Journal of Empirical Finance

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1 Journal of Emprcal Fnance (3) 4 6 Contents lsts avalable at ScVerse ScenceDrect Journal of Emprcal Fnance journal homepage: Samplng nterval and estmated betas: Implcatons for the presence of transtory components n stock prces Perre Perron a,, Sungju Chun b, Cosme Vodounou c a Boston Unversty, Unted States b Korea Insurance Research Insttute, Republc of Korea c AFRISTAT, Observatore Economque et Statstque d'afrque Subsaharenne, Mal artcle nfo abstract Artcle hstory: Receved December Receved n revsed form 5 September Accepted 7 October Avalable onlne 6 October JEL classfcaton: C C C58 G Keywords: Mean reverson CAPM Stock returns Transtory components Frm sze Contnuous tme models We provde a theoretcal framework to explan the emprcal fndng that the estmated betas are senstve to the samplng nterval even when usng contnuously compounded returns. We suppose that stock prces have both permanent and transtory components. The dscrete tme representaton of the beta depends on the samplng nterval and two components labeled permanent and transtory betas. We show that f no transtory component s present n stock prces then no samplng nterval effect occurs. However, the presence of a transtory component mples that the beta s an ncreasng (decreasng) functon of the samplng nterval for more (less) rsky assets. In our framework, assets are labeled rsky f ther permanent beta s greater than ther transtory beta and vce versa for less rsky assets. Smulatons show that our theoretcal results provde good approxmatons for the estmated betas n small samples. We provde emprcal evdence about the presence of negatve seral correlaton and mean reverson n the returns of the portfolos consdered. We dscuss why our model s better able to provde an explanaton for ths samplng nterval effect than other models n the lterature. Elsever B.V. All rghts reserved.. Introducton The Captal Asset Prcng Model (CAPM) has been the object of numerous studes over the past thrty years. Developed by Sharpe (964) and Lntner (965), t s based on the assumpton that nvestors are rsk-averse and construct ther portfolos accordng to a mean varance crteron. The basc relaton says that n equlbrum there exsts a lnear relaton between the return of a gven asset or portfolo and the return of the market portfolo. An emprcal feature that has attracted some attenton s the fact that the estmated beta (or systematc rsk of an asset or portfolo) s senstve to the samplng nterval used to compute the returns. Ths nterval effect has been analyzed n relaton wth another anomaly, the sze effect, whch shows a sgnfcant relaton between returns and the market values of frms. Banz (98) has analyzed ths effect and showed that the smaller a frm s the hgher s ts expected return. For the nterval effect, the We wsh to thank Xaokang Zhu and Zhongjun Qu for ther help and comments on prevous drafts. Perron acknowledges the fnancal support from the Socal Scences and Humantes Research Councl of Canada (SSHRC), the Natural Scences and Engneerng Councl of Canada (NSERC), the Fonds pour la Formaton de Chercheurs et l'ade à la Recherche du Québec (FCAR) and the Natonal Scence Foundaton under grants SES and SES Correspondng author at: Department of Economcs, Boston Unversty, 7 Bay State Road, Boston, 5, USA. Tel.: ; fax: E-mal address: perron@bu.edu (P. Perron) /$ see front matter Elsever B.V. All rghts reserved.

