On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt?

Transcription

1 THE JOURNAL OF FINANCE VOL LIII, NO. 2 APRIL 1998 On Stable Factor Structures n the Prcng of Rsk: Do Tme-Varyng Betas Help or Hurt? ERIC GHYSELS* ABSTRACT There s now consderable evdence suggestng that estmated betas of uncondtonal captal asset prcng models (CAPMs) exhbt statstcally sgnfcant tme varaton. Therefore, many have advocated the use of condtonal CAPMs. If we succeed n capturng the dynamcs of beta rsk, we are sure to outperform constant beta models. However, f the beta rsk s nherently msspecfed, there s a real possblty that we commt serous prcng errors, potentally larger than wth a constant tradtonal beta model. In ths paper we show that ths s ndeed the case, namely that prcng errors wth constant tradtonal beta models are smaller than wth condtonal CAPMs. LINEAR FACTOR MODELS such as the uncondtonal captal asset prcng model (CAPM) and the arbtrage prcng theory (APT) have been the cornerstone of theoretcal and emprcal fnance for decades now. Supported by semnal papers, lke those of Sharpe (1964), Lntner (1965), Merton (1973), and Ross (1976), they are the most wdely used tools to value the return on rsky assets. Although the theory mantans a lnear and stable relatonshp between rsk factors and returns, there s now consderable emprcal evdence documentng tme varaton n market betas and other factor payoffs. Ths s perhaps not so surprsng because the theoretcal underpnnngs of the uncondtonal arbtrage-prcng theory reveal that tmenvarant lnear factor structures are only obtaned when one mposes strong assumptons on the underlyng probablty dstrbutons and nvestors atttudes toward rsk. 1 In practce, many portfolo managers constantly update and reestmate factor returns, and ndeed Harvey (1989), Ferson and Harvey (1991, 1993), *Department of Economcs, Pennsylvana State Unversty, and CIRANO. We would lke to thank Wayne Ferson and Campbell Harvey for ther nvaluable help n provdng ther data. We are also most grateful to two referees and to the edtor, René Stulz, for ther very helpful comments. Benot Durocher provded excellent research assstance. We would also lke to thank Clff Ball, Rav Bansal, Wayne Ferson, René Garca, Erc Jacquer, Erc Renault, Perre Séquer, S. Vswanathan, and partcpants at the CREST Compagne Bancare Conference, the 1995 Western Fnance Assocaton and Amercan Fnance Assocaton Meetngs, the Brazlan Econometrc Socety Meetngs, HEC Pars, and Penn State for ther comments. 1 Several general equlbrum developments of the uncondtonal CAPM and APT have been advanced; for example, see Huberman (1982), Chamberlan and Rothschld (1983), Ingersoll (1984), Connor (1984), Connor and Korajczyk (1989), among others. 549

2 550 The Journal of Fnance and Ferson and Korajczyk (1995) fnd that estmated betas exhbt statstcally sgnfcant tme varaton. There appears to be a consensus now about the falure of the statc CAPM; for example, see the wdely cted work of Fama and French (1992) and the dscusson t generated. Some advocate that the statc CAPM should be replaced by some form of tmevaryng beta condtonal CAPM (see, e.g., Jagannathan and Wang (1996)). If we succeed n capturng the dynamcs of beta rsk, we are sure to outperform constant beta or so-called uncondtonal CAPM and APT models. But what f we do not succeed n correctly specfyng tme-varyng betas? Then, f beta rsk s nherently msspecfed, there s a real possblty that we commt serous prcng errors that potentally could be bgger than wth a constant beta model. In ths paper, we show that ths s not purely an abstract speculaton or remote possblty. Indeed, we show that n many cases the prcng errors wth constant beta models are smaller than those wth tme-varyng beta models. The msspecfcaton of the latter appears to be serous enough that tme-varyng beta models do not help but actually hurt. The models we consder are several condtonal CAPM and APT dynamc asset prcng models. Ferson (1985), Ferson and Harvey (1991, 1993), Harvey (1991), Ferson and Korajczyk (1995), and Dumas and Solnk (1995), among others, have appled these models to prce nternatonal equtes, bonds, and sze-sorted and ndustry-based portfolos, as well as forward currency contracts. How do we fnd out that these tme-varyng beta models are msspecfed? One way, perhaps somewhat roncally, t s to nvestgate the exact same ssue that motvated the replacement of the uncondtonal CAPM and APT by condtonal models. Indeed, msspecfcaton n beta rsk dynamcs s almost always revealed by nonconstancy of the beta rsk model parameters (just as msspecfcaton of the CAPM s revealed by tme varaton n the betas). Hence, we test whether there are structural shfts n the parameters of condtonal CAPM and APT models and fnd overwhelmng evdence for structural breaks. Ths means that condtonal CAPM and APT models prevously presented n the lterature are msspecfed. In Secton I we dscuss the mpact of msspecfcaton on prcng errors explanng ntutvely ways to assess msspecfcaton through testng for parameter constancy. Secton II presents a bref revew of the condtonal CAPM and APT and the nonlnear APT models of Bansal and Vswanathan (1993) and Bansal, Hseh, and Vswanathan (1993). Although nonlnear APT models do not provde explct prcng formulas, t s of nterest to examne ther constancy and compare them wth the other models. Usng a unform data set across all models, we report n Secton III a comprehensve emprcal study wth monthly NYSE stock returns. The results reveal some serous specfcaton problems. On that bass we proceed wth the analyss of prcng errors. In Secton IV we report results showng a strong domnance of the tradtonal CAPM n predctng returns for sze-sorted and ndustrybased portfolos. Secton V concludes the paper.

