SPURIOUS REGRESSION, SPURIOUS CORRELATION, AND DIVIDEND YIELD RETURN PREDICTABILITY

Transcription

1 JOHN G. POWELL JING SHI* TOM SMITH ROBERT E. WHALEY SPURIOUS REGRESSION, SPURIOUS CORRELATION, AND DIVIDEND YIELD RETURN PREDICTABILITY Absrac Dividend yield reurn predicabiliy has been nominaed as one of he new facs in finance, hus jusifying widespread use of he dividend yield as a sandard conrol variable in financial ime series analysis. This paper provides srong evidence ha reurn predicabiliy of he dividend yield is a spurious resul, even when he dividend yield definiion is broadened o include share repurchases. The paper demonsraes how a spurious regression problem ha is due o dividend persisence is compounded by a spurious correlaion problem when he dependen and independen variables in reurn predicabiliy regressions are raios composed of common componen variables, and i uilizes an analyical formula as well as a simulaion procedure o ake accoun of hese problems. The paper s resuls indicae ha ha he spurious correlaion problem (in ineracion wih spurious regression) is he major conribuing facor o incorrec inference in dividend yield reurn predicabiliy sudies, hus implying ha exreme care should be aken when using raios as predicor or explanaory variables in ime series regression. The paper finds ha sandard dividend behaviour explanaory models are also affeced by he spurious regression problem, and inroduces a reformulaed Linner firs difference dividend behaviour model ha is no subjec o spurious regression. This version: Ocober, 005 Please do no quoe, copy or cie wihou permission of he auhors. *Corresponding auhor. School of Finance and Applied Saisics, The Ausralian Naional Universiy, Canberra ACT 000, Ausralia Telephone: , Fax: , Jing.Shi@anu.edu.au. Powell is from Deparmen of Finance Banking and Propery, Massey Universiy, Palmerson Norh, New Zealand; Smih and Shi are from he School of Finance and Applied Saisics, The Ausralian Naional Universiy, Canberra, Ausralia; Whaley is from Fuqua School of Business, Duke Universiy, USA. We have benefied from and are graeful for commens by seminar paricipans a Massey Universiy, Jiangxi Universiy of Finance and Economics, Guangxi Universiy, he 3 h Annual Conference on Pacific Basin Finance Economics and Accouning, and he 005 Accouning and Finance Associaion of Ausralia and New Zealand.

2 SPURIOUS REGRESSION, SPURIOUS CORRELATION, AND DIVIDEND YIELD RETURN PREDICTABILITY Evidence of dividend yield reurn predicabiliy has been presened so widely and consisenly ha reurn predicabiliy is someimes considered o be one of he new facs in finance (Cochrane, 999), hus jusifying widespread use of he dividend yield as a sandard business cycle conrol variable in financial ime series analysis. This paper provides srong evidence ha reurn predicabiliy of he dividend yield is a spurious resul, regardless of wheher he definiion of he dividend yield variable used o predic reurns is updaed by broadening i o include cash disribued o invesors via share repurchases. The paper demonsraes how a spurious regression problem ha is due o regression variable persisence is compounded by a spurious correlaion problem because he dependen and independen variables in reurn predicabiliy regression equaions are raios consruced from common componen variables. Sandard dividend behaviour explanaory models are also shown o be srongly affeced by spurious regression ha is compounded by he regression model dependen and independen variables being raios wih common componen variables. A simulaion procedure is uilized o ake accoun of he spurious regression and spurious correlaion problems, and he paper derives a reformulaed dividend explanaory model o indicae how his economeric problem can be avoided. A firs hin ha reurn predicabiliy of he dividend yield is spurious can be obained by examining how he dividend yield predics reurns relaive o oher persisen variables. A sandard approach o es for dividend yield reurn predicabiliy is o regress he ime + sock index rae of reurn ( r + ) agains he ime dividend yield ( using he regression equaion D / P )

3 where r P + D P D = β + β + ε P P, () D is he level of real annual dividends during he welve monhs preceding ime and P is he real sock index level a ime. The adjused R saisic (hereafer, R ) for regression equaion () for he annual CRSP Value-weighed NYSE Index during he ime period 97 o 004 is close o %, for insance, and he resuls are usually inerpreed in relaion o a correlaion beween he curren level of dividends and subsequen reurns (see, e.g., Campbell, Lo and MacKinlay, 997). I can be noed, however, ha if variaion in dividends plays an imporan role in he reurn predicabiliy relaionship hen subsiuing a consan level of dividends (c) for he dividend erm ( D ) in he numeraor of he dividend yield variable ( D / P ) in regression equaion () should reduce or eliminae reurn predicabiliy in he following alered regression equaion r P + D P c = β + β + ε P P, () where c he is he uncondiional average of he dividend level during he sample period. Surprizingly, he R for equaion () acually rises o above 3% for he value-weighed NYSE Index. This unexpeced resul is a firs indicaion ha a spurious effec migh be occurring and is a very srong hin of a spurious regression problem. To explore his possibiliy, he persisence properies of he independen variables in regression equaions () and () are examined. They are found o be highly persisen. This persisence combines wih reurn auocorrelaion o creae spurious reurn Lewellen (004) poins ou ha dividend yield reurn predicabiliy could also be due o emporary mispricing, an inerpreaion ha will be discussed in Secion 3. 3