2 P. Perron et al. / Journal of Emprcal Fnance (3) emprcal studes show that changes n the samplng nterval used nduce a bas n the estmate of the systematc rsk whose magntude depends on the sze of the frms as measured by ther market value. Accordng to Pogue and Solnk (974), Roll (98) and Renganum (98), the beta s underestmated for small frms and overestmated for large frms when usng daly data. Such a bas s attrbuted to the small frequency at whch the assets of small frms are transacted (Scholes and Wllams, 977; Dmson, 979) and more generally to frctons n the exchange process (Cohen et al., 983). Accordng to Cohen et al., prces adjust followng the arrval of nformaton and the adjustment delays are related to the sze of frms. Accordngly, for large frms wth greater tradng volume, the adjustment delays are shorter than for small frms whose tradng volume s smaller. The nfrequent exchange for small frms s accompaned wth the non-synchronzaton of ndvdual prces n relaton to the market ndex whch nduces an ntertemporal correlaton between returns and an autocorrelaton n the market returns. In ths study, we shall not be concerned about such relatons holdng at very short samplng ntervals where market mcrostructure effects are operatve. Rather, we shall concentrate on ranges of samplng ntervals where these market mcrostructure effects are not present; for example constructng returns from monthly to annual ntervals. It s possble that ntraday perodcty or seasonalty may have an mpact on returns at longer horzons, a full nvestgaton of whch s outsde the scope of ths paper. However, we beleve that such effects are unlkely. For nstance, Andersen and Bollerslev (997) document a strong ntradaly perodc pattern n the volatlty. However, ther model used to descrbe ths feature mples a reducton n the overall level of the nterdaly return autocorrelatons (pp. 7 8). To our knowledge, there s no evdence of spllover from hgh frequency ntradaly features to temporal dependence n returns at horzons between one month and a year, whch are consdered n ths paper. To be precse about the termnology, we use the followng defntons. Frst, P (th) denotes the dvdend-renvested prce ndex for securty measured over a samplng nterval of h perods. The relatve prces P (th)/p ((t )h) are then the h-perod returns. For a gven portofolo wth, say, N securtes the buy-and-hold returns are gven by N [P (th)/p ((t )h)] and the contnuously compounded returns by log(n P (th)/n P ((t )h)). On a theoretcal level, Levhar and Levy (977) and Hawawn (98) show a relaton between the beta and the samplng nterval n the case where the returns are computed usng the relatve prces P(th)/P((t )h) to defne the h-perod returns,.e. the buy-and-hold returns. In such cases, the nterval effect s smply due to an accountng ssue. Handa et al. (989, 993) show clearly that an nterval effect s present emprcally and that the betas of more rsky assets ncrease as the samplng nterval ncreases, whle the betas of less rsky assets are decreasng. Ther results also show that the estmated betas approach that of the market portfolo (.e. one) when the samplng nterval gets smaller. They argue that f contnuously compounded returns are used, no such nterval effect should hold f markets are effcent. An nterestng fact s that a smlar samplng nterval effect s present emprcally when usng contnuously compounded returns. Ths was shown as early as n the study by Smth (978) and also n the more specalzed analyses of Corhay (99) and Defrère (995). To further corroborate ths emprcal fact, we provde an emprcal study along the lne of Handa et al. (989) and show that the samplng nterval effect s very smlar whether usng buy and hold or contnuously compounded returns. The purpose of our study s then to provde a theoretcal framework where nterval effects are present even when usng contnuously compounded returns. We suppose that stock prces have both permanent and transtory components. The permanent component s a standard geometrc Brownan moton wth constant volatlty whle the transtory component s a statonary Ornsten Uhlenbeck process. We derve the dscrete tme representaton of the beta whch depends on the samplng nterval and two components labeled permanent and transtory betas (to be defned explctly). We show that f no transtory component s present n stock prces then no samplng nterval effect occurs. However, the presence of a transtory component mples that the beta s an ncreasng (decreasng) functon of the samplng nterval for more rsky (less rsky) assets. In our framework, assets are labeled rsky f ther permanent beta s greater than ther transtory beta and vce versa for less rsky assets. Smulatons show that our theoretcal results provde good approxmatons for the estmated betas n small samples. Accordng to our results, the presence of a transtory component s the crucal element to explan the nterval effect, wthout t no such effect should be present. Ths transtory component s smlar to that used by Fama and French (988) and Poterba and Summers (988) to explan the presence of negatve seral correlaton n returns at long horzons. Our theoretcal results and the presence of the nterval effect emprcally found can be perceved as ndrect evdence for the presence of a transtory component n stock prces. We nevertheless provde emprcal evdence to that effect. Frst, we consder estmates of smple ARMA(,) processes for the varous portfolos when estmated usng 6 or months samplng ntervals. These show evdence of negatve seral correlaton n the portfolos' returns. Second, we evaluate long-horzon regressons of the type consdered by Fama and French (988) based on to year returns and show that when tested jontly the parameter estmates support the presence of mean reverson. We dscuss the mplcaton of our results n relaton to the prevous lterature, n partcular the work of Lo and MacKnlay (99). Gven the evdence presented, we beleve that our smple model wth transtory components s better able to explan the full pattern of the estmates of the betas usng varous samplng ntervals. As evdenced by the lterature that followed the work of Lo and MacKnlay (99) t s also consstent wth the lead-lag pattern and, moreover, able to explan the pattern of the estmates of the betas for both small and large frms, whle the framework of Lo and MacKnlay (99) can only provde an explanaton for the decrease n the estmated betas for small frms. The paper s structured as follows. Secton presents the emprcal results showng the presence of a samplng nterval effect for estmated betas contnuously compounded as well as buy and hold returns. Secton 3 defnes the basc model n contnuous tme and derves ts dscrete tme representaton. In Secton 4, we dscuss the propertes of the beta n relaton to the samplng nterval and the lmtng behavor of ts estmate. Secton 5 provdes smulaton evdence that supports the theoretcal results. Secton 6 presents emprcal evdence that documents the presence of transtory components and mean reverson n the varous

3 44 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 portfolos consdered. Secton 7 provdes a dscusson of our results n relaton to the prevous lterature. Secton 8 offers concludng comments. Techncal dervatons and detals on varous computatons are contaned n an appendx.. The emprcal facts In ths secton, we document the presence of the samplng nterval effect on estmated betas usng contnuously compounded returns. The setup s bascally the same as that used n Handa et al. (989) wth an extended sample. Hence, we provde only a bref summary of the procedure and refer the reader to that paper for more nformaton. We use all stocks lsted on the Center for Research n Securty Prces (CRSP) monthly tape. Ths ncludes all the New York Stock Exchange (NYSE) frms for the perod 96: to 8: and the Amercan Stock Exchange (AMEX) securtes for the perod 964: to 8:. We rank all securtes accordng to ther begnnng-of-year equty market values and dvde them nto equal portfolos (wth the portfolo labeled contanng the smallest 5% frms and the portfolo labeled contanng the largest 5% frms). Ths rankng and groupng s repeated every year. We consder estmatng market-model betas for sx samplng ntervals:,, 3, 4, 6 and months. For all ntervals, we use the equal-weghted sample mean returns as the market portfolo proxy. The betas are estmated usng a 5 year overlappng wndow. Table Sample means of market-value portfolo betas and standard errors for dfferent return measurement ntervals (NYSE and AMEX, 96: 8:). Sample means: Buy-and-hold returns Sample means: Contnuously compounded returns Portfolo Intervals (months) Portfolo Intervals (months) MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV Standard errors: Buy-and-hold returns Standard errors: Contnuously compounded returns Portfolo Intervals (months) Portfolo Intervals (months) MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV MV