3 D D Stable Factor Structures n Prcng Rsk 551 I. Predctng Asset Returns and Specfcaton Errors Let us concentrate on a very smplfed verson of the condtonal CAPM to set the stage for our dscusson: t 1 6Z t # b t E@r Mt 1 6Z t #, (1) where b t s the parameterzed tme-varyng market beta and Z t s a set of nstruments. The excess return from t to t 1 on the market portfolo s measured by r Mt 1, and r t 1 s the excess return on any asset or portfolo of assets. Once we admt that beta vares through tme we must specfy laws of moton for b t. A condtonal CAPM does precsely that; namely, wth a sngle nstrument t mples the followng: b t E@~r Mt 1 d M Z t!~r t 1 d Z t!6z t # E@~r Mt 1 d M Z t! 2. (2) 6Z t # From equaton (2) we note that two fxed parameters, namely d M and d, together wth Z t, r M, and r, determne the tme varaton n b t. The two parameters are obtaned va the projecton equatons jt 1 6Z t # d j Z t j,m. (3) The queston we are nterested n s whether ths partcular (or any other) characterzaton of b t s adequate and does not yeld a systematc msprcng of rsk factors. Combnng equatons (1) and (3), we can wrte the asset prcng equaton as follows: r t 1 b t d M Z t u t 1, (4) where E~u t 1 Z t! 0. If the restrctons of the condtonal CAPM do not hold because beta rsk s nherently msspecfed, we obtan as a generc alternatve r t 1 bd t dd Mt Z t Iu t 1, (5) wth E~ Iu t 1 Z t! 0 and b t b t obtaned from equaton (2) replacng d M by d Mt and d by dd t. It should be noted that the generc alternatve n equaton (5) emphaszes the fact that the specfcaton of b t s erroneous. Other sources of msspecfcaton, such as omtted factor rsk, are, at least for the moment, not consdered. No specfc laws for dd Mt or dd t (and hence bd t ) are used at ths pont. To assess the consequences of msspecfcaton we wll compare two dfferent scenaros. The frst scenaro s one n whch both the tradtonal fxed beta CAPM and the condtonal CAPM are msspecfed. Under the second scenaro only the uncondtonal CAPM s erroneous. We study prcng errors under both scenaros, assumng that we can gnore estmaton uncer-

4 552 The Journal of Fnance tanty. When the condtonal CAPM s correctly specfed (.e., the second scenaro) t should be better at predctng asset returns. Let us formalze ths ntuton before movng to the frst scenaro where both models are msspecfed. To examne the magntude of prcng errors we consder the mean squared error (MSE). For the condtonal CAPM we have MSE 2C [ t 1 b t d M Z t # 2 Var~u t 1!, (6) where MSE jk s the mean squared error for asset under scenaro j 1, 2, and model k C for condtonal CAPM and k U for the uncondtonal one. To determne MSE 2U let us wrte equaton (2) as b t [b b~z t ;~d,d M )), where b s the (fxed uncondtonal) beta for asset. 2 Then we obtan MSE 2U [ b b~z t ;~d,d M!!!d M Z t u t 1 b d M Z t # 2 Var(u t 1! d M 2 E@b~Z t ;~d,d M!!Z t # 2 2d M Eb~Z t ;~d,d M!!Z t u t 1. The last term on the rght-hand sde of equaton (7) s zero snce u t 1 s orthogonal to any functon of Z t. Hence, we have that MSE 2U s larger than MSE 2C by a factor of d 2 M t ;~d,d M!!Z t # Clearly, as there s evdence of tme-varyng beta we have an obvous nterest n adequately modelng tme-varyng betas and explotng them for prcng securtes. But what f the tme-varyng betas are prone to specfcaton error? Ths s precsely the frst scenaro. Under equaton (5) we can compute MSE 1C and MSE 1U. Some algebra yelds the followng: MSE 1U MSE 1C d 2 M t ;~d M,d!!Z t # 2 (7) bd t dd Mt b d M!b~Z t ;~d M,d!!d M Z 2 t #. (8) The second term on the rght-hand sde of equaton (8) can be negatve, or postve and larger than the frst term. Hence, dependng on ts magntude and sgn, we may have MSE 1U. MSE 1c, whch means the condtonal CAPM yelds the smallest prcng error; but we can also have the reverse, MSE 1U, MSE 1C. A pror we do not really know, as t all depends on how severely the condtonal CAPM s msspecfed relatve to the uncondtonal one. In the next secton we present the condtonal APT and CAPM we wll consder to nvestgate ths queston. In Secton III we explan how to test that the dynamcs of betas are msspecfed. 2 The functonal b({) s ntroduced to hghlght the fact that n equaton (2) beta rsk s mplctly a functon of Z t and parameterzed by d M and d.

5 Stable Factor Structures n Prcng Rsk 553 II. A Revew of the Condtonal Asset Prcng Models We follow Bansal and Vswanathan (1993) closely and start from the optmal portfolo allocaton condtons of dscrete tme captal asset prcng models. 3 In an economy wth N assets we obtan the followng frst-order condtons: t 1!x t 1 6 t # ~x t 1! for 1,...,N, (9) where x t 1 s the one-perod payoff of the th asset at tme t 1 that has tme t prce ~x t 1! and where MRS~t, t 1! s the representatve agent s margnal rate of substtuton between t and t 1 consumpton. The expectaton n equaton (9) s condtonal on the nformaton set t. Equaton (9) also holds when we replace MRS~t, t 1! by ts projecton on the space of all * one-perod payoffs. Let us denote ths projecton as P t 1. Hansen and Jagannathan (1991) show that ths projecton can be expressed as a lnear combnaton of the N asset one-perod payoffs represented by the vector x t t 1 # N 1 : * P t 1 N j 1 ( a jt x jt 1, (10) N where the weghts a jt # j 1 satsfy ' t 1 x t 1 6 t ## 1 ~x t 1!. (11) Equatons (10) and (11) represent a fundamental relatonshp n characterzng the prcng of assets but they do not yet comprse a workable model, whch nvolves as many factors as there are assets, namely N bass portfolos. To make the model workable we need to reduce the set of factors, yet equatons (10) and (11) tell us that ths s unlkely to be attanable wth a smple fxed lnear relatonshp. Ths observaton yelded the nonlnear APT of Bansal and Vswanathan (1993) and Bansal et al. (1993) and the condtonal CAPM and APT of Ferson (1985) and Ferson and Harvey (1991, 1995), among others. We shall begn by brefly presentng the former and then contnue wth the latter class of models. For the nonlnear APT we use equaton (10) and replace the margnal rate of substtuton by ts projecton onto t 1, t 1!6 t 1 #x t 1 6 t # ~x t 1!. (12) 3 See Lucas (1978), Breeden (1979), Stulz (1981), Huang (1987), Duffe and Zame (1989) among others.

