Time Variation of Expected Returns on REITs: Implications for Market. Integration and the Financial Crisis

Transcription

1 Tim Variation of Expctd Rturns on REITs: Implications for Markt Intgration and th Financial Crisis Author Yuming Li Abstract This articl uss a conditional covarianc-basd thr-factor pricing modl and a REIT indx-nhancd four-factor modl to xamin th tim variation of xpctd rturns on REITs ovr th priod Although xpctd rturns on quity REITs ar highly corrlatd with thir own volatility, th covariancs of rturns on REITs with th stock markt prmium, small stock prmium and valu prmium subsum th rol of th volatility of REITs in xplaining xpctd rturns on REITs. Th conditional btas of REITs associatd with th stock markt prmium and th valu prmium, along with th conditional corrlation btwn th two prmiums, ar mor important than th volatility of th stock markt or othr factors in xplaining th tim variation of xpctd rturns on REITs, spcially during th rcnt financial crisis. Tsts of asst pricing rstrictions add furthr vidnc on th intgration of th ral stat markt with th gnral stock markt. 1

2 Sinc its incption in 1960, th markt for ral stat invstmnt trusts (REITs) has grown trmndously. According to th National Association of Ral Estat Invstmnt Trusts (NAREIT), REITs rprsnt mor than $1.7 trillion of ral stat dbt and quity in 014; and narly 40 million Amricans invst in REITs through thir pnsion and rtirmnt plans in that yar. Th vast siz of th markt for REITs suggsts that th tim sris proprtis of xpctd rturns on REITs hav important ramifications for portfolio choics facd by invstors and th prformanc valuation of fund managrs. Th goal of th articl is to study th tim variation of xpctd rturns on REITs. It is wll documntd that xpctd rturns on REITs xprincd larg fluctuations in th last svral dcads. Most strikingly, during th priod of th financial crisis, xpctd rturns on quity REITs surgd to mor than ight tims of thir pr-crisis lvls. Th unprcdntd fluctuations ar puzzling. Ar th tim-varying xpctd rturns on REITs compnsation for thir own volatility or th systmatic risks associatd with th stock markt or othr risk factors? What risk factors ar most important for xplaining th tim variation of xpctd rturns on REITs? This articl studis th importanc of systmatic risks in xplaining th tim variation of xpctd rturns on REITs, using a conditional covarianc-basd thr-factor asst-pricing modl with th Fama-Frnch (1993) factors. Th modl is important for studying th bhavior of rturns on REITs around th financial crisis for four rasons. First, th ral stat litratur has documntd that th Fama-Frnch thr-factor modl is mor usful than th singl-factor, markt-basd modl in capturing rturns on REITs and gnrating stabl stimats of markt btas (Ptrson and Hsih, 1997; Chiang, L and Wisn, 005). Scond, in th financ litratur, rsarchrs hav shown that th valu prmium, as on of th Fama- Frnch factors, proxis for innovations of invstmnt opportunitis. 1 Third, rcnt rsarch shows that th systmatic volatility associatd with th aggrgat stock markt is not pricd in REIT rturns (DLisl, Pric and Sirmans, 013). As a rsult, it is imprativ to considr othr factors and altrnativ masurs of systmatic risks such as th conditional covariancs in studying xpctd rturns on REITs. Fourth, th xisting litratur has focusd on th rols of firm-lvl charactristics in xplaining th bhavior of

3 rturns on REITs during th financial crisis (.g., Sun, Titman and Twit. 015), laving th fundamntal dtrminants of REIT rturns unxplord. This papr also studis a REIT indx-nhancd four-factor modl in which xpctd rturns on REITs can b rlatd to thir own volatility and thir covariancs with th Fama-Frnch factors. Undr th markt sgmntation hypothsis, xpctd rturns on REITs ar rlatd to thir own volatility. Altrnativly, undr th full intgration hypothsis, xpctd rturns on REITs ar unrlatd to thir own volatility but rlatd to thir covariancs with th Fama-Frnch factors. To tst th various hypothss, I us th asymmtric xtnsion of th multivariat GARCH-mans procss (Engl and Kronr, 1995) to modl th volatility of rturns on REITs, th volatility of th factors, and th covariancs of rturns with th factors. Using this spcification, I stimat th covarianc-basd modls, tst markt sgmntation or intgration hypothss, and xamin th tim sris proprtis of risks and rturns on REITs. Th ral stat litratur has producd mixd vidnc on th issu of markt intgration. Liu t al. (1990) adopt a two-factor modl with a stock markt portfolio and a ral stat portfolio as factors. Thy find vidnc in support of markt sgmntation hypothsis for privatly hld ral stat. Howvr, thy cannot rjct th hypothsis that th markt for quity REITs is intgratd with th gnral stock markt using th two-factor modl. Mi and Liu (1994) study th prdictability of rturns on fiv diffrnt asst portfolios including quity REITs using a multifactor latnt-variabl modl. Li and Wang (1995) us a two-factor modl with th stock markt prmium and th bond markt s dfault prmium and find no vidnc of markt sgmntation. Ling and Naranjo (1999) us th stock markt group and data on various macroconomic risk factors to tst rstrictions on risk prics across asst portfolios. Thy find that th markt for th ral stat scuritis is intgratd with th markt for non-ral-stat stocks. Sinc tsts of any markt intgration ar joint tsts of th intgration hypothsis and asst pricing or othr conomic modls, tst rsults dpnd critically on th assumd factors in th modls. 3 3

4 I study th conditional xpctd rturns on REITs from January 197 to July 013. I first obtain strong vidnc against th CAPM and a REIT-indx nhancd two-factor modl. I also find vidnc against th CAPM in favor of th Fama-Frnch thr-factor modl but find no vidnc against th thr-factor modl in favor of th REIT indx-nhancd four-factor modl. Th rsults suggst that tsting capital markt intgration basd on th thr- or four-factor modl is mor appropriat than that basd on th CAPM or th REIT indx-nhancd two-factor modl, as usd by Liu t al. (1990). Using th thr-factor modl, I find th prics of th covarianc risks associatd with th thr factors (stock markt prmium, small stock prmium, and th valu prmium) to b pricd. Using th four-factor modl, I find that th conditional covariancs of rturns on REITs with th thr factors subsum th rol of th volatility of REITs in xplaining th tim-varying xpctd rturns on REITs. Th rsults of stimating th risk prics and tsting rstrictions on th risk prics rjct th sgmntation or mild sgmntation hypothsis in favor of th hypothsis that th ral stat markt is fully intgratd with th gnral stock markt. Th rsults of stimating th Fama-Frnch thr-factor modl indicat that th btas of REITs associatd with th stock markt risk prmium and th valu prmium, along with th corrlation btwn th two factor prmiums, incras sharply to unusually high lvls during th rcnt financial crisis. Ths incrass caus xpctd rturns on REITs to surg to a fw tims of thir pr-crisis lvls. Although th volatility of REITs also riss sharply during th financial crisis, it loss its xplanatory powr for xpctd rturns on REITs bcaus systmatic risks associatd with th Fama-Frnch factors xplain not only xpctd rturns on REITs but also most of th volatility of REITs, spcially around th pak of th rcnt financial crisis. Th rmindr of th articl is organizd as follows. Th nxt sction dscribs th conditional multifactor asst pricing modls and th multivariat GARCH modl. Th monthly data of th REIT indics and risk factors ar thn dscribd. Empirical rsults of stimating th modls and analyzing th implid xpctd REIT rturns and risk componnts ar prsntd bfor conclusions. 4

