Research Division Federal Reserve Bank of St. Louis Working Paper Series

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1 Research Division Federal Reserve Bank of S. Louis Working Paper Series Does Aggregae Relaive Risk Aversion Change Counercyclically over Time? Evidence from he Sock Marke Hui Guo Zijun Wang and Jian Yang Working Paper A hp://research.slouisfed.org/wp/006/ pdf Augus 006 FEDERAL RESERVE BANK OF ST. LOUIS Research Division P.O. Box 44 S. Louis, MO 6366 The views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors.

2 Does Aggregae Relaive Risk Aversion Change Counercyclically over Time? Evidence from he Sock Marke Hui Guo* Research Division, Federal Reserve Bank of S. Louis, P.O. Box 44, S. Louis, MO 6366 Zijun Wang Privae Enerprise Research Cener, Texas A&M Universiy, College Saion, TX Jian Yang Deparmen of Accouning, Finance and MIS, Prairie View A&M Universiy, Prairie View, TX This Version: Augus 006 * Corresponding auhor: Hui Guo, phone (34) ; fax (34) ; hui.guo@sls.frb.org. Par of he work was compleed when Jian Yang was a visiing scholar a he Federal Reserve Bank of S. Louis. We hank Markus Brunnermeier, Qi Li, Sydney Ludvigson, and Rober Whielaw for helpful suggesions and commens. The views expressed in his paper are hose of he auhors and do no necessarily reflec he official posiions of he Federal Reserve Bank of S. Louis or he Federal Reserve Sysem.

3 Does Aggregae Relaive Risk Aversion Change Counercyclically over Time? Evidence from he Sock Marke Absrac Using a semiparameric esimaion echnique, we show ha he risk-reurn radeoff and he Sharpe raio of he sock marke increases monoonically wih he consumpion-wealh raio (CAY) across ime. While early sudies have commonly inerpreed such a finding as evidence of he counercyclical variaion in aggregae relaive risk aversion (RRA), we argue ha i mainly reflecs changes in invesmen opporuniies for wo reasons. Firs, we fail o rejec he null hypohesis of consan RRA afer conrolling for CAY as a proxy for he hedge agains changes in he invesmen opporuniy se. Second, by conras wih habi formaion models bu consisen wih ICAPM, we find ha loadings on he condiional sock marke variance scaled by CAY are negaively priced in he cross-secional regressions. For illusraion, we replicae he counercyclical sock marke risk-reurn radeoff using simulaed daa from Guo s (004) limied sock marke paricipaion model, in which RRA is consan and CAY is a proxy for shareholders liquidiy condiions. Keywords: Habi Formaion, Time-Varying Risk Aversion, Counercyclical Sharpe Raio, Limied Sock Marke Paricipaion, Illiquidiy Premium, ICAPM, Condiional CAPM, Nonparameric and Semiparameric Models JEL Classificaion: G, C4

4 . Inroducion In Meron s (973) ineremporal capial asse pricing model (ICAPM), he condiional excess sock marke reurn, Er, is deermined by is condiional variance, M, + σ, condiional covariance, σ MF,, wih he sae variable(s), F: M, and is () Er = + γσ + λσ, M, M, MF, where γ and λ are he prices of risk. Equaion () ness wo main explanaions of sock reurn predicabiliy. Firs, he price of he marke risk, γ, is a funcion of aggregae relaive risk aversion (RRA), which changes across ime counercyclically in habi formaion models (e.g., Consaninides (990), Campbell and Cochrane (999), Brand and Wang (003), and Menzly e al. (004)). Second, he quaniy of risk, as measured by σ M, and σ, MF, exhibis a srong counercyclical paern in he daa (e.g., French e al. (987), Schwer (989), Scruggs (998), and Guo and Whielaw (006)). Recen sudies provide enaive empirical evidence for boh hypoheses. Leau and Ludvigson (00a) find ha he consumpion-wealh raio (CAY), which is he error erm from he coinegraion relaion among consumpion, wealh, and labor income, is a srong predicor of sock marke reurns. One possible explanaion is ha, in Campbell and Cochrane s (999) habi formaion model, he scaled sock price, e.g., CAY, moves closely wih ime-varying RRA. To es his idea, Leau and Ludvigson (00b) esimae a varian of he condiional CAPM by The coefficien γ is equal o RRA in he represenaive agen model wih power uiliy funcion. Appendix A shows ha, in Campbell and Cochrane s (999) habi formaion model, hese wo measures are closely relaed o each oher in a complex manner bu hey are no idenical. We hank Sydney Ludvigson for suggesing his clarificaion. Time-varying RRA is also consisen wih a few oher hypoheses. In Chan and Kogan s (00) heerogeneous-agen model, aggregae RRA changes wih he wealh disribuion, alhough individual agens have consan RRA. Ang e al. (005) and Pos and Levy (005) argue ha invesors may be risk averse for losses bu (locally) risk-seeking for gains, and such a behavior can generae a poenially complex ime-varying paern of RRA. Many works in he loss aversion lieraure (e.g., Benarzi and Thaler (995)) also endorse he idea ha invesors mainain an asymmeric aiude owards gains versus losses.