6 554 The Journal of Fnance Then, nstead of usng the projecton onto the entre nformaton we consder a vector P t 1 of well-dversfed bass varables such b that b t 1!6 t 1 # t 1!6P t 1 b # G~P t 1!, (13) wth G~{! a well-behaved functon chosen among a class of flexble functonal forms. Usng the fact that ~x ~t, t 1!! t and normalzng equaton (12) yelds the followng set of moment condtons: b t 1!x t 1 1!Z t # 0, (14) where Z t s a set of nstruments pcked among the elements of t. Equaton (14) forms the bass of a GMM estmaton procedure for the parameters descrbng the prcng kernel G({). The set of Z t nstruments actually used n our emprcal work wll be descrbed later snce t concdes wth those used b n the condtonal APT model. The elements enterng P t 1 are the same as those used by Bansal et al. (1993) n ther one-factor model, namely b P t 1 ~1 r Mt 1,1 r ft 1!, (15) where r Mt 1 s the nomnal return on the market and r ft 1 s the nomnal yeld to maturty on the Treasury bll next perod. What remans to be specfed s a functonal form for G({). As the exact specfcaton of the nonlnear prcng kernel s unknown, Bansal et al. (1993) suggest approxmatng t wth a polynomal seres expanson, 4 namely b G~P t 1! b 0 b 1t r ft 1 j 1,2,5 ( b Mt 1 # j. (16) Wth regard to the asset x appearng n equaton (14), we shall consder a set of sze-sorted portfolos and ndustry-based classfed portfolos that wll also be used n the condtonal APT. The detals are dscussed n Secton III.C. We turn now our attenton to the condtonal CAPM consdered by Harvey (1991) to study the prcng of nternatonal assets and used by Ferson and Korajczyk (1995) to study predctable returns and rsk n the Unted States. Agan, one can start from the observaton that equatons (10) and (11) do not drectly yeld a workable model, but nstead of consderng a nonlnear prcng kernel, Harvey proposes to study expected returns for stock markets from a set of countres va ther condtonal beta wth the return on a world market portfolo. Harvey (1991) shows that one obtans a set of moment condtons sutable for Generalzed Method of Moments (GMM; see Hansen (1982)) estmaton of # N 1 and d M as follows: 4 As Bansal and Vswanathan (1993) explan, usng the ffth order rather than the thrd was partly motvated by the need to reduce collnearty between the varous powers of the expanson.

7 Stable Factor Structures n Prcng Rsk 555 E 2 ~u Mt 1 ~r t 1 Z t d! ' ' ' ~r Mt 1 Z t d M! ' Z ' t 0, (17) Z t d u Mt 1 u t 1 Z t d M! where r t t 1 # N 1, u t r t Z t 1 d, and u Mt r Mt Z t 1 d M. We also nvestgate an alternatve specfcaton for the condtonal CAPM, one suggested by Ferson and Harvey (1993), whch models tme-varyng betas as b t Z t b c to avod the weldy specfcaton of the rato of condtonal covarance and condtonal varance. It yelds the moment condtons E ~r t 1 Z t d! ' ' ' ~r Mt 1 Z t d M! ' Z ' t 0, (18) ~Z t d Z t b c Z t d M! whch, n contrast to equaton (17), s a just-dentfed set of moment condtons. In a recent paper Ferson and Korajczyk (1995) undertake a very thorough emprcal nvestgaton of rsk and return for the Unted States usng a multfactor condtonal APT. The setup s very smlar to that descrbed n equaton (17) except that the moment condtons are a bt more elaborate because of the presence of a multtude of factors. For the multfactor condtonal APT, Ferson and Korajczyk defne the followng set of moment condtons: E ' ~F t 1 r t 1 Z t ' d ' ~F t 1 Z ' t g! ' Z t Z ' ' t g!~f t 1 Z ' t g! ' b F t 1 ~r t 1 Z ' t d! ' 0, (19) where F t s a K 1 vector of factor-mmckng portfolos, b s a K 1 vector of the betas for asset, and Z t sa(l 1) vector of nstruments. In contrast to the nonlnear APT and condtonal CAPM, the model defned n equaton (19) has parameters that play a very dfferent role. It also makes hypothess testng more nterestng. Indeed, ths more elaborate model has the advantage of separatng projecton equatons and asset prcng moment condtons nvolvng condtonal betas. In equaton (19) the thrd set of moment condtons does not nvolve any new parameters, whereas n equaton (19), and equaton (18) as well, the thrd set nvolves explctly parameterzed betas. III. Emprcal Results on Parameter Stablty We turn our attenton now to the emprcal evdence regardng the structural nvarance of the three dynamc asset prcng models descrbed n the prevous secton. We devote a subsecton to each of these emprcal models. The frst subsecton brefly dscusses tests for structural shfts. Then we

