Size and Power in Asset Pricing Tests

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1 Size and Powe in Asset Picing Tests Stephen F. LeRoy Univesity of Califonia, Santa Babaa Rish Singhania Univesity of Exete June 25, 2018 Abstact We study the size-powe tadeoff in commonly employed tests of etun pedictability. We povide conditions unde which shot-hoizon dividend-gowth tests and long-hoizon etun tests ae asympotically moe poweful than shot-hoizon etun tests. Monte Calo esults show that the asymptotic powe advantages cay ove to small samples, although the easons ae diffeent. Asymptotically, dividendgowth tests ae close to unifomly most poweful. In small samples, dividend-gowth tests and long-hoizon tests have simila powe. JEL Codes: G12, G14, C32 Keywods: Powe; Dividend-gowth egessions; Long-hoizon egessions; Retun pedictability; Maket efficiency We would like to thank James Davidson and Sebastian Kipfganz fo helpful discussions. 1

2 Empiical tests of maket efficiency defined, in the context of stocks, as the poposition that asset pices equal the discounted value of expected dividends have a long histoy. In his classic pape, Fama (1970) epoted empiical tests of the implication of maket efficiency that asset etuns ae unfoecastable (Samuelson (1965)). He concluded in favo of maket efficiency. Howeve, analysts subsequently obseved that asset pice volatility appeaed to exceed the volatility consistent, unde maket efficiency, with the volatility of dividends (Shille (1981)). Also, analysts evesed Fama s oiginal conclusion, finding that asset etuns ae foecastable (Fama and Fench (1988)). The etun foecastability appeaed to be negligible at shot hoizons, but was evaluated as being much stonge at longe hoizons, like seveal yeas. 1 The fact that diffeent tests of the same null hypothesis appea to have diffeent outcomes suggests that some tests have geate powe than othes: how likely is it that diffeent tests will eject maket efficiency unde some altenative powe holding constant the ejection pobability if makets ae efficient size? Possible diffeences in powe have been discussed in the asset picing liteatue (LeRoy and Steigewald (1995), fo example), but until ecently no fomal esults along these lines had been povided. 2 Cochane (2008) poposed addessing the size-powe tadeoff head on. He concluded fom simulations conducted using data geneated by an estimated 1 Statistical issues wee aised about the conclusion that etuns ae foecastable. In paticula, the appaently damatic diffeence between the outcome of shot-hoizon and long-hoizon etun foecastability tests was convincingly citicized by Boudoukh et al. (2006), who showed that in efficient makets sampling vaiation alone should be expected to esult in estimated foecasting coefficients that incease appoximately in popotion as the hoizon is moe distant. This occus because etun foecasts ove vaying hoizons ae highly coelated when the explanatoy vaiable (usually the dividend yield) is highly pesistent. 2 Campbell (2001) is an exception. That pape investigates the powe advantages of long-hoizon etun egessions, ove shot-hoizon egessions, using appoximate slope, a measue of asymptotic powe, as a guide. It finds that long-hoizon egessions have highe appoximate slope, suggestive of highe powe. Monte Calo simulations indicate that the highe appoximate slope tanslates to powe advantages in small samples. 2

3 model that the evidence against efficiency povided by etuns foecastability alone is of maginal statistical significance. Howeve, he noted that thee exist altenative tests of maket efficiency. To be consistent with the obseved high level of pice volatility eithe etuns o dividend gowth must be significantly foecastable. Unde maket efficiency etuns ae nonfoecastable by assumption, implying that efficiency can be tested by detemining whethe dividend foecastability is sufficient to explain pice volatility. Cochane asseted that the test based on dividend foecastability has highe powe than that based on etun foecastability. Although Cochane s stated pupose was to ague fo the geate powe of one test elative to the othe, his discussion centeed almost exclusively on the behavio of test statistics unde the null hypothesis of maket efficiency. In this pape we addess the same poblem as Cochane, but do so diectly athe than taking a detou dealing with extensive analysis of estimato popeties unde the null hypothesis. We find using Cochane s model that his conclusion in favo of geate powe of dividend foecastability tests in his model is coect. This is so because, based on a nomal appoximation, the estimated coefficient of etun foecastability has geate vaiance than that of dividend gowth foecastability. This explanation beas no close elation to Cochane s discussion. Without the nomal appoximation the powe advantage of dividend foecastability tests ove etun foeastability tests still occus (in fact, appeas to be even geate), suggesting that intuition based on the nomal appoximation caies ove to settings whee nomality is not a paticulaly close assumption, as in Cochane s setting. 1 The Model We use the same log-linea model as Cochane: log etuns t+1, log dividend gowth d t+1 and next-peiod log dividend yield d t+1 p t+1 depend linealy 3