5 using CAY as he condiioning variable and find ha heir model performs subsanially beer han he uncondiional CAPM, in which RRA is consan. By conras, in Campbell s (993) ICAPM, γ and λ are consan across ime, and he scaled sock price can serve as an insrumenal variable for he hedge componen, σ MF,, in equaion (). Consisen wih his hypohesis, Guo and Whielaw (006) uncover a significanly posiive risk-reurn radeoff in he sock marke afer conrolling for CAY as a proxy for he hedge componen. This paper provides he firs aemp o evaluae he relaive imporance of hese wo hypoheses in explaining sock price movemen over he pos-world War II period. We firs esimae equaion () using he semiparameric smooh (or varying) coefficien model considered in Cai e al. (000) and Li e al. (00), in which γ depends nonlinearly on CAY in a nonparameric manner. 3 Figure summarizes he wo main findings. Firs, he solid line shows ha γ increases monoonically wih CAY in he condiional CAPM specificaion, and he relaion is saisically significan a he % level. Second, he counercyclical variaion in γ reflecs an omied variable problem. The dashed line shows ha he posiive relaion beween γ and CAY is aenuaed dramaically and becomes insignifican a he 40% level afer we also conrol for CAY as a proxy for he hedge componen. There are wo ypes of shocks in Campbell s ICAPM he discoun-rae shock and he cash-flow shock. Under some moderae condiions, he hedge componen is proporional o he condiional variance of he discoun-rae shock. One can hen use Campbell and Shiller s (988) log-linearizaion mehod o show ha he log dividend yield is a linear funcion of condiional sock marke variance and he condiional variance of he discoun-rae shock. Therefore, he scaled sock price forecass sock marke reurns because of is close relaion wih he hedge componen. For breviy, we do no provide hese derivaions here bu hey are available on reques. 3 Appendix A shows ha his specificaion is consisen wih Campbell and Cochrane s (999) habi formaion model. In heir model, he coefficien of he risk-reurn radeoff is a complex funcion of RRA, which increases monoonically wih CAY. Therefore, a posiive effec of CAY on he risk-reurn radeoff indicaes a posiive relaion beween RRA and he risk-reurn radeoff. Because he wo measures are closely relaed and idenical in some models considered here, we use RRA and he coefficien of he risk-reurn radeoff inerchangeably in he paper. Appendix A also shows ha, in Campbell and Cochrane s model, he Sharpe raio is approximaely a linear funcion of RRA. To address his issue, we also invesigae he relaion beween he condiional excess sock marke reurn and he condiional volailiy (insead of he condiional variance) and find essenially he same resuls. (See Figure A in Appendix A.)

6 For robusness, we conduc wo addiional ess. Firs, we use a nonparameric model and find a significanly posiive relaion beween RRA and he condiional sock marke variance. However, again, he counercyclical variaion in RRA disappears afer we conrol for CAY as a proxy for he hedge componen of he condiional excess sock marke reurns. Second, we find qualiaively he same resuls by using oher commonly used sock reurn predicors as insrumenal variables for ime-varying RRA. We may fail o rejec consan RRA in he ime-series daa because of a lack of power. To address his issue, we invesigae wheher he condiional CAPM helps explain he cross secion of sock reurns by using boh condiional sock marke variance and is ineracion wih lagged CAY as risk facors. The condiional CAPM performs subsanially beer in explaining he 5 Fama and French (993) porfolios sored on size and he book-o-marke (B/M) raio han does he uncondiional CAPM. However, by conras wih habi formaion models, he ineracion erm carries a significanly negaive risk premium because growh socks have larger loadings on i han do value socks. Because he ineracion erm is closely correlaed wih CAY, his seemingly puzzling resul reflecs he fac ha CAY is a proxy for he hedge agains changes in he invesmen opporuniy se. The ineracion erm loses is explanaory power afer we conrol for CAY or he Fama and French (993) B/M facor in he cross-secional regressions. Lasly, solid line in Figure shows ha we replicae he counercyclical risk-reurn radeoff in he condiional CAPM specificaion by using simulaed daa from Guo s (004) model. Because of (exogenously assumed) limied paricipaion, Guo shows ha shareholders also require an illiquidiy premium, ILL, for holding socks, in addiion o he risk premium: () r = γσ + λill + ε. M, + M, + 3

7 Two implicaions of Guo s model generae he ime-varying risk-reurn radeoff. Firs, he illiquidiy premium is posiively relaed o CAY. This is because a posiive (negaive) liquidiy shock o shareholders decreases (increases) he illiquidiy premium as well as CAY. Second, sock marke variance is a U-shaped funcion of CAY because liquidiy shocks, eiher posiive or negaive, always drive up volailiy. 4 Therefore, he risk-reurn radeoff increases monoonically wih CAY because he illiquidiy premium and he risk premium in equaion () are negaively (posiively) correlaed when CAY is low (high). To illusrae his poin, he dashed line in Figure shows ha, afer we conrol for CAY as a proxy for he illiquidiy premium, he counercyclical variaion in CAY essenially disappears because RRA is consan in Guo s model. Our resuls are consisen wih a number of recen sudies. Campbell and Vuoleenaho (004), Brennan e al. (004), and Pekova (006) find ha changes in he invesmen opporuniy se are imporan for undersanding he cross-secion of sock reurns. Leau and Wacher (006) argue ha, o joinly accoun for boh ime-series and cross-secional sock reurn predicabiliy, here mus be a weak relaion beween he discoun-rae shock and he cash-flow shock. Because he wo shocks have a perfec negaive relaion in habi formaion models, Leau and Wacher show ha hese models canno explain he B/M effec (e.g., Fama and French (993)). Li (005) finds ha he consumpion surplus in habi formaion models does no fully accoun for he predicive power of CAY for sock marke reurns. Using household-level daa, Brunnermeier and Nagel (005) show ha, by conras wih habi formaion models, wealh flucuaions do no generae ime-varying risk aversion. Many sudies, e.g., Whielaw (994), Leau and Ludvigson (003), Brand and Kang (004), Bliss and Panigirzoglou (004), Bollerslev e al. (004), Pos and Levy (005), and 4 Consisen wih his predicion, we find ha, over he pos-world War II period, he relaion beween sock marke volailiy and CAY is posiive in he firs subsample and is negaive in he second subsample. By conras, his resul is inconsisen wih habi formaion models, which predic a posiive relaion beween he wo variables. 4