8 D 556 The Journal of Fnance study the condtonal CAPM of Harvey (1991). The next subsecton covers the multfactor model of Ferson and Korajczyk (1995). We conclude wth the nonlnear APT. To facltate comparsons, we apply the three emprcal asset prcng models to the same data. Because the study by Ferson and Korajczyk (1995) has a very wde and comprehensve set of U.S. equty return seres, we use ther data set. 5 It conssts of sze-sorted returns for stocks appearng on the CRSP tapes as well as those same asset returns classfed by ndustry. All portfolos are value-weghted. A total of 12 ndustry-sorted portfolos wll be consdered n addton to the 10 sze category portfolos. We follow step by step the specfcaton of varables and nstruments descrbed by Ferson and Korajczyk. 6 A. Testng for Structural Breaks Anyone famlar wth the emprcal evdence may fnd t surprsng that there s a need to test for structural change because condtonal CAPM and APT models are typcally well supported by the data. To clarfy ths we have to elaborate on the fact that testng for structural breaks s far more strngent than the usual overdentfyng restrctons tests, often called J-statstcs, commonly used to dagnose the ft of an asset prcng model lke the condtonal CAPM. Because such models are estmated va GMM, let us proceed by specfyng the moment condtons of such a model. Takng a smple example as n equaton (17) wth a sngle asset yelds E r t 1 dd t Z t r Mt 1 dd Mt Z t t Z t 0. d t Z Mt 1 dd Mt Z t! 2 # ~r Mt 1 dd Mt Z t!~r t 1 dd t Z t! dd Mt Z The formulaton n equaton (20) represents the set of moment condtons nvolved n the GMM estmaton procedure but does not mpose the null hypothess of constant parameters. For the moment the varaton n the parameters s left unspecfed; ultmately they wll represent structural breaks as wll be dscussed shortly. The estmaton of the condtonal CAPM mposng fxed parameters d M and d wth the data generated by equaton (20) wll yeld GMM parameter estmates dn M and dn, whch are some sort of sample (20) 5 Hence, the results we wll report for Harvey s condtonal CAPM wll not exactly replcate hs orgnal study of nternatonal excess stock returns. Yet emprcal results we obtan wth Harvey s orgnal data seres are completely n lne wth those we are about to dscuss. These results are not reported but are avalable upon request. Smlarly, the results reported for the nonlnear APT model do not correspond to the orgnal emprcal work, but the conclusons we draw from our nvestgaton are nevertheless representatve. 6 We actually drectly use the data set they constructed. Ths also apples to the factormmckng portfolos that are dscussed n Secton III.C.

9 N Stable Factor Structures n Prcng Rsk 557 averages of the underlyng dd Mt and dd t. Ghysels and Hall (1990b) show formally that overdentfyng restrctons tests based on the moment condtons such as those n equaton (20) but evaluated at fxed parameter estmates dn M and d have a tendency not to reject the model. Ths problem s not just a theoretcal curosty. Indeed, we wll provde numerous examples where ths stuaton occurs n emprcal asset prcng models. Hence, the usual dagnostc tests to judge the valdty of a model are not adequate to detect systematc msprcng of asset returns because of erroneous beta dynamcs. Testng for structural nvarance of the model amounts to verfyng whether the followng hypothess holds H 0 : dd Mt d M dd t d t 1,...,T t 1,...,T. (21) A great varety of tests for structural change for models estmated by GMM exst. 7 The majorty of tests assume as an alternatve that at some pont n the sample there s a sngle structural break; for nstance, dd jt d j1 d j2 t 1,...,pT t pt 1,...,T j M,, (22) where p determnes the fracton of the sample before and after the assumed break pont. 8 If the break pont pt were known, our task would be relatvely easy to perform. For example, calculatng d j1 and d j2 and comparng both estmates to see whether they are sgnfcantly dfferent would be one way to proceed (often referred to as a Chow test). Unfortunately, n the present context we don t really want to assume p known. In recent years several procedures have been advanced to test the null hypothess (21) aganst an alternatve such as equaton (22) wth unknown break pont p. In the remander of the secton we wll explan what these procedures amount to (the techncal detals appear n the references provded n footnote 7). We use the Sup LM, or supremum LM, test proposed by Andrews (1993). One computes the supremum of all LM tests, or score tests, over all possble break ponts pt. Andrews suggests ths type of test and tabulates ts dstrbuton under the null hypothess appearng n equaton (21). The Sup LM test has the great advantage that t only uses the parameter estmates dn M 7 Relevant references nclude Andrews (1993), Andrews and Ploberger (1994), Ghysels, Guay, and Hall (1998). 8 It s worth notng that n equaton (22) all parameters are tested jontly for stablty. In several crcumstances, however, the parameters nvolved play dfferent roles; therefore, dependng on whch ones are subject to breaks, a dfferent nterpretaton should be gven. For nstance, n the multfactor models appearng n equaton (19), one has a set of parameters d and g that arse from purely ancllary statstcal assumptons regardng projecton equatons rather than wth an economc nterpretaton. To emphasze ths dstncton we wll often conduct tests nvolvng only a subset of the parameter vector.