4 P. Perron et al. / Journal of Emprcal Fnance (3) What are reported n Table are tme seres averages of estmated betas for the varous portfolos along wth ther standard errors. To properly compare the samplng nterval effect, we report results for both cases where the estmates are constructed usng buy-and-hold returns (smple returns adjusted for dvdends over the relevant nterval) and ther contnuously compounded counterpart (the logarthm of these returns). The results shown for the buy-and-hold returns bascally confrm the results n Handa et al. (989). A samplng nterval effect s clearly present especally for the extreme portfolos. For portfolo (the smallest frms) the estmated betas ncrease as the samplng nterval ncreases and for portfolo (the largest frms) they decrease. As dscussed n Handa et al. (989), ths effect s expected from a smple accountng ssue. However, such an accountng ssue s not present wth contnuously compounded returns and, n ths case, no samplng nterval effect should be present f markets are effcent. However, the results n Table clearly show that the same samplng nterval effect remans wth log-returns. For many portfolos, t s even more pronounced. These results suggest that more than a smple accountng ssue s responsble for the explanaton of the samplng nterval effect on estmated betas. The rest of ths paper ntends to provde an analytcal framework to assess potental sources for ths emprcal fact. 3. The basc model We denote by P (t), the prce of a stock or a portfolo at date t, and by P (t) the prce of the market portfolo at the same date. We suppose that each prce has two multplcatve components. One, denoted by P a (t) represents the transtory component whle the other, denoted by P b (t), s the permanent component. Hence, we have: P ðþ¼p t a ðþp t b ðþ t ð ¼ ; Þ: The assumpton of a permanent and transtory component for stock prces s frequently made (see, e.g., Poterba and Summers (988), and Fama and French (988)). It s usually motvated by the fact that t allows negatve correlaton n returns over long horzons whch has been shown to be present emprcally. Also, we denote by lower cases, the logarthm of the respectve components,.e.: p j ðþ¼lnp t j ðþ t ð ¼ ; ; j ¼ a; bþ: The contnuous tme model descrbng the tme paths of each component s ntentonally kept smple to hghlght the features of nterest and s not ntended as a precse descrpton of the behavor of stock prces at all samplng ntervals. It s ntended to be a useful approxmaton for the samplng ntervals of nterest, namely monthly to yearly, for whch postve seral correlaton due to market mcrostructure effects do not hold but for whch negatve seral correlaton s a possblty n the presence of transtory components. Accordngly, the transtory P a (t) and permanent P b (t) components are governed by the followng stochastc dfferental equatons, defned over [,N], wth N the span of the data: dp a dp b ðþ¼ t γ p a ðþdt t þ σ a dw a ðþ; t ðþ¼α t P b ðþdt t þ σ b P b ðþdw t b ðþ; t ðþ for =,, wth W j (t)=(w j (t), W j (t)), (j=a, b) where W a (t) and W b (t) are ndependent standard Wener processes,.e., contnuous tme zero mean Gaussan processes defned on [,] wth covarance functon E[W j (t)w j (s)]=mn(t,s) for j=a, b.we make the followng assumptons: h EW j ðþwj t ðþ t ¼ ρ j ; ρ j ; j ¼ a; b; γ > ; α > ; σ a > ; σ b > ; ¼ ; : The stochastc dfferental equaton descrbng the dynamcs of the transtory component specfes that the logarthm of the transtory component of prces P a (t) s an Ornsten Uhlenbeck process. Accordngly, the long-term effect of a shock on the level of the transtory component s zero and constranng the parameters γ (=, ) to be postve ensures mean reverson. On the other hand, the dynamcs of the permanent component P b (t) s governed by a geometrc Brownan moton. The parameter α here reflects mean returns. The parameters (σ j ) (=, and j=a, b) represent the varances of the nose components W j (t) and are often called dffuson components. The parameter ρ j accounts for the correlaton between the nose of the temporary components (j=a) or permanent components (j=b) of the prce of the asset (or portfolo) and the prce of the market portfolo. Such specfcatons are often encountered n the fnance lterature. For example, a geometrc Brownan moton s often postulated for rsky stock prces whle an Ornsten Uhlenbeck s used for rskless assets (e.g., the short-term nterest rate on a safe asset); see, e.g., Merton (973) and Black and Scholes (973). The assumpton of the ndependence of the Wener processes W a (t) and W b (t), allows us to wrte the system () as two sub-systems; namely dp a ðþ¼ γ t p a ðþdt t þ σ a dw a ðþ; t ðþ

5 46 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 and dp b σ b B C ðþ¼ α Adt þ σ b dw b ðþ; t ð3þ for =,. The systems () and (3) have the followng solutons: p a ðþ¼p t a ðþexpð γ tþþσ a t exp ð γ ðt sþþdw a ðþ; s ð4þ and p b ðþ¼p t B ðþþ@ α σ b C At þ σ b W b ðþ; t ð5þ for =, (see, e.g. Theorems 8.. and of Arnold (974)). These solutons show that the logarthm of the transtory component s statonary whle the logarthm of the permanent component s an ntegrated component wth a lnear trend. For smplcty and wthout loss of generalty, we suppose, n what follows, that p j ()= (=, ; j=a, b). To compare our model wth that of Poterba and Summers (988) and Fama and French (988), we need to derve the dscrete tme representaton. To that effect, we defne the samplng nterval h such that Th=N wth T the number of observatons and N the span of the data. We have the followng dscrete tme solutons for Eqs. (4) and (5). Proposton. Let p a (t) and P b (t) (=, ) be defned by Eq. (), then the dscrete tme solutons for a samplng nterval h are gven by: and p a p b ðthþ ¼ expð γ hþp a ððt ÞhÞþu ðthþ; ð6þ ðthþ p b B ððt ÞhÞ α σ b C Ah þ v ðthþ; ð7þ for =, and t=,, T; wth and u ðthþ ¼ σ a th ðt Þh exp ð γ ðth sþþdw a ðþ; s v ðthþ ¼ σ b W b ðthþ W b ððt ÞhÞ : The proof follows mmedately from the solutons (4) and (5). The errors u (th) and v (th) have mean zero, are ndependent and are dentcally normally dstrbuted. The moments of order two satsfy (for =, ): h Eu ðthþ ¼ expð γ hþ σ a ; γ Eu ½ ðthþu ðthþš ¼ exp ð ð γ þ γ ÞhÞ ρ ðγ þ γ Þ a σ a σ a ; h h; Ev ðthþ ¼ σ b Ev ½ ðthþv ðthþš ¼ ρ b σ b σ b h: We now need to defne the returns over a horzon of h perods. Supposng that dvdends are zero (or that they are renvested), the nstantaneous returns are R(t)=dP(t)/P(t). Gven the dscrete tme solutons of prces, the dscrete tme soluton for returns over h perods s defned by: R ðthþ ¼ L h ðthþþp b ðthþ ; ð8þ p a where L h s the lag operator such that Lx t =x t h. A representaton for R (th) n terms of the errors v (th) and u (th) s gven by: B R ðthþ α σ b C Ah þ v ðthþþð expð γ hþlþ ð LÞu ðthþ: ð9þ

6 P. Perron et al. / Journal of Emprcal Fnance (3) Usng the notaton R th ¼ ðr ðthþ; R ðthþþ, we can wrte R th ¼ Ψh þ η th ; where Ψ ¼ ðψ ; Ψ Þ,, η th ¼ η ðthþ; η ðthþ Ψ =α (σ b ) / and η ðthþ ¼ v ðthþþð expð γ hþlþ ð LÞu ðthþ: We can use these specfcatons to derve the followng result pertanng to the dscrete tme representaton of returns. Proposton. In dscrete tme, the returns R (th) are characterzed by an ARMA(,) process wth frst-order covarance coeffcent gven by: covðr ðthþ; R ððt ÞhÞÞ ¼ ð expð γ hþ γ Þ ; ¼ ; : ðþ σ a A) (σ b /σ a ) = / B) (σ b /σ a ) = Autocorrelaton Autocorrelaton γ =. γ =.5 γ =. γ = Month Month C) (σ b /σ a ) = D) (σ b /σ a ) = Autocorrelaton Autocorrelaton Month Month Fg.. Autocorrelaton functons of stock returns wth varous parameters.