4 on the cuent log yield d t p t : t+1 = a + β (d t p t ) + ε t+1 (1) d t+1 = a d + β d (d t p t ) + ε d t+1 (2) d t+1 p t+1 = a dp + ϕ(d t p t ) + ε dp t+1. (3) The eos ae nomal with mean zeo and ae independent ove time. At date t the vaiances and covaiances of the eo tems ae denoted σ 2, σ d, etc. As Cochane noted, the Campbell and Shille (1988) log-lineaization of the definition of the ate of etun, t+1 = ρ(p t+1 d t+1 ) + d t+1 (p t d t ), (4) whee ρ is the constant of lineaization, implies that any one of the equations (1), (2) and (3) is edundant given the othes. It follows that the coefficients obey the estiction and the eos obey the estiction β = 1 ρϕ + β d, (5) ε t+1 = ε d t+1 ρε dp t+1. (6) Both estictions play a majo ole in the discussion below. 1.1 The Null and Altenative Hypotheses The null hypothesis of maket efficiency is geneated by setting β = 0, so that etuns ae not foecastable in the population. The altenative hypothesis is geneated by setting β and β d equal to the values Cochane estimated fom the eal-wold data. Thus etuns have a foecastable component unde the altenative. In going fom the altenative 4

5 to the null the paamete edundancy (5) implies that setting β equal to zeo must be accompanied by a coesponding intevention on at least one of the othe paametes. Until Section 4 we follow Cochane in specifying the null hypothesis so that β d is changed fom the value estimated fom the data to the value that satisfies (5). The paamete ϕ and the covaiances of the eos ae the same unde the null and the altenative. 2 Asymptotic Popeties Maket efficiency can be tested eithe diectly by detemining whethe β equals 0 o indiectly by detemining whethe β d equals ρφ 1, the value implied by (5) when β is set equal to 0. These tests have the same null and altenative hypotheses, but geneally not the same size-powe tadeoff. The asymptotic size-powe tadeoff seves as a useful benchmak when evaluating the finite sample size-powe tadeoff of the two tests. 2.1 Shot-Hoizon Tests The asymptotic popeties of OLS estimatos, d and φ follow fom a vesion of the cental limit theoem; see Poposition 11.1 in Hamilton (1994) fo a detailed discussion. As the numbe of obsevations goes to infinity, T, the distibution of each OLS estimate appoaches a nomal distibution with mean equal to the population value of the coesponding paamete. The asymptotic vaiances of the estimatos follow the standad OLS fomula. Fo and d, we have T Va( i ) = σ2 i (1 φ 2 ) ; i {, d} (7) σdp 2 whee σ 2 i is the date-t vaiance of the shock ε i t, and σ 2 dp /(1 φ2 ) is the vaiance of log dividend yield. Fo the estimate of the dividend-yield autocoelation, 5

6 we have T Va( ϕ) = 1 ϕ 2. These expessions fo the vaiances ae deived in Appendix A. How do the asymptotic distibutions change as we move fom the null to the altenative? The altenative hypothesis consists of a modification in the paametes β and β d fom thei values unde the null, with the emaining paametes emaining unchanged. Theefoe, as we move fom the null to the altenative, the asymptotic distibution of shifts to eflect the change in the mean fom 0 to the value unde the altenative, which we specified to equal the value estimated fom eal-wold data. The covaiances emain unchanged. Similaly, the distibution of d shifts to eflect the value of the paamete unde the altenative. With knowledge of the asymptotic distibutions of the OLS estimates unde the null and the altenative, we can compae the size-powe tadeoff of etun and dividend foecastability tests of maket efficiency. The following poposition povides a simple way to detemine the elative powe of the two tests. Poposition 1 Fo a given size, maket efficiency tests based on dividend gowth foecastability ae asymptotically moe poweful than etun foecastability tests if and only if σ d < σ. The size-powe tadeoff depends on the elative asymptotic vaiances of the OLS estimatos. Because, fom (7), the asymptotic vaiances inheit the vaiances of the undelying eo tems, the asymptotic powe of the tests depends on the eo vaiances. If σ d < σ, the vaiance of d is lowe than that of, implying that the dividend test is moe poweful. The opposite is tue if σ d > σ. The two tests have identical size-powe tadeoffs if σ d = σ. The intuition behind Poposition 1 is demonstated in Figue 1. Because the estimates and d ae asymptotically bivaiate nomal, the level cuves unde the null and the altenative ae ellipses. The altenative hypothesis is a 45 degee tanslation of the null in the d space. Theefoe the ellipse 6

7 Div. test 5% cv ALT. β^d NULL Retun test 5% cv Figue 1: Level cuves unde the null and the altenative β^ epesenting the level cuve unde the altenative hypothesis is the same as that of the null hypothesis, but tanslated along a 45-degee line. The ellipses ae flat, eflecting the fact that the vaiance of d is less than the vaiance of. Togethe, the flatness and tanslation along the 45 degee line imply that the null and the altenative populations ae moe easily distinguished based on the d coodinate than the coodinate. This fact implies that fo given size the powe of a maket efficiency test based on d is geate than one based on. This esult is demonstated fomally in Appendix B. below. Numeical values fo the size-powe tadeoff ae pesented and discussed 7

