ESSAYS on ASSET PRICING MODELS: THEORIES and EMPIRICAL TESTS

Transcription

1 ESSAYS on ASSET PRICING MODELS: THEORIES and EMPIRICAL TESTS A Disseraion Presened o he Faculy of he Graduae School of Cornell Universiy In Parial Fulfillmen of he Requiremens for he Degree of Docor of Philosophy by Yan Li Augus 2009

2 2009 Yan Li

3 ESSAYS on ASSET PRICING MODELS: THEORIES and EMPIRICAL TESTS Yan Li, Ph. D. Cornell Universiy 2009 My disseraion conains hree chapers. Chaper one proposes a nonparameric mehod o evaluae he performance of a condiional facor model in explaining he cross secion of sock reurns. There are wo ess: one is based on he individual pricing error of a condiional model and he oher is based on he average pricing error. Empirical resuls show ha for valueweighed porfolios, he condiional CAPM explains none of he asse-pricing anomalies, while he condiional Fama-French hree-facor model is able o accoun for he size effec, and i also helps o explain he value effec and he momenum effec. From a saisical poin of view, a condiional model always beas a condiional one because i is closer o he rue daa-generaing process. Chaper wo proposes a general equilibrium model o sudy he implicaions of prospec heory for individual rading, securiy prices and rading volume. Is main finding is ha differen componens of prospec heory make differen predicions. The concaviy/convexiy of he value funcion drives a disposiion effec, which in urn leads o momenum in he cross-secion of sock reurns and a posiive correlaion beween reurns and volumes. On he oher hand, loss aversion predics exacly he opposie, namely a reversed disposiion effec and reversal in he cross-secion of sock reurns, as well as a negaive correlaion beween reurns and volumes. In a calibraed economy, when

4 prospec heory preference parameers are se a he values esimaed by he previous sudies, our model can generae price momenum of up o 7% on an annual basis. Chaper hree sudies he role of aggregae dividend volailiy in asse prices. In he model, narrow-framing invesors are loss averse over flucuaions in he value of heir financial wealh. Persisen dividend volailiy indicaes persisen flucuaion in heir financial wealh and makes socks undesirable. I helps o explain he salien feaure of he sock marke including he high mean, excess volailiy, and predicabiliy of sock reurns while mainaining a low and sable risk-free rae. Consisen wih he daa, sock reurns have a low correlaion wih consumpion growh, and Sharpe raios are ime-varying.

5 BIOGRAPHICAL SKETCH Yan Li graduaed from Peking Universiy wih a B.A. degree in economics in 997. She coninued her sudy in Peking Universiy and obained her maser degree in economics in 200. In Augus of 2003, she began her docoral sudies in economics a Cornell Universiy. iii

6 ACKNOWLEDGMENTS I am graeful o my advisors, Yongmiao Hong and Ming Huang, for heir kindness, suppor and guidance, and o my commiee member, Yaniv Grinsein, for his kindness, suppor and guidance. iv

7 TABLE OF CONTENTS BIOGRAPHICAL SKETCH...iii ACKNOWLEDGMENTS... iv TABLE OF CONTENTS... v LIST OF FIGURES...viii LIST OF TABLES... ix CHAPTER ONE: A NEW TEST of TIME-VARYING FACTOR MODELS.... Inroducion....2 Mehodology Economeric Framework Esimaion of he Model The Choice of Window Size Two Tess of a Condiional Facor Model Tes on Individual Pricing Errors Tes on Average Pricing Errors Empirical Resuls Daa Tesing he Condiional CAPM Daa-Driven Window Size Individual Pricing Errors Average Pricing Errors Tesing he Condiional Fama-French Three-Facor Model Daa-Driven Window Size Individual Pricing Errors Average Pricing Errors Condiional Models or Uncondiional Models? Conclusion REFERENCES CHAPTER TWO: PROSPECT THEORY, THE DISPOSITION EFFECT AND ASSET PRICES Inroducion The Model Financial Asses Beliefs Preference Timeline Exension: A Muli-Sock Seing Equilibrium Age-3 Invesors' Decisions Age-2 Invesors' Decisions v

8 2.3.3 Age- Invesors' Decisions Evoluion of Sae Variables Marke Clearing Condiion Benchmark Case: Sandard Risk Neural Uiliy Numerical Resuls and Inuiions Calibraing Technology Parameers Implicaions of Diminishing Sensiiviy Disposiion Effecs Momenum Turnover Equiy Premiums Implicaions of Loss Aversion Reversed Disposiion Effecs Reversal Turnover...04 equiy premium Quaniaive Analysis and Tesable Predicions Quaniaive Analysis: How Successful is Prospec Theory? Tesable Predicions Conclusion... 2 REFERENCES... 3 CHAPTER THREE: DIVIDEND VOLATILITY and ASSET PRICING Inroducion Key Feaures of Hisorical Dividend Volailiy Dividend Volailiy Clusering Time-Varying Dividend Volailiy The Model Seup Equilibrium Prices Mehodology of Numerical Compuaion Model Resuls Calibraing Parameer Values Price-dividend Raio Funcion f Simulaion Resuls Sock Reurns and Sock Volailiy Auocorrelaions of Reurns and Price-Dividend Raios Reurn Predicabiliy Time-varying Sharpe Raios Srucural Break and Equiy Premiums Conclusion REFERENCES APPENDIX Appendix.A: Disribuion of individual pricing errors vi

9 Appendix.B: Derivaion of he asympoic disribuion of average pricing errors Appendix.C: Goodness of fi for condiional models versus uncondiional models... 7 Appendix 2.A. Sensiiviy Analysis Appendix 2.A.2 Heerogeneiy, Aggregaion and Price Impacs Appendix 3: Consrucing Dividend Time Series... 8 vii

