Yuming Li. V o l. 3 8 J R E R. N o

Transcription

1 T i m Va r i a t i o n o f E x p c t d R t u r n s o n R E I Ts : I m p l i c a t i o n s f o r M a r k t I n t g r a t i o n a n d t h F i n a n c i a l C r i s i s A u t h o r Yuming Li A b s t r a c t In this study, I us a conditional covarianc-basd thr-factor pricing modl and a ral stat invstmnt trust (REIT) indxnhancd 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. 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 2014; and narly 40 million Amricans invstd 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 valuations of fund managrs. In this study, I valuat 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 thir prcrisis 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 J R E R V o l. 3 8 N o

2 3 2 2 L i th most important factors influncing th tim variation of xpctd rturns on REITs? I xamin th importanc of systmatic risks and how thy influnc th tim variation of xpctd rturns on REITs using a conditional covarianc-basd thrfactor asst-pricing modl with th Fama-Frnch (1993) factors. Thr ar four rasons th modl is important for studying th bhavior of REIT rturns around th tim of th rcnt financial crisis. First, th Fama-Frnch thr-factor modl can b mor usful than th singl-factor, markt-basd modl in capturing REIT rturns and gnrating stabl stimats of markt btas (Ptrson and Hsih, 1997; Chiang, L, and Wisn, 2005). Scond, th valu prmium, as on of th Fama- Frnch factors, proxis for innovations of invstmnt opportunitis. 1 Third, th systmatic volatility associatd with th aggrgat stock markt is not pricd in REIT rturns (DLisl, Pric, and Sirmans, 2013). As a rsult, it is imprativ to considr othr factors and altrnativ masurs of systmatic risks, such as th conditional covariancs, in studying REITs xpctd rturns. Fourth, th rols of firm-lvl charactristics and how thy affctd REIT rturn bhavior during th financial crisis has bn xamind (.g., Sun, Titman, and Twit, 2015), laving th fundamntal dtrminants of REIT rturns largly unxplord. I also analyz a REIT indx-nhancd four-factor modl in which xpctd rturns on REITs could 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 REIT rturns, th volatility of th Fama-Frnch factors, and th covariancs of rturns with th Fama-Frnch factors. 2 Using this spcification, I stimat th covarianc-basd modls, tst th 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, Hartzll, Grig, and Grissom (1990) adopt a two-factor modl with a stock markt portfolio and a ral stat portfolio as factors. Thy find vidnc in support of th 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 L (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 ral stat scuritis is intgratd with th markt for non-ralstat stocks. Sinc tsts of any markt intgration ar joint tsts of th intgration

3 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s hypothsis and asst pricing or othr conomic modls, tst rsults dpnd critically on th assumd factors in th modls. 3 I study th conditional xpctd rturns on REITs from January 1972 to July I first obtain strong vidnc against th capital asst pricing modl (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, Hartzll, Grig, and Grissom (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 REIT rturns 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 th xpctd rturns on REITs to surg to a fw tims 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 th 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 rmaindr of th articl is organizd as follows. In th nxt sction, I dscrib 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. M o d l s 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 J R E R V o l. 3 8 N o

4 3 2 4 L i 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 implications of th modl for th tim variations of th xpctd rturns and volatility of REITs. T h C o v a r i a n c - b a s d M o d l Considr th following covarianc-basd pricing modl: K t1 it j t1 it jt j1 E (R ) cov (R, F ), (1) whr R it is th xcss rturn for priod t on th ith asst for i 1, 2..., N, F jt 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 2, quation (1) is th conditional multifactor pricing modl of Mrton (1972). Th xcss rturn rfrs to th rturn on an asst minus th risk-fr 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 t b th N 1 vctor of unxpctd rturns with th ith lmnt it and th N N conditional varianccovarianc 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, (2) 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 I tt, 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.

5 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Th conditional varianc-covarianc matrix H t GARCH spcification: follows th asymmtric BEKK H CC AH A B B D 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)/2 paramtrs, and t1 is a N 1 vctor with ith lmnt givn by i,t1 i,t1 if i,t1 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 (2) (3) ar multivariat xtnsions of th bivariat modl usd by Guo, Savickas, Wang, and Yang (2009), who find significant asymmtric ffcts but insignificant diffrncs btwn th asymmtric BEKK 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 I us 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 (2006). Although most cofficint matrics ar assumd to b diagonal, th conditional covariancs and corrlations in th modl 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 t1 Nt j Nj,t j1 K j Nj,t j1 var[e (R )] var h var( h ) intraction trms. (4) J R E R V o l. 3 8 N o