10 558 The Journal of Fnance and dn obtaned from the full sample. Ths saves an enormous amount of computer tme by avodng all the (nonlnear) GMM parameter estmatons over the varous subsamples. Because a great number of asset prcng models wll be tested, computatonal effcency has strong appeal. One may wonder by now why we focus exclusvely on tests havng a sngle break pont as alternatve. Surely, there are many other types of structural nstabltes, such as cases wth several breaks or wth gradual movements n the d k parameters. Constructng tests aganst all possble types of nstabltes s smply mpossble both statstcally and practcally. Fortunately, however, the stuaton s not hopeless because the sngle unknown break pont statstcs have power aganst a large class of parameter nstablty patterns far beyond what appears explctly as an alternatve n equaton (22). Therefore, examnng (only) sngle break pont tests goes a long way toward our goal. Before turnng to the results we provde some detals about the GMM estmaton procedures common to all the models. Frst, the nstruments are farly standard: n addton to a constant we have (1) the one-month T-bll, (2) the dvdend yeld of the CRSP value-weghted NYSE stock ndex, (3) a detrended stock prce level, (4) a measure of the slope of the term structure, (5) a qualty-related yeld spread n the corporate bond market, and (6) a January dummy. Detals of these seres appear n Ferson and Korajczyk (1995) and n Harvey (1991). The data are monthly and cover a sample from January 1927 untl January The standard devatons and J-statstcs n a GMM procedure crtcally depend on the covarance estmator. Our results are based on the estmator proposed by Andrews and Monahan (1992), a procedure that appears to have the best samplng propertes among those currently avalable. Fnally, regardng the Sup LM test we should note that one has to specfy a sample range over whch to compute the supremum. In all our computatons we 0.8T#. The choce of 20 percent trmmng s motvated by the length of the sample. B. Stable Factors n the Condtonal CAPM In Panel A of Table I we report emprcal results of the condtonal CAPM descrbed n equatons (17) for the 12 ndustry-based portfolos. Panel B pertans to model (18), and we refer to t as explct beta model because b t s estmated as Z t b c. The model n Panel A s called the mplct beta model. The frst row reports the J-statstc. The remanng seven rows n the table report the Sup LM test for parameter stablty, each row representng an nstrumental varable. Because two parameters are assocated wth each nstrument n the mplct beta condtonal CAPM, one appearng n the vector d M, the other n d, we have a jont test for the two parameters assocated wth each nstrument. For Panel B of Table I the statstcs apply to three parameters, as an element of b c s added wth each nstrument. Moreover, there are no J-statstcs reported n Panel B as model (18) s just dentfed. Table II has exactly the same structure, except that results for sze-sorted portfolos are consdered.

11 Table I Stable Tme-Varyng Beta Models wth the Condtonal CAPM: Industry-Based Classfcaton Panel A reports Implct Beta models and Panel B reports Explct Beta Models descrbed n Secton II. The set of nstruments are lsted n the left column of the table and descrbed below. The data are the same as n Ferson and Korajczyk (1995) and cover monthly observatons from January 1927 untl January T-bll s the return on a one month T-bll, Dv. yeld s the dvdend yeld of the CRSP-weghted NYSE stocks, Mark. port. s the detrended stock prce level, Maturty spr. s a measure of the slope of the term structure, Rsk spr. s a qualty-related yeld spread n the corporate bond market, and January s a dummy varable set to one f the month s January. The mplct beta model s descrbed by equaton (17) n Secton II, the explct beta model by equaton (18). The portfolos are NYSE value-weghted and ndustry-based. The entres to the tables are Sup LM tests for structural change. Ther crtcal values appear n Andrews (1993, Table 1). Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. The J-test s the overdentfyng restrctons test, whch s ch-squared wth 7 degrees of freedom. Industry 1 Industry 2 Industry 3 Industry 4 Industry 5 Industry 6 Industry 7 Industry 8 Industry 9 Industry 10 Industry 11 Industry 12 Panel A: Implct Beta Functon Models J-test ** *** * *** Constant ** *** ** ** * * T-bll * *** * * *** ** * * Dv. yeld * *** ** *** ** * Mark. port * *** * ** * Maturty spr * *** ** ** Rsk spr *** * ** January Panel B: Explct Beta Functon Models Constant *** *** *** *** *** *** *** *** ** *** *** *** T-bll *** ** *** ** *** *** *** *** * *** *** *** Dv. yeld *** *** *** ** *** *** ** *** * *** *** *** Mark. port *** *** *** *** *** *** *** *** ** *** *** *** Maturty spr *** *** *** ** *** *** *** *** * *** *** *** Rsk spr *** ** *** *** *** *** *** *** ** *** *** *** January ** * *** *** *** ** *** Stable Factor Structures n Prcng Rsk 559

12 Table II Stable Tme-Varyng Beta Models wth the Condtonal CAPM: Sze-Sorted Portfolos Panel A reports Implct Beta models and Panel B reports Explct Beta Models descrbed n Secton II. The set of nstruments are lsted n the left column of the table and descrbed below. The data are the same as n Ferson and Korajczyk (1995) and cover monthly observatons from January 1927 untl January T-bll s the return on a one month T-bll, Dv. yeld s the dvdend yeld of the CRSP-weghted NYSE stocks, Mark. port. s the detrended stock prce level, Maturty spr. s a measure of the slope of the term structure, Rsk spr. s a qualty-related yeld spread n the corporate bond market, and January s a dummy varable set to one f the month s January. The mplct beta model s descrbed by equaton (17) n Secton II, the explct beta model by equaton (18). The portfolos are NYSE value-weghted and ndustry-based. The entres to the tables are Sup LM tests for structural change. Ther crtcal values appear n Andrews (1993, Table 1). Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. The J-test s the overdentfyng restrctons test, whch s ch-squared wth 7 degrees of freedom. Sze 1 Sze 2 Sze 3 Sze 4 Sze 5 Sze 6 Sze 7 Sze 8 Sze 9 Sze 10 Panel A: Implct Beta Functon Models J-test * * * * Constant * * ** * *** T-bll * * ** * * ** * ** * ** Dv. yeld * * ** ** *** Mark. port * * * *** Maturty spr ** * *** Rsk spr * ** January * * * * Panel B: Explct Beta Functon Models Constant * * * * T-bll * * ** * *** Dv. yeld * * ** * * ** * ** * ** Mark. port * * ** ** *** Maturty spr * * * *** Rsk spr ** * *** January * ** 560 The Journal of Fnance

