Inernaional Review of Business Research Papers Vol. 4 No.3 June 2008 Pp.256-268 Undersanding Cross-Secional Sock Reurns: Wha Really Maers? Yong Wang We run a horse race among eigh proposed facors and eigh proposed condiioning variables for explaining he cross-secion of sock reurns. The purpose is o beer undersand which facors, in combinaion wih which condiioning variables, seem robus in explaining cross-secional daa, and o seek an economic inerpreaion of he specificaions ha appear mos promising. We find ha a consumpion growh facor, condiioning on lagged business income growh, is he mos successful in explaining cross-secional variaion of average quarerly reurns in he 25 Fama-French porfolios. Field of Research: Empirical Finance, Asse Pricing. Inroducion Many facor models, wih a variey of condiioning variables, have been proposed o explain he cross-secion of sock reurns. While some are rejeced, a number of recen papers have claimed beer success. A sill open quesion, however, is wha bes explains he cross-secion variaion of average reurns, and why? Previous ess are based on differen daa ses and differen mehods, making i difficul o compare heir performance direcly. In his paper, we run a horse race among several facor models and several condiioning variables, comparing heir explanaory power for he cross-secion of sock reurns. Yong Wang, School of Accouning and Finance, Hong Kong Polyechnic Universiy, Email: afywang@ine.polyu.edu.hk.
Wang 257 Ferson and Harvey (999), and Hodrick and Zhang (200) also compare several condiional asse pricing models. While hey consider several specific models, we compare all possible models consiued by combining previously proposed facors and condiioning variables. In addiion we include several new facors and condiioning variables. A significan difference is in he choice of condiioning variables. The earlier horse races use only one condiioning variable a a ime, even for muli-facor models, while we choose differen condiioning variables for differen facors. In condiional asse pricing models, condiional variables are used o capure he ime variaion of risk premium. Condiioning variables are ofen chosen from hose variables ha can predic fuure marke reurns. Those variables help o capure he variaion of aggregae marke risk premium, ime-varying price of marke risk. So hey are poenial condiioning variables for CAPM, which uses marke excess reurn as facor. However, hey are no necessarily good condiioning variables for oher facors. Differen risk facors have differen risk premiums. The variable ha can predic aggregae marke risk premium may no capure he risk premium variaion relaed o oher risk facors. Differen risk facors may need differen condiioning variables. Neverheless, all previous ess use only one condiioning variable in one model, no maer how many facors in he model. They all focus on variables ha predic marke reurns, which are poenial condiioning variables for marke risk facor. In his paper we choose differen condiioning variables for differen facors, and consider some new condiioning variables. 2. Lieraure Review I is well documened in he lieraure ha he CAPM fails o explain he cross-secion of sock reurns. Roll (977) poins ou ha he reurn on he value-weighed porfolio of socks is unlikely o be an adequae proxy for he reurn on aggregae wealh, which may be a major cause of he unsaisfacory performance of he CAPM. Many papers have pursued his basic insigh. Fama and French (993) propose a hree-facor model wih he marke excess reurn, he reurn on a Small Minus Big size porfolio (SMB), and he reurn on a High Minus Low book-o-marke raio porfolio (HML) as facors. This hree-facor model is saisically successful in explaining he cross-secion of 257
