Stock Returns, Dividend Yield, And Book-To-market Ratio

Journal of Banking & Finance 31 (2007) 455–475

www.elsevier.com/locate/jbf

Stock returns, dividend yield,and book-to-market ratio

Xiaoquan Jiang a,*, Bong-Soo Lee b,c

a Department of Finance, College of Business, University of Northern Iowa, Cedar Falls, IA 50614, United Statesb Department of Finance, College of Business, Florida State University, Tallahassee, FL 32306, United States

c KAIST Graduate School of Finance, Korea Advanced Institute of Science and Technology, Seoul, Korea

Received 17 February 2006; accepted 12 July 2006Available online 16 October 2006

Abstract

A dividend yield model has been widely used in previous research that relates stock market val-uations to cash flow fundamentals. Given controversies about using dividends as a proxy for cashflows, a loglinear book-to-market model has recently been proposed. However, these models relyon the assumption that dividend yield and book-to-market ratio are both stationary, and empiricalevidence for this is, at best, mixed. We develop a new model, the loglinear cointegration model, thatexplains future profitability and excess stock returns in terms of a linear combination of log book-to-market ratio and log dividend yield. The loglinear cointegration model performs better than the logdividend yield model and the log book-to-market model in terms of cross-equation restriction testsand forecasting performance comparisons. The superior performance of the loglinear cointegrationmodel suggests that the linear combination may be a better indicator of intrinsic fundamentals thanthe dividend yield or the book-to-market ratio separately.Published by Elsevier B.V.

JEL classification: C52; G12

Keywords: Present value model; Dividend yield; Book-to-market ratio; Cointegration

0378-4266/$ - see front matter Published by Elsevier B.V.

doi:10.1016/j.jbankfin.2006.07.012

* Corresponding author.E-mail addresses: [email protected] (X. Jiang), [email protected] (B.-S. Lee).

mailto:[email protected]

mailto:[email protected]

456 X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475

1. Introduction

In explaining fluctuations in stock market valuation levels, Campbell and Shiller’s(1988) dividend yield model has been widely used. The Campbell–Shiller model relatesthe dividend–price ratio to a present value of expected future returns and future dividendgrowth rates: high prices should eventually be followed by high future dividends, lowfuture returns, or some combination of the two. This model is useful and convenientfor empirical implementation but relies on the stability of corporate dividend policy, whichis often suspected for various reasons. In particular, many firms, especially those that arehigh-tech and high-growth, do not pay regular cash dividends until later in their life cycle.1

Instead, share repurchases have recently become very popular. Therefore, the dividendyield model which uses regular cash dividends may be less attractive.2

Vuolteenaho (2000, 2002) developed an alternative, loglinear book-to-market model.To replace dividends in the loglinear dividend yield model, he introduces the clean surplusaccounting relation: Book value this year equals book value last year plus earnings lessdividends. His model relates the current book-to-market ratio to expected future profit-ability, interest rates, and excess stock returns. The model implies that the book-to-marketratio can be (temporarily) low if the future cash flows are high and/or the future excessstock returns are low.

Both models provide a very useful framework in understanding stock price fluctuationin terms of cash flow fundamentals or profitability. In particular, Vuolteenaho’s log book-to-market model is attractive in that it does not rely on possibly unstable corporate divi-dend policy. However, the loglinear book-to-market model relies on the assumption thatthe difference of log book value and log market value is stationary, even though both seriesare non-stationary. That is, log book value and log market value are assumed to be coin-tegrated with a cointegrating vector [1,�1]. However, empirical evidence on this propertyof the variables is very weak.3

In this paper, we propose a new model, called a loglinear cointegration (LLCI) model,that explains future profitability and excess stock returns in terms of a linear combinationof (or spread between) log book-to-market ratio and log dividend yield. The LLCI modelshows that a linear combination of the log book-to-market ratio and log dividend yieldcan be written as a present value of all expected future returns and returns on equity(accounting returns or profitability). Furthermore, we show that the LLCI model per-forms better than either the log dividend yield model or the log book-to-market modelin terms of cross-equation restriction tests and various forecasting performancecomparisons.

The intuition behind the LLCI model is simple and straightforward. Previous studiesfind that both dividend yields and book-to-market ratios have some predictive powerfor stock returns (e.g., Fama and French, 1988, 1989, 1993; Campbell and Shiller, 1988;

1 Fama and French (2001) document that the percent of firms paying cash dividends among NYSE, AMEX,and NASDAQ non-financial, non-utility firms fell from 66.5 in 1978 to 20.8 in 1999.