10 LIST OF FIGURES Figure.: Condiional CAPM alphas and he Diff measure for he value-weighed S-B, V-G, and W-L Figure.2: Condiional CAPM alphas and he Diff measure for he equally-weighed S-B, V-G, and W-L Figure.3: Auocorrelaion funcion of Diff for value-weighed porfolios Figure.4: Auocorrelaion funcion of Diff for equally-weighed porfolios Figure.5: Condiional FF alphas and he Diff measure for he value-weighed S-B, V-G, and W-L Figure.6: Condiional FF alphas and he Diff measure for he equally-weighed S-B, V-G, and W-L Figure 2.: Timeline Figure 2.2: Diminishing Sensiiviy Drives he Disposiion Effec Figure 2.3: Loss Aversion Drives he Reversed Disposiion Effec Figure 3.: Dividend Volailiy, GDP Growh and Recessions Figure 3.2: Smoohed Probabiliy of a High Volailiy Regime and Hisorical Dividend Growh Rae Figure 3.3: Gain and Loss Funcion Figure 3.4: Price-Dividend Funcions f Figure 3.5: Hisorical Price-Dividend Raios v.s. Condiional Dividend Volailiy Figure 3.6: Condiional Momens of Sock Reurns Figure 3.7: Disribuion of he Condiional Sharpe Raios viii

11 LIST OF TABLES Table.: Summary saisics for value-weighed size, B/M, and momenum porfolios, Table.2: Summary saisics for equally-weighed size, B/M, and momenum porfolios, Table.3: Opimal daa-driven window size Table.4: Tes of individual pricing errors Table.5: Tes of average condiional CAPM alphas for he value-weighed porfolios, Table.6: Tes of average condiional CAPM alphas for equally-weighed porfolios, Table.7: Tes of average condiional FF alphas for value-weighed porfolios, Table.8: Tes of average condiional FF alphas for equally-weighed porfolios, Table.9: Tes on Goodness of Fi Table 2. Technology Parameer Values Table 2.2 Implicaions of Diminishing Sensiiviy Table 2.3 Implicaions of Loss Aversion Table 2.4 Quaniaive Analysis Table 2.5 Sensiiviy Analysis w.r. Dividend Growh Rae Volailiy σ(θ + ). Table 3. Dividend Volailiy Esimaes Table 3.2 Calibraed Parameers Table 3.3 Asse Prices and Annual Reurns ( ) Table 3.4 Auocorrelaions of Reurns and Price-Dividend Raios Table 3.5 Reurn Predicabiliy Regressions ( ) Table 3.6 Srucural Break and Asse Prices Table 2.A Resuls for a Decision Inerval of Six Monhs Table 2.A2 Resuls for Using Purchase Prices as Reference Poins ix

12 CHAPTER A NEW TEST of TIME-VARYING FACTOR MODELS. Inroducion Tess of ime-varying facor models have caugh a lo of aenion in recen lieraure. On he one hand, abundan empirical evidence have shown ha beas from a facor model are ime-varying (e.g., Fama and French, 997; Lewellen and Nagel, 2006). On he oher hand, ess of he uncondiional CAPM, one of he mos imporan facor models, fails o explain he cross secion of sock reurns (e.g., Fama and French, 993). As a resul, a lo of research effors have been devoed o exploring he performance of a condiional facor model by allowing beas and expeced reurns o vary over ime. A long-sanding approach o esing a ime-varying facor model is o allow facor loadings o depend on observable sae variables (e.g., Shanken, 990; Leau and Ludvigson, 200). 2 Recenly, Lewellen and Nagel (2006; henceforh, LN) don use sae variables bu divide daa ino non-overlapping small windows such as monhs, quarers, half-years or years, and direcly esimae he ime series of alphas and beas from shor-window regressions. 3 If he condiional CAPM holds period-by- This chaper is based on a join paper wih Liyan Yang. 2 Numerous sudies rely on sae variables in he esimaion of CAPM and oher asse pricing models. See, for example, Campbell (987), Ferson, Kandel and Sambaugh (987), Ferson and Harvey (99, 993), Cochrane (996), Wang (2003), Pekova and Zhang (2005), and Sanos and Veronesi (2006). 3 I is arguable wheher dividing daa ino small windows is a righ way o condiion on informaion. In his paper, we don aemp o paricipae in his debae and follow Lewellen and Nagel (2006) by assuming ha invesors

13 period, hen he average pricing errors from small window regressions should be equal o zero. Conrary o some oher recen sudies (e.g., Jagannahan and Wang, 996; Leau and Ludvigson, 200; Sanos and Veronesi, 2006; Lusig and Van Nieuwerburgh, 2005), LN find ha condiioning doesn improve he performance of he simple and consumpion CAPMs. The ime-series es proposed by LN possesses a special advanage over radiional cross-secional ess which ignore imporan resricions on crosssecional slopes. However, i also has is own limiaion. As argued in Boguh, Carlson, Fisher, and Simuin (2008; henceforh, BCFS), he procedures in LN can lead o poenially large biases in alphas, which arises when he division of windows is oo fine. In oher words, he es of LN is subjec o a small sample bias. Afer correcing for his small sample bias wih sandard insrumens, BCFS manage o obain much smaller alphas for momenum porfolios, leading hem o conclude ha he condiional CAPM is superior o he uncondiional CAPM in explaining momenum porfolios. In essence, he small sample bias in LN will evenually vanish as he window size increases. However, when esimaed using daa from a large window size, beas will generally no be sable. This makes he es subjec o he undercondioning bias, which occurs when empirical ess of a condiional model fail o accoun for he invesor's ime-varying informaion se (e.g., Hansen and Richard, 987; Jagannahan and Wang, 996). Therefore, he ideal es would rely on an opimal window size which akes ino accoun boh informaion ses change gradually and hus beas are sable wihin cerain ime periods. 2