6 3 2 6 L i 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: 2 j var(h Nj,t) K VRj. (5) var h j1 j Nj,t Th varianc ratios do not ncssarily sum to on across factors bcaus of th intraction trms. E q u i v a l n t B t a - b a s d R p r s n t a t i o n Th covarianc-basd pricing modl in quation (2) can b asily transformd into th familiar conditional bta-basd pricing modl. To s this, considr th following conditional factor modl for disturbanc trms: [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 it for th ith asst is givn by: 1 it cov t1( it, F)cov t t1(f t, F) t hit H Kt, (7) whr h it (h i1,t,..., h ik,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: E (F ) H, (8) t t1 t Kt whr ( 1,..., K ). Equation (8) stats 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 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s (7) and th risk prmium vctor in quation (8), th covarianc-basd pricing modl in quation (2) is transformd into th bta-basd pricing modl: E t1(r it) i i,titt ithkt. (9) 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 H var ( ). (10) ii,t it Kt it t1 it In quation (10), th first componnt on th right-hand 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. D a t a a n d M t h o d Givn th limitd availability of historical REIT 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 quity and mortgag REIT indics ar obtaind from th NAREIT for th sampl priod from January 1972 to July Approximatly as of th nd of 2013, stock xchang-listd quity REITs account for 70% of all U.S. listd REIT assts, whil quity REITs rprsnt 90% of th approximatly $700 billion quity markt capitalization of th listd REIT marktplac. Thr ar approximatly 150 listd quity REITs, almost all of which ar tradd on th NYSE. Thr ar 26 listd rsidntial mortgag REITs with a markt capitalization of $42.3 billion. As a snsitivity chck, I also us 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 As notd in prior studis, compard with th Wilshir Indx, th NAREIT Equity REIT Indx allows for gratr comparability with prvious REIT 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 th bnchmark modl ar th factormimicking portfolios of Fama and Frnch (1993) with K 3: F (MKT, SMB, HML), (11) t J R E R V o l. 3 8 N o

8 3 2 8 L i 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 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) (12) 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: F R (MKT, SMB, HML, REIT). (13) Th xpctd xcss rturns on th REIT portfolio in th altrnativ modl ar givn by quation (2) 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 th 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 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 (2) (3) for ach modl spcification ar stimatd using th mthod of quasi-maximum liklihood. Th standard rrors ar robust to non-normality of disturbanc trms. Th R 2 for ach xcss rturn sris is computd as th varianc of th implid xpctd xcss rturns givn by th right-hand-sid sid of quation (2) 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 is usd to compar 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

9 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s markt, REIT is not a pricd factor ( 2 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, Savickas, Wang, and Yang (2009), I tst thr typs of rstrictions on risk prics across portfolios using th fourfactor modl. Th first is to rplac th risk pric vctor of on of th four portfolios with 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. I also tst th hypothsis that ach of th four portfolios in quation (12) is pricd sparatly in sgmntd markts. To this nd, I supprss th covarianc trms in quation (2) for th first momnt and stimat th following indpndnt (diagonal) on-factor modl for ach of th four portfolios: R it i i,t ItT ihii,t it. (15) I thn tst th indpndnt on-factor modl against th unrstrictd four-factor modl. R s u l t s E s t i m a t s o f t h I n d p n d n t O n - F a c t o r M o d l 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 th papr. Th rsults ar rportd in Exhibit 1. To consrv spac, th stimats of th constant matrix C ar not rportd in th papr. J R E R V o l. 3 8 N o

10 3 3 0 L i Exhibit 1 Estimats of an Indpndnt On-Factor Modl for th REIT Indics Paramtr MKT SMB HML REIT Panl A: NAREIT quity REIT, % (0.399) (0.233) (0.185) (0.245) 93, % (0.230) (0.168) (0.170) (0.212) 2.947* (1.203) (1.779) 5.906* (1.932) 1.975* (0.863) Diag. A 0.931* (0.009) 0.812* (0.009) 0.915* (0.006) 0.898* (0.019) Diag. B 0.244* (0.017) 0.422* (0.030) 0.304* (0.017) 0.212* (0.022) Diag. D 0.218* (0.023) (0.147) 0.163* (0.044) 0.323* (0.040) R 2 (VR) (0.811) Panl B: Wilshir quity REIT, % (0.127) (0.274) (0.157) (0.119) 93, % (0.200) (0.211) (0.199) (0.322) (1.055) (3.104) 5.607* (1.725) 1.712* (0.526) Diag. A 0.944* (0.001) 0.806* (0.062) 0.898* (0.010) 0.923* (0.004) Diag. B 0.237* (0.010) 0.400* (0.048) 0.337* (0.020) 0.223* (0.014) Diag. D 0.170* (0.019) (0.101) 0.113* (0.050) 0.274* (0.023) R 2 (VR) (0.749) Panl C: NAREIT mortgag REIT, % (0.358) (0.231) (0.198) (0.390) 93, % 0.801* (0.267) (0.176) (0.143) 1.159* (0.296) (1.481) (1.838) 4.700* (2.060) (1.365) Diag. A 0.940* (0.011) 0.806* (0.035) 0.911* (0.005) 0.891* (0.015) Diag. B 0.282* (0.076) 0.396* (0.032) 0.321* (0.024) 0.234* (0.046) Diag. D (0.112) (0.086) 0.182* (0.043) 0.375* (0.044) R 2 (VR) (0.001) Nots: Th xcss rturn for ach of th four portfolios is givn by quation (15): R it i i,t I tt i h ii,t it, whr i is th ith intrcpt for th priod bfor dat T (1993:01) and it is th incrmntal intrcpt for th priod aftrwards. Th conditional varianc-covarianc matrix H t follows th asymmtric BEKK GARCH spcification: H t CC AH t1 A B t1 t1 B D t1 t1 D, whr A, B and D ar diagonal matrics and C is a lowr triangular matrix, and t1 is a vctor with ith lmnt givn by i,t1 t,t1 if i,t1 0 and zro othrwis. Th xcss rturns for ach modl includ xcss markt rturns (MKT), rturns on small stocks minus rturns on big stocks (SMB), rturns on high book-markt stocks minus rturns on low book-tomarkt stocks (HML), and xcss rturns on th REIT indx. Th R 2 is th prcntag of th varianc of ralizd xcss rturns xplaind by th varianc of xpctd xcss rturns. VR is th varianc of a componnt of th xpctd xcss rturn on th REIT indx associatd with a factor dividd by th varianc of th xpctd xcss rturn on th REIT indx. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl.