7 48 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 In partcular, for small γ or h we have the approxmaton: covðr ðthþ; R ððt ÞhÞÞ γ h σ a ; ¼ ; ; ðþ so that when ether γ or h approaches, the returns R (th) are..d.. For the autocorrelatons, we have ð expð γ hþþ σ a γ h σ corrðr ðthþ; R ððt ÞhÞÞ ¼ b þ expð γ hþ σ a ; ¼ ; : ðþ γ h σ b and, for small γ or h we have the followng approxmaton γ h σ a corrðr ðthþ; R ððt ÞhÞÞ σ b ; ¼ ; ; þ σ a σ b so that when ether γ or h approaches, the returns R (th) are..d. Ths proposton shows that stock returns have the negatve autocorrelaton that s a functon of structural parameters γ, h and (σ a /σ b ). Fg. presents the plots of the autocorrelaton functons over the monthly horzons h [,7]. Frst, the mean-reverson parameter γ determnes the shape of the functon over h. When γ s set to., the autocorrelaton s monotoncally decreasng n h over a 6 year perod. As γ ncreases to., t becomes a convex functon that exhbts ts largest negatve autocorrelaton over the 5 year nterval. These values of γ are lkely to descrbe the data well as Fama and French (988) document an emprcal evdence that stock returns for the market and decle portfolos have the negatve autocorrelaton wth the U-shaped pattern around 3 5 year perods. When γ s set to., the autocorrelaton reaches ts lowest level too early before the one-year horzon. The rato of the varances of transtory and permanent components, (σ a /σ b ), determnes the extent to whch stock returns are autocorrelated. The top two graphs n Fg. are drawn wth the rato beng half and one. For γ =.5 or., the autocorrelaton has values of about.7 ((σ b /σ a ) =/) or. ((σ b /σ a ) =) around a year horzon. These values are consstent wth the emprcal evdence documented by Fama and French (988). However, ther largest negatve autocorrelaton values are at most.8 and.3, respectvely. Fama and French (988) report that for the NYSE market portfolo returns, the autocorrelaton has the largest negatve value around. wth the sample perod from 94 to 985 or around.4 wth the perod from 96 to 985. Settng hgher values of the rato of the varances, we can obtan mnmum autocorrelaton values that are emprcally consstent. When the rato s set to, the mnmum autocorrelaton s.. However, the rato has to be set to n order to acheve.4 for the mnmum autocorrelaton that corresponds to the estmates obtaned n the sample from 96 to 985. Nevertheless, these fgures n general show that our model wth approprate parameter values satsfes the qualtatve propertes suggested by the emprcal fndngs n Fama and French (988). In summary, the proposton shows that our model satsfes the same qualtatve propertes as that of Poterba and Summers (988). In partcular, t mples negatve correlaton n returns that become stronger as the horzon h ncreases but that ths correlaton s neglgble for short horzons. Also, when the transtory component s null (σ a = or γ =) ths correlaton dsappears and the returns are..d.. In ths study, we wsh to consder the behavor of the estmator of the systematc rsk (the betas) when the samplng nterval s allowed to vary. To that effect, we shall adopt dfferent asymptotc frameworks whereby ether h decreases to keepng the span N fxed, or keepng h fxed and lettng the span N ncreases. 4. Estmates of beta: Asymptotc propertes and mplcatons We start by defnng the noton of the systematc rsk beta mpled by the model and ts lmt value as the samplng nterval ncreases or decreases. After a dscusson of the populaton value, we turn to the characterzaton of the estmates. 4.. Populaton values of betas Defnton. Let R th ¼ ðr ðthþ; R ðthþþ be defned by Eq. (9). For a samplng nterval h, the systematc rsk s defned by: β h ¼ cov ð R ðthþ; R ðthþþ : ð3þ varðr ðthþþ

8 P. Perron et al. / Journal of Emprcal Fnance (3) In partcular, f h, we use the notaton β =lm h β h and f h, we use β b ¼ lm h β h. We have the followng representaton of β h as a functon of the samplng nterval h and the parameters of the model. Proposton 3. Let R th ¼ ðr ðthþ; R ðthþþ be defned by Eq. (9) and β h by Eq. (3). We have: ð Þ expð γ hþ γ þγ Þh exp ð γ hþ γ h β h ¼ ρ bσ b σ b þ ρ a σ a σ a exp γ h ð a þ σ σ b : ð4þ If h, we have: β ¼ ρ bσ b σ b þ ρ a σ a σ a a ; ð5þ þ σ σ b.4 β b = β a β b < β a γ > γ γ = γ γ < γ β h.9 β h γ > γ.4.5 γ = γ γ < γ h. 5 5 h β b > β a.3. β h..9 γ > γ γ = γ γ < γ h Fg.. True values of β h as a functon of h.