4 predicabiliy because he regression error erm inheris auocorrelaion from he dependen variable, hus causing sandard error esimaes o be downward biased and a significan overall relaionship o appear when none acually exiss (see Ferson, Sarkissian and Simin, 003a; Foser, Smih and Whaley, 997). Simulaions indicae ha he spurious effec of regressing reurns agains a persisen dividend yield variable is srongly reinforced by a spurious correlaion ha is due o reurns on he lef hand side of equaion () and he dividend yield on he righ hand side being raios ha are consruced from he same underlying variables (he share index level and he dividend level; see also Pearson, 897; Kronmal, 993; Kim, 999). The simulaion procedure is adaped o accoun for he ineracion beween he spurious correlaion and spurious regression problems by simulaing uncorrelaed share index ( P ) and dividend ( D ) series ha have he same auo-correlaion properies as he acual daa. The simulaed share index and dividend series are hen used o consruc reurn series ( r+ ( P + + D + P)/ P)) and dividend yield series ( D / P ) ha are poenially spuriously correlaed (raher han simulaing reurns and dividend yields direcly), and he consruced reurn series are regressed agains he consruced dividend series o provide a simulaed cu-off R benchmark. The influence of spurious correlaion on dividend yield reurn predicabiliy regression resuls is also demonsraed using an analyical formula derived in Kim (999) ha provides a spurious correlaion benchmark when correlaing raios wih a common divisor. The paper reveals ha spurious correlaion due o common regression variable componens herefore combines wih he Campbell, Lo and MacKinlay (997) demonsrae ha sock index reurns are auocorrelaed, for insance, and Ferson, Sarkissian and Simin (003a) poin ou ha even if reurns are no highly persisen, underlying expeced reurns are persisen, so here is a risk of spurious regression when esing reurn predicabiliy. 4

5 persisence properies of he dividend yield o accoun for he apparen dividend yield reurn predicabiliy, rahe r han (as widely argued) any propery of he dividend level ha is relaed o risk, reurn or mispricing. Evidence ha dividend yield reurn predicabiliy is spurious is also shown o be jus as srong even when he definiion of dividends is broadened o include disribuions o invesors via share repurchases, a componen of shareholder cash disribuions ha has become increasingly imporan in recen decades (see also Boudoukh e al, 004; Grullon and Michaely, 00). Given he srong persisence properies of dividends, i is no surprising o find ha sandard dividend behaviour explanaory models such as Linner (956) are also affeced by he spurious regression problem, since he dependen and independen variables in hese models are all very persisen (see also Ferson, Sarkissian and Simin, 003a). To correc his problem, he paper demonsraes how he Linner behavioural model of dividends can be reformulaed and exended using Marsh and Meron (987) so ha he model is no subjec o spurious regression. The srucure of he paper is as follows. Secion I conains a review of he dividend yield reurn predicabiliy and dividend behaviour lieraure. The spurious regression lieraure is hen reviewed o indicae he poenial problems ha migh be presen wih he reurn predicabiliy and dividend behaviour lieraure. Secion II shows ha somehing funny is going on in he reurn predicabiliy lieraure by revealing ha oher highly persisen explanaory variables can be used o improve upon sandard dividend yield reurn predicabiliy resuls, and hen uses simulaion procedures o show ha all he resuls are spurious. The conribuion of spurious correlaion o he dividend yield reurn predicabiliy spurious regression problem is also oulined. Secion III 5

6 demonsraes how sandard dividend behaviour regression equaions are also subjec o spurious regression. Secion IV correcs his problem by reformulaing he Linner (956) dividend model. Secion V provides a brief conclusion o he paper. I. LITERATURE REVIEW A. Dividend Yield Reurn Predicabiliy The lieraure on sock reurn predicabiliy is exensive, wih he dividend yield arguably being he bes-known of he many variables ha are found o have forecasing power for sock reurns. The dividend yield is usually measured as he raio of he previous year s dividend paymens o he curren share price index level. This process removes he influence of srong seasonal effecs in dividend paymens, bu i also arificially raises auocorrelaion in monhly and quarerly overlapping dividend yield series. Many sudies find ha dividend yields predic he cross-secional and imes series variaion of sock reurns (see, e.g., Fama and French, 988; Harvey, 989; Ferson and Harvey, 99; Whielaw, 994; Pesaran and Timmermann, 995; Poniff and Schall, 998; Bossaers and Hillion, 999; Cremers, 00; Boudoukh e al, 004; Lewellen, 004; and Touros, Valkanov, and Yan, 004). 3 Table I summarizes a number of he imporan reurn predicabiliy sudies ha focus exclusively or almos exclusively on he dividend yield. Dividend yield reurn predicabiliy had been documened so frequenly ha i was recenly nominaed as one of he new facs in finance (Cochrane, 999), 3 Goyal and Wech (003, p. 639) sae This empirical regulariy - ha dividend raios seem o predic equiy reurn ranks amongs he mos imporan findings of academic finance, and i shows no signs of subsiding. For insance, a ciaion search liss more han 00 published aricles ciing he Fama and French (988) aricle alone. 6

7 wih he dividend yield now ending o be included whenever business cycle conrol variables are employed in financial ime series analysis. 4 [Table I abou here] A number of recen sudies have also responded o challenges o findings of dividend yield reurn predicabiliy. Lewellen (004) finds evidence in favour of reurn predicabiliy using he naural logarihm of he dividend yield once a small sample bias correcion is improved o more precisely accoun for a coefficien esimae bias ha is induced by a srong correlaion beween he dividend yield slope coefficien and he dividend yield s auocorrelaion. Campbell and Yogo (005) find ha he Lewellen (004) bias-correced es has poor power relaive o Bonferroni inequaliy probabiliy confidence inerval ess when persisence does no equal uniy, bu never-he-less find evidence in suppor of dividend yield reurn predicabiliy using a Uniformly Mos Powerful (UMP) es. The Campbell and Yogo (003) UMP es subracs from he dividend yield dependen variable he innovaions in he yield ha are correlaed wih reurns in order o reduce he independen variable s noise and consequenly o increase he power of he reurn predicabiliy regression coefficien significance es. Boudoukh e al (004) demonsrae ha a broadly defined dividend yield variable ha includes share repurchases sill predics reurns even in recen decades when share repurchases have become imporan, whereas he significance of he radiional dividend yield for reurn predicabiliy has declined markedly in recen decades (see also Roberson and Wrigh, 003). 4 A search of he Journal of Finance and he Journal of Financial Economics indicaes ha mos recen financial ime series sudies ha employ business cycle conrol variables include he dividend yield as one of he conrol variables. 7