5 Modls In this sction, I dscrib th conditional multifactor asst pricing modl and th multivariat GARCH modl. Mrton (1973) shows that, in an intrtmporal stting, invstors nd to hdg against changing invstmnt opportunitis. As a rsult, th xpctd xcss rturn on any asst is a function of its covariancs with rturns on th markt portfolio and a numbr of hdging portfolios. Ross (1976) dvlops an arbitrag pricing modl in which asst rturns ar gnratd by a fw common risk factors, and xpctd asst rturns ar functions of btas and risk prmiums associatd with th factors. Th covarianc and bta-basd modls offr similar implications for xpctd rturns. In this sction, I first prsnt th covarianc-basd pricing modl for th purpos of mpirical stimations and thn discuss th factor modl and quivalnt bta-basd rprsntation of th asst pricing modl for th purpos of analyzing th mpirical implications of th modl for th tim variations of th xpctd rturns and volatility of REITs. Th Covarianc-Basd Modl Considr th following covarianc-basd pricing modl: K t1 it j t1 it jt j1 E ( R ) cov ( R, F ), (1) whr R is th xcss rturn for priod t on th ith asst for i 1,, N, it jt F is th jth risk factor for j 1,, K, and j rprsnts th pric of th covarianc risk of ach asst with th jth risk factor. For simplicity th risk prics ar assumd to b constant. Th xpctd rturns and covariancs in quation (1) ar both conditional on information at tim t-1. Equation (1) is spcializd to th conditional CAPM if th rturn on th markt portfolio is th singl factor (K=1). If K, quation (1) is th conditional multifactor pricing modl of Mrton (197). Th xcss rturn rfrs to th rturn on an asst minus th 5

6 riskfr rat, or th diffrnc btwn rturns on two portfolios, such as th small and big stock portfolios. In th rmaindr of th papr I assum that th first K xcss rturns ar rturns on K factor-mimicking portfolios (K < N). Lt ε t b th N 1 vctor of unxpctd rturns with th ith lmnt it and th N N conditional varianc-covarianc matrix H t, whos ( i, j) th lmnt is h. ij, t Th xcss rturn in quation (1) is thn givn by K it i i, T tt j ij, t it j1 R I h, () whr i is th ith intrcpt for th priod bfor dat T and it is th incrmntal intrcpt for th priod aftrwards. I includ th indicator (dummy) variabl It T, which taks th valu of unity for t>t and zro othrwis, to dtct changs in rturns rsulting from a possibl structural brak or chang in rgims unxplaind by th pricing modl. Th conditional varianc-covarianc matrix H t follows th asymmtric BEKK GARCH spcification: H C' C A'H A B'ε ε 'B D'η η 'D, (3) t t1 t1 t1 t1 t1 whr A, B and D ar N N cofficint matrics, C is a lowr triangular matrix with N( N 1) / paramtrs, and η t1 is a N 1 vctor with ith lmnt givn by i, t1 i, t1 if it, 1 0 and zro othrwis. With th last trm capturing th asymmtric ffcts of ngativ shocks on volatility, quation (3) is th asymmtric xtnsion of th BEKK spcification proposd by Engl and Kronr (1995). Equation (3) is also a spcial vrsion of th asymmtric dynamic covarianc (ADC) modl proposd by Kronr and Ng (1998). Equations ()-(3) ar multivariat xtnsions of th bivariat modl usd by Guo, t al. (009), who find significant asymmtric ffcts but insignificant diffrncs 6

7 btwn th asymmtric BEEK spcification and th asymmtric ADC modl whn th modls ar applid to th stock markt prmium and th valu prmium. Th spcification hr is appaling bcaus it dirctly imposs positiv dfinitnss on th varianc-covarianc matrix. Howvr, th stimation bcoms difficult if th multivariat GARCH modl is applid to multipl assts. Sinc this papr uss four assts (N = 4), it is ncssary to mak simplifying assumptions to limit th dimnsion of paramtr spac. For this rason, I assum that A, B and D ar diagonal matrics, following D Santis and Grard (1997) and Hardouvlis, Malliaropulos and Pristly (006). Although most cofficint matrics ar assumd to b diagonal, th conditional covariancs and corrlations in th modl hr ar functions of a long history of past innovations. This fatur allows us to xamin tim variations of th volatility as wll as th systmatic risks of th REIT indx. Sinc th xpctd rturn and volatility of th REIT portfolio ar modld along with thos of th factor-mimicking portfolios in th abov systm, th dynamics of risks and th xpctd rturn on th REIT portfolio can b compard with thos of th factors. Th varianc of th xpctd rturn on REIT (asst N) can b dcomposd into componnts associatd with ach of th factors: K K t1 Nt j Nj, t j Nj, t j1 j1 (4) var[ E ( R )] var( h ) var( h ) intraction trms. Th intraction trms includ covariancs btwn jhnj, t and khnk, t ( j k). Th contribution of ach risk factor to th varianc of th xpctd xcss rturns is thn masurd by th following varianc ratios: 7