Wang 258 sock reurns. Jagannahan and Wang (996) include he reurn on human capial as a new facor, using labor income growh as a proxy and condiioning on some macroeconomic variable. Heaon and Lucas (2000) inroduce proprieary business income growh as a new facor, and find ha a linear asse pricing model wih proprieary business income has beer performance han he similar model ha includes only labor income. The Consumpion-based CAPM (CCAPM) developed by Breeden (979) claims ha wih complee markes only aggregae consumpion risk should be priced. The CCAPM has even less power han he CAPM o explain he cross-secion of average asse reurns (Campbell, 996 and Cochrane, 996). Several researchers have ried o resurrec he CCAPM by decomposing consumpion ino is componen pars. Piazzesi, Schneider and Tuzel (2007) develop a wo-facor model in which aggregae consumpion risk is ransformed ino wo facors: non-housing consumpion growh and expendiure share on non-housing consumpion. Lusig and Van Nieuwerburgh (2005) differeniae housing consumpion and non-housing consumpion as well. In heir model wih housing collaeral hey use consumpion growh on food and apparel and a renal price growh scaled by housing expendiure share as wo facors. One reason for he failure of saic model ess may be ha correlaion srucures and risk premiums are ime varying. This implies ha ime series averages may no reflec he rue condiional relaionships. Condiional asse pricing models use condiioning variables o capure he ime series variaion of risk premiums, hereby improving cross-secional predicabiliy. Several papers have exended he asse pricing lieraure in his direcion. In empirical sudies of condiional asse pricing models, condiioning variables are ofen chosen from hose variables ha can predic business cycle or forecas he marke excess reurns. Jagannahan and Wang (996) use he "defaul premium", he yield spread beween BAA- and AAA-raed bonds, as a proxy for he condiional marke risk premium, noing ha previous lieraure shows ha i predics he business cycle and forecass marke excess reurns. Dividend yields and erm spreads have been shown o have ime series predicive abiliy o forecas sock marke reurns, so hey are also widely used as poenial condiioning variables in cross-secional ess. 258
Wang 259 Leau and Ludvigson (200a) show ha flucuaions in he aggregae consumpion-wealh raio are srong predicors of excess reurns on he marke. Leau and Ludvigson (200b) use he log consumpion o aggregae wealh raio (cay) as a condiioning variable for boh he CAPM and CCAPM. Sanos and Veronesi (2006) show ha lagged values of labor income o consumpion raio (s) can predic sock reurns. They also find ha boh CAPM and CCAPM condiioning on s have beer performance in he ess of cross-secional implicaion han heir uncondiional counerpars. Lusig and Van Nieuwerburgh (2005) develop a model wih housing collaeral in which he raio of house wealh o human wealh (housing collaeral raio, my) shifs he condiional disribuion of asse prices and consumpion growh. They find ha a decrease in he house collaeral raio predics higher reurns on socks, and heir condiional model can explain eighy percen of he cross-secional variaion in annual reurns on size and book-o-marke porfolios. Piazzesi, Tuzel and Schneider (2007) build a model wih housing consumpion. They find ha he non-housing expendiure o oal expendiure raio (α) is boh a good forecasing variable for marke reurns and a good condiional variable for heir condiional Consumpion-Housing CAPM. In his paper, we reconsider he facors and condiioning variables idenified in previous sudies. Since hey all seem o have some explanaory power for he cross-secion of asse reurns, a naural firs sep is o examine he exen o which hey are correlaed. Surprisingly, we find very lile correlaion among hem, suggesing ha hey do no all proxy for he same aggregae shocks. To explore which facors, in combinaion wih which condiioning variables, really maer o cross-secional reurns, we run a horse race among eigh proposed facors and eigh condiioning variables. 3. Mehodology According o firs fundamenal heorem of asse pricing, in he absence of arbirage here exiss a sochasic discoun facor, or pricing kernel, m, which prices every asse correcly. Tha is, he following equaion holds: E (m + R + ) = () where m + is he sochasic discoun facor a ime for cash flows arriving a +, R + is he gross one period reurn for asse i realized a ime +. 259