8 Lundblad (006), have documened counercyclical variaion in he risk-reurn radeoff. These auhors ofen inerpre such a finding as evidence of ime-varying RRA. The evidence presened here suggess ha his inerpreaion could be misleading because by ignoring he hedge componen, he specificaions in hese sudies poenially suffer from an omied variable problem. Also, hese sudies do no invesigae he effec of he ime-varying RRA on he crosssecion of asse reurns, which we find o pose a serious challenge o habi formaion models. The remainder of he paper is organized as follows. We describe he daa in Secion and presen he esimaion resuls of he linear specificaion in Secion 3. We provide he nonlinear esimaion resuls in Secion 4 and he cross-secional evidence in Secion 5. We discuss heoreical implicaions in Secion 6 and offer some concluding remarks in Secion 7.. Daa Condiional sock marke variance is no direcly observable in he daa. In his paper, we follow Meron (980) and Anderson e al. (003) and use realized variance consruced from daily excess reurns as a proxy for condiional sock marke variance. Compared wih he GARCH model (e.g., see Bollerslev e al. (99)), his specificaion has several desirable properies for he purpose of his paper. Firs, he CAY variable he main focus of our analysis is reliably available only a he quarerly frequency; however, he GARCH model is only appropriae for he reurn daa of much higher, e.g., daily or weekly, frequencies. Second, a direc measure of condiional variance allows us o easily adop he semiparameric and nonparameric models. Third, French e al. (987) argue ha full-informaion maximum likelihood esimaors such as GARCH are generally more sensiive o model misspecificaion 5

9 han insrumenal variable esimaors. 5 More imporanly, as we show below, our resuls appear o be sensible, inuiive, and consisen wih predicions of economic heory. Tha said, we acknowledge ha realized variance is no necessarily an efficien measure of condiional variance. To address his issue, we also use monhly implied variance consruced from opions conracs on he sock marke index as a measure of condiional variance and find qualiaively he same resuls. The implied variance daa are he same as hose used in Guo and Whielaw (006), which span he period November 983 o May 00. We mainly use quarerly daa because he CAY variable is reliably available only a he quarerly or lower frequency. Also, Ghysels e al. (005) argue ha realized variance is a funcion of long disribued lags of pas daily reurns; herefore, i is likely o be more precisely esimaed a he quarerly frequency han he monhly frequency. We obain he CAY variable from Marin Leau a New York Universiy. Realized sock marke variance (MV) is he sum of squared daily excess sock marke reurns in a quarer. We use he daily sock marke reurns consruced by Schwer (989) before July, 96, and he daily CRSP (he Cener for Research in Securiy Prices) value-weighed sock marke reurns aferwards. Because he daily risk-free rae daa are no direcly available, we assume ha he risk-free rae is consan wihin each monh and calculae he daily risk-free rae by dividing he monhly CRSP risk-free rae by he number of rading days in he monh. The daily excess marke reurn is he difference beween he daily risk-free rae and he daily marke reurn. For robusness, we also use some oher commonly used sock reurn predicors as proxies for ime-varying RRA (see, e.g., Campbell (987) and Fama and French (989)). The defaul premium (DEF) is he yield spread beween he Baa- and Aaa-raed corporae bonds. The 5 Bollerslev e al. (99, p. 4) also poin ou ha he esimaion of a parameric GARCH-in-mean model can be severely biased in he presence of he model misspecificaion, especially when allowing for ime-varying parameers. Time-varying parameers also grealy inensify he concern abou he unclear heoreical properies of 6

10 dividend yield (DY) is he raio of he dividend paid in he pas one year o he end-of-period sock price for he S&P 500 socks. The erm premium (TERM) is he yield spread beween 0- year Treasury bonds and 3-monh Treasury bills. The sochasically derended risk-free rae (RREL) is he difference beween he risk-free rae and is average in he previous monhs. TERM is available over he 953:Q o 004:Q4 period and all he oher variables are available over he 95:Q4 o 004:Q4 period. Figure 3 plos MV and he oher sock reurn predicors, wih he shaded areas denoing business recessions daed by he Naional Bureau of Economic Research (NBER). All he variables are quie persisen and exhibi srong cyclical paerns. While RREL ends o decrease during business recessions, he oher variables move counercyclically. The visual inspecion is confirmed by he summary saisics presened in panel A of Table. All he variables are serially correlaed, wih he auocorrelaion coefficiens ranging from 40% for MV o 97% for DY. Also, while RREL is negaively correlaed wih a business cycle indicor, BCI, which is equal o for he recession quarers and 0 oherwise, he correlaion is posiive for all he oher variables. Panels B and C illusrae similar paerns in he wo subsamples. Table reveals an unsable relaion beween MV and some oher financial variables. MV and CAY are negaively correlaed in he full sample (panel A) and he second subsample (panel C); however, he relaion is posiive in he firs subsample (panel B). We find a similar paern for DY and RREL, which are posiively correlaed wih MV in he firs subsample (panel B) and he relaion becomes negaive in he second subsample (panel C). These resuls are inconsisen wih Campbell and Cochrane s (999) habi formaion model, which predics ha MV is posiively correlaed wih DY and CAY. As we discuss in Secion 6, hese resuls, which are consisen wih he limied sock marke paricipaion model by Guo (004), are imporan for maximum likelihood esimaor (or is varians such as quasi-maximum likelihood esimaor) in he mulivariae GARCH model (see, e.g., Engle and Kroner (995)). 7

11 undersanding he counercyclical variaion in he risk-reurn radeoff in he sock marke. Paye (006) also finds ha financial variables have raher weak forecasing power for realized sock marke variance a he business cycle frequency. For robusness, in his paper, we assume ha condiional sock marke variance is a linear funcion of realized variance only Linear Specificaions We use Guo and Whielaw s (006) linear specificaion as he benchmark model, in which he excess sock marke reurn ( r, ) is a linear funcion of condiional sock marke M + variance ( σ M, ) and financial variables ( X ) ha are proxies for he hedge componen: (3) r = α + γσ + λ X + ε, M, + M, + where α is a consan and ε + is he error erm. Panel A of Table presens he ordinary leas-squared (OLS) esimaion resuls of equaion (3) obained from quarerly daa. Row shows ha realized sock marke variance, MV, is posiively relaed o he one-quarer-ahead excess sock marke reurns bu he relaion is only marginally significan. Afer we also include CAY in he forecasing regression as a proxy for he hedge componen, he posiive effec of MV on he expeced sock reurn becomes significan a he 5% level (row 4). Guo and Whielaw (006) poin ou ha hese resuls reflec an omied variable problem. MV and CAY are boh posiively relaed o fuure sock marke reurns, alhough hey are negaively relaed o each oher in he full sample (panel A, Table ). Thus, he poin esimae of MV is downward biased if we do no conrol for CAY in he forecasing 6 Guo and Whielaw (006) assume ha condiional sock marke variance is a linear funcion of MV, CAY, and RREL. However, hey find ha some of heir resuls are sensiive o such a specificaion because of insabiliy in he relaion beween he condiional variance and CAY (p. 458). 8