8 Only a few sudies indicae ha here is no a srong saisical relaionship beween dividend yields and sock reurns, including Black and Scholes (974) and Goezmann and Jorion (993). Bossaers and Hillion (999) use a number of saisical model selecion crieria o examine he predicabiliy of sock reurns using dividend yields and find in-sample predicabiliy bu no ou-of-sample forecasing power. Goyal and Welch (003) find he predicive power of he dividend yield is presen in pre- bu no pos-990 daa (see also Leau and Ludvigson, 005). Valkanov (003) ess he dividend yield s abiliy o forecas reurns based upon resricions implied by he loglinearized dynamic Gordon growh model, and finds no evidence of reurn predicabiliy a o 6 year reurn horizons. Sambaugh (999) finds ha reurn predicabiliy disappears when he bias induced by a correlaion beween he regression error and innovaions in he auoccorelaed dividend yield regressor is accouned for using Bayesian auocorellaion priors. Nelson and Kim (993) use simulaions o show ha his bias has an imporan influence on saisical inference in samples ha include pre- World War II reurns. Ferson, Sarkissian and Simin (003a) indicae ha he dividend yield s predicive power for monhly reurns is quesionable when accoun is aken of he spurious regression problem combined wih daa mining. Dividend yields have also been used o predic long horizon reurns (e.g. Fama and French, 988; Campbell, Lo and MacKinlay, 997). Fama and French (988) esimae regressions of reurns on he lagged dividend yield using pos-war NYSE index daa for reurn horizons from one monh o four years. They observe ha he dividend yield explains a significan proporion of muliple year reurns, and he explanaory power of he dividend yield increases wih he reurn horizon. Hodrick (99) uses hree 8

9 alernaive mehods of conducing inference and measuremen for long-horizon forecasing and finds ha changes in dividend yields forecas significanly persisen changes in expeced sock reurns. Theoreical explanaions for he predicive power of he dividend yie ld have been developed using he dividend discoun model (see, e.g., Campbell and Shiller, 988a and 988b; Fama and French, 988; Donaldson and Kamsra, 996; and Campbell, Lo and MacKinlay, 997; Lamon, 998; Bansal and Yaron, 004). Campbell and Shiller (988a, 988b) and Fama and French (988) argue ha he dividend discoun model implies ha a high curren dividend level relaive o he share price index level predics some combinaion of eiher higher expeced fuure reurns or lower fuure dividends. A high curren dividend yield herefore forecass higher fuure reurns if expeced fuure dividends are held consan (e.g., under he perhaps srong assumpion of all else being equal ), hus providing a heoreical basis for dividend yield reurn predicabiliy. Alhough he dividend yield reurn predicabiliy lieraure is vas and heoreical explanaions for predicabiliy are also well-known, his lieraure has no been direcly conneced o he equally well-known lieraure on behavioural explanaions of dividends (see, e.g., Linner, 956; Marsh and Meron, 987). A review of he dividend behaviour lieraure provides insighs as o why dividend yield reurn predicabiliy regression models are likely o be srongly affeced by dividend persisence. B. Dividend Behaviour The saring poin for a discussion of behavioural models of dividends is he Linner (956) speed of adjusmen model (see, eg., Marsh and Meron, 987). Linner 9

10 (956) argues ha corporae managers feel ha hey have a duy o pay ou a proporion of earnings o shareholders, bu are relucan o increase dividends oo quickly in reacion o an increase in earnings in case he earnings increase urns ou o be emporary and he dividend increase subsequenly has o be reversed. The Linner (956) model of dividends herefore implies ha he curren change in dividends is equal o a arge dividend payou minus las period s dividend ( D ) imes a speed of adjusmen facor, plus a consan. The arge payou is equal o he curren level of railing annual earnings ( E ) imes a long run payou raio arge. The curren dividend level ( D ) does no adjus insananeously o he arge payou level, hus avoiding he siuaion where an increase in earnings is only emporary and would have o be reversed in fuure years. Linner s dividend model (equaion () in Linner, 956) herefore saes ha he ime dividend is given by D = θ + θ E + θ D + ε, (3) 0 where is in years, ε is he error erm, θ/( θ) is he long-run arge payou raio, and ( θ ) is a speed-of adjusmen facor whose value will be closer o zero he more slowly ha firms adjus heir dividends o heir long-run arge level. Linner (956) repors an R in excess of 90% when esing his model (regression equaion (3)) using aggregae dividends and earnings daa. Grullon and Michaely (00) use he Linner (956) model o generae dividend forecas errors for companies in order o es wheher share repurchases are subsiues for dividends, hus updaing he Linner (956) resuls 0

11 for individual companies, bu hey do no es he Linner (956) model using aggregae dividends (see also Fama, 974). 5 I can be noed ha curren ime earnings have no ye been observed when he curren dividend level is decided upon and declared, since companies repor heir earnings afer he end of he quarer whereas dividends are decided upon and announced prior o heir paymen each quarer. This problem can be eliminaed in he Linner model by lagging earnings by eiher a quarer or a year so ha he ime dividend level choice is modelled only in relaion o informaion ha is observable a ime : D = θ + θ E + θd + ε. (4) 0 Underlying he Linner model is he idea ha dividends would no be adjused o changes in earnings ha are only emporary, so Marsh and Meron (987) inroduce he concep of permanen earnings ino dividend behaviour models and argue ha dividends will only be deermined in relaion o permanen earnings. They furher argue ha, in an efficien marke, he curren share price index level is equal o he presen value of all fuure permanen earnings. The rae of change of share prices can hen be shown o be equal o he rae of change of expeced permanen earnings when i is assumed ha he long-run discoun rae is consan, hus implying a log linear relaionship beween dividend and price changes ha leads o regression equaion () in Marsh and Meron (987): D D P+ D D + log + = ψ0+ ψlog + ψlog + ε+ D P P P. (5) 5 The average adjused R hey obain of 45.7% is no direcly comparable wih he Linner (956) resuls because Grullon and Michaely (00) esimae he Linner model using changes in dividends (equaion () in Linner, 956), no he dividend level regression model ha Linner (956) ess (see equaion (3) above), bu he parameer esimaes obained are similar o he Linner (956) resuls.