8 2.2 Long-Hoizon Tests Cochane also investigated the powe of long-hoizon foecastability of etuns and dividend gowth. He identified these with β = β /(1 ρφ) (8) and βd = β d/(1 ρφ). The coefficient β /(1 ρφ) is elabeled β because it is the foecastability coefficient associated with long-hoizon etuns j ρj t+j, and similaly fo the dividend gowth coefficient. Eq. (8) implies that β = 1 + βd, fom which it follows that the long-un etun foecastability and dividend-gowth foecastability tests ae equivalent (and, in paticula, have the same powe fo given size). Cochane noted and discussed this esult. Consequently it is sufficient to compae the size-powe tadeoff fo that fo, which we now do. with The size-powe tadeoff fo long-hoizon tests depends on the asymptotic distibutions of unde the null hypothesis and the altenative, so we begin by detemining those distibutions. This is done in Appendix A. As T, the distibution of conveges to a nomal distibution with mean β. The asymptotic vaiance of is given by T Va( ) = It is seen that the asymptotic vaiance of ( ) 1 φ2 σ 2 + 2ρβ σ,dp + ρ2 β 2. (9) (1 ρφ) 2 σdp 2 1 ρφ σdp 2 (1 ρφ) 2 depends on the population value of β. The pesence of β in (9) implies that the asymptotic vaiance of is geneally diffeent unde the null and the altenative. Poposition 1 connecting the powe of and d tests with the volatilities of and d depended on these vaiances being equal unde the null and the altenative, implying that Poposition 1 does not apply diectly to. Calculating the asymptotic size-powe tadeoff, while still possible, is moe involved. Unde the null hypothesis of unfoecastable etuns, β = 0, the vaiance 8

9 of the long-hoizon estimato is the vaiance of the shot-hoizon estimato scaled up. We have T Va( ) = NULL 1 (1 ρφ) 2 T Va( ), (10) fom eqs. (7) and (9). The vaiance of the long-hoizon estimato unde the altenative hypothesis is geate than o less than its vaiance unde the null, depending on the contibution of the tems in (9) involving β. The tem ρ 2 β 2 /(1 ρφ) 2 is stictly positive. The sign of emaining tem depends on the coelation between etun shocks and dividend-yield shocks, denoted η,dp. If that coelation is positive, σ,dp is positive and the vaiance unde the altenative is geate than the vaiance unde the null. If the coelation is stongly negative, the vaiance unde the altenative is lowe than that unde the null. Fomally, thee exists a theshold η, which is negative, such that the asymptotic vaiance of unde the altenative when η,dp = η. We have unde the null is equal to its vaiance σ dp η = ρ β,alt. 1 ρφ 2σ If η,dp < η (as will be the case in the data), the vaiance unde the altenative is lowe than the vaiance unde the null. η,dp > η. The opposite would be tue if The following two popositions show that the coelation η,dp plays an impotant ole in detemining how the powe of long-hoizon tests compaes to the powe of tests. Poposition 2 Fo a given size, long-hoizon tests and shot-hoizon etun foecastability tests have equal asymptotic powe if η,dp = η. The intuition fo Poposition 2 is simple: when η,dp = η the asymptotic distibution of is the same as the distibution of a scala multiple of, so 9

10 Shot hoizon test statistic 8 NULL ALT % β^ Long hoizon test statistic scaled up by (1 ρφ) 12 9 NULL ALT 6 69% β^ (1 ρφ) Figue 2: Why long-hoizon tests have highe powe than shot-hoizon tests the two tests have equal powe. Asymptotically, the means of the estimatos and equal thei population values β the mean of and β, espectively. Theefoe, is the mean of scaled up by 1/(1 ρφ), unde both the null and the altenative. When η,dp = η, the vaiance of unde the altenative equals its vaiance unde the null. By (10), the standad deviation of, equal unde the null and the altenative, is the standad deviation of scaled by 1/(1 ρφ). Because the two estimatos ae nomally distibuted unde the null and the altenative, the asymptotic distibution of is the same distibution as that of a scala multiple of. It follows that tests involving and have equal powe when η,dp = η. Now assume that η,dp < η as the data indicate; the conclusions epoted below would be evesed fo η,dp > η. The asymptotic distibution of unde the null continues to be the same as the distibution of scaled by 10