11 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Th rsults of stimating th indpndnt on-factor modl for th NAREIT Equity Indx ar shown in Panl A of Exhibit 1. Th rsults for th Wilshir and th mortgag REIT indics ar rportd in Panls B and C, rspctivly. Throughout th rst of th papr I us January 1993 as a cutoff dat T for th indicator variabl as som structural chang in th REITs markt has bn idntifid aftr th 1990s (.g., Ling and Ryngart, 1997; Ling and Naranjo, 1999; Glascock, Lu, and So, 2000). In Panls A and B, and 93 for ach portfolio ar insignificant at th 5% lvl. Th stimats of 93 ar 0.432% with a standard rror of for th markt indx MKT and with a standard rror of 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% lvl. In Panl C, th alphas 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% lvl. In Panl A of Exhibit 1, th stimatd risk prics ( j ) for MKT, HML, and th NAREIT Equity REIT portfolio ar 2.947, 5.906, 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 th HML risk pric is significant at th 5% 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, Savickas, Wang, and Yang (2009), 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, my rsults suggst a significant positiv rlation btwn th xpctd rturn and th volatility for th quity REIT indics but not for th mortgag REIT indx. My rsult contradicts that of Dvany (2001), 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 only xamin 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 Panls A C 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), (SBL), (HML), and (REIT), 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 0.244, 0.422, 0.304, and 0.212, 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 (MKT), (HML), and (REIT), which ar J R E R V o l. 3 8 N o

12 3 3 2 L i significant at th 1% 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 2 s ar all 1.6% or lss, indicating th low prdictiv powr of th volatility of ach portfolio for its own xpctd rturn. Th varianc ratios (VRs) 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 du to its own volatility, xcluding th indicator variabl for th changing alpha. 5 E s t i m a t s o f O n - t o F o u r - F a c t o r M o d l s : N A R E I T I n d x Th rsults of stimating th modls in quations (2) (3) ar rportd in Exhibit 2 (NAREIT Equity 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 ar similar to thos rportd in Exhibit 1. Panl A of Exhibit 2 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 0.402% with a standard rror of 0.185, whil th stimat for REIT is with a standard rror of Thus, both MKT and REIT xhibit significant alphas in th scond half of th sampl priod sinc Th pric of th markt risk () is also significant, with an stimat of and a standard rror of Th R 2 s for th portfolios ar or lowr. Panl B of Exhibit 2 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 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 2 s associatd with th man quations for th MKT, SMB, HML, and REIT ar 0.009, 0.001, 0.003, and 0.025, rspctivly, suggsting that th two-factor modl provids mor xplanatory powr for th tim variations of MKT and REIT than th CAPM. Th varianc ratios associatd with MKT and REIT ar and 0.475, rspctivly, 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

13 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 2 Estimats of Covarianc-Basd Modls for th NAREIT Equity REIT Indx Paramtr MKT SMB HML REIT Panl A: On-factor modl (MKT), % (0.128) (0.103) 0.448* (0.094) (0.137) 93, % 0.402* (0.185) (0.149) (0.132) 0.462* (0.229) 2.976* (0.925) Diag. A 0.926* (0.013) 0.832* (0.041) 0.907* (0.007) 0.896* (0.015) Diag. B 0.251* (0.030) 0.400* (0.043) 0.314* (0.022) 0.208* (0.034) Diag. D 0.232* (0.046) (0.101) 0.178* (0.071) 0.345* (0.053) R 2 (VR) (0.735) Panl B: Two-factor modl (MKT, REIT), % (0.116) (0.094) 0.388* (0.087) (0.118) 93, % 0.345* (0.177) (0.137) (0.140) 0.378* (0.153) (0.882) 1.749* (0.779) Diag. A 0.928* (0.002) 0.831* (0.009) 0.908* (0.004) 0.895* (0.003) Diag. B 0.251* (0.011) 0.401* (0.018) 0.316* (0.014) 0.211* (0.015) Diag. D 0.226* (0.018) 0.157* (0.059) 0.170* (0.034) 0.335* (0.014) R 2 (VR) (0.067) (0.475) Panl C: Thr-factor modl (MKT, SMB, HML), % (0.129) 0.312* (0.141) (0.141) (0.139) 93, % (0.162) (0.138) (0.176) (0.230) 3.597* (0.841) 3.504* (1.490) 6.516* (2.124) Diag. A 0.926* (0.006) 0.831* (0.036) 0.910* (0.008) 0.898* (0.010) Diag. B 0.246* (0.022) 0.389* (0.042) 0.314* (0.026) 0.204* (0.028) Diag. D 0.235* (0.024) 0.192* (0.053) 0.159* (0.045) 0.337* (0.016) R 2 (VR) (0.392) (0.023) (0.221) Panl D: Four-factor modl (MKT, SMB, HML, REIT), % (0.181) (0.169) (0.155) (0.229) 93, % (0.200) (0.154) (0.128) (0.250) 3.733* (1.633) 3.440* (1.583) 6.649* (2.362) (1.724) Diag. A 0.925* (0.013) 0.831* (0.044) 0.910* (0.013) 0.898* (0.019) Diag. B 0.246* (0.028) 0.389* (0.056) 0.313* (0.031) 0.203* (0.031) Diag. D 0.235* (0.037) (0.137) 0.160* (0.063) 0.338* (0.046) R 2 (VR) (0.439) (0.023) (0.239) (0.003) J R E R V o l. 3 8 N o