8 VR j j var( h ) var( hnj, t ) K. j1 j Nj, t (5) Th varianc ratios do not ncssarily sum to on across factors bcaus of th intractions trms. Equivalnt Bta-Basd Rprsntation Th covarianc-basd pricing modl in quation () can b asily transformd into th familiar conditional bta-basd pricing modl. To s this, considr th following conditional factor modl for disturbanc trms: β' [ F E ( F )], (6) it it t t1 t it Undr th assumption that th idiosyncratic componnt it is conditionally uncorrlatd with ach of th factors, th K 1 vctor of β it for th ith asst is givn by β' cov (, F' )cov ( F, F' ) h' H, (7) 1 1 it t1 it t t1 t t it Kt whr h it ( hi 1, t,, hik, t )' and H Kt rprsnts a submatrix of H t containing th first K rows and columns. Substituting th vctor of th factor-mimicking portfolios F t as xcss rturns into quation (1) implis factor risk prmiums givn by γ ( ), t Et 1 Ft HKtλ (8) whr λ ( 1,, K )'. Equation (8) says that factor risk prmiums ar functions of th conditional varianc-covarianc matrix of th factors. In othr words, th risk prmium on ach factor is not only rlatd to its own volatility, but also rlatd its covarianc with othr factors. Givn th bta matrix in quation (7) and th risk prmium vctor in quation (8), th covarianc-basd pricing modl in quation () is transformd into th bta-basd pricing modl: 8

9 E ( R ) I β' H λ. (9) t1 it i i, T tt it Kt Th conditional bta-basd pricing modl in quation (9) suggsts that th tim variation of th xpctd REIT rturn is rlatd to thos of btas, variancs and corrlations btwn pairs of factors. Similarly, th conditional factor modl in quation (6) implis th following varianc dcomposition: h β' Η β var ( ). (10) ii, t it Kt it t1 it In quation (10), th first componnt in th right sid rprsnts th systmatic varianc for priod t and th scond componnt rprsnts th idiosyncratic varianc. Lik th xpctd rturns, th systmatic varianc varis with th conditional btas, variancs and corrlations btwn pairs of factors. Data and Mthod Givn th limitd availability of historical REITs data dating back to th 1970s, this study uss monthly rturns on REITs along with th monthly data of th Fama-Frnch factors providd by Knnth Frnch. 4 Monthly rturns on th Equity and Mortgag REIT indics ar obtaind from th National Association of Ral Estat Invstmnt Trusts (NAREIT) for th sampl priod from January 197 to July 013. Approximatly as of th nd of 013, stock xchang-listd Equity REITs account for 70 prcnt of all U.S. listd REIT assts, and Equity REITs rprsnt 90 prcnt of th approximatly $700 billion quity markt capitalization of th listd REIT marktplac. Thr ar approximatly 150 listd Equity REITs, almost all of which ar tradd on th Nw York Stock Exchang. Thr ar 6 listd rsidntial Mortgag REITs with a markt capitalization of $4.3 billion. As a snsitivity chck, th study also uss monthly rturns on th Wilshir (quity) REIT indx availabl from th Wilshir Associats. Th sampl priod for th Wilshir REIT indx is from January 1978 to July 013. As notd in prior studis, compard with th Wilshir indx, th NAREIT Equity REIT indx allows for gratr comparability with prvious REIT 9

10 studis, whil at th cost of a highr lvl of survivorship bias bcaus th NAREIT indx contains mor REITs with smallr markt capitalizations. In what follows, I assum that th factors in our bnchmark modl ar th factor-mimicking portfolios of Fama and Frnch (1993) with K 3: F (MKT,SMB,HML)', (11) t whr MKT is th monthly rturn on th stock markt portfolio in xcss of th on-month U.S. Trasury bill, SMB is th monthly rturn on a portfolio of small stocks minus and th rturn on a portfolio of big stocks, and HML is monthly rturn on a portfolio of stocks with high book-to-markt ratios (valu stocks) minus and th rturn on a portfolio of stocks with low book-to-markt ratios (growth stocks). Whil SMB is th siz prmium masuring th prformanc of small stocks rlativ to big stocks, HML is th valu prmium capturing th prformanc of valu stocks rlativ to growth stocks. For th as of rprsntation, dfin R (MKT,SMB,HML,REIT)' (1) as a N 1 vctor of xcss rturns (N = 4). To tst th (mild) sgmntation hypothsis against th full intgration hypothsis, I considr a four-factor modl with th REIT as th fourth factor-mimicking portfolio. Undr this stting, th vctor of th factors is th sam as th vctor of xcss rturns: FR (MKT,SMB,HML,REIT)'. (13) Th xpctd xcss rturns on th REIT portfolio, in th altrnativ modl ar givn by quation () with K 4. If th hypothsis, 4 0, is rjctd, th xpctd xcss rturn on th REIT indx is rlatd to its own conditional varianc and th vidnc is against full intgration hypothsis in favor of th markt sgmntation or mild sgmntation hypothsis. Othrwis, if th hypothsis, 4 0, is not rjctd but th risk prics on th Fama-Frnch portfolios ar significantly diffrnt from zro, thn th vidnc is in 10

11 favor of full intgration. Lastly, if th hypothsis, 4 0, is rjctd and th risk prics on th Fama- Frnch portfolios ar also significant, thn th vidnc is in support of mild sgmntation. Th multivariat GARCH-in-mans quations ()-(3) for ach modl spcification ar stimatd by using th mthod of quasi-maximum liklihood. Th standard rrors ar robust to non-normality of disturbanc trms. Th R for ach xcss rturn sris is computd as th varianc of th implid xpctd xcss rturns givn by th right sid of quation () dividd by th varianc of th ralizd xcss rturn. I tst th hypothss of markt intgration and modl spcifications using th liklihood ratio (LR) statistic which compars th fit (th maximizd valu of th log liklihood function) of a spcial (null) modl with th fit of an altrnativ modl. I first tst th singl-factor CAPM against a two-factor modl with MKT and REIT as th risk factors. Undr th joint hypothsis that th CAPM is corrctly spcifid and th REITs markt is intgratd with th gnral stock markt, REIT is not a pricd factor ( 0 ). I also tst th CAPM against th Fama-Frnch thr-factor modl or th Fama-Frnch thr-factor modl against th four-factor modl. Following Ling and Naranjo (1999) and Guo, t al. (009), I tst thr typs of rstrictions on risk prics across portfolios using th four-factor modl. Th first is to rplac th risk pric vctor λ of on of th four portfolios with λ δ and stimat th lss rstrictd four-factor modl. Th scond is to assum unrstrictd risk prics ij (for all i, j 1,,4 ) and stimat th unrstrictd four-factor modl: 4 it i i, T tt ij ij, t it j1 R I h. (14) I tst th four-factor modl with qual risk prics across portfolios against th four-factor modl with lss rstrictd or unrstrictd risk prics. If th REIT indx and th Fama-Frnch portfolios ar pricd in intgratd markts, th two typs of rstrictions should not b rjctd. 11