12 The Marsh and Meron (987) model explains roughly half as much of he variaion in aggregae dividends as does Linner s equaion (3) above (see Linner, 956; Marsh and Meron, 987; as well as he Resuls secion). An alernaive explanaion of aggregae dividend behaviour is provided by Shiller s (983) rend-auoregressive model. 6 The model implies ha deviaions in aggregae dividends follow a rend-auoregressive process whereby half of he deviaion in dividends from rend disappears wihin hree years, wih he underlying rend being explained by ongoing (no anicipaed) earnings growh. Changes in curren dividends are herefore deermined by a ime rend as well as pas deviaions from rend in Shiller s model, raher han anicipaed earnings growh, bu empirical ess indicae ha he rend - auoregressive model provides (a mos) half as much explanaory power as he Marsh and Meron (987) model (see Marsh and Meron, 987). C. Spurious Regression Behavioural models of dividends share in common he use of lagged dividend erms o explain curren dividend choices, a feaure which implies ha dividends are highly persisen. The recen lieraure on spurious regression poins ou ha he use of persisen independen variables can lead o spurious regression resuls when he dependen variable is also a leas parially persisen, since error erms in he regression equaion inheri auocorrelaion from he persisen dependen variable (Ferson, Sarkissian, and Simin, 003a). This auocorrelaion in he error erm leads o biased sandard error esimaes and can herefore indicae a significan overall relaionship when 6 For an exensive heoreical and empirical review of dividend policy and dividend behaviour a he individual firm level, see Allen, Bernardo and Welch (000).

13 none exiss, especially when daa ses are mined for poenially significan explanaory variables (Ferson, Sarkissian, and Simin, 003a; Foser, Smih and Whaley, 997). Behavioural models of dividends are herefore likely candidaes for spurious regression due o heir srong persisence properies. I is imporan o noe ha he spurious regression problem does no disappear as he sample size is enlarged, unlike he persisen regressor coefficien esimae bias oulined by Sambaugh (999) and Nelson and Kim (993), so i is a poenial problem for dividend behaviour sudies regardless of he sudy sample ha is employed (Ferson, Sarkissian, and Simin, 003a). Reurn predicabiliy regressions ha use he dividend yield as a predicor variable are also likely candidaes for spurious regression because he numeraor of he independen variable, he dividend level, is highly persisen and will consequenly make he dividend yield variable highly persisen as well. I is herefore likely ha persisence migh be responsible for he apparen reurn predicabiliy of he dividend yield, raher han any propery of dividends relaed o risk, reurn or mispricing, and in his siuaion oher similarly persisen variables migh also possess apparen reurn predicabiliy, as will be illusraed below. II. DIVIDEND YIELD RETURN PREDICTABILITY A saring poin for examining wheher dividend yield reurn predicabiliy is a spurious resul is o reproduce sandard dividend yield reurn predicabiliy resuls and compare hem o he reurn predicabiliy resuls ha can be obained using oher persisen explanaory variables. The oal reurn and capial reurn sock indices used in he sudy are he annual CRSP Value-weighed index and he annual Sandard and Poor 3

14 500 Index for he ime period 96 o 004. All sock index daa are obained from CRSP and all series are convered from nominal o real values by dividing by he Consumer Price Index (CPI) obained from Rober Shiller s web page. The dividend yield on he index a ime equals he level of real dividends during he welve monhs preceding ime ( D ) divided by he real sock index level a ime (P ). 7 Panel A of Table reproduces sandard dividend yield reurn predicabiliy resuls by regressing annual ime + raes of reurn agains a consan as well as he annual ime dividend yield (see, e.g., Fama and French, 988): r P + D P D = β + β + ε P P. () The Fama and French (988) resuls and he resuls of oher sudies are replicaed in Panel A of Table II, wih he regression R equalling.95% for he CRSP Valueweighed Index and.05% for he S&P 500 Index. 8 [Table II abou here] The reurn predicabiliy resuls documened in Panel A of Table II are ofen explained in erms of a heoreical relaionship beween curren dividend yields and subsequen reurns ha is derived from he dividend discoun model (see, e.g., Campbell and Shiller, 988a and 988b; Fama and French, 988). I can be noed ha his heoreical jusificaion for reurn predicabiliy relies upon a high curren dividend level relaive o he share index level predicing higher fuure reurns, so subsiuing a consan 7 Numerous dividend yield reurn predicabiliy sudies also use monhly or quarerly reurns wih overlapping dividend yield observaions. The use of overlapping observaions inroduces excess dividend yield auocorrelaion. This well-known overlapping observaion problem is avoided wih he use of annual daa. 8 Regressions are es imaed using Ordinary Leas Squares and -saisics are esimaed using Newey-Wes (987) sandard errors. 4