14 he undercondiioning bias and he small sample bias. The firs goal of his paper is o propose such a es. 4 Following LN, we don' rely on sae variables, bu assume ha he invesor's informaion se is relaively sable wihin cerain ime periods. Raher han dividing he windows arbirarily, we use a nonparameric mehod o find he daa-driven window size, such ha wihin he window (i) invesors informaion ses don grealy change; (ii) here are sufficien observaions o achieve esimaion efficiency. In oher words, our esimaion aims o minimize boh he undercondiioning and he small sample biases. We find ha he opimal window size varies grealy across differen porfolios. For insance, in he es of he condiional CAPM, he opimal window size varies from as shor as 47 days o as long as 333 days for differen value-weighed porfolios. To compare our resuls wih hose obained by LN, we also esimae he model using heir non-overlapping window approach. We find ha he esimaes from LN's mehod are very sensiive o he window size. When he window size changes from one monh o hree monhs, he monhly average pricing error can differ by as much as %! More imporanly, differen window esimaes can also lead o differen inferences. Therefore, arbirarily fixing he window size as hree monhs or six monhs for all porfolios may lead o unreliable esimaes and inconsisen inferences. The second goal of his paper is o consider a more general nonlinear relaionship beween asse reurns and facor reurns. Ang and Chen (2002), 4 In a conemporaneously proposed paper, Ang and Krisensen proposed a similar es o ours. 3

15 Ang, Chen and Xing (2006), and Hong, Tu and Zhou (2007) show ha many securiies covary differenly when he marke goes down from when i goes up, providing evidence of payoff nonlineariy 5. Our empirical sudies are based on a nonparameric mehodology ha avoids he misspecificaion beween asse reurns and facor reurns, and hence is immune from poenial nonlineariy biases. In esing he condiional CAPM, Wang (2003) also uses a nonparameric mehod o avoid nonlineariy biases, bu his focus is on he nonlinear relaionship beween beas (risk premia) and sae variables ha represen condiioning informaion. We, in conras, don' rely on sae variables; we are concerned wih he nonlinear relaionship beween asse reurns and facor reurns. Our esimaion mehod possesses furher advanages. Firs, we use an overlapping window esimaion, which allows a gradual change in beas raher han a drasic change hrough he non-overlapping window esimaion as in Grundy and Marin (200) and LN. Moreover, previous sudies on ime-varying beas by Campbell and Vuoleenaho (2004), Fama and French (2006), and LN, among ohers, assume ha beas are consan wihin subsamples, hereby ignoring he variaions in he beas wihin each window. Our mehod esimaes he condiional alphas and beas a every poin in ime, and hence direcly capures he variaions ha are overlooked by hese sudies. Anoher advanage is ha our esimaion is conduced in he spiri of generalized leas 5 The nonlinear relaionship considered in Ang and Chen (2002), Ang, Chen and Xing (2006), and Hong, Tu and Zhou (2007) depends on realized daa, hence i is an ex-pos relaionship. Our nonlinear relaionship is ex-ane because a ime we don' observe he realizaion a +. 4

16 squares, which pus more weigh on recen daa han on remoe daa, hus improving esimaion efficiency. Our esimaion mehod applies o any ime-varying facor models. In his paper, we focus on wo models: he condiional CAPM and he condiional Fama- French hree-facor model (993; henceforh, he FF model). These wo models have been widely used in empirical applicaions bu wheher hey are able o explain he cross secion of sock reurns are in he spoligh of curren research. Afer esimaing hese wo models, we propose wo ess o examine heir performance in explaining asse-pricing anomalies. The null hypohesis is ha if a condiional model holds a every poin in ime, hen he pricing error should be zero a all ime periods. Our firs es focuses on individual pricing errors, i.e., we examine wheher a condiional model holds a any given ime. The unique advanage of his es is ha i enables us o idenify he exac ime periods in which a condiional model holds or fails. Invesigaing he ime periods when a model holds migh sharpen our undersanding of he condiions under which a model beer applies, and examining he periods in which a model fails migh help us idenify he missing facors o furher improve he model. Our second es looks a he average pricing error. Tha is, if a condiional model holds, hen he average pricing error should be zero. Under a general assumpion of heeroskadasic innovaions, we derive he asympoic disribuion of he average pricing error, which urns ou o follow a normal disribuion. For value-weighed porfolios, our resuls show ha he condiional CAPM fails o explain any of he asse-pricing anomalies. For hese porfolios, 5

17 he condiional FF model explains he size and value effec quie well, hus sanding in sharp conras o he resuls in Ferson and Siegel (2003) among ohers; i also helps o explain he momenum porfolios, bu unlike Wang (2003), we srongly rejec he model. For equally-weighed porfolios, i's raher difficul for eiher he condiional CAPM or he condiional FF model o explain reurn variaions. In addiion o evaluaing he condiional models from an economics poin of view, i.e., wheher hey are able o explain asse-pricing anomalies, we also perform a saisical es o evaluae he goodness of fi for he condiional versus he uncondiional models. We are ineresed in which model, he condiional or he uncondiional, is closer o he rue daa-generaing process. Our resuls show ha he condiional models invariably ouperform heir uncondiional counerpars for all porfolios, implying ha he condiional models fi he daa beer. The paper proceeds as follows. Secion.2 inroduces he mehodology used o esimae and es he condiional models. Secion.3 describes he daa and presens he empirical resuls for he condiional CAPM, he condiional FF model, and he es on goodness of fi. Secion.4 concludes he paper..2 Mehodology In his secion, we inroduce a nonparameric approach o esimaing and esing a condiional facor pricing model, for which he condiional CAPM and he condiional FF model are special cases. We firs define he economeric specificaion of a condiional facor pricing model. We hen discuss how o 6

18 choose he opimal window for esimaing i. Finally we propose wo ess o evaluae is performance in explaining he cross secion of sock reurns..2. Economeric Framework If a condiional facor model holds, hen we have he following relaionship E R ) β E ( f ), () ( i, + = i, + where R i, + is he excess reurn for porfolio i a ime +, and f + sands for he facors a ime + in he corresponding facor model. For he CAPM, he marke excess reurn is he only facor, so f + = R m, + ; for he FF model, here are wo addiional facors SMB and HML oher han he marke facor, so ( R SMB HML ) f. The noaion E ( ) indicaes he condiional + = m, +, +, + expecaion, given a common public informaion se I a ime. In order o esimae (), economericians mus know he invesor's informaion se I, bu a significan pracical obsacle is ha I is unobservable. Sandard empirical mehods use sae variables observable o invesors, such as he dividend yield or erm spread, o proxy I, and specify bea as a linear funcion of lagged insrumens (e.g., Shanken, 990). This mehod herefore requires ha he sae variables be he righ ones in he invesor's informaion se. I is raher difficul, however, o idenify which sae variables are he righ ones. In recen lieraure, an alernaive approach has been proposed for doing away wih sae variables and esimaing facor loadings direcly from shor-windows. Grundy and Marin (200) use monhly reurns in he window from o + 5 o 7