12 regression. 7 Similarly, he effec of MV becomes significanly posiive a he % level afer we conrol for DEF, DY, RREL, and TERM in he forecasing equaion (row 5). Overall, row 6 shows ha, in he linear specificaion, CAY appears o be a beer proxy for he hedge componen han he oher financial variables. Panel B of Table repors very similar resuls for he monhly implied variance daa. In paricular, he poin esimae of he RRA in he full-fledge specificaion (row ) is abou 3, which is almos idenical o ha obained from he quarerly daa (row 6). This resul provides confidence ha realized variance provides a reasonably good measure of condiional sock marke variance. To summarize, consisen wih Guo and Whielaw (006), we find a posiive risk-reurn radeoff afer conrolling for he hedge componen, for which CAY is a good proxy. We hen invesigae wheher he coefficien γ in equaion (3) changes counercyclically across ime. We firs esimae a varian of he condiional CAPM, in which we follow Leau and Ludvigson (00b) by assuming ha risk reurn radeoff is a linear funcion of sae variables: (4) r = α + ( γ + γ X ) σ + ε. M, + 0 M, + Appendix A shows ha, in Campbell and Cochrane s (999) habi formaion model, he riskreurn radeoff is a complex nonlinear funcion of ime-varying RRA, which is closely relaed wih he sae variables, for example, CAY. In paricular, when CAY is high, consumpion is closer o is habi level, invesors are more risk averse, and hus expec a higher risk-reurn radeoff. Recall ha, as discussed in foonoe, equaion (4) is also consisen wih several oher economic heories, in which RRA is ime varying. In his paper, we focus on wheher he riskreurn radeoff is ime-varying and do no disinguish hese alernaive hypoheses. Noe ha he linear specificaion migh be a bi oo resricive and we relax his assumpion in he nex secion. 7 Secion 6 shows ha omiing CAY from he forecasing regression can also generae an upward bias in he poin esimae of MV when CAY and MV are posiively correlaed, as in he firs subsample (panel B, Table ). 9

13 We repor he GMM (generalized mehod of momens) esimaion resuls in Table 3. Because Table shows ha he cyclical variables are closely correlaed wih each oher, we include only one of hem in a regression. For example, for he column under BCI, we assume ha RRA is a linear funcion of a consan and he business cycle indicaor BCI. However, o improve he esimaion efficiency, we include all he cyclical variables and a consan in he insrumenal variable se. We use Hansen s (98) J-es o evaluae he goodness of fi for each specificaion. Panel A of Table 3 shows ha, consisen wih Leau and Ludvigson s (00b) finding, here appears o be srong suppor for he hypohesis ha RRA moves counercyclically in quarerly daa. The relaion beween RRA and CAY is posiive and saisically significan a he % level (row 3). The condiional CAPM accouns for abou 8% of variaion in quarerly excess sock marke reurns, which is very similar o ha of he unresriced linear specificaion repored in row 4, Table. This resul reflecs he fac ha CAY and is ineracion erm wih MV (as in equaion 4) are closely correlaed, wih a correlaion coefficien of 76%. No surprisingly, he over-idenifying resricion es does no rejec he model a any convenional significance level, indicaing he condiional CAPM provides a good descripion of he daa. Panel A of Table 3 also shows ha he relaions beween RRA and all he oher insrumenal variables have expeced signs and are saisically significan a he % level for TERM, he 5% level for BCI, MV, DY, and he 0% level for RREL. However, because Table shows ha CAY is a beer predicor of sock marke reurns, he over-idenifying resricion es overwhelmingly rejecs he specificaions wih hese variables as he proxies for RRA. Panel B of Table 3 shows ha we find similar resuls by using he monhly implied variance daa. The relaion beween RRA and CAY is posiive and significan a he % level, and we fail o rejec he condiional CAPM a any convenional significance level. However, because he sample of 0

14 he monhly implied variance daa is relaively shor, we do no precisely idenify he effec of he oher variables on RRA. Noeworhy, we need o inerpre he resuls repored in Table 3 wih cauion. By ignoring he hedge componen, he specificaion in equaion (4) poenially suffers from an omied variable problem, which could bias he risk-reurn radeoff esimae. As menioned above, in quarerly daa, CAY is closely relaed o is ineracion wih MV. Therefore, he ineracion erm in equaion (4) is found o be significanly posiive possibly because of is close correlaion wih CAY a proxy for he hedge componen. To address his issue, we add CAY o he condiional CAPM as a conrol for he hedge componen: (5) r = α + ( γ + γ X ) σ + λcay + ε. M, + 0 M, + Noe ha including he oher insrumenal variables as proxies for he hedge componen does no change he resuls in any qualiaive manner because Table shows ha hey provide lile informaion abou fuure sock reurns beyond CAY. Table 4 presens he esimaion resuls of equaion (5). For quarerly daa (panel A), he relaion beween RRA and CAY becomes saisically insignifican a any convenional level, alhough i is remains posiive. Ineresingly, he relaions beween RRA and all he oher sae variables are also saisically insignifican afer we conrol for CAY as a proxy for he hedge componen. Also, panel B shows ha we find very similar resuls by using he monhly implied variance daa. Lasly, for robusness, we assume ha ime-vary RRA is a linear funcion of all he sae variables. These variables are joinly significan in he condiional CAPM specificaion (equaion 4); however, he join explanaory power becomes saisically insignifican a he convenional level afer we conrol for he hedge componen (equaion 5). For breviy, we do no repor hese resuls here bu hey are available on reques. To summarize, he counercyclical risk-reurn radeoff appears o be mainly explained by he hedge agains changes in he