Wang 260 In his paper, we consider linear facor models, in which sochasic discoun facor is a linear funcion of a consan and a k vecor of facors, f +. Le model parameers be [ a, b ' ], hen he pricing kernel is = (2) m + a + b ' f + Each linear facor model can be idenified by he specific f +. Cochrane (996) provides one mehod o incorporae he condiioning informaion ino asse pricing models. He assumes he coefficiens in pricing kernel equaion (2) are linear funcions of condiioning variables. In a one facor model, a = a + a z, b = b + b z, he pricing kernel becomes: 0 0 m + (a 0 + az ) + ( b 0 + bz ) f + = a 0 + az + b 0f + + bz f + = (3) Then he condiional model can be rewrien as scaled facor model wih consan coefficien: E [(a 0 + az + b 0f + + bz f + )R i,+ ] = (4) In his paper, we esimae and es condiional models in his form. In previous sudies of condiional asse pricing models, condiioning variables are ofen chosen from hose variables ha can predic fuure marke reurns. Those variables help o capure he variaion of aggregae marke risk premium, ime-varying price of marke risk. So hey are naural condiioning variables for CAPM, which uses marke excess reurn as facor. However, hese variables are no necessarily good condiioning variables for oher facors, since hey may no reflec he ime variaion of risk premium relaed o oher facors. In general, differen risk facors have differen risk premiums. The variable ha can predic one facor risk premium may no capure he risk premium variaion relaed o oher risk facors. Differen risk facors may need differen condiioning variables. In his paper, we choose differen condiioning variables for differen facors. We use Generalized Mehod of Momens (GMM) o esimae and es he condiional models as in equaion (4). For linear facor models, define he pricing error vecor: 260
Wang 26 g(b) = E(b' F R -) (5) where b is parameer vecor, F is facor vecor, and R is asse reurn vecor. The GMM esimaes are formed by choosing b o minimize he weighed sum of pricing errors: min J = g(b)' W g(b) (6) where W is he weighing marix. There are wo weighing marices widely used in he lieraure, he opimal weighing marix and Hansen-Jagannahan weighing marix. The opimal weighing marix is proposed by Hansen (982), W=S ¹, where S is he covariance marix of g(b). This weighing marix is opimal in he sense ha he esimaed parameers have he smalles asympoic covariance. Hansen and Jagannahan (997) propose anoher weighing marix, E[RR ] ¹, which is he inverse of he second momens of asse reurns. When Hansen-Jagannahan weighing marix is used, J measures he minimum disance from he pricing kernel in he model o he se of rue pricing kernels, which is ofen called HJ-disance. While he opimal weighing marix changes wih differen models, Hansen-Jagannahan weighing marix E[RR ] ¹ is invarian across models given he se of es asse reurns. Using a common weighing marix gives a uniform measure of performance across models, so HJ-disance is suiable for model comparisons. 4. Resuls We use quarerly daa in he U.S. marke. For es porfolio reurns and facors, he sample period is from 952Q3 o 2002Q2. Condiioning variables are one quarer lagged, so he sample period is from 952Q2 o 2002Q. There are a oal of 200 observaions for each variable. The es asse reurns are he value-weighed reurns on he 25 Fama-French porfolios. These porfolios are he inersecions of 5 porfolios formed on size and 5 porfolios formed on he raio of book equiy o marke equiy. The porfolios include all NYSE, AMEX, and NASDAQ socks. We consider eigh facors: Marke excess reurn (R m ), Consumpion growh ( c), Fama-French facors (SMB and HML), Labor income growh ( y), Proprieary income growh ( prop), Scaled renal 26
