those by the simple r-test of significance of AO' Fro m Table 4. we see for model tx): in Table 4. that 'HI is qu ite clearly rejected . Thi s confirms thai asset pr icin g Th e point estima tes and standard e rro rs of param eters model (4) may not he supported hy our data. However. becau se fi and y are similar in the two model s. Th eir values arc close H ~ is not rejected. it fo llow s that. if the additional co mpone nt to tho se obt ained from qu adrat ic market mod el ( I) . In partie unexpl ain ed by mod el (4) COI11l': S fro m an omi tted fac tor, then ular, the estimates of parame ter y confi rm that sma ll (la rge) its sensitivities should he homogen eou s ac ro ss portfoli os in our port folio s have sig nifica ntly negat ive (pos itive) coskewncss co sample. \Ve conclude that size is unlike ly to have explanatory efficients. Parameter 1J is signiticantly negati ve in both model s. Po\VCf for e xpected ex cess ret urn s when co skcwness is taken as ex pec ted. hut the implied estimate for the risk premium for into account. M oreover. the contributio n to expected excess rc co skcw ncss, f~ . is not statistically significa nt in e ither mod el. turn s of the uncxplaincd component. deduced fro m the estima te However. the estimate in mod el (8). J:; = - 7,439. has at lea st of param eter AD . is quite mo de st. approxim atel y .4 on a annual the expected negati ve sig n. Using this es timate. we deduce that percent age basis. No te in particul ar that this is less than half the for a portfolio with coxkcwn css y = - .0 1 (a mod erate-sized co ntribution du e to cos kew ncss for portfol ios of mod est size. portfolio. such as portfoli o :' or 4 ). the coskewncss contri bution As ex plained in Sec tion 2.2. AD > 0 may be due to the use of a to the expec ted excess return on an annual percentage basis is risk-free rate lower than the actu al rate faced hy inve stors, approx imate ly .9. Thi s value increases to 1.5 for the sma llest 4.2.3 lvli,\'.\pccijicatioll From Neglected Coskrwness. As portfolio in our dataset. a lready ment ioned in Section 2. we arc a lso interested in invcs \\le test the empirica l validity of asse t prici ng model (4) tigal ing the consequences on asset pric ing tests of erroneou sly in our sample hy testing hyp oth esis 'HI aga inst the alterna negle cting cos kcwness. Th e results present ed so far sugges t tive. 'HI . Th e ALS test statistic ~ j. given in (2 1) assume the that marke t model (2 ) is misspcci tlcd. because it docs not ta ke value ~ j. = 16.27. wh ich is not significam at the 5lh' level. even into account the qu adratic mark et return. Indeed . when tested though ver y c lose to the cr itical value X .~5( 9) = 16 .90 . Thus aga inst qu adrat ic mark et mo de l ( I). it is strongly rejected . For there is SOI11l' evi de nce that usse t pricin g mod el (4 ) may not be compa riso n. we report the estima tes of parameter s a and f3 in sati sfied in our sample. In oth er wo rds , an additio na l compo nent mark et model (2 ) in Tabl e 5. The Ii coe ffic ients in Tah le 5 are other than covaria nce and coskcwness to mark et may be pre sent c lose to those obt ai ned in the qu adrati c mark et model repo rted in expec ted excess returns. To test for the hom ogeneit y of this in Tuhlc I. Th erefore. neg lecting the qu ad rat ic market returns Barone Adesi, Gagliardini, and Urga : Coskewness in Asset Pricing 48 1 Table 5. Estimates of Model (2) tcst .... model (4) through the ort hogonality con di tio ns (9). In ~) . Portfolio i iii /l, addition. we investigate the effects on the A LS sta tistic in- duccd by the non normality of e rror E, or by the misspcc itica- .080 1.102 (.39) (23.97) tion o f the return-generating procc ( I). 