15 invesmen opporuniy se bu no he counercyclical variaion in RRA. Equaion (A9) in Appendix A shows ha, in Campbell and Cochrane s (999) habi formaion model, he Sharpe raio is approximaely a linear funcion of RRA. To address his issue, we use condiional volailiy insead of condiional variance in equaions (4) and (5) and find essenially he same resuls. For example, he Sharpe raio is posiively and significanly relaed o CAY; however, he relaion becomes insignifican a any convenional level afer we conrol for CAY a proxy for he hedge componen. For breviy, hese resuls are no repored here bu are available on reques. 4. Nonlinear Specificaions Asse pricing heories do no provide unambiguous guidance for he funcional form of he empirical specificaion, and i is a bi oo resricive o assume ha RRA and he hedge componen are linear funcions of sae variables, as in equaion (5). In paricular, Ghysels (998) argues ha a parameric asse pricing model wih a known funcional form may yield misleading resuls if he funcional form is misspecified. I is emping o use fully nonparameric models because hey are robus agains he funcional form misspecificaion; however, hey also have some drawbacks. Firs, i may no esimae he condiional mean wih high accuracy. Second, i ofen canno be esimaed wihou running ino a serious curse of dimensionaliy problem, when he daa are raher limied, as in our sudy. This is because he rae of convergence of many nonparameric esimaors worsens dramaically as he number of covariaes increases. For example, i appears ha he number of quarerly daa in his paper can meaningfully allow for no more han one covariae in he nonparameric esimaion. To address hese issues, we adop several popular classes of semiparameric nonlinear specificaions, which are well suied for capuring he poenially complex nonlineariy wihou

16 much loss of generaliy. In general, he semiparameric models have he advanage of allowing for more appreciable flexibiliy in funcional forms han does a parameric linear or nonlinear model. A he same ime, hey can gain more esimaion efficiency han nonparameric models wih (correcly) imposed lineariy resricions on some componens of he model. Also, hese models can avoid much of he curse of dimensionaliy problem ha plagues fully nonparameric models, which ofen render (meaningful) nonparameric model esimaion (and inference) infeasible for he limied amoun of economic daa. Lasly, hese models end o be easier o inerpre and hus could be more informaive han fully nonparameric models. In addiion o he general appealing saisical properies, he semiparameric models considered here are paricularly suiable for he main purpose of he paper. The mulifacor asse pricing models, as in equaion (), are no ineresed in general ineracions beween differen risk facors, which can be bes capured by a fully nonparameric model. Insead, we are ineresed in wheher he prices of risk facors are poenially nonlinear funcions of some sae variables, e.g., CAY, as suggesed by finance heories. As we show below, one can illusrae his dependence in an inuiive manner by using he semiparameric smooh coefficien model (Cai e al., 000; Li e al., 00), which allows for a sae variable o affec RRA in a nonparameric nonlinear manner. For robusness, we also consider semiparameric parially linear and addiive models (and a nonparameric model in he one-facor conex), in which he price of marke risk does no depend on any sae variable. We obain essenially he same conclusion by using boh classes of semiparameric nonlinear models. See Appendix B for more deails on esimaion of hese models and associaed model specificaion ess. 3

17 4.. Semiparameric Smooh Coefficien Model To address he poenial nonlineariy in boh he risk and hedge componens, we firs adop he following smooh coefficien model: (6) r = γ ( X ) σ + λ( X ) + ε, M, + M, + where he coefficiens γ ( X ) and λ( X ) are unspecified smooh funcions of sae variables X. The model is quie general and ness hreshold regression models, smooh ransiion regression, and many oher regime-swiching models as special cases. Due o he relaively small number of observaions, we can allow for only one sae variable in he coefficiens γ ( X ) and λ ( X ). This limiaion is innocuous because our main focus is o es wheher CAY proxies for ime-varying RRA or he hedge componen, as suggesed by finance heories. Similar o Li e al. (00), we esimae he erm γ ( X ) nonparamerically using a local consan esimaor. We use he normal disribuion as he kernel funcion, in which he smoohing parameer or he bandwidh of he window of he kernel esimaion is deermined by popular leave-one-ou leas square cross-validaion mehod. We firs es he null hypohesis of a consan risk-reurn radeoff (7) r = α + γσ + ε M, + M, + agains he general smooh coefficien model, as in equaion (6). This es, which is equivalen o a semiparameric varian of he omied variable es as discussed in Fan and Li (996), addresses wheher sae variables X provide addiional informaion abou fuure sock marke reurns beyond condiional sock variance ha eners he equaion linearly as implied by he CAPM. To evaluae he relaive performance of he wo models, we use he boosrap version of he goodness-of-fi es saisic advocaed by Cai e al. (000), which can be undersood as a ype of generalized likelihood raio ess. Panel A of Table 5 shows ha CAY provides imporan 4

18 informaion abou fuure sock marke reurns beyond he condiional sock marke variance, and such a relaion is saisically significan a he % level. We hen invesigae wheher he effec of he sae variables comes from heir roles as he condiioning variables for ime-varying RRA, as in habi formaion models: (8) r = α + γ( X ) σ + ε. M, + M, + The benchmark or null model remains o be he condiional CAPM wih consan RRA, as in equaion (7). Panel B of Table 5 shows ha we rejec he linear one-facor model and accep he alernaive of he model wih sae-variable-dependen RRA for CAY a he % level. Moreover, he solid line in Figure shows ha he esimaed RRA increases monoonically wih CAY and he relaion is srikingly close o being a linear one. These resuls confirm ha he specificaion of RRA as a linear funcion of CAY (equaion 4) provides a good descripion of he expeced sock marke reurns, as repored in row 3, Table 3. Ineresingly, he esimaed RRA is negaive when CAY is low bu becomes posiive when CAY is high. Many early sudies, e.g., Campbell (987), Glosen, Jagannahan, and Runkle (993), Whielaw (994), Leau and Ludvigson (003), and Brand and Kang (004), have also documened a negaive risk-reurn radeoff. Noe ha he negaive RRA poses a challenge o habi formaion models because hey predic a posiive risk-reurn radeoff. Nex, we show ha he seemingly puzzling finding reflecs an omied variable problem. Table 4 shows ha he ime-varying risk-reurn radeoff migh reflec he counercyclical variaion in he hedge componen. To address his issue, in panel C of Table 5, we invesigae wheher he counercyclical variaion in RRA remains saisically significan afer we conrol for he hedge componen, which is a linear funcion of he sae variable: (9) r = γ ( X ) σ + λx + ε. M, + M, + 5