15 level of dividends (c) for he dividend erm ( D ) in he numeraor of he dividend yield variable ( D / P ) in regression equaion () should reduce or eliminae reurn predicabiliy in he following alered regression equaion: r P + D P c = β + β + ε P P, () where c is he uncondiional average of he dividend level during he sample period. The resuls for regression equaion () are oulined in Panel B of Table II. Surprizingly, he R acually rises o above 3.% for he S&P 500 Index and 3.3% for he Value-weighed CRSP Index. The resuls for his dividend yield from a consan dividend variable ( c / P ) are herefore so srong ha hey provide an indicaion ha somehing funny is going on in previous dividend yield reurn predicabiliy sudies since, ineresingly, he dividend yield from a consan dividend variable ( c/ P ) has higher explanaory power han does he sandard dividend yield variable ( D / P ). This indicaes ha dividends are unlikely o be providing he explanaory power in reurn predicabiliy regressions. The Table II resuls insead sugges ha he observed empirical relaionship beween subsequen reurns and he dividend yield migh be spurious. To invesigae his possibiliy, he persisence properies of he dividend, dividend yield, price index, dividend yield from a consan dividend, and reurn series are invesigaed in Table III. Table III confirms ha he real dividend series are highly persisen, as are he dividend yield series. The dividend yield from a consan dividend series are even more persisen han he dividend yield series, hus indicaing ha persisence migh have an imporan influence on reurn predicabiliy, since predicabiliy increases when more persisen variables are used as predicor variables in Table II. Table III indicaes ha reurns are 5

16 sufficienly auocorrelaed o sugges ha regressing subsequen raes of reurn agains highly persisen dividend yield explanaory variables could lead o spurious regression. [Table III abou here] To follow up his possibiliy, a simulaion procedure is uilized ha provides he cu-off R ha would be obained from a regression for which he dependen and independen variables are uncorrelaed bu have he same auocorrelaion properies as he acual daa, hus aking accoun of he poenial for spurious regression (see also Foser, Smih and Whaley, 997; Ferson, Sarkissian and Simin, 003a). The simulaion procedure is closely relaed o he procedure used in Nelson and Kim (993), wih he imporan difference being ha dependen variable auo-correlaion is direcly incorporaed, hus moving he analysis from he small sample persisen regressor coefficien esimae bias examined by Nelson and Kim (993) o he spurious regression problem examined by Ferson, Sarkissian and Simin (003a). To obain he cu-off R using simulaion, he momens and he serial correlaion properies of he regression variables are firs esimaed for each daa series, as described in he Appendix. Uncorrelaed dependen and independen variables wih he same seria l correlaion properies and sample momens are hen simulaed for a ime period equal o he sample lengh (97 o 004), and a regression is run on hese simulaed series. The process is repeaed 0,000 imes, and he R s are recorded for each regression and ranked from lowes o highes. The 95 h percenile R is hen repored as he 5% cu-off R and is compared o he acual R obained using he original daa o assess he overall significance of he esimaed regression relaionship (see Foser, Smih and Whaley, 997; Ferson, Sarkissian and Simin, 003a). 6

17 A second se of cu-off R saisics is also obained by simulaion o ake accoun of he poenial effec of daa mining since, as was menioned earlier, he dividend yield is only one of many variables ha have been explored for evidence of reurn predicabiliy (see Cremers, 00, as well as Lo and MacKinlay (990); Foser, Smih, and Whaley (997); Sullivan, Timmermann and Whie (999) and (00); Ferson, Sarkissian and Simin, 003a). Daa mining for explanaory variables reinforces he spurious regression problem, since highly persisen explanaory variables are more likely o display apparen significance. The second se of cu-off R saisics use Bonferonni correcion inervals o ake accoun of he number of series ha are examined in he search for poenially significan relaions, wih a conservaive adjusmen facor of five being used, hus represening he assumpion ha a leas five explanaory variables have been used in he search for sock marke index reurn predicabiliy (Foser, Smih and Whaley, 997). This modificaion is equivalen in an operaional sense o requiring a one percen level of overall significance raher han a five percen level due o he number of explanaory variables ha have been searched for reurn predicabiliy. The eighh and ninh columns of Table II repor he cu-off R and he cu-off R wih daa mining ha are obained using he simulaion procedure so ha i can be compared wih he R repored in he sevenh column of Table II. For insance, he cu- off R (cu-off R wih daa mining) repored for he Value-weighed CRSP Index in Panel A of Table II indicae ha a R of 3.95% (7.4%) would be expeced o be obained by regressing subsequen reurns agains uncorrelaed independen variables ha are equally as persisen as he dividend yield variable ( D / P ), whereas a R of only.95% is acually obained. The R of.95% does no exceed he criical cu-off R levels, hus 7

18 implying ha he Value-weighed dividend yield reurn predicabiliy regression R is insignifican (see Foser, Smih and Whaley, 997; Ferson, Sarkissian and Simin, 003a). The cu-off R levels repored in Table II herefore indicae ha all of he dividend yield reurn predicabiliy regression resuls are due o spurious regression since he repored R s never exceed he cu-off R s (see he eighh column and ninh columns of Table II). The simulaion procedure reas he variables in regression equaion (), r P + D P D = β + β + ε P P, () as if hey are independen. This independence assumpion is no appropriae since reurns on he lef hand side of equaion () and he dividend yield on he righ hand side boh come from he same underlying variables (he share index level P and he dividend level D). The simulaion procedure is herefore modified o recognize he dependency of boh he reurn and dividend yield variables on he share index and dividend levels (see he Appendix for deails of he modified simulaion procedure). Raher han simulaing he reurn series r + as he dependen variable using he properies of he reurn series and simulaing an uncorrelaed dividend yield series ( D / P) using he properies of he dividend yield series o calculae he cu-off R, uncorrelaed dividend ( D ) and share index ( P ) series are insead simulaed using he esimaed properies of hese series. Dividend yield ( D / P) and reurn ( r+ ( P + + D + P)/ P) observaions are hen calculaed using he simulaed dividend and share index values, a regression is run using he consruced simulaed variables, and he modified cu-off R is hen repored using he 95 h percenile R obained from he modified simulaion regression procedure. 8