9 5 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 and f h, β b ¼ ρ bσ b : ð6þ σ b The expresson (4) dffers from those of Levhar and Levy (977) and Hawawn (98) who present the rato of an asset's beta computed over h perods relatve to that over one perod as a functon of h and ntertemporal cross correlatons. The relaton (4) suggests that f the transtory component s not present n the assets' prces (σ a =σ a =), the true value of beta s ndependent of the samplng nterval and concdes wth β b defned as the lmt of β h when h ncreases. Hence, wthout the transtory component the samplng nterval does not affect the value of the beta. For that reason, we shall refer to β b as the permanent-beta. By analogy, we refer to β a ¼ ρ a σ a =σ a as the transtory-beta. It s the value that the beta would take n the absence of a permanent component when h s small. Usng ths notaton, we see that the true value of beta, β h, s a functon of β b, β a and h gven by: β h ¼ β b þ β a σ a =σ b þ σ a =σ b exp ð γ hþ expð γ hþ ðγ þγ Þh exp ð γ hþ γ h : For small values of the samplng nterval h, we have β ¼ σ b β þ σ a β b a σ b þ σ a ; ð7þ whch shows that beta s a lnear combnaton of the permanent and transtory betas. It s nterestng to remark that when permanent and transtory betas are equal ( β a ¼ β b ), we have β ¼ β b so that the betas are the same at short and large samplng ntervals. However, when γ γ, the betas need not be the same for all samplng nterval so that non-monotonctes are possble. Fg. presents the graph of β h as a functon of h for selected cases. It shows that f β a b β b (resp. β a > β b ) then β h s a strctly ncreasng (resp. decreasng) functon of h for any γ >. However, when β a ¼ β b, the true value β h s ndependent of h f γ =γ and s a non-monotonc functon of h f γ γ (the non-monotoncty s due to the fact that when the permanent betas are the same, dfferent mean-reverson parameters have a short-term effect on the betas and as the samplng nterval ncreases they go back to the same value). Note that f γ s very small, β h =β and there s no samplng nterval effect. 4.. Propertes of estmates of betas We now turn to the propertes of estmates of β h. When usng log returns, the regresson assocated wth the Captal Asset Prcng Model (CAPM) for any gven samplng nterval h s gven by wth R ðthþ ¼ α h þ β h R ðthþþeth ð Þ; ð8þ σ b B C σ b α h α Ah = B C β α Ah = ; and eth ð Þ ¼ η ðthþ β h η ðthþ: The ordnary least-squares estmate of β h s: ^β h ¼ T t¼ R ðthþ R R ðthþ R T t¼ R ðthþ R ; ð9þ where R ¼ T T t¼ R ðthþ for =,. The parameter values are descrbed n Table (see Secton 5. for more detals). For the case β a ¼ β b, t corresponds to portfolo P, when β a > β b to portfolo P and when β a b β b to portfolo P4. For each case, the followng values of γ and γ are used: γ =γ =., γ =.6 and γ =. (case γ >γ ) and γ =. and γ =.6 (case γ bγ ).

10 P. Perron et al. / Journal of Emprcal Fnance (3) Proposton 4. For any samplng nterval h, we have, as T : wth T = ^βh β h d Nð; V h Þ; ðþ V h lm Tvar ^β h β h T ¼ σ a β γ þ σ b h þ σ a exp ð γ h Þ σ b h σ a exp ð γ hþ exp ð γ hþ γ h γ þ σ b h : ðþ If h, we have: σ b a β þ σ σ b a þ σ V lm V h ¼ h þ σ a ; and f h, V lm V h ¼ β b h ρ b σ b =ρ b; whch corresponds to the asymptotc varance for any fxed h n the absence of a transtory component. The proof of ths result s qute standard and omtted. What t bascally says s that the estmated betas wll be close to the true betas as defned by β h. Hence, we can approxmate the behavor of the estmated betas by the behavor of the populaton values as h vares. Of nterest also, s the fact that when h s large, the varance of the estmated beta s drectly proportonal to the permanent beta. Ths last convergence result for ^β h n conjuncton wth Proposton 3 concernng the behavor of β h as a functon of h has the followng mplcatons: In the absence of a transtory component, there s no samplng nterval effect on the estmated betas, for large enough sample szes; If the transtory and permanent betas are equal there wll, to a frst approxmaton (.e., wth a large enough sample), also be no samplng nterval effect f the mean reverson parameter of the transtory component s the same for the asset and the market portfolo; Wth a transtory component n prces and a dfference between the permanent and transtory betas, the lmt of ^β h when h ncreases (whch corresponds to the permanent beta) can be less than or greater than β (the lmt as h goes to zero) whch s a lnear combnaton of the permanent and transtory beta. The sgn of the dfference wll depend on the sgn of the dfference between the permanent and transtory beta. We can explan the systematc bas E^β h β n terms of some relatons whch have a drect lnk wth the sze of a frm. To make explct these relatons, we frst defne some concepts. Defnton. We say that there s under-evaluaton of the beta when β b bβ. Conversely, we say that there s over-evaluaton when β b > β : These defntons only nvolve the lmtng values β and β b. Intutvely, the samplng nterval h can be nterpreted for an nvestor as the horzon of the nvestments' proftablty (Levhar and Levy (977)). Under the hypothess that the nvestors often choose a short horzon for such purposes, the beta correspondng to the true horzon would be β. In general, β s a lnear combnaton of the permanent and transtory betas. If these are equal, then the beta at a short horzon (β ) s the same as the beta at a long horzon ( β b ) and ncreases n the samplng nterval nvolve no bases. Consder now the case where the transtory beta s smaller than the permanent one ( β a b β b ). Ths mples that the systematc rsk s larger for the long term than over the short term. Hence, followng Banz (98), we may nterpret ths case as applyng to small frms. Thus, for small frms, the lmt of ^β h ncreases f the samplng nterval ncreases and there s over-evaluaton of the betas (ths follows snce, f β b > β a, we have, takng the lmt as h, β b > β h > β > β a Þ. The case wth the permanent beta smaller than the transtory beta s one where short-term consderatons account for more of the long-term rsk. We may thus expect ths case to apply to less rsky or larger frms. Thus for large frms, we have the opposte relaton, namely ^β h decreases wth an ncrease n the samplng nterval and there s under-evaluaton of the betas. Our framework, ndeed, helps to provde an alternatve nterpretaton of the relaton between estmated betas or systematc rsk and the sze of the frms. For short horzons, we should expect the co-movement of returns to be roughly smlar for small and large frms. At longer horzons, thngs are qute dfferent for the two groups when transtory components are present. Wth a