19 esimae he facor loading in monh. LN argue ha (a)s long as beas are relaively sable wihin a monh or quarer, simple CAPM regressions esimaed over a shor window---using no condiioning variables---provide direc esimaes of asses' condiional alphas and beas. (P. 29) Like Grundy and Marin (200) and LN, we also dispense wih sae variables. Since he invesor's informaion se is ime-varying, we can le ime index her informaion se, and so he condiional alphas and beas change wih ime. More specifically, R α i, + βi, f+ + i,,,2,..., (2) i, + = ε + = T E ε i, + + =. i, where ( f ) 0 respecively. α and β i, are porfolio i 's alpha and bea a ime, Unlike Grundy and Marin (200) and LN, we apply a daa-driven mehod o obain he opimal esimaion window size. As will be seen from he empirical resuls laer on, he esimaes of a model from LN's mehod vary grealy as he window size enlarges or shrinks. The monhly average pricing error can differ by as much as.00% when he window size changes from one monh o six monhs. Moreover, i is possible o arrive a oally differen inferences based on differen window esimaes. Dividing daa ino arbirary windows may herefore lead o inconsisen and unreliable conclusions. Anoher imporan difference from BCFS is ha we allow a more general relaionship beween asse payoffs and facor payoffs. BCFS assume an 8

20 asymmeric nonlinear relaionship beween asse reurns and facor reurns, i.e., beas are differen for up and down markes. They demonsrae ha wih his special srucure, he bea esimaed in any window can covary wih conemporaneous marke reurns, generaing large biases in LN. In general, however, he rue relaionship beween asse reurns and facor reurns is more complicaed han merely asymmeric. Adjusmens based on a paricular srucure, as assumed by BCFS, could poenially lead o large biases as well. Our esimaion of (2) doesn' impose any special srucure beween R i, + and f +, hereby avoiding he misspecificaion beween asse reurns and facor reurns. Anoher advanage of our mehod is ha we direcly capure he variaion of beas over ime. In pracice, new informaion keeps arriving, and he invesor keeps adjusing her porfolio according o he changing informaion ses. Beas herefore keep changing. Campbell and Vuleennaho (2004), Franzoni (2004), Adrian and Franzoni (2005), Fama and French (2006), and LN, among ohers, consider he variaion of beas only across differen non-overlapping windows, bu ignore he variaion of beas wihin each window. Our esimaion, on he oher hand, uilizes overlapping windows, permiing coninuous informaion updaing and hus capuring he gradual changes in beas..2.2 Esimaion of he Model To esimae (2), we firs find an opimal window size, o be discussed in he nex subsecion. Wih he opimal window in hand, a every ime, we use he daa wihin his window o obain he condiional alpha and bea corresponding o ime. Our goal is o choose parameers o minimize he following local 9

21 sum of squared residuals: min a 0, b 0, a, b Th s Th R i,s i,s i,s f s 2 k s s.. i,s s a T 0 a s, T i,s s b T 0 b s, T k s s k. h Th (3) Here, s is a paricular daa poin wihin he window, so R, is he porfolio i 's i s excess reurn a ime s, and f s is he facor reurn a ime s. T is he oal sample size, and h is he opimal window size. Thus, afer fixing a ime poin, we use observaions from [Th] o + [Th] o esimae α i, and β i,, where [Th] denoes he ineger par of Th. To simplify noaion, we drop he porfolio index i and he ime index for α ( ) and β ( ). Noe ha α ( ) and β ( ) are funcions of T raher han, because, as shown by Robinson (989), i's necessary o le hese funcions depend on he sample size T in order o achieve asympoic consisency. In essence, we are approximaing he unknown funcions () α and β () wih a firs-order Taylor expansion wihin he window, hus inroducing four unknown coefficiens a 0, b 0, a and b. There are wo main approaches o esimaing α () and () β in he nonparameric lieraure: local consan smoohing and local linear smoohing. If a and b are zero, hen α ( ) and β ( ) are consans wihin he esimaion window, which corresponds o he local consan smoohing mehod; on he oher hand, if a and b are no zero, hen α ( ) and β ( ) are differen even wihin each esimaion window, which corresponds o 0

22 he local linear smoohing mehod. These wo approaches yield qualiaive similar resuls, so o save space, we repor he esimaion resuls of (2) based on he local consan smoohing mehod only. The ime- condiional alpha and he condiional bea β i, are he esimaes for a 0 and b 0, respecively. 6 α i, k ( ) is a weighing funcion saisfying cerain saisical properies. 7 In our empirical work, we presen resuls based on he following Epanechniov kernel 3 2 k ( u) = ( u )( u 4 ), which has been proven o achieve he highes esimaion efficiency. This kernel funcion also gives higher weigh o observaions close o he poin a which he condiional alpha and bea are esimaed and discouns he observaions far away from, which is consisen wih he idea ha recen daa conain more relevan informaion han remoe daa. We also ry wo oher popular kernels in he nonparameric lieraure, he uniform kernel and he Daniel kernel. Our main resuls are robus o he choice of hese alernaive kernels. 6 A a differen ime, we use differen daa o esimae a 0 and b 0. Therefore, a 0 and b 0 are ime-varying. 7 The kernel funcion k ( ) is a pre-specified symmeric probabiliy densiy 2 funcion such ha (i) k ( u) du =, (ii) k ( u) udu = 0, (iii) k( u) u du <, and + k 2 ( u) du <