19 The modified simulaion procedure leads o a considerably higher simulaed cu- off R (see he modified cu-off R repored in he final column of Table II). The modified cu-off R for he Value-weighed CRSP Index dividend yield regression rises sharply o 9.77%, for insance, and he modified cu-off R for he S&P 500 Index rises o 9.5% (see he final column of Panel A of Table II). The modified cu-off R s repored in Table II herefore make i very clear-cu ha dividend yield reurn predicabiliy is a spurious resul since he acual R s obained for he dividend yield regressions which range from.95% o.05% are an order of magniude smaller han he modified cu-off R levels. The increase in he modified cu-off R s in Table II highlighs he influence of he dividend yield and raes of reurn variables sharing a common denominaor (he share price index level P ), an effec ha can creae a spurious correlaion even when all variables ha make up he numeraors and he common denominaor in a regression are independen (Pearson, 897; Kronmal, 993). This spurious correlaion proble m arises when all variables excep for he consan in a rue regression equaion are divided by a common variable, ofen in an (incorrec) aemp by he researcher o conrol for a common confounding influence. Kronmal (993, page 38), summarizing Friedlander (980), oulines how his leads o a biased leas squares esimae of he independen coefficien when he rue regression consan is non-zero (see also Pearson, 897; Tanner, 949; Neyman, 95; and Friedlander, 980). More imporanly, i can also lead o a significan esimae of he overall relaionship beween he dependen and independen regression variables even when all of he componen variables making up he numeraors and he denominaor of he dependen and independen regression variables 9

20 are uncorrelaed. Inuiively, he common denominaor on boh sides of he regression model can inroduce a correlaion beween he independen and he dependen variable if he regression consan is no divided by he same conrol variable. Kronmal (993) documens how he spurious correlaion problem leads o incorrec inferences in empirical sudies ha examine he cross-secional relaionship beween weigh size and body size as well as he quesion of wheher females lose lung capaciy a a faser rae han males as hey age. Pearson (897) firs noed he spurious correlaion problem and derived an approximae correlaion formula ha provides a reference poin spurious correlaion benchmark when correlaing raios wih a common divisor. When random variables X, Y, and Z are uncorrelaed hen Pearson s approximae formula for he correlaion r( X/ Z, Y/ Z) beween raios wih a common divisor Z is r ( X/ Z, Y/ Z) = V Z ( VX + VZ )( VY + VZ), (6) where V indicaes a variable s coefficien of variaion (sandard deviaion divided by expeced value). Pearson s approximae formula indicaes ha he raio correlaion will be posiive unless Z is a consan; if he variables X, Y and Z are independenly and idenically disribued hen he approximae formula correlaion value will be.5, even hough X, Y and Z are independen! Pearson s (897) approximae formula derivaion ignores higher order erms, so Kim (999) derives an exac formula for he correlaion r( X/ Z, Y/ Z) beween raios of uncorrelaed variables wih a common divisor Z: 0

21 r ( X/ Z, Y/ Z) = V / Z [ V ( + V ) + V ][ V ( + V ) + V ] X / Z / Z Y / Z / Z. (7) Kim s exac formula is more precise han Pearson s approximae formula when he coefficien of variaion of he common divisor (Z) differs significanly from he coefficien of variaion of he inverse of he common divisor (/Z). Kim (999) provides heoreical examples o demonsrae ha he correlaion beween raios composed of hree independen variables can be arbirarily close o one, and also documens how he spurious correlaion problem can affec inferences concerning he empirical relaionship beween birh raes and deah raes in 97 counries in he year 990. Applicaion of equaion (7) indicaes ha a correlaion of.09 would be expeced by chance beween birh raes and deah raes even if birhs, deahs and populaion are uncorrelaed, hus providing a benchmark o assess wheher birh raes are conemporaneously relaed o deah raes. The Kim (999) spurious correlaion formula can explain why he modified cu- off R s which ake accoun of boh spurious correlaion and spurious regression are so much higher han he cu-off R s in Table II. Table IV repors implied modified cu-off R for reurn predicabiliy regression model () ha are obained using analyical correlaion formula (7). 9 The Table IV resuls reveal ha he implied modified cu-off R s obained using analyical formula (7) are 4.5% for he CRSP Value-weighed index and 7.56% for he S&P 500 index, hus providing a benchmark for he influence 9 Correlaion formula (7) can be convered o an implied modified cu-off R for reurn predicabiliy regression () using he equaion, ( R )( T )/( T K), where R is esimaed using equaion (7) and is he square of he correlaion beween dividend yields ( D / P ) and subsequen reurns ( r+ ( P + + D + P)/ P), T is he number of observaions, and K is he number of independen variables in he regression model.

22 of spurious correlaion (in isolaion) on dividend yield reurn predicabiliy regression resuls. The implied modified cu-off R s ha are obained using he analyical formula closely mach he modified cu-off R s obained by simulaion of 9.77% for he CRSP Value-weighed index and 9.5% for he S&P 500 index (see Table II), hus indicaing ha he modified cu-off R s repored in Table II are precisely wha should be expeced! The modified simulaion procedure akes accoun of he ineracion beween spurious correlaion and spurious regression whereas he analyical formula accouns for spurious correlaion only, hus explaining he difference beween he analyical and simulaion resuls, bu he wo ses of resuls are remarkably close. 0 Boh he simulaion procedure and analyical formula cu-off R s grealy exceed he implied R s of.95% for he CRSP Value-weighed index and.05% for he S&P 500 index ha are esimaed using he acual daa, hus srongly reinforcing he findings ha dividend yield reurn predicabiliy is a spurious resul (see Tables II and IV). Table IV also implies ha concerns abou spurious regression in dividend yield reurn predicabiliy sudies can largely be aribued o he spurious correlaion problem. [Table IV abou here] This paper is he firs o documen he imporance of he spurious correlaion problem in a financial ime series regression seing wih auo-correlaed regression variables. The difference beween he modified cu-off R and he cu-off R idenifies he influence of he spurious correlaion problem on he overall regression relaionship. For insance, he cu-off R in Panel A of Table II indicaes ha in he absence of a 0 The Kim (999) analyical formula provides a benchmark for spurious correlaion in isolaion by assuming ha variables X, Y, and Z are uncorrelaed, whereas hey are no uncorrelaed in he modified simulaion procedure, hus explaining why he wo ses of resuls can differ.