12 I also tst th hypothsis that ach of th four portfolios in quation (1) is pricd sparatly in sgmntd markts. To this nd, I supprss th covarianc trms in th quation () for th first momnt and stimat th following indpndnt (diagonal) on-factor modl for ach of th four portfolios: R I h (15) it i i, T tt i ii, t it. I thn tst th indpndnt on-factor modl against th unrstrictd four-factor modl. Empirical Rsults Estimats of th Indpndnt On-Factor Modl I first invstigat th univariat rlation btwn th xpctd rturn and volatility to chck th xistnc of th rlation for ach portfolio. To this nd, I first stimat th indpndnt on-factor modl in quation (15). I us th multivariat GARCH modl in quation (3) for th scond momnts instad of four univariat GARCH modls to gain stimation fficincy from th corrlatd rsiduals and as comparisons with altrnativ spcifications of th man quations in th rst of papr. Th rsults ar rportd in Exhibit 1. To consrv spac, th stimats of th constant matrix C ar not rportd throughout th papr. Cofficints that ar significant at th 5 prcnt lvl ar highlightd in bold and th robust standard rrors ar rportd in th parnthss. Th rsults of stimating th indpndnt on-factor modl for th NAREIT Equity indx ar shown in panl A. Th rsults for th Wilshir and th Mortgag indics ar rportd in panls B and C, rspctivly. Throughout th rst of th papr I choos January 1993 as a cutoff dat T for th indicator variabl as th prvious rsarchrs not som structural chang in th REITs markt aftr 1990s (.g., Ling and Ryngart. 1997; Ling and Naranjo, 1999; Glascock, Lu, and So, 000). In panls A and B, and 93 for ach portfolio ar insignificant at th 5 prcnt lvl. Th stimats of 93 ar 0.43 prcnt 1

13 with a standard rror of 0.30 for th markt indx MKT and with a standard rror of 0.1 for th NAREIT Equity REIT indx in panl A. Ths stimats ar vry clos to two standard rrors from zro and significant at th 10 prcnt lvl. In panl C, th alpha bfor 1993 ar rlativly small and insignificant for ach portfolio but th stimatd 93 for th MKT and Mortgag REIT portfolios ar rlativly larg and significant at th 1 prcnt lvl. In panl A th stimatd risk prics ( ) for MKT, HML and th NAREIT Equity REIT portfolio ar j.947, and 1.975, rspctivly and thy ar two standard rrors away from zro. Similar rsults ar obtaind for HML and th Wilshir indx in panl B. Howvr, only HML risk pric is significant at th 5 prcnt lvl in panl C. Th risk pric associatd with th Mortgag REIT indx is mrly with a standard rror of Th significanc of MKT and spcially HML in th modls hr ar in contrast to th rsults of stimating pooling univariat GARCH modls rportd by Guo t al. (009), partly bcaus of th fficincy gains from using th multivariat rathr than univariat GARCH modls for th scond momnts. Th positiv and significant rlation btwn th xpctd rturn on th stock markt and its volatility is in contrast to a wak or ngativ risk-rturn rlation rportd by Campbll (1987), Glostn, Jagannathan and Runkl (1993), and Whitlaw (1994). Most importantly, th rsults hr suggst a significant positiv rlation btwn th xpctd rturn and th volatility for th Equity REIT indics but not for th Mortgag REIT indx. Th rsult hr contradicts that of Dvany (001), who finds th xpctd quity REIT rturns to b unrlatd to thir own volatility. Givn th wak rsults for th Mortgag REIT indx, in th rst of th papr I will only study th xpctd rturns on th quity REIT indics. Nxt, with th xcption of th asymmtric GARCH cofficint for SMB, th GARCH paramtrs ar two standard rrors away from zro. Sinc th rsults tnd to b similar across th thr panls in Exhibit 1, I only discuss th stimats in panl A. Th stimatd diagonal lmnts of matrix A (which link scond momnts to thir laggd valus) ar (MKT), 0.81 (SBL), (HML), and (REIT), 13

14 implying high prsistnc of th volatility of ach factor and th REIT rturn. Th stimatd diagonal paramtrs of matrix B (which link scond momnts to past innovations) ar 0.44, 0.4, and 0.1, indicating sizabl ffcts of past innovations on th scond momnts. Mor intrstingly, th diagonal paramtrs of matrix D (which masur th asymmtric ffcts of ngativ shocks) ar 0.18 (MKT), (HML) and 0.33 (REIT), which ar significant at th 1 prcnt or lowr lvl. By comparing th sizs of th stimatd paramtrs of B and D, I not that ignoring th asymmtric ffcts of ngativ shocks would gratly undrstimat th impacts of ngativ nws on th volatility of MKT, HML and spcially REIT. Finally th stimatd R s ar all 1.6 prcnt or lss, indicating low prdictiv powr of th volatility of ach portfolio for its own xpctd rturn. Th varianc ratios (VR) in brackts ar for th NAREIT Equity REIT indx and for th Wilshir indx but only for th Mortgag REIT indx. Hr th VR masurs th prcntag of th xpctd rturn on ach portfolio xplaind by its own volatility, xcluding th indicator variabl for th changing alpha. 5 Estimats of On- to Four-Factor Modls - NAREIT Indx Th rsults of stimating th modls in quations ()-(3) ar rportd in Exhibit (NAREIT quity indx) and Exhibit 3 (Wilshir indx). Panls A-D prsnt rsults for th on- to four-factor modls whr risk prics ar rstrictd to b qual across portfolios. Panl E rports th rsults of stimating a four-factor modl with unrstrictd risk prics givn by quation (14). Th stimatd GARCH cofficints in all panls hr ar similar to thos rportd in Exhibit 1. Panl A of Exhibit prsnts th rsults for th CAPM. Th stimatd for HML is with a standard rror of 0.094, suggsting that th valu prmium is not xplaind by th CAPM. Mor intrstingly, th stimatd 93 for MKT is 0.40 prcnt with a standard rror of and th stimat for REIT is 0.46 with a standard rror of 0.9. Thus both MKT and REIT xhibit significant alphas in th scond half of th sampl priod sinc Th 14