14 3 3 4 L i Exhibit 2 (continud) Estimats of Covarianc-Basd Modls for th NAREIT Equity REIT Indx Paramtr MKT SMB HML REIT Panl E: Four-factor modl with unrstrictd risk prics (MKT, SMB, HML, REIT), % (0.455) (0.312) (0.210) (0.494) 93, % (0.280) (0.195) (0.202) (0.306) MKT (2.562) (4.719) (5.993) (2.830) SMB (4.659) (3.369) (6.265) (6.236) HML (2.520) (3.851) 6.366* (2.390) (3.886) REIT (3.441) (8.405) (6.964) (2.840) Diag. A 0.929* (0.007) 0.830* (0.022) 0.915* (0.005) 0.900* (0.014) Diag. B 0.247* (0.028) 0.410* (0.033) 0.302* (0.023) 0.198* (0.036) Diag. D 0.230* (0.037) (0.087) 0.173* (0.055) 0.332* (0.039) R 2 (VR) (0.143) (0.393) (0.155) (1.149) Nots: For Panls A D, th xcss rturn for ach of th four portfolios is givn by quation (2): K R it i i,titt j1 jhij,t it with rstrictd risk prics j. For Panl E, th xcss rturn 4 is givn by quation (14): R it i i,titt j1 ijhij,t it with unrstrictd risk prics ij. Th conditional varianc-covarianc matrix H t in ach panl follows th asymmtric BEKK GARCH spcification: H t CC AH t1 A B t1 t1 B D t1 t1 D. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl. hr contradicts th findings of Liu, Hartzll, Grig, and Grissom (1990), who cannot rjct th markt intgration hypothsis for quity REITs using a similar two-factor modl. Panl C of Exhibit 2 rports stimats in th thr-factor modl. Only th associatd with SMB is significant at th 5% lvl, suggsting that th modl is capabl providing xplanatory powr for th 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 ), for SMB (std. rr ), and (std. rr ) 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 2 s associatd with th man quations for th thr risk factors and REIT ar rspctivly 0.012, 0.032, 0.018, and 0.027, which ar highr than th R 2 in th on- and two-factor modls. Thus, tim-varying conditional covariancs provid rlativly mor xplanatory powr for th tim variations of SMB and REIT than th tim variations of MKT and HML. Consistnt with th stimats of th risk prics, th stimatd varianc ratios suggst that SMB provids xplanatory powr for only 2.3% of th tim variation

15 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 3 Estimats of Covarianc-Basd Modls for th Wilshir REIT Indx Paramtr MKT SMB HML REIT Panl A: On-factor modl (MKT), % 0.395* (0.137) (0.138) (0.089) (0.156) 93, % (0.211) (0.178) (0.117) 0.552* (0.171) 0.633* (0.276) Diag. A 0.944* (0.006) 0.862* (0.027) 0.885* (0.022) 0.925* (0.009) Diag. B 0.237* (0.031) 0.355* (0.058) 0.361* (0.046) 0.215* (0.048) Diag. D 0.166* (0.071) (0.084) (0.119) 0.291* (0.069) R 2 (VR) (0.046) Panl B: Two-factor modl (MKT, REIT), % (0.132) (0.099) (0.099) (0.132) 93, % (0.179) (0.130) (0.137) (0.159) (1.024) 2.207* (0.798) Diag. A 0.944* (0.002) 0.866* (0.007) 0.883* (0.007) 0.924* (0.003) Diag. B 0.239* (0.010) 0.354* (0.016) 0.364* (0.017) 0.226* (0.013) Diag. D 0.154* (0.021) 0.148* (0.070) (0.072) 0.272* (0.016) R 2 (VR) (0.000) (0.834) Panl C: Thr-factor modl (MKT, SMB, HML), % (0.331) (0.223) (0.249) (0.321) 93, % (0.354) (0.190) (0.209) (0.370) (0.796) (0.853) 2.872* (1.285) Diag. A 0.942* (0.008) 0.804* (0.042) 0.895* (0.005) 0.921* (0.017) Diag. B 0.239* (0.023) 0.404* (0.043) 0.347* (0.016) 0.225* (0.024) Diag. D 0.178* (0.024) (0.091) (0.065) 0.288* (0.028) R 2 (VR) (0.003) (0.174) (0.352) Panl D: Four-factor modl (MKT, SMB, HML, REIT), % 0.331* (0.160) (0.145) (0.159) (0.150) 93, % (0.235) (0.153) (0.154) 0.582* (0.242) (0.410) (0.868) 2.869* (1.259) (0.235) Diag. A 0.942* (0.008) 0.804* (0.062) 0.896* (0.017) 0.921* (0.019) Diag. B 0.239* (0.022) 0.404* (0.056) 0.345* (0.035) 0.225* (0.036) Diag. D 0.179* (0.043) (0.081) (0.079) 0.288* (0.046) R 2 (VR) (0.023) (0.183) (0.340) (0.037) J R E R V o l. 3 8 N o