2 .050 1.188 (.3 1) (32.34) 5.1 Experiment 1 3 .092 1.183 (.67) (38.09) 4 .088 1.167 The data-generating proce...... u...cd in experiment I is given by (.67) (39.65) 5 .148 1.135 r, = a + {Jr.H., + Yl/ .\I., + E,. 1 = J. .. .. 450. (1.36) (46.43) 6 .044 1.110 where rv. = R\I., - Rf .,. and qst» = R .~, . , - RF.,. with (.48) (53.69) 7 .076 1.105 N,H., ..... iid ;\:(/1.H. (J ,~, l. (1.00) (64.39) 8 .069 1.083 e, - iid:-.'(tl. 1:I. Ce,) independent of IR,I.I). (1.05) (72.67) 9 .034 1.017 Rr ., = Rf . a ronslant. (.71) (92.41) 10 .005 .933 and (.10) (88.18) a iI Y All'- NOTE Th.s table repons for each portfolio I. j ", 1. 10 the PML- SUR astmates ollhe = + coetf!Cl8nts" fl, of the tradlltooal mar kEl l model The value... o f the parameters are chosen 10 he eq ual to the est i r.I " . + #,r"" + ~ , , ' - l ....T. - l . N mates obta ined in the em pirica l analys is reported in Section 4 . where r" R I R, , and ril l R"" R" . R. j IS the return 01 portfolio. lI'I month I. and Specifica lly. p and y are the third and fourth co lumns in Ta R.. , (R, ,I .s the marke t return Ithe nsk-tree return) . /·s tatISUCs are reported In paren theses ble 1. ma trix 1: is taken from Table '2. iJ = - 14 .995 fro m Ta hle J. liM = .52. aM = 4.4 1. and R,,· = .4. corres ponding to the docs not see m to have dram atic consequences for the cs tim ..l avcruge of the risk- fret: return in our dataset. Different values tion of parameter fL However. we ex pec t the co nse quences of of parameter Ao arc used in the simulations. \Ve refer to this thi s mis spcc ification to he serious for inference. Indeed . we data-generat ing process as DGPI. Unde r DGPI. whe n ).0 = O. have see n ea rlier that the cos kcw ncss coefficie nts are correlate d quadratic equilibrium mod el (4) is satisfied. When i.o =F O. cqui with size. with sma ll portfolios havin g negative mark et coskew Iihr ium mo del (4) is not correctly spcc itlcd. and the misspec ness and large port foli os having pos itive mar ket co- kcwncvs. ification is in the form o f an additional co mpo nent. whic h i.... Thi s feature suggcsts that sill' can have a spurious explanatory homogen eous across portfo lio.... ".. orrcsponding to Illud eI (X) . powe r in the cro ...... -scction of expected excess returns because it However, for any value of ).0. quadratic ma rket mod el ( I ) is is a proxy for omitted coskewncss. Therefore. as anticipated in well specified. Sectio n 2. the abi lity of size to explain expected excess rcturn-, \Ve perform a Monte Carlo simulation ( IO,(XX) replications) could he due to rn isvpccification of models neglecting co-kew for different values of ).0 and report the rejection trequc ncie .... of nc...s risk . the two test vtuti... ti cs.~} and ~? \ I M . at the nominal size of .05 in Finally. it is interesting to compare our findi ngs with tho ...e Tab le 6. The second row. i.o = O. reports the empirical tes t sizes. reponed by Barone Adcvi (ll.JX5). whose investigation covers Hoth ... tat i-tics co ntro l size quite we ll in finite samples. at least the period 1931 -1975. We see tha t the sign of the premium for sample si/ c T = 450. The subsequcm rows. corresponding for covkewncs-, has not changed over time. with aloo ...ct .... hav to i.o #- O. report the powerof the two test statistks again ... t alter ing negative co-kcwnc...... commanding. no t surprisingly. higher native ... corresponding to unexplained components in expected expected returns. In contrast. both the sign of the premium for excess returns. \\ hich are