2 Given the recent tax law change in 2003 in favor of dividends, the recent trend that share repurchases areincreasing relative to dividends may reverse itself.

3 Even Vuolteenaho (2000) acknowledges that there is marginal evidence against the presence of a unit root inthe book-to-market ratio. See Table A.3 in Vuolteenaho (2000).

X. Jiang, B.-S. Lee / Journal of Banking & Finance 31 (2007) 455–475 457

Hodrick, 1992; Pontiff and Schall, 1998; Vuolteenaho, 2000, 2002; Ali et al., 2003a). How-ever, the assumption of stationarity of these two variables is often suspected. If so, theymay share a common trend, and a linear combination of these variables may yield a betterpredictive power for stock returns. Since book value is closely related to earnings, the coin-tegration between log book-to-market and log dividend yield seems consistent with thecomovements (or cointegration) of earnings, dividends, and stock prices (e.g., Lee,1996, 1998).

The LLCI model can be thought of as an extension of the models in Campbell andShiller (1988) and Vuolteenaho (2000, 2002). Therefore, the LLCI model shares all thebenefits of their loglinear properties and has additional interesting features. First, givenmixed evidence on the previous loglinear models’ assumptions (about the stationarity oflog dividend yield and log book-to-market), the LLCI model exploits possible cointegra-tion between log dividend yield and log book-to-market variables in an explicit manner.

Second, loglinear models’ dynamic implications can be summarized by cross-equationrestrictions on vector autoregression (VAR) coefficients. As a result of taking into accountthe cointegration relation, the LLCI model tends to perform better in the cross-equationrestriction tests than the other two loglinear models. In addition, the LLCI model tends tooutperform the other two loglinear models in in-sample fit of excess returns and in out-of-sample forecast performance tests.

Third, the LLCI model incorporates both dividend yield and book-to-market ratio intoa closed form present value relation that explains expected future profitability and stockreturns. For stock return forecasts, some studies find that dividend yields have predictivepower while others find that book-to-market ratio is informative. The former is related toa finance approach based on the conventional dividend discount model, while the latter isrelated to an accounting approach based on the accounting clean surplus relation. Assuch, the LLCI approach provides an integration of the two approaches.

The paper is organized as follows. In Section 2, we briefly introduce the loglinear div-idend yield model and the loglinear book-to-market model and propose a loglinear coin-tegration model. Section 3 describes data and reports the results of various unit root testsfor variables in the three loglinear models. In Section 4, as a means of testing implicationsof the three loglinear models, we implement cross-equation restrictions tests. Section 5presents the results of estimation of returns based on each model, and Section 6 reportsout-of-sample forecast performance. Section 7 presents forecast performance based on abootstrapping method. Section 8 examines cointegration among dividends, book value,and market value, and Section 9 concludes the paper.

2. Loglinear models

2.1. Model 1: Loglinear dividend yield model

The realized log gross return on a portfolio, held from the beginning of time t to thebeginning of time t + 1, can be written as

logð1þ RtÞ � logðP t þ DtÞ � logðP t�1Þ; ð1Þwhere Rt is the realized return during the period t, Pt is the real price of a stock or stockportfolio measured at the end of time period t, and Dt is the real dividend paid on the port-folio during period t.


Campbell and Shiller (1988) assume that the ratio of the price to the sum of price anddividend is approximately constant through time at the level q. That is, q = P/(P + D),where P and D are the mean values of stock price and dividend, respectively. By usinga Taylor approximation, they derive the following equation:

rt ¼ qpt þ ð1� qÞdt � pt�1 þ k; ð2Þwhere the lowercase letters represent logs of the corresponding uppercase letters (e.g.,rt = log(1 + Rt)). The parameter q is slightly smaller than 1, and k is a constant term. Theyrewrite Eq. (2) in terms of the dividend–price ratio dt = dt � pt and the dividend growthrate Ddt:

rt ¼ k þ dt�1 � qdt þ Ddt: ð3ÞSolving forward by imposing a transversality condition, ignoring a constant term, and

taking the conditional expectation, they obtain

dt ¼ Et

X1j¼0

qj½rtþjþ1 � Ddtþjþ1�" #

: ð4Þ

Eq. (4) states that the spread, the log dividend–price ratio, is an expected discounted valueof all future returns less dividend growth rate discounted at the discount rate q. In otherwords, the log dividend–price ratio is a present value of all expected future one-period‘growth-adjusted discount rates’, rt+j � Ddt+j. Therefore, the log dividend–price ratio pro-vides the optimal forecast of the present value of all expected future returns less future div-idend growth rates.