23 An imporan issue in he nonparameric lieraure is he boundary problem. Simply pu, here are no symmeric daa for esimaing he models in he boundary areas. For insance, if we wan o esimae he model a ime =, we have daa only afer = bu lack daa before =. Following he lieraure, we use a reflecion mehod o obain pseudodaa lef boundary when [ ] 2 boundary when T T + [ Th] Th, and Ri = Ri, 2T +. 8 R i, = Ri,, f f,, T f = for he = f 2, for he righ In our empirical implemenaion, for each porfolio i, we firs esimae is opimal window size and hen, a every ime, we solve he minimizaion problem (3) o obain he condiional alpha α i, and he condiional bea β i,. Since he opimal window size serves o minimize he undercondiioning and he small sample biases, le us now urn our discussion o how o find i..2.3 The Choice of Window Size When we dispense wih sae variables and assume he invesor's informaion se o be relaively sable in adjacen periods, he opimal window size approximaes he righ amoun of informaion o be used in he esimaion. To reduce he undercondiioning bias, we wan he window size o be as small as possible. If h chosen is oo large, he informaion se may have already changed wihin he window. As a resul, if we esimae he model according o his large window, we are more likely o miss he variaions in risk and are 8 For a robusnes check, we also esimae he model only for he inerior poins which have symmeric daa. 2

24 herefore subjec o he undercondiioning bias. 9 On he oher hand, o miigae he small sample bias, we would like he window size o be as large as possible. If h chosen is oo small, here will be oo few observaions wihin he window, so ha he daa are oo noisy o yield reliable esimaes. In his case, we run he risk of he small sample bias. Therefore, he opimal window size ough o minimize boh he undercondiioning and he small sample biases. This is exacly wha he exensive nonparameric lieraure has been cenering on. We obain he opimal window size from a sandard nonparameric mehod called he cross-validaion mehod. Define he leave-one-ou esimaors and ˆ β from he following regression 0 i, ˆ0 α i, min a 0, b 0, a, b Th s Th, s R i,s i,s i,s f s 2 k s s.. i,s s a T 0 a s, T i,s s b T 0 b s, T k s s k, h Th (4) wih ˆ0 α and ˆ β i, 0 being he esimaes for a i, 0 and b 0 for porfolio i a ime. The only difference beween (4) and (3) is ha when doing he minimizaion problem a ime, we exclude he daa poin a in (4). The opimal window size h is hen chosen o minimize 9 An exreme example is a model esimaed using all observaions, which corresponds o he larges window size. In his case, we simply esimae he uncondiional model, oally ignoring he predicable variaions in risk. 3

25 T CV h R i, 0 i, 0 i, f 2. Inuiively, for any porfolio i, we firs fix an arbirary window size. A every ime, we use all daa wihin his window excep he daa a ime o do he minimizaion in (4), obaining he prediced value and predicion error corresponding o. Inuiively, since he daa in he viciniy of ime conain similar informaion o he daa a ime, we can use hem predic he ime- observaion. We do his for all ime periods ( =, 2,... T ), and sum up all he predicion errors denoed by CV (h). The opimal window is chosen o minimize CV (h). For any given porfolio, he opimal window size obained from he leave-oneou cross-validaion mehod is he same for all ime periods. I is possible ha beas migh change faser in some periods han in ohers, hus a ime-varying window size migh seem o be needed. We leave his for fuure research. Since exising sudies relying on he simple window approach use a uniform window size, in order o beer compare our resuls wih he lieraure, we sick o he uniform window size in his paper..2.4 Two Tess of a Condiional Facor Model Our null hypohesis is ha a condiional facor model holds a each poin in ime. If facors hemselves are excess reurns, as is he case wih he condiional CAPM and he condiional FF model, hen esing his hypohesis is equivalen o esing α ( ) = 0 for all 0. Firs, we es if he individual pricing T 0 As has been menioned, o achieve esimaion consisency as proved by 4

26 errors are zero, i.e., α ( ) = 0 for any. Second, we es if he average pricing error is equal o zero, i.e., α ( ) = 0. T T T = T.2.4. Tes on Individual Pricing Errors Under usual echnical assumpions, Cai (2007) shows ha he individual alpha obained from (2) follows an asympoic normal disribuion. Le τ = T, under he null hypohesis ha alphas are equal o zero a every poin, he inerior alphas follow d ( τ ) N( 0, ν Σ( )), Th ˆ α 0 τ (5) where T is he sample size, and h is he opimal bandwidh or he window size. Since he effecive daa used o esimae α (τ ) is Th, ˆ α ( τ ) converges a he rae of Th. The deails for he variance of αˆ ( τ ) are provided in Appendix.A. The asympoic behavior of he esimaed boundary alphas is differen from ha of he inerior ones. Bu he boundary alphas are no paricularly ineresing in our conex, 2 and hey also make up only a small proporion of Robinson (989), he pricing errors are funcions of / T insead of. The inerior alphas are hose corresponding o he ime periods which doesn' suffer he boundary problem. 2 One scenario in which he boundary alphas are paricularly ineresing is when he condiional alphas (also beas) are funcions of, for example, he marke reurn raher han ime. In his case, he boundary alphas correspond o he pricing errors under exreme marke condiions, such as marke crashes or marke frenzies. 5

27 he esimaed ime series of he condiional alphas. To save noaion and space, we repor he resuls only for inerior alphas. Incorporaing he boundary alphas won change our resuls dramaically Tes on Average Pricing Errors If wo models are boh rejeced a, for example, 80% of he ime periods, he es on individual pricing errors alone canno ell us which model is relaively beer. Thus we need o urn o he second es, which focuses on he implicaion ha if a condiional facor model holds, hen he average pricing error should be equal o zero. This measure is also adoped by LN. The average pricing error is T ˆ ˆ α = α T = T. In Appendix.B, we derive he asympoic disribuion of αˆ when he random error process { } T ε is heeroskedasic. We find i follows a normal disribuion: = d T ˆ α N( 0, V ), (6) where V is he asympoic variance and equals he (,)h elemen of Ω 0 j= E ( X X ε ) Ω + j ε, wih X (, f ) + j 0 =, Ω = E( X X ) 0, and j denoing he lag order. Even hough we use a nonparameric esimaion mehod, he asympoic variance V resembles he sandard Newey-Wes esimaor. In implemenaion, we use he corresponding sample momens o esimae V. 6