23 spurious correlaion problem (and daa mining) hen we would expec by chance an R of 3.95% for he CRSP Value-weighed reurn predicabiliy regression. The spurious correlaion problem is herefore insrumenal in helping o conribue almos 6% o he modified cu-off R of 9.77% for he Value-weighed reurn predicabiliy regression. I can herefore be argued, based upon he resuls of Table II, ha he spurious correlaion problem (in ineracion wih spurious regression) can be viewed as he major conribuing facor o incorrec inference in dividend yield reurn predicabiliy sudies. The Table II spurious regression reurn predicabiliy resuls can also be relaed o he persisen regressor coefficien bias lieraure (see, e.g., Sambaugh, 999; Lewellen, 004). Table II focuses on he significance of he overall relaionship beween dividend yields and subsequen reurns o deermine wheher he observed relaionship is spuriously affeced by dependen variable auocorrelaion, and does no focus on wheher he dividend yield coefficien esimaes are biased, since he paper does no focus on he coefficiens per se. Sill, i can be useful o examine how he resuls relae o recommendaions made in he reurn predicabiliy lieraure o overcome he spurious regression and coefficien bias problems. To make his comparison, i can firs be noed ha if high dividend yields really do predic higher subsequen reurns, hen an increase in he dividend yield should also forecas an increase in subsequen reurns. Ferson, Sarkissian and Simin (003b) sugges ha esing he relaionship beween reurns and changes in predicor variables away from heir railing moving average can overcome he spurious regression problem, since derending predicor variables will creae a less persisen independen variable. Ferson, Sarkissian and Simin (003b) poin ou ha he reurn predicabiliy coefficien bias lieraure does no address he ineracion beween spurious regression and daa mining. 3

24 Subsequen reurns can herefore be regressed agains changes in he dividend yield from is railing moving average, wih a welve monh moving average lag lengh being uilized (as recommended by Ferson, Sarkissian and Simin, 003b). Lewellen (004) implemens a somewha relaed regression model o es for reurn predicabiliy, since he uses he naural logarihm of he dividend yield as he regression independen variable; he naural logarihm of he dividend yield can be inerpreed as a change in he dividend yield away from a dividend yield of one. Campbell and Yogo (003) also recommend derending dividend yields when esing for reurn predicabiliy. Subsequen real reurns are herefore regressed agains he derended dividend yield using he regression model r P + D P = β + β X + ε, (8) P where r + denoes he annual real index reurn a ime +, X denoes he sochasic derended dividend yield calculaed as X D D j =, (9) P τ P j=,..., τ j D is he level of real annual dividends during he welve monhs preceding ime, and P is he real sock index level a ime. While differen numbers of lags could be used in he derending, a -monh lag is used, as recommended by Ferson, Sarkissian and Simin (003b). Resuls for reurn predicabiliy regression model (8) are repored in Table V. Ineresingly, he resuls imply ha an increase in he dividend yield acually foreshadows lower, no higher, reurns (alhough he overall relaionship is clearly insignifican). Campbell and Yogo (003) recommend subracing from he dividend yield independen variable he innovaions in he yield ha are correlaed wih reurns o obain a less noisy independen variable, hus eliminaing some of he noise and increasing he power of he es. 4

25 Regression model (8) has no been esed before, bu he Table V resuls can be compared o Lewellen s (004) finding ha he reurn predicabiliy regression coefficien of he naural logarihm of he dividend yield is acually negaive wihin sub-periods for he monhly CRSP equally-weighed and value-weighed Indices when esimaed using he Sambaugh (999) bias-adjusmen. 3 The Table V resuls again reinforce he Table II findings ha he observed relaionship beween dividend yields and subsequen reurns is spurious, and hey also make i exremely unlikely ha predicor coefficien bias adjusmens would lead o a significan overall relaionship, since bias adjusmens do no normally reverse he sign of he coefficiens. The Table V resuls, combined wih he Table II resuls, furher sugges ha a spurious correlaion effec influences he esimaed relaionship beween dividend yields and subsequen reurns, since he reversal of he dividend yield regression coefficien sign beween Table II and Table V is consisen wih spurious correlaion effecs oulined in Kronmal (993). 4 The Table V resuls also cas severe doub on he emporary mispricing explanaion of reurn predicabiliy, since he Table V resuls imply ha an increase in mispricing would predic higher, no lower, subsequen reurns. [Table V abou here] A final challenge o he spurious regression conclusion obained from Table II is he argumen ha ha dividend yield reurn predicabiliy can be shown o be a nonspurious resul if he definiion of dividends is expanded o include share repurchases 3 Lewellen (004) repors very low adjused R s (less han.04) when regressing subsequen monhly reurns agains he naural logarihm of he dividend yield, and ends o find a significan dividend yield coefficien esimae only when imposing he assumpion of uniary dividend yield serial correlaion, an assumpion ha is clearly violaed in Table III for annual daa. 4 Kronmal (993) repors how sorks can be responsible for eiher a decrease or an increase in birhs, depending on how he dependen variable birhs is regressed agains he independen variable sorks in alernaive incorrecly specified regression models ha are subjec o spurious correlaion! 5