13 Stable Factor Structures n Prcng Rsk 561 Let us focus frst on the mplct beta condtonal CAPM appearng n Panels A of Tables I and II. It was noted that the J-statstc s a dagnostc test ll-equpped to scrutnze a model n terms of ts structural nvarance and by the same token the ablty of a model to predct the market prce for rsk. Although the results n Tables I and II are not as strkng as those that wll be reported below, they do reveal some qute extreme cases. We start wth two such cases: Industry 8 (Transportaton) n Table I and the portfolo of returns on the largest frms (Sze 10) n Table II. In both cases, accordng to the overdentfyng restrctons tests, the condtonal CAPM s not rejected. Yet, for all nstruments except the January dummy there s strong evdence of parameter nstablty. Hence, despte the favorable evdence accordng to the usual J-statstc, t s clear that the return on both portfolos cannot be satsfactorly prced wth the condtonal CAPM. Although both cases together wth Industry 4 (Basc Industres) are extreme, t s clear that Tables I and II contan many other examples. In fact, n only one case (Industry 1, Petroleum) of the 22 asset return seres s there no rejecton of the condtonal CAPM wth the J-statstc nor wth the Sup LM tests. If we were to rely only on the J-statstc we would not reject the condtonal CAPM for 6 of the 12 ndustres and for 2 of the 10 sze-based portfolos. The frst of three emprcal examples underscores several mportant ponts that motvate our study. We report a set of models that would be found emprcally acceptable, accordng to ther overdentfyng restrctons, for explanng the returns on selected portfolos wth a prcng formula based on a set of common nstruments. After elmnatng all the cases where structural breaks are found we are left wth only one case. In practcal terms ths means these nstruments do not yeld a satsfactory dynamc condtonal asset prcng model. In such crcumstances ether the model needs to be modfed or we need to search for a stable rsk factor alternatve. Before makng any conclusons on ths we turn to the other three models dscussed n Secton II to apprase ther performance. The frst s the condtonal CAPM wth the explct beta model b t Z t b c that appears n equaton (18). The results are reported n Panels B of Tables I and II. We fnd resoundng rejectons of the stablty hypothess. One may wonder what happened whle movng from equaton (17) to equaton (18). Note that the Sup LM tests now apply to three parameters nstead of just the two assocated wth each nstrument. The added parameter belongs to b c. The overwhelmng rejectons reveal that the lnear representaton Z t b c s totally nadequate. Consequently the smplfcaton, whch avods the weldy specfcaton of the covarance0varance rato, s smply not acceptable. Comparson of Panels A and B of both tables underlnes agan the scope of testng for structural nvarance of parameters to uncover msspecfcaton. Because we report evdence for structural breaks t s worth mentonng that the estmated break ponts, that s, the sample ponts where the supremum of the LM statstcs s attaned, do not occur at the same moment; they are n fact qute dspersed. It should frst be noted that t s not clear that

14 562 The Journal of Fnance the supremum s attaned at the true break pont (formal proofs for the general case of nonlnear dynamc system of equatons are to our knowledge nonexstent). In the event they yeld an unbased estmaton of break ponts we must conclude that no specfc event caused the rejecton of parameter stablty. C. Stable Factors n the Condtonal Multfactor APT To descrbe the emprcal results let us return to equaton (19) and recall the nterpretaton of each of the parameters. There are essentally three sets of moment condtons, the frst two defne condtonal expectatons (lnear projectons) of asset returns and the thrd relates to the multfactor beta model. Hence, breaks n the parameters d and g reflect a msspecfcaton of the statstcal models for the predctable dynamcs n returns or factor mmckng portfolos. In contrast, the hypothess of fxed condtonal betas s more a fundamental and crucal assumpton from an asset prcng perspectve (see, e.g., Ferson (1990, Table VIII) on ths ssue). We wll frst dscuss the two sets of projecton equatons and then turn to the rsk-prcng equaton. We begn wth the emprcal results regardng the stablty of the coeffcents d n equaton (19) obtaned from projectng the sx nstruments plus constant on sze-sorted and ndustry-based portfolo returns. Before dscussng the Sup LM test results we need to be more specfc about the specfcaton of the factors n the condtonal APT model. Two alternatve sets of rsk factors are examned. The frst set conssts of economc varables smlar to those of Chen, Roll, and Ross (1986) and Ferson and Harvey (1991). Among those factors s the market return measured by the S&P 500. The latter s also the market portfolo for the condtonal CAPM. Mmckng portfolos are constructed usng ndvdual common stocks for the fve factors. The second approach s motvated by many prevous studes of the APT and uses the asymptotc prncpal components method of Connor and Korajczyk (1989) to estmate the common factors. We compute the results for both factor confguratons but report only the economc factors. We do not report explctly the results wth the prncpal component factors because they are qute smlar except that one typcally fnds even more evdence for structural breaks. Tables III and IV cover the emprcal results for the condtonal APT wth a combnaton of ndustry-based and sze-sorted portfolos. The J-statstcs appear at the top of each table and the Sup LM statstcs are lsted n the rows labeled d all and d j, j 1,...,7. 9 The tests correspondng to d all are jont tests for all seven nstruments (the frst beng a constant); the others measure each nstrument ndvdually. Let us focus on the results n Table III, whch cover the 12 ndustres selected by Ferson and Korajczyk. Accordng to the J-statstc we would almost never reject the model. 9 To smplfy the notaton n Tables III and IV we should note that the ndex j to d j s not to be confounded wth ndex d n (19). The latter refers to asset and represents the entre vector (d all n the tables), but d j s an element of d all.