11 5 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 longer horzon, we should expect small frms to be more rsky than the market snce they have more chances of undergong bg changes (ether bankruptcy or a large growth). Ths translates nto our framework n sayng that ther permanent (or long-horzon) beta s larger than ther transtory (or short-term) beta. For large or well-establshed frms, we can expect that f some short-term movement n return occurs, t s more lkely to be smoothed out n the future (for example a negatve shock s less lkely to send a large frm nto bankruptcy and a postve one to make ther value double n a year). Ths translates n our framework n sayng that ther permanent beta s less than ther transtory beta. 5. Smulaton experments In ths secton, we verfy f the theoretcal results obtaned provde an adequate descrpton of the fnte sample propertes of the estmates of the betas and f these are robust to varous changes n the parameters. As nterestng cases for the smulatons, we consder the three cases depcted n Fg., namely: Case : β b ¼ β a. If the permanent and transtory betas are dentcal, then the lmt of ^βh s ndependent of h when the coeffcents γ (whch control the degree of mean-reverson) are equal (γ =γ ). However, f γ γ, the lmt of ^βh s a non-monotonc functon of h. Case : β b b β a. If the permanent beta s less than the transtory beta, the lmt of ^β h decreases as the samplng nterval ncreases and there s under-evaluaton of the betas. Case 3: β b > β a. If the permanent beta s greater than the transtory beta, ^β h ncreases wth an ncrease n the samplng nterval and there s over-evaluaton of the betas. 5.. Calbraton of the model To calbrate the model, we frst start by normalzng β to. Ths leads us to retan values of β a ¼ ρ a σ a =σ a and β b ¼ ρ b σ b =σ b whch satsfy for case the nequalty β b bb β a, and for case 3, the nequalty β a bb β b. For case, we have the equalty β b ¼ β a ¼ when γ =γ. We select the values of ρ a, σ a, σ a, ρ b, σ b, and σ b to have fve base cases, see Table. The frst, P, specfes that the asset or portfolo has a permanent and a transtory beta whch are equal β b ¼ β a ¼. For the second portfolo, P, the permanent beta s much less than the transtory beta ( β b ¼ :5 and β a ¼ :35). For the thrd portfolo, the dfference between the transtory and permanent betas s reduced ( β b ¼ :9 and β a ¼ :35). For portfolos 4 and 5, the specfcatons are the same as for portfolos and 3 except that we nterchange the values for β a and β b. The values retaned for the coeffcents γ are.,.6 and.. The value. s consdered to llustrate the effect of a weak reverson to the mean for the transtory component. Here, we can no longer really consder that component as transtory snce t s nearly ntegrated, and we would expect to have results correspondng to the no-transtory component case. The other values are such that they mply autoregressve coeffcents of.98 and.95 selected by Poterba and Summers (988) wth monthly data (. ln(.98) and.6 ln(.95)). Gven the absence of any emprcal results gvng nformaton on the relatve magntude of γ (the mean-reverson coeffcent for a stock or portfolo) and γ (the mean-reverson coeffcent for the market portfolo), we set γ =γ n the base case. However, gven that returns are ARMA(,) statonary processes for any fxed h, ts lkely that the effect of a shock on the transtory component of prces becomes neglgbly faster than for the market portfolo for some types of assets and slower for others. Hence, we also assess the extent to whch the results are senstve to settng γ /γ b or γ /γ >. The samplng nterval h consdered are h=,,, 4, 8,, 4, and and we set T= (other values of T gave smlar qualtatve results). For a gven samplng nterval h, we smulate T ndependent realzatons of the processes u (th)=(u (th), u (th)) and v (th)=(v (th), v (th)) from a multvarate N(,Ω) dstrbuton where Ω s the varance covarance matrx of the process u (th) orv (th) (see Proposton ). We then construct the processes η (th) (=, ) and deduce from them the returns R (th) and R (th) and estmate β h from Eq. (9). We repeat ths procedure 3 tmes to obtan the mean of the estmator. Table Selected parameter values. Portfolo P P P3 P4 P5 ρ a a σ a σ ρ b b σ b σ β a β b

12 P. Perron et al. / Journal of Emprcal Fnance (3) Table 3 Mean of the estmated beta. h γ =γ =.. P P P P P γ =γ =.6. P P P P P γ =γ =.. P P P P P The fnte sample mean of β h n the base case The results are presented n Table 3 for the cases γ =γ =., γ =γ =.6 and γ =γ =.. In general, the results support the theoretcal fndngs of Secton 4. If the transtory and permanent betas are equal (P), there s ndeed no samplng nterval effect for any value of the mean-reverson coeffcents γ and γ. However, when the transtory beta s greater than the permanent beta, β a > β b, the mean of the estmated beta decreases as h ncreases whle the opposte holds when β a b β b. Ths rate of decrease (when β a > β b ) or ncrease (when β a b β b ) s faster the larger s the dfference between β a and β b. The dfferences are also more mportant when the mean-reverson coeffcent ncreases (.e., from. to.6). When the mean-reverson coeffcents are set to., we see that the bas practcally dsappears. Ths s to be expected, snce wth such small value there s no longer a temporary component snce t s almost ntegrated Senstvty analyses To study the senstvty of the results to changes n varous parameters, we consder, as a bass for reference, the case where γ =γ =. and the dfference between β a and β b s large, for example β a > β b wth β a ¼ :35 and β b ¼ :5 (Table wth P). We performed smulatons usng dfferent cases as a bass for reference and the conclusons are smlar. The senstvty of the results s analyzed n three drectons n relaton to the sub-groups of parameters (γ, γ ), (ρ a, σ a, σ a ) and (ρ b, σ b, σ b ). The strategy s to vary the parameters of one group whle keepng the others constant. We frst consder varatons n the parameters (γ, γ ) and n partcular on the effect of specfyng γ >γ or γ bγ. Secondly, we analyze the effect of changng the parameters of the varance covarance matrx of the transtory component keepng constant β a ¼ ρ a σ a =σ a. Fnally, we examne the effect of changes n the parameters (ρ b, σ b, σ b ), related to the permanent component, keepng constant β b ¼ ρ b σ b =σ b. Consder frst the effect of changes n the parameter γ. Table 4 presents the mean of the estmated betas as a functon of h for dfferent values of γ (resp. γ ) when γ (resp. γ ) s fxed, the reference curve correspondng to the case γ =γ =.. We observe that the monotoncally decreasng behavor of the estmated betas s not affected by alternatve choces of the mean-reverson parameters. Consder now the effect of changes n the parameters (ρ a,σ a,σ a ) and (ρ b,σ b,σ b ). We analyze jontly the effect of changes n the parameters ρ a, σ a and σ a (resp. ρ b, σ b and σ b ) keepng β a ¼ ρ a σ a =σ a (resp β b ¼ ρ b σ b =σ b ) and the rato σ a /σ b fxed (the Table 4 Effect of the mean-reverson coeffcents on the mean of the estmated beta. h γ =. γ = γ =.6 γ = γ =3 γ = γ =. γ = γ =. γ = γ =. γ =