28 .3. Empirical Resuls.3. Daa Our daa are obained from Professor Kenneh French's websie. 3 From he 25 size-b/m porfolios, we form six size and B/M porfolios. S is he average reurn of he five porfolios in he lowes size quinile, B is he average reurn of he five porfolios in he highes size quinile, and S-B is he difference. G is he average reurn of he five porfolios in he low-b/m quinile, V is he average reurn of he five porfolios in he high-b/m quinile, and V-G is he difference. The hree momenum porfolios are direcly obained from Professor Kenneh French's websie, where we le W sand for he reurn of he winner porfolio, L for he reurn of he loser porfolio, and W-L for heir difference. To compare our findings wih exising sudies, we look a boh he valueweighed and he equally-weighed porfolios, using daily daa from 963 o The long ime series of daily daa no only provide rich informaion abou he underlying informaion srucure, bu also help improve esimaion efficiency. Moreover, he debae on he small sample bias of LN s procedure also focuses on daily daa. For a robusness check, we also conduc ess using monhly daa and obain qualiaively similar resuls no repored here. 3 hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml 4 LN examine he performance of he condiional CAPM using value-weighed porfolios, while BCFS focus on momenum porfolios which are equallyweighed. 7

29 I is well known ha nonsynchronous rading can have a grea impac on shorhorizon beas (Lo and MacKinlay, 990). Since we use high-frequency daily daa, we need o consider he microsrucure issues such as he bid-ask spread. To address hese issues, when esimaing he uncondiional models, we use Dimson (979) regressions wih he srucure suggesed by LN: R β 4 i,3 α i + βi, f + βi, 2 f + f p + ε i,, (7) 3 i, = p= 2 where p denoes lag. The esimaed pricing error for porfolio i is α i, and he esimaed bea is β + β, 2 + β, 3. i, i i Table. and Table.2 presen he summary saisics for he value-weighed and equally-weighed porfolios, respecively. The daily esimaes are muliplied by 2, he average number of rading days per monh, so ha all esimaes are expressed as monhly percenages. Wih respec o he valueweighed porfolios, excess reurns exhibi he usual cross secion paerns. Overall, he small socks ouperform he big socks (0.63% vs. 0.5%), he value socks ouperform he growh socks (0.84% vs. 0.30%), and he winner socks ouperform he loser socks (.2% vs %). Excep for he size porfolios, he uncondiional CAPM alphas are all significan, implying ha he uncondiional CAPM fails. In line wih prior research (e.g., Fama and French, 993), he uncondiional FF model improves upon he uncondiional CAPM because he alphas for he size and B/M porfolios are much smaller. However, he alphas for he B/M porfolios are sill significan. For he momenum porfolios, he uncondiional FF alphas are highly significan, and hey are also 8

30 of he same magniude of he uncondiional CAPM alphas, indicaing ha he uncondiional FF model doesn' help o explain he momenum effec. Table.2 shows ha he equally-weighed porfolios display some ineresing paerns. Firs, he size effec is very pronounced. The equally-weighed S-B has an excess reurn of.05%, compared o only 0.3% for he valueweighed S-B. The uncondiional CAPM alpha is also much higher:.04% for equally-weighed S-B vs. 0.07% for value-weighed S-B. The uncondiional FF alpha for equally-weighed S-B is.00%, which is close o he uncondiional CAPM alpha, indicaing ha he uncondiional FF fails o explain he size effec in he equally-weighed porfolios. This is no surprising because he equallyweighed porfolios pu more weigh on small socks, which, as shown in Fama and French (996), he uncondiional FF model doesn' explain quie well. Second, momenum porfolios have a very differen paern from he one usually observed in he monhly daa. The loser porfolio acually earns a higher average reurn han he winner porfolio (2.08% vs..7%), implying ha he equally-weighed momenum sraegy is no profiable a daily horizon. Neiher he CAPM nor he FF model is able o accoun for he reurn variaions in he momenum porfolios. We now allow he facor loadings o vary over ime, and invesigae wheher he condiional versions of he CAPM and he FF model are able o accoun for he reurn variaions in hese porfolios. To correc for he impac of nonsynchronous rading, we also append wo lags in he esimaion of (2): 9

31 R ( ) 4 β3 T α + β f + β 2 f + f p + ε i,,,2,.... (8) T T T 3 p= 2 i, = = T Thus, porfolio i 's pricing error a ime is α ( ) ( ) β ( ) ( ) β + +. T 2 T β 3 T T, and is condiional bea a is.3.2 Tesing he Condiional CAPM In subsecion.3.2., we repor he daa-driven window size for he condiional CAPM obained from he cross-validaion mehod described in subsecion.2.3. In subsecions and.3.2.3, we esimae he condiional CAPM and evaluae is performance hrough he wo ess proposed in subsecion Daa-Driven Window Size If he invesor opimally rebalances her porfolios according o changes in her informaion se, hen he realized daa srucure should reflec changes in he underlying informaion srucure and, as a resul, he esimaed window size serves as a proxy for he sabiliy of he informaion srucure. A larger window size implies ha he relaionship beween asse reurns and facor reurns, or he underlying informaion srucure, is generally more sable. Consequenly, beas will change less frequenly wih a larger window han wih a smaller window. 20

32 Table.: Summary saisics for value-weighed size, B/M, and momenum porfolios, The able repors he average excess reurns, he uncondiional CAPM alphas and he uncondiional FF alphas for value-weighed size, B/M, and momenum porfolios using daily daa. The uncondiional CAPM alphas are obained from he regression in (7) by leing f = Rm, and he uncondiional FF alphas are obained from he regression in (7) by leing f = ( R SMB HML) m. Average reurns and alphas are expressed in percenage monhly. Bold values denoe esimaes greaer han wo sandard errors from zero. Size B/M Mom S B S-B G V V-G L W W-L Panel A: Excess reurns Ave Sd. err Panel B: Uncondiional CAPM alphas Es Sd. err Panel C: Uncondiional FF alphas Es Sd. err