15 Table III Stable Factors Structures n the Condtonal APT: Industry Classfcaton wth Economc Varables Factors The J-test s ch-squared wth 30 degrees of freedom. The entres to the table are Sup LM tests. Ther crtcal values appear n Andrews (1993, Table 1). d all tests for all coeffcents of d together (7 parameters), d ~ 1,2,...,7! tests for all coeffcents of d one by one (1 parameter each). g c ~ 1,2,...,5! tests for all coeffcents of g column by column (7 parameters by column). b all tests for all coeffcents of b together (5 parameters), b ~ 1,2,...,5! tests for all coeffcents of b one by one (1 parameter each). T-bll s the return on a one month T-bll, Dv. yeld s the dvdend yeld of the CRSP-weghted NYSE stocks, Mark. port. s the detrended stock prce level, Maturty spr. s a measure of the slope of the term structure, Rsk spr. s a qualty-related yeld spread n the corporate bond market, and January s a dummy varable set to one f the month s January. The Condtonal APT model s descrbed by equaton (19) n Secton II. Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. Industry 1 Industry 2 Industry 3 Industry 4 Industry 5 Industry 6 Industry 7 Industry 8 Industry 9 Industry 10 Industry 11 Industry 12 J-test ** ** 33.5 Sup LM d all d ** * d d ** d ** d * d 6 8.3* *** * d g 1c 77.9*** 76.4*** 66.9*** 62.6*** 99.2*** 69.5*** 62.7*** 62.5*** 61.4*** 81.6*** 64.8*** 71.7*** g 2c 170.6*** 112.0*** 108.4*** 95.7*** 142.0*** 95.8*** 91.9*** 92.5*** 90.4*** 131.3*** 105.2*** 97.9*** g 3c 32.2*** 27.8*** ** 22.9** 21.0* *** 22.7** ** 16.2 g 4c 31.0*** 23.0** 51.2*** * *** 23.2** g 5c 43.1*** * *** * b all 36.8*** 17.6* *** 22.2** * 17.8* *** 19.8** b ** 7.8* b ** 9.1** *** b *** *** 2.2 b *** * b *** *** 11.0** *** 10.7** *** 4.4 Stable Factor Structures n Prcng Rsk 563

16 Table IV Stable Factors Structures n the Condtonal APT: Sze Classfcaton wth Economc Varables Factors The J-test s ch-squared wth 30 degrees of freedom. The entres to the table are Sup LM tests. Ther crtcal values appear n Andrews (1993, Table 1). d all tests for all coeffcents of d together (7 parameters), d ~ 1,2,...,7! tests for all coeffcents of d one by one (1 parameter each). g c ~ 1,2,...,5! tests for all coeffcents of g column by column (7 parameters by column). b all tests for all coeffcents of b together (5 parameters), b ~ 1,2,...,5! tests for all coeffcents of b one by one (1 parameter each). T-bll s the return on a one month T-bll, Dv. yeld s the dvdend yeld of the CRSP-weghted NYSE stocks, Mark. port. s the detrended stock prce level, Maturty spr. s a measure of the slope of the term structure, Rsk spr. s a qualty-related yeld spread n the corporate bond market, and January s a dummy varable set to one f the month s January. The Condtonal APT model s descrbed by equaton (19) n Secton II. Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. Sze 1 Sze 2 Sze 3 Sze 4 Sze 5 Sze 6 Sze 7 Sze 8 Sze 9 Sze 10 J-test ** 53.5*** 50.7** 46.5** *** Sup LM d all *** 23.2** *** d * * 10.5** d ** ** 7.9* 11.2** 3.8 d ** ** 11.7** d * * 10.2** d ** ** ** 10.5** d * * * d g 1c 64.3*** 59.7*** 65.1*** 61.7*** 64.4*** 75.4*** 66.5*** 57.2*** 67.5*** 75.2*** g 2c 103.0*** 105.0*** 109.7*** 115.0*** 141.6*** 126.4*** 110.4*** 98.8*** 108.1*** 108.9*** g 3c 31.3*** 26.0** * 30.0*** 24.0** 19.7* ** 26.2** g 4c 33.3*** 28.8*** ** 25.5** 27.5*** 24.1** ** 22.0** g 5c *** 22.8** *** b all ** 18.6** 17.5* 18.1* 39.2*** b * b * b ** *** b * ** 12.9*** 7.1 b ** *** 564 The Journal of Fnance

17 Stable Factor Structures n Prcng Rsk 565 Based on the Sup LM tests for d all and the ndvdual d we fnd very lttle evdence for breaks among the ndustry-based portfolos (Table III) except for Industry 5 (Food0Tobacco) but much more evdence for breaks n the sze-based classfcaton (Table IV). The stuaton s qute dfferent wth the parameters n the second block of moment condtons, where there s very strong evdence aganst the null hypothess. The second set of moment condtons n equaton (19), lke the frst, nvolves projectons on the set of nstruments descrbed before, to extract the predctable part of the K 1 vector F t. Because ths s a multvarate process predcton wth K 5 we focus on tests for each column whch project the entre set of nstruments on each of the fve factor-mmckng portfolos. Hence, we use the notaton g c, 1,...,5, to denote the tests assocated wth each of the column vectors. 10 The overwhelmng evdence of breaks n g c means t s very dffcult to predct the returns on the factor-mmckng portfolos. So far we have dscussed only those parameters related to the two blocks of projecton equatons for the portfolo return and the factors. The remanng coeffcents pertanng to the condtonal betas are more mportant from an asset prcng pont of vew. We fnd n many cases strong evdence of nstablty, notably n Industres 1 (Petroleum), 2 (Fnance0Real Estate), 5 (Food0Tobacco), 6 (Constructon), 8 (Transportaton), 11 (Servces), 12 (Lesure), and to a lesser extent Industry 9 (Utltes). For the sze-sorted portfolos n Table IV we seem to reject constant beta entres partcularly for the large szes of NYSE stocks. D. Stable Factors n the Nonlnear APT Last we examne the nonlnear APT proposed by Bansal et al. (1993). As n Secton III.B wth Harvey s model, we do not attempt to exactly replcate ther data and estmates. Instead, for comparson, we use the Ferson and Korajczyk (1995) data set of szed-sorted and ndustry-based portfolos to estmate the nonlnear APT specfed n equatons (14) and (16) usng the same set of nstruments. Ths means we have seven nstruments, ncludng a constant, to specfy the moment condtons n equaton (16). Because there are fve parameters n equaton (16) we have two overdentfyng restrctons. The results are reported n two tables, one coverng the asset returns for each of the 10 sze-sorted portfolos and the other contanng the ndustrybased portfolos. The followng tests appear n Tables V and VI: (1) tests for the stablty of each of the fve parameters n the nonlnear APT separately, (2) two jont tests, one nvolvng the parameters of the nonlnear part, namely b 2M and b 5M, and one nvolvng the jont set of fve beta parameters. The results n Table V show that there are clearly problems wth the small sze categores. All other sze categores appear to be well ftted by a stable nonlnear APT 10 We could not perform an overall test for the entre matrx g nvolvng 35 coeffcents as no crtcal values are avalable for that many coeffcents. For reason of space we do not report ndvdual tests nor tests assocated wth a partcular nstrument n ths case.