26 (Boudoukh e al, 004; Roberson and Wrigh, 003). This is especially imporan in recen decades, since he decline in he radiional dividend yield and in dividend yield reurn predicabiliy has occurred a he same ime as share repurchases have been subsiued for radiional dividends. 5 Table VI herefore reproduces he reurn predicabiliy regression coefficiens and R s repored in Table of Boudoukh e al (004) using he broadly defined oal payo u dividend yield variable ha includes share repurchases. In addiion, Table VI also repors he modified cu-off R ha is obained from he modified cu-off simulaion procedure using he parameers of he Boudoukh e al (004) oal payou dividend and share price index series. Once again, he modified cu-off R s repored in Table VI make i very clear-cu ha dividend yield reurn predicabiliy is a spurious resul, no maer wha dividend yield series is used, since he acual R s repored in Table VI of 5.96% for he full sample and 0.3% for he shorer 96 o 985 sample are considerably smaller han he modified cu-off R levels. [Table VI abou here] The resuls of Tables II, IV, V, and VI, aken ogeher, imply ha he spurious effec of regressing reurns agains a highly persisen explanaory variable such as he dividend yield is srongly reinforced by a spurious correlaion effec. To furher explore he origins of his spurious reurn predicabiliy effec, models of dividend behaviour are also esed for spurious regression. 5 Noe ha, unlike he radiional dividend yield variable used in Panel A of Table II, he dividend yield from a consan dividend reurn predicabiliy variable used in Panel B of Table II is no affeced by his recen subsiuion away from dividends. 6

27 III. DIVIDEND BEHAVIOUR RESULTS The persisence properies of dividend yields ha conribue o spurious reurn predicabiliy imply ha dividend behaviour models are also likely o be subjec o spurious regression, since he dependen and independen variables in he Linner (956) and Marsh and Meron (987) dividend behaviour models are all very persisen (see also Ferson, Sarkissian and Simin, 003a). These dividend behaviour models are examined in Table VII. The Table VII resuls indicae ha he Marsh and Meron (987) model is no subjec o spurious regression, whereas spurious regression appears o play an exremely imporan role in alernaive versions of he Linner (956) model which use highly persisen lagged dividend and earnings erms o explain subsequen dividend levels or dividend changes. Panel A of Table VII reveals ha curren log reurns play a very imporan role in explaining subsequen dividend changes in he Marsh and Meron (987) model (recall equaion (5)): D D P+ D D + log + = ψ0+ ψlog + ψlog + ε+ D P P P The R s of 33.95% for he CRSP Value-weighed index and 9.84% for he S&P 500 index grealy exceed he modified cu-off R s of 9.3% and.04% (respecively) in Panel A of Table VII, hus implying ha he resuls are no due o spurious regression. [Table VII abou here] The Linner (956) model of he ime dividend level choice (recall equaion (4)),. D = θ + θ E + θd + ε, 0 7

28 is esed using he S&P 500 index only due o earnings daa availabiliy (see also Arno and Asness, 003). 6 Panel B of Table VII reveals ha he esimaed independen variable coefficiens for regression model (4) are very similar o hose found in Linner (956), even hough he daa se is exended by four decades. The resuls imply ha lagged earnings and (especially) lagged dividends explain almos all of he variaion in he curren dividend level. The R exceeds 90%, as in Linner (956), and slighly exceeds he modified cu-off R, hus implying ha he original Linner model appears o somewha survive he spurious regression problem. The very high modified cu-off R repored for he Linner model in Panel B of Table VII suggess ha he spurious regression problem has an influence on he Linner model regressio n, even if i is no solely responsible for he resuls, hus indicaing he poenial need for a reformulaion of he original Linner model. A sep owards a reformulaion of he Linner regression model is already provided in Linner (956), where he Linner dividend behaviour heoreical model is originally presened in erms of changes in dividends, no he dividend level as in regression model (4) above (see equaion () in Linner, 956, and see also Grullon and Michaely, 00; Fama, 974). Resaing he Linner dividend change model (Linner (956) equaion ()) in erms of informaion ha is observable a ime leads o he following regression model: D D = λ + λe + λ D + ε (0) 0 6 The Sandard and Poor 500 earnings daa se is obained from Rober Shiller s web sie. This paper follows he recen pracice of using repored earnings o es he Linner model due o daa availabiliy, raher han operaing or cash earnings. 8

29 Resuls for he Linner (956) dividend change regression model (0) are presened in Panel C of Table VII and are consisen wih he resuls for he Linner (956) dividend level regression model (4) presened in Panel B. Having changes in dividends raher han he dividend level as he dependen variable in he Linner regression model grealy reduces he R (from 9.9% down o only 3.3%) as well as he modified cu-off R (from 84.48% o 4.8%) in Panel C, so he R is exceeded by he modified cu-off R, hus indicaing he Panel C resuls are spurious. The following secion akes furher seps o reformulae he Linner dividend behaviour model enirely in erms of firs differences on boh sides of he regression equaion so ha i can remain as a simple and inuiive alernaive o (or complemen for) he Marsh and Meron (987) dividend behaviour model wihou being subjec o spurious regression. IV. AN ALTERNATIVE DIVIDEND BEHAVIOUR MODEL SPECIFICATION A source of persisence in he Linner (956) dividend level model ha is likely o have a very imporan effec on he model s ime series regression properies is a common ime rend in boh he dependen and independen regression variables (recall equaion (4)): D = θ + θ E + θd + ε, 0 A sandard soluion o his problem is a reformulaion enirely in erms of firs differences (see also equaion () in Linner, 956). To derive he Linner model in erms of firs differences, firs noe ha equaion (4) also implies: D = θ + θ E + θ D + ε. () 0 9