2.2. Model 2: Loglinear book-to-market model

In accounting literature, an alternative valuation model, the residual income model(RIM), has become popular recently primarily due to its formalization by Ohlson(1990, 1991, 1995) and Feltham and Ohlson (1995) (see also Ohlson, 2005).4 The RIMmaintains that the current stock price equals the current book value of equity plus thepresent value of expected future residual income (or abnormal earnings), which is definedas the difference between accounting earnings and the previous period book value multi-plied by the cost of equity. Jiang and Lee (2005) examine the empirical validity of the div-idend discount model and the RIM, and find that the RIM performs better in the variancebounds test and the VAR-based cross-equation restrictions test (see also Ali et al., 2003b).

In finance literature, the book-to-market equity ratio has been widely used as a risk fac-tor since Fama and French (1992, 1993, 1995, 1996) carefully reexamine the book-to-mar-ket effect.5 They show that book-to-market ratio is related to relative distress.6 However,

4 Frankel and Lee (1998) find that fundamental value (based on a residual income model)-to-price ratio is agood predictor of long-term cross-sectional returns. Ali et al. (2003b) find that the predictive power of this ratiofor future returns is more consistent with a mispricing explanation than a risk-proxy explanation. Dechow et al.(1999) provide evidence to support information dynamics of residual income model.

5 Examples include Fama and French (1992, 1993, 1995, 1996), Pontiff and Schall (1998), Kothari et al. (1995),Breen and Korajczyk (1993), Ali et al. (2003a) and Vuolteenaho (2000, 2002).

6 In contrast to a popular interpretation that book-to-market is a proxy for a state variable associated withrelative financial distress, in an attempt to explain the value effect, Zhang (2005) shows that the value anomalyarises naturally in the neoclassical framework with rational expectations based on costly reversibility andcountercyclical price of risk.


Ali et al. (2003a) provide a market mispricing explanation for the book-to-market effect.7

Kothari and Shanken (1997) find evidence that both book-to-market and dividend yieldtrack the variation in expected stock returns over time. However, Kothari et al. (1995)and Breen and Korajczyk (1993) argue that there is a survivorship bias in the data usedto test these new asset pricing specifications.8

Vuolteenaho (2000, 2002) proposes an alternative, accounting-based, approximate pres-ent value model.9 He derives the loglinear book-to-market-ratio model by using a log-lin-earized RIM which is based on the clean surplus relation, allowing for time-varyingdiscount rates. In deriving the model, Vuolteenaho assumes that the difference of log bookvalue (bvt) and log market value (mvt) is stationary even though both series are non-sta-tionary. That is, bvt and mvt are cointegrated with a cointegrating vector [1,�1], and thelog book-to-market ratio, ht, is stationary. He also assumes that the log dividend–priceratio is stationary.

Let BVt, MVt, Xt and Dt be the book value of equity, market value of equity, earnings,and dividends, respectively. Then, the book-to-market ratio can be written as:

BVt

MVt¼ ð1þ X t=BVt�1ÞBVt�1 � Dt

ð1þ ðDMVt þ DtÞ=MVt�1ÞMVt�1 � Dt

¼ ð1þ X t=BVt�1 � Dt=BVt�1Þðð1þ DMVt þ DtÞ=MVt�1 � Dt=MVt�1Þ

BVt�1

MVt�1

: ð5Þ

Using first-order Taylor series approximations, solving forward a difference equation,and taking the expectations, Vuolteenaho approximates this nonlinear relation of thelog book-to-market ratio, ht = log (BVt/MVt), as a linear model:

ht�1 ¼XN

j¼0

qjEtðrtþj � ftþjÞ �XN

j¼0

qjEtðartþj � ftþjÞ þ kt; ð6Þ

where q is a parameter, q < 1, art is the logROE (i.e., art = log(1 + Xt/BVt�1), ft is the logone plus the interest rate, (rt � ft) is the excess log stock return, and kt is the one-periodapproximation error. Eq. (6) states that the log book-to-market ratio is an infinite dis-counted sum of expected future excess stock returns less profitability (art+ j � ft+j). Thebook-to-market ratio can be (temporarily) low if future cash flows are high and/or futureexcess stock returns are low.