19 The benchmark model is ha he expeced excess sock marke reurn is a linear funcion of condiional sock marke variance and he hedge componen, as in equaion (3). Consisen wih he resuls repored in Table 4, panel C of Table 5 shows ha we fail o rejec he null hypohesis of no relaion beween RRA and CAY a he 40% significance level afer conrolling for CAY as a proxy for he hedge componen. The dashed line in Figure shows ha, alhough he esimaed RRA sill increases wih CAY, he relaion is dramaically weaker han he case wihou he conrol for he hedge componen, as illusraed by he solid line in Figure. Ineresingly, afer we conrol for CAY as a proxy for he hedge componen, he esimaed RRA is always posiive and falls ino a igh range 0.9 o 3.3. The poin esimae also falls comforably wihin he plausible range o 0, as advocaed by Mehra and Presco (985). Therefore, allowing for ime-varying RRA does no change Guo and Whielaw s (006) main finding of a posiive risk-reurn radeoff in any qualiaive manner. Many finance heories, e.g., Campbell and Cochrane s (999) habi formaion model and Guo s (004) limied paricipaion model, predic a posiive relaion beween CAY and fuure excess sock marke reurns; however, such a relaion does no have o be linear. To address his issue, we allow for he possible nonlinear presence of he hedge componen, which is modeled as a nonparameric funcion of a single sae variable, λ ( X ) : (0) r = γσ + λ( X ) + ε. M, + M, + As a saring poin, we assume ha RRA is consan in equaion (0) bu will relax his assumpion laer. The benchmark model is ha he expeced excess sock marke reurn is a linear funcion of condiional variance and he hedge componen, as in equaion (3). Panel D of Table 5 shows ha we fail o rejec he null hypohesis of he linear presence of he hedge 6

20 componen for CAY a he 50% significance level. Similarly, he solid line in Figure 4 shows ha he effec of CAY on he expeced excess sock marke reurn is essenially linear. We hen compare he linear specificaion of equaion (3) wih he general smooh coefficien specificaion in equaion (6). Panel G of Table 5 shows ha, again, we canno rejec he linear specificaion a any significance level for he CAY variable. Also, he esimaed coefficiens γ ( X ) and λ( X ) are essenially he same as hose ploed in Figure (dashed line) and Figure 4 (solid line), respecively. Lasly, for compleeness, we also compare he specificaions in equaions (0) and (9) wih he general smooh coefficien specificaion in equaion (6) and find no evidence of nonlineariy in eiher he risk (panel E) or he hedge (panel F) componen. To summarize, he linear specificaion of equaion (3), as adoped in Guo and Whielaw (006), has explanaory power for he expeced sock marke reurn almos idenical o ha of he more elaborae nonparameric smooh coefficien model. This finding suggess ha one can use he simple linear specificaion wihou much loss of generaliy. We find similar resuls by using he oher financial variables as proxies for he imevarying RRA. Panel A of Table 5 shows ha DEF, DY, and TERM provide imporan informaion abou fuure sock marke reurns beyond condiional sock marke variance. Consisen wih he resuls repored in Table 3, panel B shows ha DEF, DY, and TERM have a significan effec on RRA in he one-facor model. Also, he esimaed RRA moves counercyclically in all cases. (For breviy, his resul is no repored here bu available on reques) However, by conras wih he resuls repored in Table 4, panel C shows ha heir effecs in RRA remain saisically significan (DY and TERM) or marginally significan (DEF) afer we conrol for he hedge componen, which is a linear funcion of hese sae variables. There are wo reasons for he difference. Firs, consisen wih he finding in Boudoukh e al. (997) and Harvey (988), panels D and F of Table 5 show ha here is a significan nonlinear 7

21 relaion beween TERM (as a proxy for he hedge componen) and he expeced sock marke reurn. Afer we conrol for he nonlinear effec of he hedge componen on he expeced reurn, panel E of Table 5 shows ha we fail o rejec he null hypohesis of no relaion beween TERM and RRA a he 7% significance level. Second, Table shows ha DEF and DY alone do no capure all he variaion in he hedge componen. To address his issue, we augmen he proxy for he hedge componen wih addiional sae variable(s): () r = γ ( X ) σ + λ X + λ X + ε M, +, M,,, + or () r = γ ( X ) σ + λ ( X ) + λ X + ε. M, +, M,,, + We hen es he augmened models wih ime-varying RRA agains he augmened benchmark model wih consan RRA: (3) r = γσ + λ X + λ X + ε. M, + M,,, + The models in equaions () and () allow for poenial nonlinear dependence of RRA on one sae variable (i.e., DEF and DY), which is of cenral ineres. Also, while he firs model (equaion ) allows for he linear presence of boh iself and one or all of he oher sae variables as proxy for he hedge componen, he second model (equaion ) allows for he nonlinear presence of iself and he linear presence of one or all of he oher sae variables as proxy for he hedge componen. When we use all he sae variables as arguably he bes empirical proxy for he hedge componen, we fail o rejec he null hypohesis of no dependence of RRA on DEF or DY a he 0% significance level in boh specificaions. For breviy, hese resuls are no repored here bu are available on reques. 8