16 3 3 6 L i Exhibit 3 (continud) Estimats of Covarianc-Basd Modls for th Wilshir REIT Indx Paramtr MKT SMB HML REIT Panl E: Four-factor modl with unrstrictd risk prics (MKT, SMB, HML, REIT), % (0.999) (0.303) (0.188) (3.351) 93, % (1.842) (0.796) (0.994) (0.533) MKT (4.297) (9.335) (20.336) (1.900) SMB (0.959) (20.932) (13.008) (0.321) HML (6.617) (15.320) (12.203) (0.400) REIT (8.953) (1.633) (12.706) (2.006) Diag. A (0.040) (0.133) (0.012) 0.922* (0.080) Diag. B (0.101) (0.064) (0.103) 0.214* (0.051) Diag. D (0.089) (0.223) (0.056) (0.245) R 2 (VR) (0.048) (0.160) (0.026) (0.466) Nots: For Panls A D, th xcss rturn for ach of th four portfolios is givn by quation (2): K R it i i,titt j1 jhij,t it with rstrictd risk prics j. For Panl E, th xcss rturn 4 is givn by quation (14): R it i i,titt j1 ijhij,t it with unrstrictd risk prics ij. Th conditional varianc-covarianc matrix H t in ach panl follows th asymmtric BEKK GARCH spcification: H t CC AH t1 A B t1 t1 B D t1 t1 D. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl. of th xpctd REIT rturn, but MKT and HML provids significantly mor at 39.2% and 22.1%, rspctivly. Th fact that th thr-factor modl provids mor xplanatory powr for th REIT rturns than th CAPM is consistnt with th findings of Ptrson and Hsih (1997) and Chiang, L, and Wisn (2005). Panl D of Exhibit 2 displays th rsults of stimating th four-factor modl, which includs REIT as th fourth factor. I find that nithr nor 93 is significant at th 5% 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 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 R 2 s for th first thr factors ar similar to thos in th thr-factor modl. Th R 2 (0.027) for th REIT rturn is slightly lowr. Th varianc ratios associatd with th four factors ar 0.439, 0.023, 0.239, and 0.003, rspctivly, suggsting that vry littl tim variation in xpctd xcss rturn on REIT is associatd with its volatility. Finally, I xamin th rsults in

17 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s 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 th 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. E s t i m a t s o f O n - t o F o u r - F a c t o r M o d l s : W i l s h i r I n d x Exhibit 3 prsnts th rsults of stimating th on- to four-factor modls in which th NAREIT Equity REIT Indx is rplacd by th Wilshir REIT Indx. Th rsults ar mostly similar to thos rportd in Exhibit 2. 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 is significant lik in Exhibit 2. 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% lvl. Risk prics associatd with othr factors ar not as significant as in Exhibit 2. 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 provids full xplanatory powr for th tim variation of th xpctd rturns on REIT. Th covarianc of REIT with HML is mor important for subsuming th rol of th REIT volatility. Lik th rsults in Exhibit 2, th stimatd risk prics in Exhibit 3 provid support for th full intgration hypothsis. T s t s o f M o d l S p c i f i c a t i o n s a n d M a r k t I n t g r a t i o n 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, along with 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% whn ithr REIT indx is usd. Th p-valu associatd with 93 in th modl is also lss than 5% whn th NAREIT Equity REIT Indx is usd. Thus, th hypothsis that xcss rturns ar zro for th full sampl priod or incrmntal rturns for th post-1993 priod ar zro is rjctd. Nxt, and 93 in ithr th on- or two-factor modl ar jointly significant at th 5% lvl, rjcting both modls whn ithr REIT indx is usd. Sinc thr is strong J R E R V o l. 3 8 N o

18 3 3 8 L i Exhibit 4 Spcification Tsts of Covarianc-basd Modls NAREIT Equity REIT Wilshir REIT Numbr of Factors p-valu p-valu Log- Log- Liklihood 93 Liklihood 93 Panl A: Log-liklihood and joint significanc of alphas Zro factor * * On: MKT * * 0.004* Two: MKT, REIT * * 0.456* Thr: MKT, SMB, HML Four: MKT, SMB, HML, REIT Rstrictd risk prics Unrstrictd risk prics * Hypothsis DF LR p-valu DF LR p-valu Panl B: Liklihood ratio (LR) tsts Zro vs. thr-factor * On- vs. thr-factor * Thr- vs. four-factor Four-factor: qual vs. diff. prics MKT SMB HML * * REIT Four-factor Rstrictd vs. unrstrictd * * Indpndnt vs unrstrictd * * Nots: Th indpndnt modl rfrs to th modl in Exhibit 1. Othr modls ar dscribd in Exhibits 2 3. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl. 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% lvl. For th NAREIT Indx, th stimats of 93 ar also not significant in thr- and four-factor modls at th 10% lvl. For th Wilshir Indx, a similar