7 Ali et al. (2003a) find, among other things, that the book-to-market effect is greater for stocks with higheridiosyncratic volatility. Griffin and Lemmon (2002) also document that high book-to-market cannot be explainedas a risk factor, instead it is related to mispricing. Bali and Wu (2005) find that the loading of book-to-market isnot significant.

8 Lo and Mackinlay (1990) raised the issue of data snooping in general. However, Kim (1997) finds that book-to-market equity still has predictive power after carefully considering the potential issue of data snooping:selection bias and errors-in-variables bias. Ferson and Harvey (1999) emphasize the importance of conditioninginformation in testing these new multifactor asset pricing models, while Harvey and Siddique (2000) propose anasset pricing model that incorporates conditional skewness.

9 Campbell and Vuolteenaho (2004) propose a two-beta model that captures a stock’s risk in two risk loadings,cash-flow beta and discount-rate beta. The return on the market portfolio can be split into two components, onereflecting news about the market’s future cash flows and another reflecting news about the market’s discountrates.


2.3. Model 3: Loglinear cointegration model

In the spirit of Campbell and Shiller (1988) and Vuolteenaho (2000, 2002), we beginwith the definitions of market and accounting returns (i.e., ROE):

rt � log½ðP t þ DtÞ=P t�1� ¼ logðP t þ DtÞ � lnðP t�1Þ; ð7Þart � logð1þ X t=Bt�1Þ ¼ log½ðBt þ DtÞ=Bt�1� ¼ logðBt þ DtÞ � lnðBt�1Þ; ð8Þ

where rt is the log of one plus the real return on a stock held from time t � 1 to time t, art isthe log of one plus the return on equity or accounting return, and Bt is book value. Using aTaylor expansion and ignoring a constant term, we obtain

rt ¼ qpt þ ð1� qÞdt � pt�1; and ð9Þart ¼ q1bt þ ð1� qÞdt � bt�1; ð10Þ

The parameters q = P/(P + D) and q1 = B/(B + D) are constants where P, D, and B arethe mean values of stock price, dividend, and book value, respectively.

Solving forward, taking the conditional expectation, ignoring a constant term andimposing the transversality condition, we obtain the log book-to-market, bpt, as

bpt ¼1

q

X1j¼1

qjEtrtþj �X1j¼1

qjEtartþj þ ðq� q1ÞX1j¼1

qjEtdbtþj

" #; ð11Þ

where dbt denotes the log dividend-to-book value ratio. In Vuolteenaho (2000, 2002), thelast term in Eq. (11) is ignored. However, Lamont (1998) finds that the dividend payoutratio contains primary information about short run variations in stock returns. Now,we approximate the log dividend-to-book value ratio, dbt, by using an AR(1) process,

dbt ¼ udbt�1 þ et; ð12Þwhere E(et) = 0 for all t. It follows that

EtðdbtþjÞ ¼ ujdbt: ð13ÞBy substituting Eq. (13) into (11), we obtain a loglinear cointegration model:

st ¼1

q

X1j¼1

qjEtðrtþj � artþjÞ; ð14Þ

where st is a spread between (or a linear combination of) bpt and dpt: st =q((1 + k)bpt � kdpt), with k = (q � q1)u/(1 � qu).10 Eq. (14) implies that the linear com-bination of the book-to-market ratio and dividend yield is high if future returns are high,and/or if the future profitability (or cash flows) is low. It explains the log spread, st, as apresent value of all expected future ‘‘fundamental adjusted discount rates,’’ rt+j � art+j.

11

This represents the combined effect of expected future discount rates and accounting

10 A more detailed derivation of Eq. (14) is available from the authors upon request.11 Comparison of (14) with the log book-to-market ratio model of Vuolteenaho (6) indicates that the

approximation error in (6) amounts to the last term on the right-hand-side of (11). As such, the approximationerror in (6) may be safely ignored when the mean values of book equity and market equity are equal (i.e., q = q1).In Vuolteenaho’s model, the error term can be safely ignored for the purpose of variance decompositions.However, the error term may matter in deriving cross-equation restrictions and in-sample and out-of-sampleforecasts based on the model.


returns on the spread. Thus, the LLCI model combines and extends the residual incomemodel and dividend discount model by taking into account the possible cointegration.Whether this loglinear cointegration model turns out to be a useful extension remainsan empirical issue to which we now turn our attention.