22 Lasly, we invesigae he relaion beween expeced excess sock marke reurns and condiional sock marke variance, as implied by Campbell and Cochrane s habi formaion (see equaion (A9) in Appendix A). The resuls are essenially he same as hose repored above. For example, we find ha he Sharpe raio is posiively relaed o CAY and such a relaion is saisically significan a he % level. However, i becomes insignifican a he over 40% level afer we conrol for CAY as a proxy for he hedge componen. Figure A in Appendix A shows ha here is a srong posiive relaion beween he Sharpe raio and CAY (solid line); and i is aenuaed dramaically afer we conrol for he hedge componen (dashed line). These paerns are essenially he same as hose in Figure. 4. Volailiy-Dependen Risk Aversion The full-fledged semiparameric smooh coefficien wo-facor model is quie general because i allows for he effec of boh he risk and hedge componens on he expeced reurn o vary across business cycles. However, i does no adequaely address he possibiliy of imevarying risk aversion driven by volailiy regimes shif, which may or may no be he same as he sae-variable-dependen risk aversion. To address his issue, we consider a raher general addiive wo-facor model, (4) r = g( σ ) + λ( X ) + ε, M, + M, + where we sill allow he sae variable as proxy for he hedge componen o have poenially nonlinear effecs on he expeced sock reurn, as in he smooh coefficien model. The difference beween he addiive model (equaion 4) and he smooh coefficien model (equaion 6) lies in he specificaion of volailiy. In he addiive model, ime-varying RRA is modeled as an unspecified funcional form in volailiy. Such an issue of poenial volailiy-dependen risk aversion is also considered by Mayfield (004), Bliss and 9

23 Panigirzoglou (004), and Lundblad (006); and heir specificaions can be nesed in he wofacor model in equaion (4). Neverheless, unlike he smooh coefficien model, he addiive model does no allow for he poenial ineracion beween he sae variable and he volailiy. Hence, hese wo classes of nonparameric models are designed o capure differen ypes of nonlineariy, boh of which have been invesigaed in he exising lieraure. Again, we sar wih esing he general addiive wo-facor model (equaion 4) agains he CAPM wih consan RRA (equaion 7). We also use he boosrap version of he goodnessof-fi es saisic advocaed by Cai e al. (000) o evaluae he relaive performance of he wo models. 8 Panel A of Table 6 shows ha, in he cases of CAY and TERM, here is again evidence agains he adequacy of he linear CAPM model, which could be due o eiher he nonlinearly priced risk componen (as driven by he volailiy-dependen risk aversion) or he linearly or nonlinearly priced hedge componen. Nex, recognizing he possibiliy of rejecion due o inadequacy of capuring volailiydependen risk aversion in he linear one-facor model (equaion 7), we consider a one-facor CAPM model wih poenially volailiy-dependen risk aversion as he alernaive specificaion: (5) r = g( σ ) + ε. M, + M, + Several recen sudies have invesigaed specificaions ha are similar o ha in equaion (5). Bliss and Panigirzoglou (004) consider wo equal-sized subsamples corresponding o periods of high and low volailiy and examine wheher he esimaed RRA differs across he wo subsamples. Mayfield (004) uses a more sophisicaed model o allow for wo regimes of sock marke volailiy bu assumes he same RRA in boh saes of volailiy. Lundblad (006) no only allows for wo regimes of sock marke volailiy bu also allows for he differen values of RRA in he wo regimes. Our model is more general han hese specificaions by observing ha 0

24 g( σ ) can approximae for s, γ ( σ ) σ, where S denoes differen regimes of volailiy (e.g., s, s, high versus low) as deermined by differen hreshold levels of volailiy. Noe ha γ ( σ ) σ s, s, allows for boh muliple (raher han wo) regimes in volailiy and differen risk aversion coefficiens in each regime. 9 Panel B of Table 6 shows ha we can rejec he linear one-facor model and accep he alernaive specificaion of he one-facor CAPM wih volailiy-dependen risk aversion a he 5% level. Solid line in Figure 5 plos he fied dependen variable from he nonlinear one-facor model agains condiional variance, and he slope of he curve represens he risk aversion coefficien. I is clear ha he slope of he nonparamerically fied curve is generally upward, and no downward, indicaing a posiive risk-reurn radeoff. Ineresingly, Our resul appears o verify he exisence of roughly wo regimes of volailiy, as assumed in Mayfield (004). When condiional sock marke variance is relaively low, he slope is fla, indicaing weak risk aversion. However, when sock marke variance is higher, he upward slope becomes seeper and hus suggess sronger risk aversion. This finding is consisen wih he resuls repored in row of Table 3, which shows ha he risk-reurn radeoff increases wih condiional or realized sock marke variance. Bu i differs from ha in Bliss and Panigirzoglou (004), who find an inverse relaion beween sock marke variance and heir opion-based measures of RRA. One possible reason is ha hese auhors use a relaively shor sample spanning he period 983 o 00, as opposed o he 953 o 004 period used here. 8 For parially linear and addiive model specificaion ess in Table 6, we also implemen anoher goodness of fi es due o Dee (999) and Fan and Huang (00), and find ha he resuls are qualiaively he same. 9 Mayfield (004) uses a wo-facor model, and our poin here would beer apply o our addiive wo-facor model wih such nonparameric funcion in volailiy.

25 To address he concern abou he poenial omied-variable problem, we allow for a linear presence of he hedge componen (proxied by one sae variable) in he model of volailiydependen risk aversion: (6) r = g( σ ) + ( α + λx ) + ε. M, + M, + The benchmark is he linear wo-facor model, as in equaion (3). Panel C of Table shows ha we fail o rejec he null hypohesis of no volailiy-dependen risk aversion a any convenional significance levels across all he five sae variables considered afer allowing for he linear presence of he hedge componen. In paricular, he dashed line in Figure 5 shows ha he posiive relaion beween g( σ ) and M, σ becomes very close o being a linear one afer we M, conrol for CAY as a proxy for he hedge componen. Lasly, we esimae he addiive model, which allows for nonlinear presence of boh risk and hedge componens, as specified in equaion (4). The benchmark model is again he linear wo-facor model, as in equaion (3). The resul (Panel D, Table 6) confirms no rejecion of he linear wo-facor model excep for TERM. Noe ha he rejecion of he linear model for TERM reflecs is nonlinear effecs on he expeced sock reurn as a proxy for he hedge componen. Overall, consisen wih he smooh coefficien model, he resul suggess ha he Guo and Whielaw s (006) specificaion of he expeced excess sock marke reurn as a linear funcion of condiional variance and CAY provides a reasonably good descripion of he daa. The disappearance of volailiy-dependen risk aversion in he wo-facor model could again be a manifesaion of he omied variable bias in he one-facor model and can be well explained by he model of Mayfield (004). Specifically, Mayfield (004) heoreically demonsraes ha changes in invesmen opporuniies can be roughly proxied by unpredicable, sae-dependen changes in he level of sock marke volailiy. Neverheless, as his model is only a special case of Meron s ICAPM, he explanaory power of he sae-dependen volailiy

