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.