11 1/(1 ρφ), but the vaiance unde the altenative is lowe than unde the null. The following poposition shows that fo η,dp < η the specification of the altenative hypothesis detemines which test is moe poweful. Poposition 3 Suppose η,dp < η. Fo a given size, let β denote the citical value of the shot-hoizon etun foecastability test. Given the size, 1. long-hoizon tests ae moe poweful than shot-hoizon etun tests if and only if β,alt > β. Shot-hoizon tests ae moe poweful if and only if the opposite inequality holds. 2. The two tests have equal powe if β,alt = β. To develop intuition fo Poposition 3 multiply by (1 ρφ) to undo the scale effect. The asymptotic distibution of (1 ρφ) is identical to the distibution of unde the null: nomal with mean zeo and vaiance given by (7). Unde the altenative, the two estimatos ae nomally distibuted with equal means, but not equal vaiances. Which test is moe poweful depends on whethe it is easy to distinguish the altenative fom the null; this is the case when the altenative is fa away fom the null. In that case, the test with the lowe vaiance has geate powe because it geneates moe values close to the altenative, and theefoe fa away fom the null. Convesely, suppose that the altenative is close to the null, making it difficult to distinguish between the two. In that case, the test with the highe vaiance has geate powe because it geneates moe values that ae fathe away fom the null. To see this, conside the case in which the vaiance of extemely low. In that setting the pobability of accepting the null if the data ae geneated unde the altenative would be high, possibly highe than would occu if the data wee geneated unde the null. The theshold that detemines whethe o not its easy to distinguish the null and the altenative is the citical value β. If β,alt 11 is > β it is easy

12 to distinguish the null and the altenative. Theefoe the long-hoizon test, having lowe vaiance than the shot-hoizon test, povides the moe poweful test. If β,alt < β the shot-hoizon test is moe poweful. If β,alt = β, we obtain a knife-edge esult: the powe of both the tests equals 0.5. The elative vaiance of the estimatos does not matte in that case. The coelation η,dp plays no ole in detemining which test is moe poweful. The fact that the elative powe of shot-hoizon vs. long-hoizon tests depends on the citical value of the shot-hoizon test means that the powe odeing may be diffeent depending on whethe one is conside a 1%, 5% o 10% ejection pobability. We will see below that the long-hoizon test has highe powe than the shot-hoizon test unde 5% o 10% ejection pobabilities, but lowe powe in the 1% case. Figue 2 shows why the powe of the long-hoizon test elative to the shot-hoizon test depends on whethe β,alt is geate than o less than β. 2.3 Likelihood Ratio Tests Thee is no eason to focus on efficiency tests based on eithe o d to the exclusion of the othe, as we have done. We can also conduct a likelihood atio test, so that fo any specified pobability of Type I eo the bounday between the acceptance and ejection egions consists of pais, d that have equal likelihood atios. In the case of the bivaiate nomal the boundaies sepaating these egions consist of staight lines taveling fom nothwest to southeast. The Neyman-Peason lemma states that likelihood atio tests have maximal powe fo given size. 3 Empiical Implementation The asymptotic distibution may not be a good appoximation of the finite sample distibution. In eal-wold data, the dividend yield is highly autocoelated, which educes the effective sample size fo the estimation. Given 12

13 that we only have about 80 yeas of eal-wold data, with the lowe effective sample size the cental limit theoem might tuns out to yield only a coase appoximation of the finite sample distibution. In this section we epot the application of Monte Calo simulations to compute both the size-powe tadeoffs in both the asymptotic case and the finite-sample case. In the asymptotic case the simulations wee used to estimate the paametes that detemine powe, as discussed in Section 2. In the finite-sample case we diectly estimated powe fom the simulated egession coefficients. The two sets of esults diffe because in the latte case thee is no appeal to nomality, and the elevant distibutions exhibit depatues fom nomality. The computed powe estimates ae shown in Table 1. We set the values of paametes used to geneate the simulations to match estimates of thei countepats in the data. To facilitate compaison between ou pape and Cochane s, we take the empiical estimates fom Cochane (2008), instead of e-estimating the model using the latest available data. The Monte Calo execise consists of 50,000 daws with each daw consisting of 80 dates, ageeing with the length of the dataset fo Cochane s eal wold estimation. The constant of log-lineaization is ρ = , calculated fom mean log dividend yield in the data. 3 In each un, we daw two time seies {ε d t } and {ε dp t } and impute the value of {ε t } using (6). The shocks ae dawn fom a bivaiate nomal distibution with mean zeo and vaiance-covaiance matix given by σ2 d σ dp,d σ 2 dp = (11) Having obtained the time seies of shocks, we ceated time seies of log etuns, log dividend gowth and log dividend yield using (1)-(3). The autocoelation φ of the log dividend yield is equal to in both the null and the altenative hypotheses. As noted, the coefficients β and β d depend 3 We have ρ = e E(p d) /(1 + e E(p d) ). 13

14 on whethe the we assume null o the altenative hypothesis holds in the simulated data. If the null of maket efficiency holds, then we have β = 0. Unde the altenative, we have β = The coesponding values of the dividend foecastability coefficient follow fom (4). We have β d = and β d = in the null and the altenative, espectively. Fo each un, we obtained the OLS estimates and d. Fo the test we computed the 95% citical value as the value β of such that 5% of the 50,000 simulated values of ae geate than β. Powe was then computed as the popotion of daws unde the altenative that ae geate than β. The calculation fo d was simila. We also computed 10% and 1% citical values and the coesponding figues fo powe. The conclusions follow: We find that dividend-gowth foecastability is moe poweful than etun foecastability in testing maket efficiency. We aleady knew that this would be tue in the asymptotic case fom the fact that the volatility of etun shocks exceeds the volatility of dividend-gowth shocks. The fact that the finite-sample esults suppot a simila conclusion suggests that the outcome does not depend citically on the nomality assumption. The powe advantage of the dividend gowth tests is much moe ponounced in the finite-sample case than in the asymptotic case. This pobably eflects the fact that the simulated distibution of is skewed, wheeas the simulated distibution of d is not. Figue 3 shows these distibutions unde the null and the altenative. It also displays the 1%, 5% and 10% citical values. The figue makes clea why skewness esults in a loss of powe. As discussed in Stambaugh (1999), the finite-sample distibution of the OLS estimato depends on whethe o not shocks to the pedicto vaiable (dividend-yield) ae coelated with shocks to the egessand 14