18 Table V Stable Factors Structures n Nonlnear APT: Sze Classfcaton The Nonlnear APT s descrbed by equatons (14) and (16) n Secton II. The nstruments are the same as for the Condtonal APT, namely (1) the return on a one month T-bll, (2) the dvdend yeld of the CRSP-weghted NYSE stocks, (3) the detrended stock prce level, (4) a measure of the slope of the term structure, (5) a qualty-related yeld spread n the corporate bond market, (6) a dummy varable set to one f the month s January, and (7) a constant. The entres to the table are Sup LM tests where b represents tests for each parameter separately. b 2M & b 5M tests these two parameters jontly. b all tests all parameters together. Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. Sze 1 Sze 2 Sze 3 Sze 4 Sze 5 Sze 6 Sze 7 Sze 8 Sze 9 Sze 10 b ** * b 1f * b 1M b 2M ** b 5M ** b 2M & b 5M ** b all *** *** The Journal of Fnance

19 Table VI Stable Factors Structures n Nonlnear APT: Industry Classfcaton The Nonlnear APT s descrbed by equatons (14) and (16) n Secton II. The nstruments are the same as for the Condtonal APT, namely (1) the return on a one month T-bll, (2) the dvdend yeld of the CRSP-weghted NYSE stocks, (3) the detrended stock prce level, (4) a measure of the slope of the term structure, (5) a qualty-related yeld spread n the corporate bond market, (6) a dummy varable set to one f the month s January, and (7) a constant. The entres to the table are Sup LM tests where b represents tests for each parameter separately. b 2M & b 5M tests these two parameters jontly. b all tests all parameters together. Rejectons at 10 percent appear n the table wth *, at 5 percent wth **, and at 1 percent wth ***. Industry 1 Industry 2 Industry 3 Industry 4 Industry 5 Industry 6 Industry 7 Industry 8 Industry 9 Industry 10 Industry 11 Industry 12 b * ** *** * b 1f *** ** * b 1M *** b 2M *** * * b 5M *** b 2M & b 5M *** b all *** ** *** * Stable Factor Structures n Prcng Rsk 567

20 568 The Journal of Fnance model. Ths s far better than the condtonal APT of the prevous secton, because for almost all portfolos the model seems acceptable. It only fals to explan returns on very small frms whch probably are more affected by nformed tradng and dosyncratc events. The nonlnear APT also appears qute successful f one looks at ndustry-based portfolos. In Table VI we can see that for at least half of the 12 ndustres there s no nstablty accordng to the Sup LM tests. The ndustres where the model fals are: Industry 1 (Petroleum), 4 (Basc Industres), 5 (Food and Tobacco), 6 (Constructon), and to some extent perhaps 9 (Utltes) and 10 (Servces). IV. Do Tme-Varyng Betas Help or Hurt? The emprcal results n the prevous secton revealed that the tmevaryng beta condtonal CAPM and APT models do not seem to capture very well the temporal dynamcs of betas. Consequently, they msprce rsk. Is ths msprcng serous, so serous that we are bound to make larger errors n comparson to fxed beta models? Ths s a queston of relatve msprcng of one (msspecfed) model aganst another one. To address ths we wll compute the n-sample root mean squared error (RMSE) of the condtonal CAPM and APT models appearng n equatons (17) and (19), respectvely, and compare them wth the RMSE of the fxed beta model, specfed through the followng moment condtons 11 : E r Mt 1 Z t d M r t 1 bz t d M Z t 0. (23) Note that the model n equaton (23) uses the expected return E~r Mt 1 6Z t! nstead of the actual return as a factor so that t nvolves the same condtonng nformaton set Z t as the condtonal CAPM and APT and has no nformatonal advantage that would make comparsons of forecasts dffcult The RMSE calculatons for the condtonal CAPM and APT are based on the prcng error as defned n Ferson and Harvey (1993), equaton (8). Note that we report results for the explct beta functon condtonal CAPM. The results for the model n equaton (17) are smlar to those obtaned wth equaton (18) and are therefore omtted. 12 The model n equaton (23) corresponds to equaton (7) n Harvey (1991). It should be noted that the comparson of root mean squared errors does not nvolve the nonlnear APT. Indeed, although the results n the prevous secton are favorable wth regard to the stablty of the prcng kernel, they do not yeld straghtforwardly a predcton model. Ths problem arses n any approach that drectly tres to specfy a prcng kernel such as consumpton-based asset prcng methods (e.g., see Hansen and Sngleton (1982) or Epsten and Zn (1991)). The man reason we do not engage n a comparson nvolvng the nonlnear APT s that the model needs to be augmented n nontrval ways wth predcton formulas to generate predctons of returns. Such augmentaton would nvolve qute a number of auxlary assumptons and noveltes that would take us far beyond the scope of the present paper. We are grateful to S. Vswanathan for havng gven thought to several of our queres regardng predcton formula for nonlnear APT models.