3. Data and preliminary empirical results

3.1. Data

For empirical estimation and tests of the three loglinear models, we employ theannual S&P industrial index for the sample period of 1946–2004, which is obtained fromStandard & Poor’s Analysts’ Handbook 2005. We choose the S&P Industrials Index,instead of the S&P 500 Index, in part because the former is available for a longer period.The price index is the end of calendar year price. Dividend is the total amount of cash div-idends for both common and preferred stocks. Earnings are basic earnings per shareadjusted to remove the defect of all special items from the calculation; they reflect earningsper share which exclude the effect of all non-recurring events. Book value represents thecommon and preferred shareholder’s interests. It includes capital surplus, common stock,non-redeemable preferred stock, redeemable preferred stock, retained earnings, and trea-sury stock. All variables are deflated by the consumer price index.12

3.2. Tests for cointegration

One of the major assumptions in the loglinear book-to-market model of Vuolteenaho(2000, 2002) is that the difference of log book value (bvt) and log market value (mvt) is sta-tionary even though both series are non-stationary. That is, bvt and mvt are cointegratedwith a cointegrating vector [1,�1], and the log book-to-market ratio, ht, is stationary.He also assumes that the log dividend–price ratio is stationary. As a means of evaluatingthe empirical validity of these assumptions, we implement the augmented Dickey–Fuller(ADF) and Phillips and Perron (PP) tests of unit root for the variables in the loglinear mod-els. While these two procedures test for the null hypothesis of a unit root in a variable, wealso implement the KPSS (Kwiatkowski et al., 1992) tests for the null of stationarity.

Although we fully recognize the problems associated with the power of various unitroot tests, we include these tests for two reasons. First, the standard theory of inferencein regressions with stochastic regressors requires that all variables be stationary. If weregress expected excess returns on variables with a unit root, the conventional standarderrors may be misleading. Second, the estimation and test results are likely to be sensitiveto the stationarity of variables because all loglinear models either assume (Campbell andShiller, and Vuolteenaho) or imply (loglinear cointegration model) the stationary of thevariables on the left-hand-side. When the right-hand-side variables are stationary, havinga non-stationary variable on the left-hand-side would be inconsistent.

We present the unit root test results in Table 1. First, we test for a unit root in the right-hand-side variables in the three loglinear models: dividend growth-adjusted return

12 For a robustness check, we have used the DJIA data for the sample period of 1920–2004, and replicatedTables 1–5. The results are very similar to those using the S&P index. To save space, we do not report the results.

Table 1Unit root and stationarity tests for loglinear models

Variables q ADF PP KPSS (mu) KPSS (tau)

rt � Ddt 1 �6.395*** �7.603*** 0.080 0.0552 �4.806*** �7.674*** 0.087 0.0593 �3.965*** �7.747*** 0.090 0.0614 �2.968** �7.716*** 0.088 0.060

rt � Dbdt 1 �6.232*** �7.186*** 0.061 0.0212 �6.494*** �7.243*** 0.078 0.0283 �6.259*** �7.542*** 0.114 0.0414 �3.150** �8.104*** 0.174 0.064

rt � art 1 �5.253*** �6.439*** 0.104 0.0942 �3.592*** �6.428*** 0.111 0.1013 �3.067** �6.427*** 0.112 0.1034 �2.815* �6.427*** 0.110 0.101

dpt 1 �0.870 �0.712 1.854*** 0.343***2 �0.738 �0.665 1.283*** 0.246***3 �0.965 �0.666 0.994*** 0.195**4 �1.483 �0.718 0.823*** 0.164**

bdpt 1 �3.020* �3.049** 0.470** 0.365***2 �2.176 �2.953** 0.374* 0.294***3 �1.440 �2.808* 0.320 0.255***4 �0.850 �2.747* 0.279 0.224***

bpt 1 �1.394 �1.147 1.858*** 0.330***2 �1.098 �1.069 1.283*** 0.237***3 �1.461 �1.067 0.993*** 0.188**4 �1.443 �1.089 0.820*** 0.159**

st 1 �3.229** �3.128** 0.620** 0.139*2 �2.344 �3.016** 0.473** 0.1093 �2.943** �3.056** 0.395* 0.0944 �2.753* �3.101** 0.351* 0.086