13 54 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 Table 5 Effect of the coeffcent of the varance covarance matrx on the mean of the estmated beta. h Base case: γ =γ =., σ a /σ b =.558, (ρ b,σ b,σ b )=(.,.5,.), and σ a =.558. ρ a =. σ a = ρ a =.4 σ a = Base case: γ =γ =., σ a /σ b =.558, (ρ a,σ a,σ a )=(.7,3.5,.558), and σ b =.. ρ b =.4 σ b = ρ b =.7 σ b = reference values are those n Table for the case P). From the results, presented n Table 5, we see that the samplng nterval effect remans. Concernng the parameters ρ b, σ b and σ b, the results (compared to the reference case) show that varatons n these parameters do not sgnfcantly affect the mean of the estmated betas. However, f σ b s very large relatve to σ b, there can be large dspersons n small samples when h s small. 6. Emprcal evdence We have shown that the presence of transtory components n stock returns could generate the emprcal patterns n the estmates of the betas computed usng dfferent samplng ntervals. In order to make our argument more convncng, t s useful to see f there s enough evdence of such transtory components n the data for the varous portfolos consdered. Ths s a delcate ssue. Our model s (by desgn) smple to keep the man features of nterest and abstract from many others that can affect the stochastc propertes of the returns. Ths s especally the case for short horzons for whch t s well documented that returns are postvely correlated for small frms due to features such as thn tradng. Hence, we certanly do not and cannot clam that our model s a full and adequate descrpton of the data. But we can stll try to assess whether mean reverson s ndeed present by lookng at medum to long-horzons. 6.. ARMA(,) estmates and mpled mean reverson In our model, stock prces have transtory components and the errors of the permanent and transtory components are..d., hence the demeaned returns r (th) follow ARMA(,) processes. More specfcally, demeaned stock returns for portfolo are Table 6 MLE estmates and lkelhood rato test results for the ARMA(,) model. ( ϕ L)R (th)=( θ L)ε (th) Portfolo Horzons of 6 months Horzons of months ^ϕ ^θ LR test ^ϕ ^θ LR test MV [.4] [.] MV [.5] [.5] MV [.8] [.] MV [.] [.] MV [.7] [.5] MV [.7] [.5] MV [.7] [.9] MV [.9] [.4] MV [.3] [.4] MV [.7] [.] MV [.] [.9] MV [.7] [.] MV [.6] [.7] MV [.8] [.6] MV [.7] [.3] MV [.4] [.4] MV [.5] [.8] MV [.3].7.83.[.57] MV [.] [.3] MV [.5].6..98[.6] EW [.5] [.9] Note: Entres n bold ndcate sgnfcance at least at the % level. The p-values are gven n parenthess.

14 P. Perron et al. / Journal of Emprcal Fnance (3) generated as, ð ϕ L Þr ðthþ ¼ ð θ LÞε ðthþ; where ϕ exp( γ h) and the MA parameter gven by θ ¼ þ ϕ þ ϕ σ b γ h σ a " Þ # ϕ σ a þ ð þ ϕ γ h σ b Þ σ b σ a þ ϕ σ b : ðþ ð ϕ ϕ γ h Also, ε (th)s..d wth varance σ ε =(/θ )((/γ)ϕ ( ϕ )(σ a ) +(σ b ) ). In ths secton, we ft ths ARMA(,) model to the demeaned returns of the portfolos measured usng samplng ntervals of 6 and months and test the null hypothess of..d. returns versus ths ARMA(,) usng a lkelhood rato test. The exact specfcatons of the estmaton procedure and the test are presented n the computatonal appendx. Table 6 presents the results. The ARMA(,) parameter estmates and the test statstcs are n bold f they are sgnfcant at least at the % level. Note frst that n all cases (except MV at the month nterval), the estmate of the MA parameter s larger than that of the AR parameter mplyng negatve seral correlaton n returns. We can ndeed fnd sgnfcant evdence of mean-revertng behavor n many of the MV portfolos. For the sem-annual returns, we are able to reject the null at the % sgnfcance level for 6 portfolos, ncludng the market portfolo. The test statstcs are sgnfcant at the 5% level for the frst two smallest portfolos. Wth annual returns, we can reject the null at the % level for 9 portfolos, ncludng the market portfolo. Note that we cannot fnd any sgnfcant evdence of mean-reverson for the fve largest MV portfolo returns, whch mght be due to the presence of a small mean-reverson parameter. The mean-reverson parameter, γ, can easly be nferred from our MLE estmates of the autoregressve parameters. For the sem-annual returns, the largest (sgnfcant) autoregressve parameter estmate s.9 and the smallest s.83. Snce γ = ln(ϕ )/h, we can nfer that the mean-reverson parameters range between. and.37. Wth annual returns, they range between.7 and.3. These values of the mean-reverson parameters mply a monthly autoregressve coeffcent rangng between.97 and.98. The followng dsplay compares the averages of the mean-reverson parameter estmates among four groups of dfferent MV portfolos. Averages of mean-reverson parameters Portfolos Intervals 6 MV MV5.3.3 MV6 MV.8. MV MV5.3.3 MV6 MV.7.7 The frst row reports the average γ for the fve smallest portfolo returns and the last row reports the average γ for the fve largest portfolo returns. The results show that the mean-reverson parameters are ndeed smaller for the largest MV portfolos but the pont estmates reman economcally mportant. 6.. Long-horzon regressons In ths secton, we estmate long-horzon regressons to see f there s evdence of transtory components n stock prces. The long-horzon regressons consdered take the form R ðthþ ¼ a h þ ρðhþr ððt ÞhÞþε ðthþ; ð3þ where we consder values of h rangng from year to years. Tables 7a and 7b report the OLS estmates of ρ(h) n panel (a) and the correspondng t-statstc n panel (b). As s well-known, n such regressons the observatons are overlappng and the dsturbances ε (th) are serally correlated. Therefore, we adjust the standard errors n the t-statstc usng a standard heteroskedastcty and autocorrelaton consstent (HAC) estmate of the varance based on the Bartlett kernel wth the bandwdth selected usng the data-dependent procedure recommended by Andrews (99) and Andrews and Monahan (99). Note: In calculatng the average for MV6 MV for a month nterval, we do not nclude MV snce the returns are estmated as a low-order AR() process.