33 Table.2: Summary saisics for equally-weighed size, B/M, and momenum porfolios, The able repors he average excess reurns, he uncondiional CAPM alphas and he uncondiional FF alphas for equally-weighed size, B/M, and momenum porfolios using daily daa. The uncondiional CAPM alphas are obained from he regression in (7) by leing f = R, and he uncondiional FF alphas are obained from he regression in (7) by leing = ( R SMB HML) m f m. Average reurns and alphas are expressed in percenage monhly. Bold values denoe esimaes greaer han wo sandard errors from zero. Size B/M Mom S B S-B G V V-G L W W-L Panel A: Excess reurns Ave Sd. err Panel B: Uncondiional CAPM alphas Es Sd. err Panel C: Uncondiional FF alphas Es Sd. err

34 Table.3 presens he esimaed window size for he condiional CAPM and he condiional FF model. For now, we focus only on he condiional CAPM, and discuss some noable paerns of he esimaed window size. Firs, he window size is much larger for he big porfolio han for oher porfolios, 333 if value-weighed and 285 if equally-weighed, indicaing a less frequen change in beas of large socks. This is consisen wih Shanken (990), in which he T-bill rae serves as he sae variable, and beas of large socks are far less sensiive o changes in he T-bill rae han beas of small socks. Second, he window size is always larger for he value-weighed porfolios han for he equally-weighed porfolios. For example, we use 87 observaions o esimae he value-weighed loser porfolio, while we use only 33 observaions for he equally-weighed one. This is mosly likely due o he fac ha he value-weighed porfolios pu more weigh on large socks, whose underlying informaion srucure urns ou o be less volaile. We cauion ha we are no aemping o map one-o-one he window size o he underlying informaion srucure. Bu we do argue ha he daa-driven window size reveals imporan informaion abou he unknown informaion srucure. Our resuls in Table.3 show ha he esimaed window size ranges from as shor as 3 days o as long as 333 days, varying widely from porfolio o porfolio. This suggess ha fixing a window size as one monh or hree monhs for all porfolios may incur esimaion biases, which generally become larger if he underlying relaionship beween asse reurns and facor reurns changes in a more complicaed way. 23

35 Using he opimal window size, we esimae (8) wih f = Rm o ge he ime series of he condiional CAPM alphas for every porfolio. In order o evaluae wheher he condiional CAPM explains he reurn variaions, we apply he wo ess on he pricing errors proposed in subsecion Individual Pricing Errors Panel A of Figures. and.2 plo he condiional alphas for he valueweighed and equally-weighed S-B, V-G, and W-L, respecively. The condiional alphas of all porfolios flucuae grealy over ime, bu W-L displays he larges variaion, wih he daily alpha ranging from a minimum of -0.88% o a maximum of.38% if value-weighed, and from -2.% o 0.97% if equallyweighed. Unlike he exising sudies, ours obains he condiional alpha a every poin in ime, which enables us o invesigae wheher he condiional CAPM holds a any given ime. Based on he disribuion of he individual pricing error in (5), we can calculae he sandard error sd, for α, he condiional alpha a ime. Define he difference beween α and.96sd,.96 imes he corresponding ime sandard error, as follows: Diff =.96 α. (9) sd The sign of Diff indicaes wheher we should accep or rejec he condiional CAPM a he 5% significan level. If evidence for rejecing he condiional CAPM a ime ; if Diff is posiive, hen we don' have he Diff is negaive, hen we find evidence ha indicaes he failure of he model a ime. The series of 24

36 Diff for S-B, V-G, and W-L are ploed in Panel B of Figures. and.2. These graphs show ha for all porfolios, Diff end o be negaive mos of he ime, implying ha he condiional CAPM may hold for only a small fracion of he ime periods. To examine he persisence of he model's explanaory power, we plo he auocorrelaion funcion of he Diff measure for he value-weighed S-B, V-G and W-L in Panel A of Figure.3 and he corresponding equally-weighed ones in Panel A of Figure.4. These figures show ha he auocorrelaion of Diff generally declines o zero in an AR() fashion, because he informaion srucure is more sable wihin adjacen ime periods. Today's informaion srucure, for example, is mos like yeserday's, so ha if we rejec (accep) he model oday, i's mos likely ha we rejeced (acceped) he model yeserday. As we move furher away from oday, he similariy in informaion srucure ypically declines, and i becomes less likely for us o rejec (accep) he model, given ha we rejec (accep) i oday. Afer cerain periods, he informaion srucure may have oally changed, sharing no commonaliy wih oday's informaion srucure, which explains why he auocorrelaion usually drops o zero afer cerain lags. 25

37 Table.3: Opimal daa-driven window size This able repors he esimaed window size using he cross-validaion mehod described in subsecion window size. The window size is measured in erms of days. For example, a window size of 60 days means ha when esimaing he model a day, we use he 60-day daa from 30 o Size B/M Mom S B S-B G V V-G L W W-L Panel A: Value-weighed Porfolios condiional CAPM condiional FF Panel B: Equally-weighed Porfolios condiional CAPM condiional FF

38 To ge a quaniaive idea of he overall performance of he condiional CAPM, for each porfolio, we calculae he fracion of he ime when he condiional CAPM holds, he resuls of which are shown in Table.4. 5 Panel A shows ha, for he value-weighed porfolios, he condiional CAPM performs bes for L, holding 22.3% ou of all ime periods; i performs wors for S, holding 5.89% ou of all ime periods. In oher words, ou of he 44 years of daa we are considering, he condiional CAPM roughly holds 9.8 years for he loser porfolio and 7.0 years for he small porfolio. Ineresingly, examining he performance of he condiional CAPM period-by-period reveals ha i fails mos ofen for size porfolios, raher han, as generally assumed, momenum porfolios. In fac, among all he porfolios, he condiional CAPM seems o perform bes for momenum porfolios, yielding 22.3% for L, 22.2% for W, and 7.25% for W-L. Panel B of Table.4 shows ha, for he equally-weighed porfolios, he condiional CAPM works bes for G, holding 20.25% ou of all ime periods, and wors for S, holding only 2.76% ou of all ime periods. Therefore, he small porfolios, boh value-weighed and equally-weighed, represen he greaes challenge o he condiional CAPM. Overall, he condiional CAPM holds for fewer periods for he equally-weighed porfolios han for he valueweighed ones, which is especially rue for momenum porfolios. For example, i holds 4.24% of he ime for he equally-weighed L, much less han 22.3% for he value-weighed L. 5 A saisical es needs o be consruced o rigorously evaluae he ime periods in which a model holds. Here we propose his preliminary inuiive measure. 27