15 pric of th markt risk ( ) is also significant with an stimat of.976 and a standard rror of Th R s for th portfolios ar or lowr. Panl B of Exhibit rports th rsults of stimating th two-factor modl in which th xpctd xcss rturn on ach portfolio is rlatd to its covarianc with MKT and REIT. Whil and 93 hr ar qualitativly similar to thos in panl A, th pric of markt risk bcoms insignificant, with a cofficint of and a standard rror Howvr, th risk pric associatd with REIT is significant with a cofficint of and a standard rror of This suggsts that th xpctd xcss rturn on REIT is rlatd to its own volatility, but is not rlatd to its covarianc with MKT. Th risk prmiums on MKT and othr factors ar rlatd to thir covariancs with th rturns on REIT in this stting. Th R s associatd with th man quations for th MKT, SMB, HML and REIT ar 0.009, 0.001, and 0.05, rspctivly, suggsting that th two-factor modl xplains mor tim variations of MKT and REIT than th CAPM. Th varianc ratios associatd with MKT and REIT ar and 0.475, implying that th tim variation of th xpctd xcss rturn on REIT is attributd mor to its own volatility than its covarianc with MKT. Th rsults of stimating th CAPM and two-factor modls thrfor suggst that th markt for REITs is sgmntd from th gnral stock markt. Th vidnc hr contradicts th findings of Liu, t al. (1990) who cannot rjct th markt intgration hypothsis for quity REITs using a similar two-factor modl. Panl C of Exhibit rports stimats in th thr-factor modl. Only associatd with SMB is significant at th 5 prcnt lvl, suggsting that th modl is capabl of xplaining rturns from othr portfolios, including REIT, unlik th on- and two-factor modls. Th stimats of th risk pric paramtrs ( ) ar for MKT (std. rr. = 0.841), for SMB (std. rr. = 1.490) and (std. rr. =.14) for HML, implying that th risk prics associatd with MKT and HML ar positiv and significant whil th risk pric associatd with SMB is ngativ and significant. Th R s associatd with th man quations for th thr risk factors and REIT ar rspctivly 0.01, 0.03, and 0.07, 15

16 which ar highr than th R in th on- and two-factor modls. Thus tim-varying conditional covariancs xplain rlativly mor tim variations of SMB and REIT than tim variations of MKT and HML. Consistnt with th stimats of th risk prics, th stimatd varianc ratios suggst that SMB xplains only.3 prcnt of th tim variation of th xpctd REIT rturn, but MKT and HML xplain 39. and.1 prcnt, rspctivly. Th fact that th thr-factor modl dos a bttr job than th CAPM in xplaining th REIT rturns is consistnt with th findings of Ptrson and Hsih (1997) and Chiang, L and Wisn (005). Nxt, panl D of Exhibit displays th rsults of stimating th four-factor modl, which includs REIT as th fourth factor. Non of or 93 is significant at th 5 prcnt lvl. In th prsnc of th REIT factor, th risk prics associatd with MKT, SMB and HML ar similar to th risk prics stimatd in th thr-factor modl and still significant, but th risk pric for REIT is not statistically significant, indicating that th thr factor-mimicking portfolios subsum th rol of th REIT volatility in xplaining th xpctd REIT rturn. Th stimatd risk prics ar against th sgmntation or mild sgmntation hypothsis but consistnt with th full intgration hypothsis. Th similar to thos in th thr-factor modl. Th R s hr for th first thr factors ar R (0.07) for th REIT rturn is slightly lowr hr. Th varianc ratios associatd with th four factors ar 0.439, 0.03, 0.39 and 0.003, suggsting that vry littl tim variation in xpctd xcss rturn on REIT is associatd with its volatility. Finally, I xamin th rsults in Panl E whr risk prics on all four factors ar not rstrictd to b qual across portfolios. Just lik th rsults in panl D, and 93 ar insignificant. Howvr, only th risk pric associatd with HML portfolio on its own volatility is mor than two standard rrors away from zro. Th risk prics associatd with MKT, SMB and REIT ar not significant on ach factor, including REIT. Th lack of significanc of risk prics suggsts that th unrstrictd four-factor modl is likly ovr-paramtrizd. Estimats of On- to Four-Factor Modls - Wilshir Indx 16

17 Exhibit 3 prsnts th rsults of stimating th on- to four-factor modls in which th NAREIT quity REIT indx is rplacd by th Wilshir REIT indx. Th rsults ar mostly similar to thos rportd in Exhibit. In th on-factor modl (CAPM), th alpha on REIT in th post-1993 priod is positiv and significant and th pric of th stock markt risk is significant. Th risk pric associatd with REIT in th two-factor modl hr is significant lik that in Exhibit. In th thr-factor modl, non of th alphas is significant and th risk pric associatd with HML is positiv and significant at th 5 prcnt lvl. Risk prics associatd with othr factors ar not as significant as in Exhibit. In th four-factor modl, th risk pric associatd with REIT is not significant, unlik that in th two-factor modl. Th rsults ar vidnc against th hypothsis that MKT fully xplains tim variation of th xpctd rturn on REIT. Th covarianc of REIT with HML hr is mor important for subsuming th rol of th REIT volatility. Lik th rsults in Exhibit, th stimatd risk prics in Exhibit 3 ar in support of th full intgration hypothsis. Tsts of Modl Spcifications and Markt Intgration To obtain mor insights into th rlativ prformanc of th modls in Exhibits 1-3, I prsnt rsults on th tsts of modl spcifications and markt intgration in Exhibit 4. Panl A rports th maximizd valu of th log liklihood function for ach modl and th p-valus associatd with tsting th joint significanc of or 93 on four portfolios. I also includ a modl with zro risk prics (no factor) to chck th significanc of th risk prmiums or xcss rturns on REITs. First, th maximizd valu of th log liklihood function incrass as mor factors ar includd or th modls bcom lss rstrictd. In th zro-factor modl, th alphas masur avrag portfolio rturns. Th p-valu associatd with in th modl is lss than 5 prcnt whn ithr REIT indx is usd. Th p- valu associatd with 93 in th modl is also lss than 5 prcnt whn th NAREIT Equity REIT indx is usd. Thus th hypothsis that xcss rturns ar zro for th full sampl priod or incrmntal rturns 93 for th post-1993 priod ar zro is rjctd. Nxt, and in ithr th on- or two-factor modl ar 17