19 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s rsult is found in th thr-factor modl but th stimat of 93 is significant at th 10% lvl for th rstrictd four-factor modl or at th 5% lvl for th unrstrictd four-factor modl. Panl B of Exhibit 4 rports rsults of th liklihood ratio (LR) tsts. Using th NAREIT Equity REIT Indx, th tsts rjct th zro-factor modl and th onfactor modl in favor of th thr-factor modl, as th p-valus ar lss than 1%. 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 th 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 rsults rval that th risk prics associatd with REIT ar not significantly diffrnt from th othr common factors at th 5% 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 rsults of th tst of th rstrictd four-factor modl against th unrstrictd four-factor modl and th tst of th indpndnt on-factor modl (Exhibit 1) against th unrstrictd four-factor modl. Th rsults indicat rjction of th rstrictd modl in favor of th unrstrictd modl at th 5% lvl, mostly bcaus of th diffrnt risk prics associatd with HML. Th rsults also indicat rjction of th indpndnt on-factor modl in favor of th unrstrictd four-factor modl, which suggsts that th covarianc trms provid xplanatory powr for th xpctd rturns and th markts for th four portfolios ar intgratd. Similar to what Ling and Naranjo (1999) find, th rsults of th rstrictions on risk prics across portfolios, gnrally provid support for th hypothsis that th scuritizd ral stat markt is intgratd with th gnral capital markt. C o m p a r i s o n w i t h Vo l a t i l i t y - b a s d M o d l s 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 4T tt j jj,t 4t j1 R I h, (16) whr K 1, 2 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. J R E R V o l. 3 8 N o

20 3 4 0 L i Exhibit 5 Estimats of Volatility-basd Modls for th NAREIT Equity REIT Indx Paramtr MKT SMB HML REIT Panl A: On-factor modl (MKT), % (0.429) (0.183) (0.212) (0.471) 93, % (0.190) (0.139) (0.202) (0.336) (1.410) (1.533) 5.815* (2.918) (1.471) Diag. A 0.933* (0.008) 0.817* (0.014) 0.913* (0.005) 0.904* (0.014) Diag. B 0.239* (0.021) 0.409* (0.034) 0.312* (0.016) 0.201* (0.050) Diag. D 0.224* (0.023) 0.161* (0.079) 0.141* (0.058) 0.340* (0.036) R 2 (VR) (0.001) Panl B: Two-factor modl (MKT, REIT), % (0.391) 0.358* (0.175) (0.133) (0.353) 93, % 0.424* (0.177) (0.140) (0.165) (0.212) (1.552) (1.324) 5.827* (1.187) (1.186) 2.089* (0.979) Diag. A 0.932* (0.009) 0.811* (0.010) 0.915* (0.004) 0.899* (0.017) Diag. B 0.242* (0.022) 0.421* (0.025) 0.305* (0.010) 0.210* (0.032) Diag. D 0.220* (0.033) (0.166) 0.161* (0.039) 0.324* (0.054) R 2 (VR) (0.034) (0.990) Panl C: Thr-factor modl (MKT, SMB, HML), % (0.136) (0.173) (0.140) (0.214) 93, % 0.329* (0.154) (0.182) (0.123) 0.406* (0.143) (0.934) (1.429) 5.763* (1.440) (0.712) (1.929) (2.530) Diag. A 0.933* (0.002) 0.810* (0.062) 0.913* (0.002) 0.904* (0.004) Diag. B 0.241* (0.009) 0.415* (0.058) 0.313* (0.010) 0.202* (0.018) Diag. D 0.224* (0.016) 0.157* (0.092) 0.138* (0.045) 0.339* (0.017) R 2 (VR) (0.000) (0.274) (0.055) 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