hornogcncou... across portfolios. Note size and. con...cqucntly, the link between covkcwncvs and sill' that such additional component... . with i.o = .033. were found arc inverted. Although it appears to be difficult to discrimi in the. empirical ana ly ... is reported in Section 4 . Table 6 ... how-, nate stuti... tically between a .... tructural si/.e effect and reward that the power of the ALS ... t a t i st i c ~ ) i... considerably higher for co ... kcwncw. according to the cri terion of Kim and Zhang than that of the G ~ I ~ I ... tatistic ~ .~ \ I \ I . This is due to the fac t ( 191.J1.Ja.h) the size effect is more like ly explained by neglected that the A LS statistic ~ l use :1 well-... pcciticd alternative given coxkcwncsv . by ( I ). whereas the alternative for the G ~ 1 ~ 1 statistic ~ P \ I !\ f is left unspecified, 5. MONTE CARLO SIMULATIONS 5.2 Experiment 2 In thi ...... cc tio n we repo rt the res ults of a seri c ... of Mont e Ca rlo simulations undert aken to assess the importance of spec Unde r DGPI. residu als £1 arc no rma l. Wh en residual s e, arc ifyi ng the ret urn -gen erat ing process to ob ta in reli ab ly power not norma l. the alterna tive used hy the ALS s ta t is t ic s~} [i.e.. ful stm io,;,\ical tests. We compare the finite-sample propcni c ... model ( 1)1is still correc tly ....pccifico . becau se P~1L estimators (loo i/e and power ) o f two statistics for testin g the asset pric arc used to co nstruct ~} . However, the se es timators are not ef ing model with coskcwness (4) : Ihe ALS s la t is t ie ~) in (2 1). licient. In e xperiment 2 \\'e invcs tigatc the effec t of nonnormal · whic h lests mmiel (4) hy thc restr iction s imposed on thc relurn ity of re.... idu al s € , on the AL S test stati.... lic. Th e data-generating genl'rating prore...... ( I ). and a G MM tes t statistic ~Y i\ l \ l. which proccss lIscd in thi s ex pe rime nt. ca lled DGP2. is cq ual to DGPI 482 Journal of Business & Economic Statistics , October 2004 Table 6. Rejection Frequencies in Experiment 1 Tetxe 8. Rejection Frequencies in Expe riment 3 ~ f unde r DGP4 ~ + under OGP3 (homoscedasticity) (conditional neterosceossticity) o .0404 .0559 .03 .0505 .4641 o .0587 .0539 .06 .0712 .9746 .03 .3683 .1720 .10 .12 17 .9924 .06 .9333 .5791 .15 .2307 .9945 .10 .9855 .9373 NOTE This table reports the rerecton l requer'l(:l9S of the GMM stansnce ~F" [derived NOTE' This table reports Ihe rejection frequencies al lhe ALS statls tlCs ~ ; [In {2 111 1or test- from {gIl and Ihe AlS steusucs tl lin (21)) for lestlng the asset pricing model with ccskew- '"9 (41. ness (4 ), at the .05 con fidence level. in excerereot 3. In this expenment we ccosoar tWOdala -gen eratlng at Ihe .05 con fidence level , In axpe nrnantt , The dala-generallng process (called DGP1 ) used in proc ess es (called DG P3 and DG P4j haVing the same uoccnononer venence 01 the resKluals F,. thIs exper iment IS given by bur sucn thaI the residuals ,I , are conditionally neteresceoasuc In on e case and rorrosceesstc in the other case. Spec ifically. DG P3 ISthe same as DGP1 (see Table 6). buf the encvenone ,I , I, : a + {J r"" + yq"' r+ If j . 1=1 , .. , 450. toncw a con ditIOnally normal. multrvartate ARCH{I ) process Without cross-enects. where rw r = R"u - R~ . I and q.." '" Rt" - R, ,. With 6, ....iidN(O.l:), (. r) Independenl 01(R.",) , The metro Sl = [w,l;s chosen as in Table 2. and (I "" .2. DG P4 is Similar to DGP 1 (eee Table 6) . Wllh lid r ormarmnovanons whose unconditiona l vanancemetnx ISthe same as the unco ndition al R",r = R" a constant. varianc e 01,1, in DG P3 Thus unde r DGP 4.lhe anernatwe of the Al S erauarcs ISwe lt speci fied . but not under DG P3 ">d a = r~y + A a· heteroscedasticity of errors e .. \Ve thus consider two data Param eters fJ and yare the tturd and fOUfth columns in Table 1. the matnx r IS taken Irom Table 2. r'::z - 14,995 from Table 3. /1", = .52. 