Wang 262 price change (A logρ), and Non-housing expendiure raio change ( logα). Some of hem are from radiional models, such as he marke excess reurn in CAPM. Ohers, such as housing facors, are proposed in more recen sudies. TABLE. Facor Correlaion Coefficiens R m c SMB HML y prop A lgρ lgα R m.00 c 0.23.00 SMB 0.42 0.3.00 HML -0.37-0.02-0..00 y 0. 0.72 0.0-0.05.00 prop 0.2 0.56 0.0 0.06 0.54.00 A lgρ 0.0-0.05 0.03 0.04 0.04-0.0.00 lgα 0.02 0.28 0.7 0.06 0.0 0.5-0.7.00 Table repors he correlaion coefficiens beween hese facors. Consumpion growh ( c), labor income growh ( y) and proprieary income growh ( prop) are correlaed, wih correlaion coefficiens 0.72, 0.56, 0.54. All oher correlaion coefficiens have raher small absolue value. Overall he correlaions beween hese facors are no high. We consider eigh variables as poenial condiioning variables: Defaul premium (Defaul), Term spread (Term), Dividend yield (Dividend), Consumpion o wealh raio (cay), Labor income o consumpion raio (s), Housing collaeral raio (myfa), Non-housing expendiure raio (α), and Business income growh (big). Seven of hem have appeared in previous sudies, and we add business income growh o he lis. All condiioning variables are lagged one quarer comparing wih facors. Table 2 provides he daa summary of condiioning variables. The correlaions beween proposed condiioning variables are moderae. The highes correlaion coefficien is only 0.58. Mos of hem are jus around 0. and 0.2. TABLE 2. Condiioning Variable Correlaion Coefficiens Defaul Term Dividend cay s myfa α big Defaul.00 Term 0.5.00 Dividend 0.46 0.09.00 262
Wang 263 cay 0.0 0.4 0.38.00 s 0.22-0. 0.47-0.2.00 myfa 0.4 0.00 0.58 0.2-0.5.00 α -0.0-0.05 0.5-0.0-0.09 0.29.00 big 0.07-0.5 0.0 0.03 0.05-0.04 0.00.00 Mos of facors and condiioning variables have low correlaions wih each oher, suggesing ha hey may represen differen risks ha should be priced. All of hese facors and condiioning variables have been found o be imporan in previous sudies. I is naural o ask if some facors are more imporan han ohers and if some condiioning variables are beer han ohers. Since mos previous ess are based on differen daa ses and differen mehods, i is difficul o compare heir performances. We run a horse race among hese proposed facors and proposed condiioning variables, using same daa ses and same mehod o compare heir performances in explaining he cross-secion of sock reurns. TABLE 3. Compare Condiioning Variables for One Facor Facor Condiioning Variable HJ-disance T J p-value c big 0.40 3.30 0.78 c Defaul 0.53 56.90 0.02 c s 0.56 6.60 0.00 c Term 0.56 62.0 0.00 c cay 0.56 62.0 0.00 c Dividend 0.56 62.20 0.00 c myfa 0.56 62.40 0.00 c α 0.56 63.70 0.00 Firs we compare all possible one facor condiional models. For each facor, we ry eigh differen condiioning variables, and hen choose he one ha has smalles pricing error. Table 3 gives an example of comparing performance of one facor condiioning on eigh differen condiioning variables. The resuls are ranked by HJ-disance. For consumpion risk facor c, business income growh (big) is he bes condiioning variable, wih he smalles HJ-disance. We repea he same procedure for every facor, choose he bes condiioning variable for each, and repor he resuls in Table 4. The resuls are also 263
Wang 264 ranked by HJ-disance. According o Jagannahan and Wang (996), he asympoic disribuion of T J is weighed sum of 2 i.i.d. random variables of χ²() disribuion. The p-value is esimaed by simulaion. Compared wih oher scaled facors, c condiioning on big has he bes performance. The condiional CCAPM wih he bes condiioning variable big can no be rejeced, and he pricing error is no significanly differen from zero. The marke excess reurn facor has he wors performance in erms of highes pricing errors. Term spread is he bes condiioning variable for marke risk facor. Even wih he bes condiioning variable, he condiional CAPM sill can be rejeced under 0% significance level. TABLE 4. One-Facor Models wih he Bes Condiioning Variable Facor Condiioning Variable HJ-disance T J p-value c big 0.40 3.30 0.78 y big 0.43 37.40 0.65 A lgρ Defaul 0.45 4.0 0.4 HML Defaul 0.46 42.40 0.5 SMB big 0.48 46.60 0.08 prop big 0.49 48.30 0.2 lgα Dividend 0.50 