15 (etuns o dividend-gowth). If the shocks ae coelated, then the finite-sample distibution is skewed. In ou setting, shocks to etuns ae highly negatively coelated with shocks to dividend yield, which geneates the skewness obseved in. In contast, shocks to dividendgowth ae close to being uncoelated, which is why the distibution of d does not exhibit skewness. Fo the most pat powe is much highe in the asymptotic case than in the finite-sample case. This is as expected: finite-sample vaiability esults in high citical values, implying highe pobability of acceptance of the null when data ae geneated unde the altenative. Howeve, the long-hoizon tests ae an exception: powe is about the same in the asymptotic and finite-sample cases. This may eflect the fact, shown in Figue 4, that skewness is much lowe fo long-hoizon etuns than shot-hoizon etuns. As expected fom the Neyman-Peason lemma, the likelihood-atio tests have geate powe than any of the othe tests, both asymptotically and in finite samples. The asymptotic dividend tests have powe that is only slightly lowe than the likelihood atio tests. This esults fom the fact that the citical values of the two tests ae vey close, so the two tests have almost exactly the same ejection egions. Similaly, the finite-sample dividend gowth foecastability tests have about the same powe as the likelihood atio tests (except in the case of the 1% test), again because the citical lines of the two tests ae close. This is illustated in Figue 5. As noted in Section 2, asymptotically the long-hoizon test has highe powe unde the 5% and 10% tests, but lowe powe unde the 1% test, compaed to the etuns test. The fact that these diffeences ae not 15

16 ponounced coesponds to the analysis of Boudoukh et al. (2006), also based on asymptotic distibutions, that the two test statistics ae highly coelated due to the high pesistence of dividend yields. Poposition 3 helps us undestand this elationship between size and powe fo the two tests. We have that the theshold η equals Because η,dp = 0.70 in the benchmak paameteization, we have that η,dp < η. As Poposition 3 showed, the asymptotic powe of the long-hoizon test depends on whethe the citical value of the β test is geate than o less than β,alt = The citical value of the 10% and 5% β tests ae and Theefoe the long-hoizon test has highe powe than the β test. The citical value of the 1% β test is > 0.097, and theefoe the 1% β has highe asymptotic powe. The finite-sample long-hoizon test has much highe powe than the finite-sample shot-hoizon etuns test. Cochane epoted the same esult. Again, this appeas to eflect the fact that the long-un coefficient has lowe skewness than the shot-un etuns coefficient (see Figue 4). The powe diffeences between the asymptotic test of etun foecastability and the coesponding long-hoizon tests ae small, so we conclude that the two tests have about the same powe. This finding connects with the demonstation of Boudoukh et al. (2006) that the two tests ae vey close to being equivalent due to the high coelation of shot-hoizon and long-hoizon test statistics when the explanatoy vaiable is highly autocoelated. Howeve, in the finite-sample case the long-hoizon tests have consideably highe powe. 16

17 8 10% 5% 1% % 5% 1% NULL ALT NULL ALT d Figue 3: Simulated distibutions of and d. The distibution unde the null is shown in blue, while the distibution unde the altenative is in ed. The vetical black lines indicate 10%, 5% and 1% citical values. 10% 5% 1% 1.0 NULL ALT Figue 4: Simulated distibution of. The distibution unde the null is shown in blue, while the distibution unde the altenative is in ed. The vetical black lines indicate 10%, 5% and 1% citical values. 17

18 Asymptotic Finite Sample Size d LR test d LR test 1% % % Table 1: A compaison of statistical powe of vaious asset picing tests. 4 Tests Involving Pice Volatility It follows fom (3) that the dividend yield (the ecipocal of which is the level of stock pices divided by dividends) is geneated by the same equation with the same paamete values unde the null and the altenative. This means that the assumed model and test specification imply that pice volatility is the same whethe o not makets ae efficient. Thus the pesent setting can shed no light on the question of whethe volatility tests have geate powe than etun foecastability tests in moe geneal settings, as discussed in the intoduction. One way to alte the test so as to ende it elevant to the size-powe compaison of volatility and etun foecastability tests would be to assume that the intevention setting β to the foecastable value is offset by an alteation in ϕ athe than β d. Doing so implies that an efficiency test based on yield foecastability would eplace the test based on dividend gowth foecastability. Howeve, thee exists a citical poblem with this poposed modification: a value setting β equal to zeo is consistent with (5) only if ϕ is set equal to a value geate than 1, so that dividend yields ae nonstationay. Thee is some empiical evidence fo nonstationaity of dividend yields see Caine (1993), and Welch and Goyal (2007), fo example but without stationaity the log-lineaization undelying ou size-powe calculations is inaccuate. Thus the pesent famewok does not apply. 18