0., '" 4 ,41, and R, = .4. correspo nd ing to the generating processes having the same unconditional variance aver age 01the ns ~A re e return In our datase t. Under DG P1. when AC_ O. the quadra tICequilib of the residuals e.. but so that the residuals E , are conditionally rium mod el (4) is sansfied . Whe n AC * O. the equi librilJmmodel (4) ISnot correctly specrhed . an d the rmsspecdic auon IS In Ihe term of an additional component homogeneous across portfolios. hetcrc scedustic in one case and hornoscedastic in the other case . co rrespo nding 10 model (8) . Spec ifica lly, DGP3 is the same as DGPl, but innova tions 8 1 follow a cond itionally normal. multivariate ARCH ( I) process without cross-effects. but residuals E , follow a multi variate I-distrihution with 5 de grees of freedom . The correlation matrix is chosen so that i =j the resulting variance of residuals E, is the same as under i i'j. DGP I. The rejection frequencies of this Monte Ca rlo simula tion ( 10.000 replications) for the ALS statistic ~ j. are reported Matrix Q = l NOTE : This lable reports me rejection t-eqceoces 01 the Al S return s. We have show n that the cos kewness coefficients of the statistics ( ; [in (21 )] lor lestlng (4) . portfolios in our sample are statistica lly significant. This finding rejects the usual market model and demonstrates the validity of atme .05 conuc eoce level . in espanment 2. The data-generating the quadratic market mode l as a possible exte nsion. proc ess use d In this experim en t (called DGP2) IS me same as The analysis of the premium in expected excess returns in DGP 1 (see Table 6). bul the rescuers F, lollow a mulflvariate '· dlslrrbulion w ~h 5 de grees 01 treed om, and a correlation matnx duced by market co skewness requ ires the specification of ap such that the variance of ,I , is the sama as under DGP1 . propriate asset pricing models including co skewness among the l Barone Adesi. Gagliardini, and Urga: Coskewness in Asset Pricing 483 rewarded factors. To obtain testi ng method ologies that are more The probab ilistic measure of coskewness is de fined by powerful than a GMM approac h. we tested asse t pricin g mod els including coskewness through the restrictions that they impose covl r"i. R~ I . ,) = Elru R;, ) - El r,,;IEI R;,). on the qu ad ratic mark e t mod el. We used asy mptotic test sta which can be rewritten as tistics whos e finite-sample properti es arc valida ted by mean s of a series of Monte Ca rlo simulations. \\'e found ev ide nce of cov(r,.i. R ~ f ,,) = fI,(E[ R{r.rI - E [ R ~ r.r I E [ R ,I( . t1 ) a component in expected excess returns that is not explained by eithe r covariance or coskewness with the mark et. However. + Yi s ' a r lR~ /.I1 + EIR.;/.IIE[f,t1. this unexplained co mponent is relatively small and is consis tent In the final eq uation . the first term is a measure of the mar for instance wi th a min or rnisspeci ficati on of the ris k-free rate . ket asy mme try. the second term is essentially our measure More imp ortan t. the unexplained co mponent is ho mog e neo us of cos kewness. and the final term is equal to 0 by ass ump ac ros s portfol ios. Th is find ing implies that add itional var iables. tion ( I()). Evidently. our approach considers the second term rcpresc ruing portfoli os c haracteristics suc h as firm size. have no only. However. our claim is motivated by the negligihle effects explanatory power for expected excess return s once coskewness is taken into account. of the fi rst term. In fact. for values fI, = I and Y, = -.0 1.which firm Finall y. the homogeneity property of the unexplained cornpo are represe ntat ive for sma ll port foli os. the first term is . 