21 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 5 (continud) Estimats of Volatility-basd Modls for th NAREIT Equity REIT Indx Paramtr MKT SMB HML REIT Panl D: Four-factor modl (MKT, SMB, HML, REIT), % (0.175) 0.310* (0.144) (0.130) (0.229) 93, % 0.411* (0.142) (0.151) (0.118) 0.323* (0.107) 2.361* (0.491) (1.246) 5.891* (1.575) (0.723) (1.359) (2.241) 2.139* (0.481) Diag. A (0.005) (0.042) (0.004) (0.003) Diag. B (0.022) (0.044) (0.017) (0.019) Diag. D (0.018) (0.081) (0.041) (0.020) R 2 (VR) (0.038) (0.046) (0.003) (1.012) Panl E: Four-factor modl with unrstrictd risk prics (MKT, SMB, HML, REIT), % (0.388) (0.353) (0.192) (0.560) 93, % (0.266) (0.247) (0.170) (0.317) MKT (1.871) (5.206) (4.786) (1.940) SMB 8.281* (3.550) (2.712) (6.311) (6.408) HML (3.519) (4.522) 6.382* (2.723) (3.804) (2.921) (11.987) (5.155) 2.641* (1.118) Diag. A 0.929* (0.010) 0.830* (0.018) 0.915* (0.014) 0.900* (0.020) Diag. B 0.247* (0.031) 0.410* (0.036) 0.302* (0.030) 0.198* (0.035) Diag. D 0.230* (0.038) (0.126) 0.173* (0.051) 0.332* (0.044) R 2 (VR) (0.143) (0.394) (0.154) (1.146) Nots: For Panls A D, th factor risk prmiums on MKT, SMB and HML ar givn by indpndnt on-factor modls in quation (15): R it i i,titt ihii,t it (i 1, 2, 3) and th xcss REIT rturn is givn by quation (16): K R 4t 4 4TItT j1 jhjj,t 4t. For 4 Panl E, th xcss rturn is givn by quation (14): R it i i,titt j1 jjhjj,t it with unrstrictd risk prics ij. Th conditional varianc-covarianc matrix H t in ach panl follows th asymmtric BEKK GARCH spcification: H t CC AH t1 A B t1 t1 B D t1 t1 D. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl. th rsults of stimating a four-factor modl with unrstrictd risk prics (s nots to Exhibit 5). Although th alphas ar not significant for ach portfolio including REIT in Panl A in Exhibit 5, th cofficint is with a standard rror of 1.471, implying J R E R V o l. 3 8 N o

22 3 4 2 L i 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 with a standard rror of Similar to th covarianc btwn MKT and REIT rturns, th stock markt volatility dos not provid xplanatory powr for th positiv rlation btwn th xpctd rturn and volatility of REIT. In Panls C and D of Exhibit 5, 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 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, although th cofficint associatd with th volatility of ach othr factor is not significant. Th varianc ratios associatd with th REIT volatility in th fourfactor modls xcd unity. Thus, unlik th covarianc-basd modl, th volatility of th Fama-Frnch factors dos not provid xplanatory powr for 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, although 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 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 thrfactor 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-factor 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 th on- and two-factor modls ar significant (s Exhibit 4). Liklihood ratio tst rsults in Panl B in Exhibit 5 do not indicat rjction of th on-factor modl in favor of th thr-factor modl but do indicat rjction of th thr-factor modl in favor of th fourfactor 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 indicat that th covariancs of REIT with risk factors ar mor usful than th volatility of th factors in providing xplanatory powr for th tim variation of th xpctd REIT rturn.

23 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 6 Estimats of Volatility-Basd Modls for th Wilshir Equity REIT Indx Paramtr MKT SMB HML REIT Panl A: On-factor modl (MKT), % (0.337) (0.294) (0.195) (0.215) 93, % (0.283) (0.208) (0.170) 0.573* (0.215) (0.974) (2.678) (2.002) (1.058) Diag. A (0.009) (0.055) (0.006) (0.018) Diag. B (0.020) (0.063) (0.018) (0.028) Diag. D (0.033) (0.099) (0.073) (0.034) R 2 (VR) (0.006) Panl B: Two-factor modl (MKT, REIT), % (0.430) 0.499* (0.190) (0.182) (0.429) 93, % (0.275) (0.117) (0.150) 0.482* (0.249) (1.562) 5.027* (2.151) 5.570* (2.198) (1.746) 1.832* (0.521) Diag. A 0.945* (0.009) 0.805* (0.011) 0.898* (0.008) 0.924* (0.018) Diag. B 0.234* (0.022) 0.400* (0.030) 0.338* (0.025) 0.221* (0.033) Diag. D 0.171* (0.035) (0.091) (0.069) 0.274* (0.034) R 2 (VR) (0.036) (0.919) Panl C: Thr-factor modl (MKT, SMB, HML), % (0.293) (0.274) (0.203) (0.337) 93, % (0.163) (0.180) (0.148) 0.529* (0.236) (1.481) (2.628) 5.549* (2.060) (1.227) (2.712) (2.487) Diag. A 0.945* (0.009) 0.805* (0.060) 0.896* (0.005) 0.926* (0.019) Diag. B 0.234* (0.015) 0.397* (0.052) 0.342* (0.018) 0.213* (0.024) Diag. D 0.178* (0.029) (0.094) (0.063) 0.287* (0.031) R 2 (VR) (0.010) (0.280) (0.034) J R E R V o l. 3 8 N o