19 Retun test Div. test Long hoizon test LR test β^ β^d Figue 5: Scatte showing 1000 pais of d and with citical egions fo the etuns test, dividend gowth test, long-hoizon test and likelihood-atio test. The citical egions wee computed using all the 50,000 daws. The ed tiangle indicates the null hypothesis (b = 0). The ed squae indicates the altenative hypothesis (b = 0.097). 5 Conclusion Cochane concluded that tests based on dividend gowth foecastability povide stonge evidence against maket efficiency than those based on etun foecastability. By this he meant that the test statistic fo the dividend gowth paamete was fathe out on the tail of its distibution unde maket efficiency than that of etuns. Ou appoach, in contast, involves detemining whethe tests based on the dividend gowth paamete ae moe likely than tests based on etuns foecastability to detect depatues fom 19

20 maket efficiency if they exist. Detemining this consists of ascetaining how the size-powe tadeoffs of the two tests diffe, and why. Ou finding is that the size-powe tadeoff favos the dividend gowth test. Ou most impotant contibution hee is to povide a definitive explanation: dividend gowth tests ae moe poweful asymptotically because etuns have highe volatility than dividend gowth. Ou conclusion is the countepat in ou famewok of Cochane s conclusion that dividend gowth tests povide stonge evidence than etuns tests against maket efficiency. To the extent that ou conclusions coespond to those of Cochane, ou findings can be intepeted as suppoting his. A lage liteatue is devoted to demonstating that the appeaance of etun foecastability can occu even unde the null due to a vaiety of econometic issues elated to sampling vaiability. These issues apply as much to ou simulations as to econometic tests based on eal-wold data. That being so, they ae taken into account in simulations in detemining citical values fo maket efficiency tests. It is, in fact, likely that the low powe of finite-sample etun foecastability tests eflects the fact that econometic biases esult in high citical values fo the etun foecastability paamete. It appeas that simila poblems with the dividend-gowth foecastability tests, if they exist at all, ae less sevee. Howeve, ou asymptotic tests, which by definition ae not subject to the finite-sample issues unde discussion, imply that the highe powe of dividend gowth foecastability tests cannot be attibuted entiely to these issues. We believe that ou analysis gives stong suppot to Cochane s contention that egessions of futue etuns on cuently-obsevable vaiables do not povide the stongest evidence fo etun pedictability. Rathe, as Cochane asseted, the best evidence comes fom compaison of the behavio of dividends and stock pices: the volatility of stock pices damatically exceeds the volatility level consistent with dividend behavio unde efficient makets. This was exactly the conclusion epoted in the vaiance bounds 20

21 tests. Howeve, thee is a citical diffeence: the vaiance bounds tests took dividend volatility to be the deteminant of stock pice volatility, wheeas in Cochane s pape and hee the elevant deteminant is dividend gowth pedictability. This diffeence, if anything, geatly stengthens the evidence fom the vaiance bounds liteatue fo excessive stock pice volatility elative to the efficient makets pediction: dividend gowth foecastability is essentially zeo, implying that pice volatility should also be essentially zeo unde efficient makets. 21

22 Refeences Boudoukh, Jacob, Matthew Richadson, and Robet F Whitelaw (2006) The myth of long-hoizon pedictability. The Review of Financial Studies 21(4), Campbell, John Y (2001) Why long hoizons? a study of powe against pesistent altenatives. Jounal of Empiical Finance 8(5), Campbell, John Y, and Robet J Shille (1988) The dividend-pice atio and expectations of futue dividends and discount factos. The Review of Financial Studies 1(3), Cochane, John H (2008) The dog that did not bak: A defense of etun pedictability. Review of Financial Studies 21(4), Caine, Roge (1993) Rational bubbles: A test. Jounal of Economic Dynamics and Contol 17(5-6), Fama, Eugene F (1970) Efficient capital makets: A eview of theoy and empiical wok. Jounal of Finance 25(2), Fama, Eugene F, and Kenneth R Fench (1988) Dividend yields and expected stock etuns. Jounal of Financial Economics 22(1), 3 25 Hamilton, James D. (1994) Time Seies Analysis (Pinceton Univesity Pess) LeRoy, Stephen F, and Douglas G Steigewald (1995) Volatility. Handbook of Finance 9, Samuelson, Paul A (1965) Poof that popely anticipated pices fluctuate andomly. Industial Management Review 6(2),