1. nent in expected excess returns is not satisfied whe n coskcwncss whereas the second is - 15. If Yi is equa l to .003. as in large ~ . is neglected. Th e refore. our results have impo rtant implicat ions firm portfo lios. then the terms are eq ual to . 1 and Finally. if for testing methodologies. show ing that neglecting covkcw nc-s the port folio has a Yi = O. then the second term is also O. risk ca n ca use mis leadi ng inference. Indeed. becaus e coskew \\'e are greatly indebted to one of the referees. wh ose com ness is positively corre lated with size. a possib le j ustificat ion for ments highlighted this point. the anomalous explanatory power of size in the cross-secti on of expected returns is that it is a pro xy for omitted coskcw ncss APPENDIX B: PSEUDO-MAXIMUM LIKELI HOOD risk. Th is view is supported by the fact that the sign of the pre IN MODEL (8) mium for coskcwncss, contrary to that of size. has not c hange d in a very long tim e. 1n this appendix we consider the P~1L estimator of model (8 ). defined by maximization of (15). We lirst derive the PML equa ACKNOWLEDGMENTS tions. Th e sco re vector is given by T The authors thank the participants in the 91h Conference aLl " - I on Panel Data. the 10th Annual Co nfe rence of the European a«r .y' )' = L... II, e r e.. Fina ncial Management Association. the Co nference on Mul t=1 timoment Cap ital Asset Pricin g Models and Related Topics. T __ ,_~I...:_T'------I the Ii>: QUIRE UKI-lth Annual Seminar on Beyond Mean = L -Z:!:, e.. aW. ),n)' Variance: Do Higher Moments Mauer? the 20()~ ASSA 1=1 Meeting, and P. Balestra. S. Cain -Polli , J. Chen. C. R. Han ey. and R. Jagannathan, R. Morek, M. Roc kinger, T. Wansbcck. C. Zh ang. two ano ny mo us referees. and the ed itor. Eric Ghysels, [ ~ ' -:"::":'aL,_=-:;I PT:!: __I ® :!: I" sech L...("', -:!: )] . All of them have co ntributed. throu gh di scussion. very help ~ vcc ht E ) _ ful comments and suggestions that improved th is article. The '=1 usual discl aimer applies. Th anh are also du e to K. Frenc h. where lit = (r.\I .l .q.\I ., + a )'. £ , = r, - fl rM., - yq.\I ,1 S" ' i s ~ R. Jagannath an, and R. Kan for providing their dataset . y tJ - ).0 1. Z = (y. n. and I' is such that vec(:!:) = P vechfE ). i>: CCR Finrisk is gratefully acknowledged by the tiN two By equating the score to O. we immediately find ( 16)-( I8) . aut ho rs. \Ve now deri ve the asy mpto tic distributi on of the P~ lL esti mator in mod el (8). Unde r usual regul arit y co nditions (se c the APPENDI X A: RELATIONSHIP BETWEEN references in the main text). the asymptotic d istribution of the CROSS-MOMENT COSKEWNESS AND general P ~ IL estimator ii defined in I II) is given by THE Yi PARAMETER "filii - 0° ) .i... NIO. .Io110.10I). In thi s append ix we show ho w the paramete r Yi relates to the cox kcwncs s te rm of our quadratic market model. \Ve also whereL, (the so-called "informa tion matrix") and 10 are sym rep ort erro r es timates whe n the cross-mome nt cos kewncs s is metri c. positive-definite matrices defined by ap proximated by the param eter Yi. 2L Our basic model is [see ( 1)1 .1 0 = h.m E[--I-i) -,r (00)] T ~", T iJOao ru = Ui + fi,( RJ/,t - RF.,) + Yi ( R~ 1.t - Hr .,) + £ ;,t . and where 10 = lim E __I ilL7 (0°) __ilLT, (0° )] . 