18 jointly significant at th 5 prcnt lvl, rjcting both modls whn ithr REIT indx is usd. Sinc thr is strong vidnc against th on- and two-factor modls, infrncs on capital markt intgration basd on ths modls ar not appropriat. Howvr, for ithr REIT indx, th stimats of in th thr-factor and (rstrictd and unrstrictd) four-factor modls ar not jointly significant at th 10 prcnt lvl. For th NAREIT indx, th stimats of 93 ar also not significant in thr- and four-factor modls at th 10 prcnt lvl. For th Wilshir indx, a similar rsult is found in th thr-factor modl but th stimat of is significant at th 10 prcnt lvl for th rstrictd four-factor modl or at th 5 prcnt lvl for th unrstrictd four-factor modl. 93 Panl B rports rsults of th liklihood ratio (LR) tsts. Using th NAREIT Equity REIT indx, tsts rjct th zro-factor modl and th on-factor modl in favor of th thr-factor modl, as th p-valus ar lss than 1 prcnt. Howvr, th thr-factor modl is not rjctd in favor of th four-factor modl with ithr REIT indx as th fourth factor. Givn th abov rsults, th tsts of markt intgration hypothss ar basd on th four-factor modl. Th first is tsting th hypothsis that risk prics associatd with ach portfolio ar qual to thos associatd with th rst of portfolios. Th tst rvals that th risk prics associatd with REIT ar not significantly diffrnt from th othr common factors at th 5 prcnt lvl. This provids furthr vidnc that th markt for REITs is intgratd with th gnral stock markt. Th bottom two rows of panl B rport th tst of th rstrictd four-factor modl against th unrstrictd four-factor modl and th tst of th indpndnt four on-factor modl (Exhibit 1) against th unrstrictd four-factor modl. Th tsts rjct th rstrictd modl in favor of th unrstrictd modl at th 5 prcnt lvl, mostly bcaus of th diffrnt risk prics associatd with HML. Th tsts also rjct th indpndnt on-factor modl in favor of th unrstrictd four-factor modl, which suggsts that th covarianc trms ar important for xplaining xpctd rturns and th markts for th four portfolios ar intgratd. Similar to what Ling and Naranjo (1999) find, th tsts of th rstrictions on 18

19 risk prics across portfolios, gnrally support th hypothsis that th scuritizd ral stat markt is intgratd with th gnral capital markt. Comparison with Volatility-Basd Modls Instad of th covarianc-basd modls, an altrnativ spcification assums that xpctd asst rturns ar functions of th volatility of th factors (Flannry, Hamd, and Harjs, 1997). In this sction, I considr th altrnativ spcification for th xpctd REIT rturn. I assum that th first momnts of MKT, SMB and HML ar dscribd by th indpndnt on-factor modl givn by quation (15). Whn factor btas ar assumd to b constant, th xcss rturn in quation (9) on ach REIT indx is givn by th following: K 4t 4 4 T tt j jj, t 4t j1 R I h, (16) whr K =1, or 3. To compar th prformanc of th volatility-basd modl with th covarianc-basd modl studid arlir, I includ REIT in th right hand sid of quation (16) with K = 4. To as comparison of stimats and spcification tsts, th spcification of scond momnts rmains unchangd. Exhibit 5 (panls A-D) rports th rsults of stimating quations (15)-(16). Panl A prsnts th rsults for th on-factor modl in which only th MKT volatility is th xplanatory variabl for th REIT rturn. In panl B th REIT volatility is th scond factor. Panl C includs th volatility of MKT, SMB and HML and panl D includs th volatility of REIT as th fourth factor. Finally, panl E rports th rsults of stimating a four-factor modl with unrstrictd risk prics (s nots to Exhibit 5). Although alphas ar not significant for ach portfolio including REIT in panl A, th cofficint is with a standard rror of 1.471, implying an insignificant rlation btwn th xpctd REIT rturn and th volatility of MKT. In panl B, th cofficint on th MKT volatility is with a standard rror of but th cofficint on th REIT volatility is.089 with a standard rror of Similar to 19

20 th covarianc btwn MKT and REIT rturns, th stock markt volatility dos not xplain th positiv rlation btwn th xpctd rturn and volatility of REIT. In panls C and D, non of th stimatd associatd with th first thr factors ar two standard rrors away from zro. Howvr, th cofficint associatd with th REIT volatility is.139 with a standard rror of Unlik th covarianc-basd modl, th alphas in th thr- and four-factor modls in th post-1993 priod ar significant. Strikingly, vn for th unrstrictd modl in panl E, th cofficint associatd with th REIT volatility is significant but th cofficint associatd with th volatility of ach othr factor is not significant. Th varianc ratios associatd with th REIT volatility in th four-factor modls xcd unity. Thus unlik th covarianc-basd modl, th volatility of th Fama-Frnch factors do not xplain th positiv rlation btwn th xpctd rturn and volatility of th REIT indx. Exhibit 6 prsnts th rsults of stimating th volatility-basd modl using th Wilshir indx. Th rsults ar largly similar to thos prsntd in Exhibit 5 for th NAREIT Equity REIT indx. Th cofficint on MKT or othr factor volatility in th on- to four-factor modls is not significant but th cofficint on th REIT volatility is mor than two standard rrors away from zro in ithr th two- or four-factor modl. Exhibit 7 rports th rsults of spcification tsts of th volatility-basd modls. Th maximizd valu of th log liklihood function for ach volatility-basd modl hr is somtims highr than that in th covarianc-basd modl with th sam numbr of factors in Exhibit 4 bcaus th volatility-basd modls involv mor paramtrs. Th maximizd valu of th log liklihood function for th thr-factor modl without th REIT factor is lowr than th valu of th function for th two-factor modl including th REIT factor, suggsting poor prformanc of th thr-factor volatility-basd modl. Th stimatd alphas in th two or four modls ar jointly significant, with ithr th NAREIT indx or th Wilshir indx. Th rsults diffr from thos in th covarianc-basd modls whr alphas in only on- and two-factor modls ar significant (s Exhibit 4). Liklihood ratio tsts hr in panl B do not rjct th on-factor modl in favor of th thr-factor modl but rjct th thr-factor modl in favor of 0

21 th four-factor modl with ithr REIT indx. Lastly, thr is som significant vidnc against th hypothsis that risk prics ar zro but th vidnc is limitd only to th NAREIT indx. Ovrall, th rsults hr offr vidnc that th covariancs of REIT with risk factors ar mor usful than th volatility of th factors in xplaining th tim variation of th xpctd REIT rturn. Tim-Sris Proprtis of Expctd Rturns and Volatility of REITs Givn th similarity of th rsults obtaind from th two REIT indics, in this sction I xamin th timsris proprtis of th xpctd rturn and volatility of REITs implid by th covarianc-basd thrfactor modl for th NAREIT Equity REIT indx. Cofficints stimats in Exhibit, panl C ar assumd to b givn. Graphical Illustrations To compar th xpctd xcss rturn on th REIT indx with its volatility, th xpctd REIT rturn is plottd along with its volatility (standard dviation) in Exhibit 8. Larg variations of th xpctd xcss rturn and volatility ar obsrvd in th four dcads of our sampl priod, spcially during th financial crisis. Th sharp incras of th conditional REIT volatility during th crisis is consistnt with th rcnt vidnc providd by Sun, Titman and Twit (015). During th pak of th latst financial crisis, th xpctd rturn incrass to approximatly ight tims and th volatility incrass to approximatly four tims of thir pr-crisis lvls. Exhibit 9 illustrats th dcomposition of th REIT varianc into th systmatic and idiosyncratic componnts givn by quation (10). Similar to th xpctd REIT rturn, th total varianc dclins slightly from th first dcad to th scond and third dcads (from arly 1980s to lat 1990s) thn incrass sharply during th financial crisis. In priods of high volatility (th first and last dcads), th contribution of th systmatic varianc tnds to much highr than that of th idiosyncratic varianc. For 1