23 Shille, Robet J (1981) Do stock pices move too much to be justified by subsequent changes in dividends? Ameican Economic Review 71(3), Stambaugh, Robet F. (1999) Pedictive egessions. Jounal of Financial Economics 54(3), Welch, Ivo, and Amit Goyal (2007) A compehensive look at the empiical pefomance of equity pemium pediction. The Review of Financial Studies 21(4),

24 A Appendix: Asymptotic Distibution of Estimatos In this section we deive the asymptotic distibution of b, b d and φ. We do so using Poposition 11.1 in Hamilton (1994). Fo completeness, we estate the poposition below. Poposition 4 Let y t = c + θ 1 y t 1 + θ 2 y t θ p y t p + ε t (12) whee ε t is independent and identically distibuted with mean 0, vaiance Ω, and E(ε it ε jt ε lt ε mt ) < fo all i, j, l, m and whee the oots of I n θ 1 z θ 2 z 2 θ p z p = 0 lie outside the unit cicle. Let k np + 1 and let x t be the 1 k vecto x t [1 y t 1 y t 2... y t p]. Let π T = vec( Π T ) denote the (nk 1) vecto of coefficients esulting fom OLS egessions of each of the elements of y t on x t fo a sample of size T : π T = π 1,T π 2,T. π n,t whee [ T ] 1 [ T ] π i,t = x t x t x t y it ; t=1 t=1 24

25 and let π denote the (nk 1) vecto of coesponding population coefficients. Finally, let whee Then 1. (1/T ) T t=1 x tx t 2. π T p π; 3. ΩT p Ω; Ω T = 1 T T ε t ε t, t=1 ε t = [ ε 1t ε 2t... ε nt ] ε it = y it x t π i,t. p Q whee Q = E(x t x t); 4. T ( π T π) poduct. L N(0, (Ω Q 1 )), whee denotes the Konecke We veify that the conditions equied fo Poposition 4 hold in the model pesented hee. We have y t = [ t d t dp t ] It follows that n = 3. The eo tems ae dawn fom a multivaiate nomal distibution with vaiance-covaiance matix (11). The vecto of coefficients θ 1 is given by 0 0 b θ 1 = 0 0 b d 0 0 φ The fist two columns have zeos because the egessand is dp t 1. The coefficients on all lags geate than one ae zeo. 25

26 The condition E(ε it ε jt ε lt ε mt ) < fo all i, j, l, m is satisfied because the eo tems ae nomally distibuted. The condition that the oots lie outside the unit cicle is satisfied because b, b d and φ ae less than unity. Because we ae not egessing on past values of t and d t, we edefine k and x t accodingly. We have k = = 2. The vecto x t is given by x t = [1 dp t ] The matix Π T is given by Π T = a a d a dp b b d φ The vecto π T vec(π T ) follows. We can now calculate Q E(x t x t). We have Q = E(x t x t) = [ σ 2 dp /(1 φ2 ) whee, WLOG, we have assumed dp t is mean-zeo. Because Q is a diagonal matix, it is easy to invet. We have Q 1 = [ (1 φ 2 )/σ 2 dp We can calculate Ω = E(ε t ε t). We have ] σ 2 σ,d σ,dp Ω = σ,d σd 2 σ d,dp σ,dp σ d,dp σ 2 dp ] 26

27 The Konecke poduct Ω Q 1 follows. In paticula, we note the asymptotic vaiance of T ( b b ) is given by σ 2 /σ 2 dp (1 φ2 ). It follows that Similaly, ava( b ) = 1 T ava( b d ) = 1 T σ 2 (1 φ 2 ) σ 2 dp σ 2 d (1 φ2 ) σ 2 dp ava( φ) = 1 T (1 φ2 ) It should be noted that the expessions fo the asymptotic vaiance ae identical to the standad OLS expessions. A.1 Asymptotic vaiance of We find the asymptotic vaiance of Let h(, φ) using the multivaiate delta method. = /(1 ρ φ). The delta method uses the Taylo appoximation to deive the asymptotic distibution of h(, φ) fom knowledge of the asymptotic joint distibution of and φ. We have β T d N 0, Σ (13) φ φ 0 whee Σ denotes the 2 2 vaiance-covaiance matix. The delta method implies T [h(, φ) h(β, φ)] d N (0, [ h(β, φ)] Σ h(β, φ)) (14) 27

28 whee h(β, φ) denotes the gadient of h evaluated at the mean (β, φ), and pime denotes tanspose. We have It follows that h(β, φ) = 1 1 ρφ ρβ (1 ρφ) 2 (15) [ h(β, φ)] Σ h(β, φ) [ ρβ = 1 1 ρφ ] γ2 (1 ρφ) 2 γ,dp γ,dp γ 2 dp 1 1 ρφ ρβ (1 ρφ) 2 ( = 1 φ2 σ 2 + 2ρβ σ,dp + ρ2 β 2 (1 ρφ) 2 σdp 2 1 ρφ σdp 2 (1 ρφ) 2 ) (16) The asymptotic vaiance of the long-hoizon estimato depends on the value of β. Theefoe it is diffeent unde the null and the altenative hypotheses. Weighted cumulative etun egessions. The delta method can also be used to deive the asymptotic distibution of cumulative etun egessions that ae commonly employed in the liteatue. Geometically declining weights. Conside the egession in which futue etuns ae weighted using ρ, with the weights declining geometically. We have Let g(, ϕ) τ j=0 ρ j t+1+j = a (τ) (τ). We have that + β (τ) (d t p t ) + ε t+1,t+1+τ (17) ( ) g(, ϕ) = (1 + ϕ + ϕ ϕ τ ) = 1 (ρ ϕ) τ+1 1 ρ ϕ We compute g(, ϕ), and evaluate the esulting gadient at the asymp- 28