7 - 00 [ T iJO ilO 464 Journal 01Business & Economic Statistics. October 2004 \\'e now compute matriccsJn and 10 in model (8). The seco nd Th e asymptot ic va riance-covariance or(p'.y ' )' an d (;:j . ~) ' deri vatives of the p... cu do-log-like lihood are grvcn by is given ex plic itly in block form hy l ' L T , ,,, - I _ • 0.11 •" (J I ~ J l • 11 - , . 1 1 , "' 11 . ,. I = - "' II,II' 0 r - . [•'0 . 0 il (/f . y ' )' il (/f. y ' ) :':I ' .~., t. , T ,-./ =-"'1I0 r - ' z. .I ~ ' I = tr r+ H 'I -1 0 r ~( R ' ')"' «(}' )' L ' f p . y u • An 1= 1 + Ir,IH '(r r + H 'I- 110 z(z'r- IZj- IZ'. .I ~ ' ~ = - r,lA0 z , ,,, 11 _ , *11' • () - . 0 . and and _-,-=:-i1=L~1 ~ -:-:::-c = I' / r - I 0 r - 'I' J~ ~ ~ 1 ;h ech( r ) iI vec ht r )' 0 = t I + A'r, 1A)(Z'r- Zl- I. Finally. we consider the asymptotic d istribution of es timato r I - ~ l'l r -' 0 r - (tE'E;)r- ' I' i defin ed in ( 19). T he est ima to r I=J I p- . = TI '"L f,. - ~ I'l r -' (tE'E;)r- ' 0 r - II'. 1= 1 1= 1 where f, = (rA/ ,I' l/.\1.l)" call be seen as a co mponcm ofthc P~lL with the other ones vuni... hing in expectation. It follows (halma cstirnator on the ex tende d pscudolikclihood trices .lo and 10 ere g iven by [in the block repre sentation co rre LrtH. p. rd sponding to (fl' . v', O. AO) ' and vccht E j] TI T , .I (~ .I (~ "S] o ] = Lr tH) - " log del r,.- :; LII',- IL)'r , 1(I', - p ). , 0 = [ and 10 = .... II .... - • [ .loS 'I .IoKlo .. .. /=1 where 1.1' (H) is given in ( 15). It is easily seen that H an d ilL. \\.. here l:r> are asy mptotically ind ep endent. It follows that \ I"I ./Tt r:~ - i. ~.o)1 = Er . ~ ~ + V"I ./T(1j - ,9011. S = covls .. \'l~c h{ e{ E ;)l APPENDIX C: ASYMPTOTIC LEAST SQUARES K = \arl\echI E,E;i1. In thi ... appendix we de rive the t\LS ... tati... ti c ~ } in (2 1) an d and lin the block form corresponding to (P'. y ')' . (d. i.n)' 1 ~i in C~2) . In bot h case ... the restric tions [see (20)1 are of the form , _ [ EIII,II,10 r - I i. 0 .I r - IZ] 0 - i: Z' 0 r -1 z'r - Iz ~ IH .;o 1 =A, (a) vectll) + A ~ (a l . and where II is the N x 3 ma trix defined hy II = la P y I and ,\ 1(a) i-, ... uch that i. 0 r - ' ] q = [ z'r- I . Ada) = (I. O. -() ) 0 Iv " Ai l"l 0 1.-.. \Ve derive the we igh ting matrix So = (f)g/ilH' Qu ilg / ilO)- I. (All parameters are evaluated at true value) , Th er efore. the <1"' where 120 = 1',,( filii - H ) ). From (131, we ge t ym pto tic var iance-covariance ma trix of the P7\IL est imator '9 in il u' . model (8) is given by -a"~ Q -~- = A' E I F F' I - I A' 0 r 110 ' 0 iJ O I I , 1 J~ - l QS] . = I I + A'rJ- 'A)r. K The text .... tat istics follow. j~) ' Note that the a ... ymptotic variance-covaria nce of (fi '. y'. ;:j. It sho uld be noted that exact te sts (unde r normality ) ca n be (i.e.. .J l~ - l ) doc s not dep end o n the distribution of erro r term co nstruc ted for testing hyp oth e se s H 1 and "H. '2 agai nst 'Hf" (sec. e ., and in parti cul ar it coincides with the asympto tic vuriuncc-. c.g.. Zhou 1995: Vclu and Z ho u 1999 ). Th e se tcsb arc asy m p y '. ~ . ~» '. covariance matrix o f the f\lL estimator of (ii'. toti call y equ iva lent to the ALS te sts used in this artic le for the ir when £ 1 is normal. In contrast. asymmetries and kurtovis computational sim plici ty. An evaluation of the tin itc-sample of the di stribution of e, influ en ce the asymptotic va riance properties of the ALS test statistics is presented in Section 5. covariance matri x of "cch(!: ) and the asymptotic co variance of (fi '.9'.7i. j~) ' and vcch t I:). throu gh matrices S and K . Barone Adesi, Gaqliardmi. and Urga: Coskewnes s in As set Pricing 485 REFERENCES J.If!:lOn arha n. fL