15 56 P. Perron et al. / Journal of Emprcal Fnance (3) 4 6 Table 7a Results for the long-horzon regressons; market-value portfolo returns MV MV. Portfolo Return horzon year years 3 years 4 years 5 years 6 years 7 years 8 year 9 years years GMM WAC a. OLS estmate of ρ(h) MV. [.38] MV. [.5] MV3. [.55] MV4.4 [.6] MV5.5 [.33] MV6. [.5] MV7.5 [.3] MV8. [.56] MV9.3 [.4] MV.3 [.4].5 [.8].3 [.].7 [.37].8 [.34].5 [.].8 [.36].7 [.6].7 [.4].5 [.].4 [.].7 [.7].35 [.].8 [.6].6 [.8].5 [.9].3 [.].3 [.4]. [.6].6 [.9].5 [.9].46 [.].57 [.].5 [.].46 [.].4 [.].39 [.].45 [.].33 [.6].4 [.].35 [.4].5 [.].63 [.].59 [.].5 [.].45 [.].43 [.3].49 [.].37 [.7].5 [.].36 [.7].36 [.].5 [.].47 [.3].37 [.].35 [.].3 [.5].39 [.8].7 [.].43 [.5].7 [.]. [.3].4 [.].38 [.3].3 [.3].7 [.6].5 [.9].3 [.].6 [.7].36 [.4]. [.34].5 [.5].43 [.3].43 [.].37 [.8].34 [.].3 [.3].34 [.].36 [.8].4 [.3].8 [.8].8 [.64].37 [.4].46 [.].39 [.8].37 [.].36 [.].33 [.6].39 [.7].46 [.].33 [.4].4 [.7].33 [.33].44 [.7].34 [.3].34 [.3].34 [.9].9 [.37].35 [.7].46 [.5].34 [.8] a.3 b.9 a a a 9.86 b 6.55 c 7.8 c b 8.46 a 6.36 c c.88 b. t-statstcs of ρ(h) MV MV MV MV MV MV MV MV MV MV Note: Bootstrapped p-values are gven n brackets; bold entres ndcate sgnfcance at the 5% level. a,b, and c denote rejecton at the %, 5% and %, respectvely. We also provde the bootstrapped p-values for testng whether the ndvdual estmates ^ρ ðhþ are sgnfcantly less than zero. Ths s useful snce nference based on the standard asymptotc dstrbuton theory may not provde a good approxmaton n fnte samples. As noted by Rchardson and Stock (989), the conventonal large-sample approxmatons may fal to perform well gven the small number of effectve non-overlappng observatons n the long-horzon regressons. Second, there s a fnte-sample bas n the OLS estmates of ^ρ ðhþ. It can be shown that the emprcal dstrbuton of ^ρ ðhþ under the null hypothess that ρ(h) = tends to be downward-based and more so as the return horzon ncreases (see Kendall (954), Marrott and Pope (954), Danel ()). To deal wth these ssues, we follow the lterature such as Goetzmann and Joron (993), Kothar and Shanken (997) and Kllan (999) and construct p-values of the estmates usng a bootstrap method. We frst use the statonary bootstrap method of Polts and Romano (994) to randomly draw (wth replacement) a new sample of monthly returns {R (th); t=,, T}, where R (th) are drawn n blocks whose startng ndces and lengths are determned randomly to preserve the tme-seres dependence n returns. The block length s drawn from a geometrc dstrbuton wth a parameter q set to. and the number of bootstrap replcatons s 5. The parameter q determnes the average block length as b=/q. The results are smlar when we set b=5. The dstrbuton of ð^ρ ðhþ ρðhþþ can then be approxmated by the emprcal dstrbuton of ^ρ ðhþ ^ρ ðhþ, where ^ρ ðhþ s calculated from regressng each bootstrap sample {R (th); t=,, T} n the long-horzon regresson. By mposng the null hypothess ρ(h)=, we can test H :ρ(h)= aganst H A :ρ(h)b and compute the p-value as the proporton of draws of ^ρ ðhþ ^ρ ðhþ that are less than ^ρ ðhþ. In addton, we studentze the test statstcs by dvdng ^ρ ðhþ ^ρ ðhþ by the standard devaton of ^ρ ðhþ as advocated by Romano and Wolf (5) to mprove both sze and power. Fnally, we evaluate the statstcal sgnfcance of mean reverson for the MV portfolo returns by jontly testng whether the long-horzon regresson coeffcents are equal to zero. As ponted out by Rchardson (993), t s better to evaluate the statstcal sgnfcance of mean reverson usng a jont test. We consder the χ jont test based on the GMM framework of Rchardson and Smth (99) and the χ jont test of a set of weghted autocorrelaton test statstcs of Danel (). The detals are lad out n the computatonal appendx. The results are presented n Tables 7a and 7b. When assessng the statstcal sgnfcance of a sngle estmate, most of the HAC t-statstcs are sgnfcant for regressons constructed wth 3 5 year return horzons. For the small to md-sze MV portfolo returns, the t-statstcs are sgnfcant at the 4 5 years return horzons, whle the statstcs are sgnfcant at the 3 4 year return