29 totic means of the estimates. We have It follows that g(β, φ) = 1 (ρϕ) τ+1 1 ρφ ρβ (1 ρφ) 2 (1 (ρϕ) τ (1 + τ(1 ρϕ))) (18) [ g(β, ϕ)] Σ g(β, ϕ) = 1 [ ϕ2 (1 ) ( (ρϕ) τ+1 σ 2 + 2σ,dp ρβ [ 1 (ρϕ) τ (1 + (1 ρϕ)τ) ]) (1 ρϕ) 2 σdp 2 σdp 2 1 ρϕ + ρ2 β 2 [ 1 (ρϕ) τ (1 + (1 ρϕ)τ) ] ] 2 (19) (1 ρϕ) 2 The long-hoizon egession coesponds the limiting case τ. In that case, the vaiance expession in (19) educes to (16). Equal weights. We can also conside egessions with equal weights on futue etuns τ j=0 t+1+j = ã (τ) + β (τ) (d t p t ) + ε t+1,t+1+τ (20) In that case, the asymptotic vaiance of the T ( (τ) β τ ) is given by, ( 1 ϕ 2 (1 ) ( ) ϕ τ+1 σ 2 + 2σ,dp β (1 ϕ) 2 σdp 2 σdp 2 1 ϕ [1 ϕτ (1 + (1 ϕ)τ)] β 2 [ + 1 ϕ τ (1 + (1 ϕ)τ) ] ) 2 (1 ϕ) 2 (21) 29

30 B Appendix: Deivation of Powe Conside two estimatos 1 and 2 that ae asymptotically distibuted as bivaiate nomal. Suppose that they satisfy 1 BN µ 1,N, 2 µ 2,N γ2 1,N γ 12,N γ 12,N γ2,n 2. (22) unde the null hypothesis, and 1 BN µ 1,A, γ2 1,A γ 12,A. (23) 2 µ 2,A γ 12,A γ 2 2,A unde the altenative. Define β i as the citical value of i fo a one-tailed test of the null hypothesis with pobability of Type-I eo Unde the null hypothesis we have ( ) βi µ i,n N = 0.95, fo i = 1, 2 (24) γ i,n whee N is the CDF of the standad univaiate nomal distibution. The powe of test i equals the pobability that i β i unde the altenative, o ( ) powe of βi µ i,a i test = 1 N = 1 N γ i,a ( N 1 (0.95) γ i,n + µ ) i,n µ i,a. (25) γ i,a γ i,a Hee N 1 is the invese of N (note that fo µ i,a = µ i,n and γ i,n = γ 1,A we have powe = 0.05, as expected). It follows that test 2 is moe poweful than test 1 if and only if N 1 (0.95) γ 2,N γ 2,A + µ 2,N µ 2,A γ 2,A N 1 (0.95) γ 1,N γ 1,A + µ 1,N µ 1,A γ 1,A (26) 30

31 We can use (26) to compae the powe of vaious asset picing tests. Conside the tests involving and d. Unde the null hypothesis, the mean of [ ; d ] is equal to [0; µ d,a µ,a ]. Unde the altenative hypothesis, the mean is [µ,a ; µ d,a ]. The vaiance-covaiance matix of the estimatos and d is the same unde the null and the altenative, so we do not distinguish between the two cases and denote the estimato vaiances as γ 2 and γ 2 d. Fom (26), the d test has geate powe than the test if and only if γ d < γ. This esult holds fo any values of the paametes in (22) and (23); in the text we specified the paamete values to be those estimated by Cochane. Unde those paamete values, we have γ d < γ. Theefoe the powe of the test based on d is highe than that based on. Now conside tests involving and. Unde the null hypothesis, the mean of [ ; ] is equal to [0; 0]. Unde the altenative hypothesis, the mean is [µ,a ; µ,a ], whee µ,a = µ,a/(1 ρφ). The vaiance of does not change as we go fom the null to the altenative. Howeve, the vaiance of changes as we move fom the null to altenative. We denote the vaiances of the long-hoizon test unde the null and the altenative as γ 2,N and γ2,a. Fom (25), we have that the long-hoizon test has geate powe than the test if and only if, ( ) N 1 γ,n (0.95) µ,a γ,a γ,a 0 µ,a γ (27) This inequality holds unde the benchmak paamete values, so the longhoizon test is moe poweful than the test. 31