22 th full sampl priod, th systmatic varianc contributs an avrag of 53.3 prcnt of total varianc. Around th pak of th rcnt financial crisis (from Nov. 008 to July 009), th systmatic varianc accounts for mor than 80 prcnt of th total varianc. Exhibit 10 plots btas and factor risk prmiums. Th dcrasing pattrn of th stock markt bta in th first thr dcads, as rportd by Khoo, Hartzll and Hosli (1993), and Chiang, L and Wisn (005), is compltly rvrsd in th last dcad. Similar to th stock markt bta and othr btas, th stock markt prmium and th valu prmium tnd to b highr in th most rcnt dcad. Thus th btas and risk prmiums contribut to th incrass in th xpctd REIT rturn bginning in th arly 000s, spcially during th rcnt financial crisis. Exhibit 11 plots th volatility of ach factor and th corrlation btwn ach pair of th factors. Th lvl of th stock markt volatility during th financial crisis is similar to that during th mid-1970s and lat-1980s. Consistnt with th rsults in Ptrson and Hsih (1996), th corrlation 13 btwn MKT and HML tnds to b ngativ (clos to -0.50) and lowr than othr corrlations most of tims. Howvr, th corrlation 13 surgs quickly to positiv lvls during th mid- to lat-1970s and surgs to unusually high and positiv lvls (0.50) during th financial crisis. A comparison of Exhibits rvals that th stock markt prmium and th corrlation 13 incras mor sharply during th rcnt financial crisis than th volatility of th stock markt or othr factors. Cross-Corrlations Exhibit 1 prsnts cross-corrlations btwn pairs of th momnts implid by th thr-factor modl with th NAREIT quity REIT indx. In fact, th xpctd REIT rturn, ER ( 4 ), is highly corrlatd with its own volatility, h 44, with a corrlation of Howvr, as th rsults in Exhibit show, th thr risk factors subsum th rol of th REIT volatility in xplaining th xpctd REIT rturn. Consistnt with th rsult, th xpctd REIT rturn, ER ( 4 ), is vry highly corrlatd with th stock markt

23 prmium 1 (corrlation = 0.933), th HML bta 3 (corrlation = 0.650), and th markt bta 1 (corrlation = 0.584). As discussd arlir, th risk prmium on ach factor is proportional to th conditional factor volatility if th factors ar conditionally uncorrlatd. Sinc th avrag corrlations of rturns on th stock markt portfolio and two factor-mimicking portfolios ar rlativly small (absolut valus of avrag corrlations ar lss than 0.31), th risk prmium on ach factor should b mor rlatd to its own volatility than th volatility of othr factors. Th corrlation of th stock markt prmium 1 with its volatility h 11 is and th corrlation of th HML prmium 3 with its volatility h 33 is Th SMB prmium is ngativly corrlatd to its volatility h, as a rsult of th ngativ risk pric associatd with this factor. In addition to th factor volatility, quation (8) implis that th risk prmium on ach factor can b rlatd to its corrlations with othr factors. Exhibit 1 rports that th stock markt prmium is mor highly corrlatd with th MKT-HML corrlation 13 (corrlation = 0.766) than th stock markt volatility (corrlation = 0.610). Similarly, th corrlation of th xpctd REIT rturn with th corrlation 13 is 0.71 whil th corrlation with th stock markt volatility h 11 is only To a lssr xtnt, th HML prmium is positivly corrlatd with 13, givn th corrlation of Nxt, I xamin th implid volatility of th REIT rturn, h 44. Consistnt with th high corrlation btwn th xpctd rturn and volatility of REIT, th corrlations btwn th REIT volatility and othr momnts in Exhibit 1 tnd to b quit similar to th corrlations btwn th xpctd REIT rturns and th momnts. For xampl, th REIT volatility is mor highly corrlatd with th stock markt prmium than th HML prmium, mor highly corrlatd with th MKT-HML corrlation 13 than th stock markt volatility, and corrlatd to a larg xtnt with th MKT and HML btas. 3

24 Rgrssion Rsults As discussd in th introduction sction, prvious studis attribut th tim variation in xpctd rturns on REITs to tim-varying risk prmiums and volatility (Li and Wang 1995; Dvany, 001), or timvarying btas associatd with th stock markt portfolio or othr common factors (Khoo, Hartzll and Hosli, 1993; Liang, McIntosh and Wbb, 1995; Chiang, L and Wisn, 005). To discrn th rlativ importanc of sourcs of th tim-varying xpctd REIT rturn, I us linar approximations of quation (9) to prform linar rgrssions, in which th indpndnt variabls includ REIT btas, factor variancs and corrlations btwn pairs of th factors. To undrstand why th REIT volatility loss its xplanatory powr for th xpctd REIT rturn in th thr-factor modl studid arlir, I also prform similar rgrssions of th varianc of REIT rturn, using linar approximation of quation (10). Th cofficint stimats in th rgrssions ar prsntd in Exhibit 13 with standard rrors in parnthss. In panl A, th dpndnt variabl is th xpctd REIT rturn whil in panl B th dpndnt variabl is th varianc of th REIT indx. All standard rrors ar adjustd for htroskdasticity and rsidual autocorrlations up to 4 (two yars) lags. Any furthr incrass in th lag lngth do not affct th rsults in any significant way. As th indpndnt variabls ar stimatd from our sampl and subjct to th rrors-in-variabls problm, th standard rrors hr ar likly to b undrstimatd. As a rsult, cofficints that that significant at th 1 prcnt rathr than th usual 5 prcnt lvl ar highlightd in bold. All cofficints and standard rrors ar multiplid by 100 xcpt for thos associatd with variancs h ii. Panl A or B rports rsults of four rgrssions. Th first rgrssion rportd in column (1) includs th 1993 indicator plus all of nin indpndnt variabls (btas, variancs and corrlations btwn pairs of factors). Th scond to fourth rgrssions rportd in columns () to (4) includ thr indpndnt variabls (btas, variancs or corrlations). This tim indicator is significant for th xpctd rturn but not for th volatility (not rportd). I first discuss th rsults in column (1). Th rsults in panl A indicat that th MKT and HML btas, th variancs of MKT and SMB and th corrlation btwn MKT and 4