50.0 0.8 R m Term 0.50 50.70 0.09 Nex we compare all possible wo-facor condiional models. Since we consider eigh facors, we have oal 28 possible wo-facor combinaions. For each wo-facor combinaion, we ry differen condiioning variables for each facor, and choose he bes model ha has smalles HJ-disance. Table 5 repors he performance of facor combinaion ( c, logα) condiioning on differen condiioning variables. Each facor has eigh possible condiioning variables, so we ge 64 possible specificaions. The resuls are ranked by HJ-disance. The op specificaions all have big as condiioning variable for c. The asympoic disribuion of T J is weighed sum of 8 i.i.d. random variables wih χ²() disribuion. Those op specificaions can no be rejeced, ha is, pricing errors are no significanly differen from zero. 264
Wang 265 TABLE 5. Compare Condiioning Variables for Two Facors Facor Condiioning Variable Facor 2 Condiioning Variable 2 HJ-dis T J p-value c big lgα Dividend 0.28 6.20 0.99 c big lgα myfa 0.33 2.20 0.96 c big lgα α 0.33 2.30 0.95 c big lgα cay 0.33 2.40 0.96... c cay lgα α 0.55 59.50 0.00 c Dividend lgα cay 0.55 59.90 0.00 c α lgα cay 0.55 60.20 0.00 c cay lgα cay 0.55 60.20 0.00 We repea he same procedure, and choose he bes condiioning variables for each wo-facor combinaion. Resuls are repored in Table 6. The resuls are also ranked by HJ-disance. Those facors have good performance in one facor models sill have good performance in wo facor models. The facor c condiioning on big is sill he mos imporan facor. We also compare all possible hree-facor condiional models. As we expeced, he op specificaions all have he facor c condiioning on big and adding he hird facor does no improve he performance of wo-facor models much. We herefore do no repor he resuls of hree-facor models. The purpose of condiioning variable is o incorporae he ime variaion of risk premium. In condiional CCAPM, an asse's sysemaic risk is deermined by correlaion of is reurn wih consumpion growh condiioning on a sae variable ha reflecs ime variaion in risk premium. Business income growh (big) is such a sae variable. Heaon and Lucas (2000) find ha household wih high and variable business income hold less wealh in socks han similarly wealhy households. Since hey have more undiversifiable background risk, hey are more risk averse agains oher source of risks, and hen hey inves less in socks and demand higher compensaion for holding socks. Consequenly, when he aggregae proprieary business income increases, expeced risk premium goes up. 265
Wang 266 TABLE 6. Two-Facor Models wih he Bes Condiioning Variables Facor Condiioning Variable Facor 2 Condiioning Variable 2 HJ-dis T J p-value c big lgα Dividend 0.28 6.20 0.99 c big HML Defaul 0.33 2.30 0.95 c big y big 0.35 24.40 0.89 SMB big y big 0.35 25.00 0.85... HML Defaul lgα Dividend 0.43 37.40 0.7 prop big A lgρ Defaul 0.44 38.20 0.39 HML Defaul A lgρ Defaul 0.44 39.0 0.25 R m Term SMB big 0.46 4.60 0.08 5. Conclusion We run a horse race among eigh proposed facors and eigh condiioning variables, using same daa ses and same mehod o compare heir performances in explaining he cross-secion of sock reurns. We compare all possible linear facor model specificaions ha arise by combining hese facors and condiioning variables. Facor specific condiioning variables are used o reduce he number of parameers ha need o be esimaed. We find ha he consumpion risk facor ( c) wih lagged business income growh (big) as he condiioning variable performs he bes. References Breeden, D. 979. An Ineremporal asse pricing model wih sochasic consumpion and invesmen opporuniies, Journal of Financial Economics, vol. 7, pp. 265-296. Campbell, J. 996. Undersanding risk and reurn, Journal of Poliical Economy, vol. 04, pp. 298-345. Cochrane, J. 996. A cross secional es of an invesmen-based asse pricing model, Journal of Poliical Economy, vol. 04, pp. 572-62. Fama, E. and French, K. 993. Common risk facors in he reurn on bonds 266
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