26 regimes may well be subsumed by he sae variables, which could be beer proxies for invesmen opporuniies. 4.3 Monhly Daa We have repeaed he above analysis using monhly implied variance daa. In general, he resuls are qualiaively he same as hose found in quarerly daa. For example, in he addiive model, we find a significan nonlinear risk-reurn radeoff in he sock marke, which ends o comove posiively wih sock marke variance. Also, he counercyclical variaion in RRA disappears afer we conrol for CAY as a proxy for he hedge componen. In he smooh coefficien model, we find ha CAY provides imporan informaion abou fuure sock marke reurns beyond condiional sock marke variance. However, because of he relaively shor span, counercyclical variaion in RRA is never significan in he smooh coefficien model, even wihou he conrol for he hedge componen. For breviy, we do no repor hese resuls here bu hey are available on reques. 5. Cross-Secional Evidence We have shown ha he risk-reurn radeoff in he sock marke moves counercyclically in he condiional CAPM specificaion, and such a relaion becomes saisically insignifican afer we conrol for CAY as a proxy for he hedge componen. This resul appears o be robus because we reach he same conclusion by using hree differen specificaions. We argue ha hese resuls are consisen wih he hypohesis of ime-varying invesmen opporuniies, as in Meron s ICAPM. However, i is imporan o noe ha we canno compleely rule ou he hypohesis of ime-varying RRA, as in habi formaion models. In paricular, because of he close relaion beween CAY and is ineracion wih MV (CAY*MV), 3

27 he ime-series daa do no allow us o draw a clear-cu disincion beween he wo hypoheses. To illusrae his poin, we run a regression of sock marke reurns on a consan, MV, CAY (as a proxy for he hedge componen), and he ineracion erm beween MV and CAY (as a proxy for ime-varying RRA). We find ha, because of he mulicollineariy problem, he ineracion erm is saisically insignifican and CAY is significan only a he 0% level. Thus, here is only marginal suppor for ime-varying invesmen opporuniies. To furher differeniae he wo hypoheses, in his secion, we follow Leau and Wacher s (006) suggesion and invesigae heir implicaions for he cross-secion of sock reurns. We invesigae wheher a varian of he condiional CAPM helps explain he crosssecion of sock reurns on he 5 Fama and French (993) porfolios sored on size and he book-o-marke raio over he 95:Q o 004:Q4 period. For each of he 5 porfolios, we firs run he ime-series regression: r = α + γ MV + γ MV * CAY + ε, (7) P, + p p0 p + where r P, + is he excess reurn on he porfolio p. If loadings on he marke risk are consan across ime, as assumed in Leau and Ludvigson (00b), he coefficiens γ p0 and γ p are proporional o loadings on he marke risk. Under he null hypohesis of habi formaion models, he ineracion erm CAY*MV in equaion (7) should carry a posiive risk premium. We can also moivae equaion (7) using he equilibrium model by Zhang (005). Zhang shows ha, in he presence of adjusmen coss for invesmen, socks wih high B/M (value socks) have higher expeced reurns han socks wih low B/M (growh socks) because he former end o be more risky when he risk-reurn radeoff or RRA is high. Therefore, under he null hypohesis of habi formaion models, we expec ha value socks have higher loadings on he ineracion erm MV*CAY in equaion (7) han do value socks. Figures 6 and 7 plo loadings of he 5 Fama and French porfolios on condiional sock 4

28 marke variance MV and he ineracion erm MV*CAY, respecively. 0 Each porfolio is idenified wih a wo-digi number. The firs digi refers o size, wih denoing he smalles socks and 5 he larges socks. The second digi refers o B/M, wih denoing he lowes B/M raion and 5 he highes B/M raio. Figure 6 shows ha, consisen wih early sudies, e.g., Leau and Wacher (006), growh socks end o have higher loadings on he marke risk han do value socks wihin each size quinile. However, by conras wih habi formaion models, Figure 7 shows ha growh socks have subsanially higher loadings on he ineracion erm han do value socks wihin each size quinile. We hen invesigae wheher loadings on MV and MV*CAY help explain he crosssecion of sock reurns by using he Fama and MacBeh (973) cross-secional regression approach. Row of Table 7 shows ha he condiional CAPM accouns for over 40% of variaion in he cross-secion of sock reurns. This resul clearly indicaes ha he condiional CAPM is a subsanial improvemen over he uncondiional CAPM, which has negligible explanaory power for he 5 Fama and French porfolios (see, e.g., Leau and Ludvigson (00b)). More imporanly, he ineracion erm MV*CAY is significanly priced a he 5% level, according o Shanken s (99) correced sandard errors (as repored in squared brackes). However, here is a problem wih he condiional CAPM inerpreaion. The ineracion erm carries a negaive risk premium, as opposed o he posiive premium prediced by habi formaion models. The negaive premium reflecs he fac ha growh socks have higher loadings on he ineracion erm han do value socks (Figure 7). Therefore, consisen wih he heoreical work by Leau and Wacher (006), our empirical resuls indicae ha habi formaion models canno explain he B/M effec. To summarize, he cross-secional evidence cass doub on he hypohesis 0 In he ime-series regressions, he wo facors are saisically significan a he 5% level for mos porfolios. For breviy, we do no repor he resuls here bu hey are available on reques. Several recen sudies, e.g., Pekova and Zhang (005), Lewellen and Nagel (005), and Fama and French (005), have also cas doub on explanaory power of he condiional CAPM for he cross-secion of sock reurns. 5