24 3 4 4 L i Exhibit 6 (continud) Estimats of Volatility-Basd Modls for th Wilshir Equity REIT Indx Paramtr MKT SMB HML REIT Panl D: Four-factor modl (MKT, SMB, HML, REIT), % 0.328* (0.132) (0.259) (0.223) (0.203) 93, % 0.300* (0.166) (0.176) (0.153) (0.240) (1.077) (2.495) 5.665* (2.217) (1.110) (2.835) (2.470) 1.920* (0.581) Diag. A 0.945* (0.009) 0.800* (0.058) 0.898* (0.005) 0.925* (0.019) Diag. B 0.235* (0.016) 0.402* (0.051) 0.338* (0.016) 0.220* (0.023) Diag. D 0.170* (0.026) (0.097) (0.065) 0.270* (0.031) R 2 (VR) (0.046) (0.083) (0.006) (0.953) Panl E: Four-Factor modl with unrstrictd risk prics (MKT, SMB, HML, REIT), % (0.279) (0.183) (0.208) 1.182* (0.286) 93, % (0.222) (0.133) (0.196) 0.627* (0.238) MKT 3.386* (0.603) (5.340) 7.042* (3.835) 2.122* (0.991) HML (4.338) 7.966* (3.614) (2.938) (4.322) (1.103) * (3.757) * (2.999) 2.708* (0.318) Diag. A 0.944* (0.004) 0.839* (0.043) 0.895* (0.010) 0.935* (0.004) Diag. B 0.232* (0.021) 0.378* (0.045) 0.336* (0.024) 0.185* (0.022) Diag. D 0.176* (0.026) (0.079) 0.153* (0.046) 0.271* (0.017) R 2 (VR) (0.051) (0.658) (0.603) (0.984) Nots: For Panls A D, th factor risk prmiums on MKT, SMB, and HML ar givn by th indpndnt on-factor modl in quation (15): R it i i,titt ihii,t it (i 1, 2, 3) and th xcss REIT rturn is givn by quation (16): K R 4t 4 4TItT j1 jhjj,t 4t. For 4 Panl E, th xcss rturn is givn by quation (14): R it i i,titt j1 ijhjj,t it with unrstrictd risk prics ij. Th conditional varianc-covarianc matrix H t in ach panl follows th asymmtric BEKK GARCH spcification: H t CC AH t1 A B t1 t1 B D t1 t1 D. Robust standard rrors ar rportd in parnthss. *Cofficint significant at th 5% lvl.

25 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 7 Spcification Tsts of Volatility-Basd Modls NAREIT EREIT Wilshir EREIT Numbr of Factors p-valu p-valu Log- Log- Liklihood 93 Liklihood 93 Panl A: Log-liklihood and joint significanc of alpha On: MKT * * Two: MKT, REIT * * Thr: MKT, SMB, HML Four: MKT, SMB, HML, REIT * 0.000* * Four: Unrstrictd risk prics * Hypothsis DF LR p-valu DF LR p-valu Panl B: Liklihood ratio (LR) tsts Zro vs. thr-factor * On vs. thr factors Thr vs. four factors * * Indpndnt vs. unrstrictd * Nots: Th indpndnt modl rfrs to th modl in Exhibit 1. Othr modls ar dscribd in Exhibits 5 6. *Significant at th 5% lvl. T i m - S r i s P r o p r t i s o f E x p c t d R t u r n s a n d Vo l a t i l i t y o f R E I Ts Givn th similarity of th rsults obtaind from th two REIT indics, in this sction I xamin th tim sris proprtis of th xpctd rturn and volatility of REITs implid by th covarianc-basd thr-factor modl for th NAREIT Equity REIT Indx. G r a p h i c a l I l l u s t r a t i o n s 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 our sampl priod, spcially during th financial crisis. Th sharp incras of th conditional REIT volatility during th crisis is consistnt J R E R V o l. 3 8 N o

26 3 4 6 L i Exhibit 8 Expctd Rturns and Volatility of th NAREIT Equity REIT Indx with th vidnc providd by Sun, Titman, and Twit (2015). During th pak of th latst 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 th full sampl priod, th systmatic varianc contributs an avrag of 53.3% of total varianc. Around th pak of th rcnt financial crisis (from Novmbr 2008 to July 2009), th systmatic varianc accounts for mor than 80% of th total varianc. In Exhibit 10, I plot btas and factor risk prmiums. Th dcrasing pattrn of th stock markt bta in th first thr dcads of my sampl priod, as also rportd by Khoo, Hartzll, and Hosli (1993) and Chiang, L and Wisn (2005), 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 2000s, spcially during th rcnt financial crisis.

27 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Exhibit 9 Systmatic and Idiosyncratic Variancs of th NAREIT Equity REIT Indx In Exhibit 11, I plot th volatility of ach factor and th corrlation btwn ach pair of th factors. Th lvl of th stock markt volatility during th rcnt 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 10 and 11 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. C r o s s - C o r r l a t i o n s Exhibit 12 prsnts cross-corrlations btwn pairs of th momnts implid by th thr-factor modl with th NAREIT Equity REIT Indx. In fact, th xpctd REIT rturn, E( R 4 ), is highly corrlatd with its own volatility, h 44, with a corrlation of Howvr, as th rsults in Exhibit 2 show, th thr risk factors subsum th rol of th REIT volatility in providing xplanatory powr for th xpctd REIT rturn. Consistnt with th rsults, th xpctd REIT rturn, E( R 4 ), is vry highly corrlatd with th stock markt prmium 1 (corrlation 0.933), th HML bta 3 (corrlation 0.650), and th markt bta 1 (corrlation 0.584). J R E R V o l. 3 8 N o

28 3 4 8 L i Exhibit 10 Conditional Btas and Factor Risk Prmiums Panl A: Btas Panl B: Factor Risk Prmiums

29 T i m V a r i a t i o n o f E x p c t d R t u r n s o n R E I T s Panl A: Factor Volatility Exhibit 11 Conditional Factor Volatility and Corrlations Panl B: Factor Corrlations 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 J R E R V o l. 3 8 N o