39 Figure.: Condiional CAPM alphas and he Diff measure for he valueweighed S-B, V-G, and W-L. Panel A plos he series for he condiional alphas, which are obained from he nonparameric esimaion of (8) wih f = Rm. The condiional alphas are repored as daily percenages. Panel B plos he series of Diff which are calculaed from (9). Posiive values of Diff correspond o he periods in which he condiional CAPM is acceped while negaive values indicae he failure of he model. 28

40 Figure.2: Condiional CAPM alphas and he Diff measure for he equallyweighed S-B, V-G, and W-L. Panel A plos he series for he condiional alphas, which are obained from he nonparameric esimaion of (8) wih f = Rm. The condiional alphas are repored as daily percenages. Panel B plos he series of Diff which are calculaed from (9). Posiive values of Diff correspond o he periods in which he condiional CAPM is acceped while negaive values indicae he failure of he model. 29

41 Figure.3: Auocorrelaion funcion of Diff for value-weighed porfolios. The series of Diff are calculaed from (9). Panel A plos he auocorrelaion of Diff for he condiional CAPM, and Panel B plos he auocorrelaion of Diff for he condiional FF model. 30

42 Figure.4: Auocorrelaion funcion of Diff for equally-weighed porfolios. The series of Diff are calculaed from (9). Panel A plos he auocorrelaion of Diff for he condiional CAPM, and Panel B plos he auocorrelaion of Diff for he condiional FF model. 3

43 The ess on individual pricing errors hus show he inadequacy of he condiional CAPM o explain he dynamics of sock reurns. For every porfolio, he condiional CAPM holds less han /4 of he ime. More imporanly, o claim success, a model has o be able o price all porfolios simulaneously. If we consider he ime when he condiional CAPM holds for all hree porfolios of he value-weighed S-B, V-G, and W-L, i will be even less han 4%! Compared o exising mehods in he lieraure, hese ess on individual pricing errors possess a unique advanage, i.e., hey enable us o idenify he exac ime periods in which he condiional CAPM holds or fails. For insance, referring o Figures. and.2, we observe ha he mos exreme values of alphas for S-B, V-G, and W-L all appeared around March 200, when he echnology bubble burs, hus represening he greaes failure of he condiional CAPM. We can also idenify he periods when he condiional CAPM holds, and by invesigaing hese periods' imporan variables, such as he marke condiions and he economic siuaions, we will be able o discover he condiions under which he marke risk facor will deermine invesors' porfolio choice. This has imporan heoreical and empirical implicaions bu hasn' ye been pursued in he lieraure. We leave his for fuure research. 32

44 Table.4: Tes of individual pricing errors This able repors he proporion of ime in which he condiional CAPM and he condiional FF model are acceped. A each ime, using he disribuion in equaion (5), we calculae he es saisic for he alpha a ime, and compare i wih.65, he 5% criical value for he sandard normal disribuion. If he es saisic is less han.65, we accep he condiional model o hold a. Summing up all he periods in which he model holds and dividing by he oal number of periods gives he proporion, which is repored as he percenage. Size B/M Mom S B S-B G V V-G L W W-L Panel A: Value-weighed Porfolios (%) condiional CAPM condiional FF Panel B: Equally-weighed Porfolios (%) condiional CAPM condiional FF

45 Average Pricing Errors Now le us examine how he condiional CAPM explains he nine porfolios based on ess of he average pricing errors. We firs conduc our discussions for he value-weighed porfolios and hen for he equally-weighed. Since BCFS challenge he resuls of LN hrough he momenum porfolios, we firs look a he resuls of he momenum porfolios, and hen analyze he resuls of he size and B/M porfolios. Value-weighed Porfolios The resuls from our nonparameric mehod are presened in Panel A of Table.5. They show ha for he momenum porfolios, he average condiional alphas are -0.88% (z-sa -6.77), % (z-sa 5.40), and.72% (z-sa.47) for L, W and W-L, respecively. The esimaes for L and W are slighly smaller han he uncondiional alphas of -0.94% and 0.56%, bu he esimae for W-L is larger han is uncondiional alpha of.50%. Therefore, he condiional CAPM performs even worse han he uncondiional CAPM in explaining W-L. Moreover, all hese esimaes are highly significan, providing srong evidence ha he condiional CAPM fails o explain he momenum porfolios. BCFS poin ou ha he mehod in LN suffers poenially serious small sample biases. Bu how large are hese biases? Are hey as large as BCFS have 6 We use "z-sa" o sand for he saisics calculaed based on he normal esimae of α disribuion of (6). Tha is, z = sandard error of α. If z >. 65, we rejec he model a 5% significan level. 34

46 claimed? To answer hese quesions, we also esimae he model using he mehod in LN by choosing he non-overlapping window as N =,3, 6 monhs. Following Fama and Macbeh (973), we obain he sandard error of he esimaes from he ime series variaion of he condiional alphas. The resuls from LN's mehod are presened in Panel B of Table.5, which show ha he average condiional alphas for W-L are 2.49% (-sa 7.78) when N =,.96% (-sa 7.54) when N = 3, and.58% (-sa 6.32) when N = 6. Therefore, he LN mehod provides esimaes ha are very sensiive o he window size, where he difference in average pricing errors is as large as 0.9% (2.49%-.58%) for W-L. This sensiiviy o window size highlighs he imporance of using he daa-driven window o esimae he model. An imporan feaure of he momenum porfolios is ha hey are ypically rebalanced every monh, and he enering and exiing socks may no have similar beas. 7 Anoher shorcoming of he non-overlapping window esimaion is ha i fails o accoun for changing composiion in he momenum porfolios, because by fixing N = 6, for insance, i assumes ha beas are consan over periods of as long as six monhs. Our mehod can accoun for he high urnover in he momenum porfolios because we esimae he condiional alphas and beas coninuously a each poin in ime. 7 Grundy and Marin (200) show ha due o selecion, beas of newly added winner and loser socks vary wih he marke reurn in he formaion period. 35