25 HML ar significant at th 1 prcnt lvl for xplaining th xpctd REIT rturn. All cofficints on ths variabls ar positiv xcpt that on th SMB varianc h. Th cofficint associatd with th stock markt bta is 1.377, which far xcds th cofficint of associatd with th HML bta, suggsting that th xpctd REIT rturn is mor snsitiv to th markt bta than othr btas. Th rsults in panl B show that th stock markt and SMB btas, th varianc of MKT and th corrlation btwn MKT and HML ar significant at th 1 prcnt lvl for xplaining th REIT volatility. Comparing with th rsults in panl A, I find that th stock markt bta and varianc, along with th corrlation btwn MKT and HML, ar significant in xplaining both th xpctd REIT rturn and volatility. Th adjustd R s of th rgrssions ar 0.84 in panl A for th xpctd rturn and for th volatility in panl B. Th similar and high R s offr furthr vidnc that th systmatic risks associatd with th factormimicking portfolios xplain most of th tim sris movmnts of th xpctd REIT rturn and volatility. Nxt, I xamin rsults in columns ()-(4). Th cofficints on th btas in column () of panls A and B tnd to b gratr than thos in column (1) but th statistical significanc of th btas ar largly unchangd whn th btas ar th only indpndnt variabls. Th rsults in column (3) indicat that th variancs of th factors ar not significant at th 1 prcnt lvl for xplaining ithr th man or th varianc of th REIT rturns whn othr indpndnt variabls ar xcludd from th rgrssion. Th loss of th statistical significanc is an xampl of th missing-variabls problm. Th rsults in column (4) show that th corrlation 13 btwn MKT and HML rmains significant in xplaining th xpctd REIT rturn and volatility whn th corrlations btwn factors ar th only indpndnt variabls. Finally, th adjustd R s in columns ()-(4) ar rspctivly 0.613, and in panl A and.635, and in panl B. Th sizs of th adjustd R s along with th significanc of th cofficints suggst that th btas and factor corrlations ar mor important than th markt and factor volatility for xplaining th xpctd REIT rturn and volatility. 5

26 Conclusions This articl uss conditional covarianc-basd thr and four modls to study th tim variation of xpctd rturns on REITs. In th four-factor modl, xpctd rturns on REITs ar rlatd to thir own volatility and thir covariancs with th Fama-Frnch factors. I apply th modl to th NAREIT quity REIT indx from January 197 to July 013 and th Wilshir REIT indx from January 1978 to July 013. Estimation rsults obtaind from both data sts suggst that th covariancs of REIT rturns with th Fama-Frnch factors subsum th rol of th REIT volatility in xplaining th xpctd REIT rturns, spcially around th pak of th rcnt financial crisis (lat-008 to mid-009). Th finding diffrs from othr xplanations of th REIT rturns around th financial crisis basd on firm-spcific charactristics (.g., Sun, Titman and Twit. 015). Th vidnc suggsts that xpctd REIT rturns ar not compnsation for thir own volatility but compnsation for th risks associatd with th stock markt prmium and th valu prmium. Th rsult hr is consistnt with th notion that th markt for REITs is intgratd with th gnral stock markt. Capital markt intgration not only rquirs assts in diffrnt markts to b pricd by common factors but also rquirs th prics of risks associatd with th factors to b th sam across th markts. I find that th risk prics associatd with th REIT portfolio ar not significantly diffrnt from thos associatd with th thr factor portfolios at th 5 prcnt lvl. I also find that an indpndnt on-factor modl for ach factor or th REIT portfolio, in which th xpctd rturn on ach portfolio is rlatd only to its own volatility, is rjctd in favor of a full four-factor modl with covarianc trms and unrstrictd risk prics. Th rsult furthr supports th hypothsis that th markt for REITs is intgratd with that for th factor portfolios. Extnding th vidnc rportd in th ral stat litratur (DLisl, Pric and Sirmans, 013), this papr documnts that xpctd rturns on REITs ar insignificantly rlatd to not only th volatility of th stock markt but also th volatility of th Fama-Frnch factors. Expctd rturns on REITs in th covariancsbasd thr-factor modl, thrfor, ar mor usful than thos in th volatility-basd modls in practical 6

27 applications, including constructions of fficint portfolios for invstors and th prformanc valuation of fund managrs. Sinc xpctd REIT rturns tnd to vary with systmatic risks associatd with th common factors, spcially during th financial crisis, it is ssntial to obtain prcis and latst stimats of tim-varying covarianc risks of REITs associatd with th factors. As shown in th papr, th stimatd tim-varying covarianc risks can b transformd into stimats of tim-varying btas associatd with th factors and tim-varying factor risk prmiums. Th asymmtric multivariat GARCHin-mans modl srvs as a usful tool for obtaining dynamic stimats of xpctd rturns and risks of REITs. 1 Fama and Frnch (1996) conjctur that th valu prmium is linkd to human capital. Jagannathan and Wang (1996) find that th variability of labor incom xplains part of th valu prmium. S also Zhang (005), Ptkova (006), and Lttau and Wachtr (007) for mor xplanations of th valu prmium. Cottr and Stvnson (006) us th multivariat VAR-GARCH modl to documnt th rturn and volatility linkags btwn REIT sub-sctors and infrncs of othr U.S. quity sris. Yang, Zhou and Lung (01) apply a multivariat asymmtric gnralizd dynamic conditional corrlation GARCH modl to REITs and othr assts and find asymmtric volatility of REITs and asymmtric corrlation of REITs with stock rturns. 3 Othr rsarchrs (Myr, Chaudhry and Wbb, 1996; Glascock, Lu, and So, 000; Yunus, 01) apply tim sris tchniqus to xamin th dynamic intraction of th ral stat markt with othr markts. Ths tchniqus ar usd to obtain insights into th long-run (co)intgration of diffrnt markts unlik asst pricing modls which ar oftn usd in tsts of various rstrictions implid by th modls. 4 Sinc tmporal aggrgation would dilut crtain charactristics of tim sris such as volatility clustring, th multivariat GARCH modl mployd in th papr is bst suitd for data of highr frquncy (daily or wkly) rathr than monthly. In a latr sction I will som prform snsitivity analysis with daily data. 5 Th indpndnt on-factor modl is applid to daily data of MKT, SMB, HML or th NAREIT Equity REIT indx (or th NAREIT Mortgag REIT indx) for th priod from Fb. 4, 006 to Aug. 9, 014. For th short sampl priod, th rlation btwn th xpctd rturn on ithr REIT indx and its volatility is not significant at th 5 prcnt lvl whn ithr th diagonal or full asymmtric GARCH modl in quation (3) is usd. 7