We present the results of the unit root tests and the stationarity tests using the annual S&P industrial index data(from 1946 to 2004). dt, bdt, rt, and art are regular dividend, broad dividend, market return, and accounting return,respectively. rt � Ddt, rt � Dbdt, and rt � art are regular dividend adjusted return, broad dividend adjusted return,and accounting return adjusted return, respectively. dpt, bdpt, bpt, and st are log dividend yield, broad dividendyield, book-to-market ratio, and loglinear cointegration model’s spread, respectively. All variables are in real values.We report the augmented Dickey–Fuller (ADF), Phillips–Perron (PP), and KPSS test statistics. We consider the laglengths (q) of one to four for each variable for robustness checks. The Schwarz Bayesian criterion (SBC) chooses lag3 for bdp and s, and lag 1 for all others. For the ADF and PP unit root tests of the spreads S, critical values with 100(200) observations are 10%, �3.03 (�3.02); 5%, �3.37(�3.37); and 1%, �4.07(�4.00), respectively (see Engle andYoo, 1987, Table, p. 157). For KPSS tests, critical values are 10%, 0.347 (0.119); 5%, 0.463 (0.146); and 1%, 0.739(0.216), for mu (tau), respectively. *, **, and *** represent significance at 10%, 5% and 1% levels, respectively.


(rt � Ddt), broad dividend growth-adjusted return (rt � Dbdt, where bdt is the broadlydefined dividend), and accounting return adjusted return (rt � art). The null hypothesisof a unit root in each variable is rejected by both ADF and PP tests, in particular whenthe lag length in the test is chosen by Schwarz Bayesian information criterion (SBC). Con-sistent with this result, the null hypothesis of the stationarity of these variables is notrejected by the KPSS test for any lags chosen. This implies that the right-hand-side vari-ables in all three loglinear models are stationary.


Now we test for a unit root in the left-hand-side variables in the three loglinear models.The null hypothesis of a unit root in the log dividend yield (dpt, bdpt) and log book-to-market (bpt) series is not rejected by either ADF or PP tests for any lag lengths considered.In addition, the null of stationarity in dividend yield and book-to-market is rejected by theKPSS test for any lags considered. These indicate that the log dividend yield and the logbook-to-market series are non-stationary. This result is consistent with the findings ofCampbell and Shiller (1987).

However, the null of a unit root in the spread, st, in the LLCI model is rejected by bothADF and PP tests for any lag length except for the ADF test with two lags. Similarly, thenull of stationarity of the spread is not rejected by the KPSS test for any lags considered.This suggests that the spread in the LLCI model is stationary, and thus log dividend yieldand log book-to-market are cointegrated.13 In sum, the results in Table 1 are generallysupportive of the assertion that log book-to-market and log dividend yield are cointe-grated as implied by the LLCI model. Thus, the LLCI model seems more consistent withthe data than the loglinear dividend yield model and the loglinear book-to-market model.

4. Cross-equation restriction tests

Now we turn to the test of the implications of the three loglinear models. It is noted thatall three models are in the dynamic expectations framework and linear in the log. There-fore, we can summarize the models’ implications by cross-equation restrictions on a VARsystem of the relevant variables. For the loglinear dividend yield model, consider a bivar-iate vector autoregressive (BVAR) representation of dpt (=dt) and rt � Ddt,

dpt

rt � Ddt

� �¼

aðLÞ bðLÞcðLÞ dðLÞ

� �dpt�1

rt�1 � Ddt�1

� �þ

u1t

u2t

� �; ð15Þ

where the variables in the vector are demeaned, and a(L), b(L), c(L), and d(L) are the kth-

order polynomials in the lag operator (e.g., aðLÞ ¼Pk

j¼1ajLj�1h i

, with L being the lag

operator, LkXt = Xt�k), u1t = dpt � E[dptjdpt�s, rt�s � Ddt�s,] for s = 1,2, . . . ,k, andu2t = (rt � Ddt) � E[(rt�s � Ddt�s)jdpt�s, rt�s � Ddt�s], for s = 1,2, . . . ,k. This BVAR canbe stacked into a first-order VAR system as

dpt

��

dpt�kþ1

rt�Ddt

��

rt�kþ1�Ddt�kþ1

266666666666664

377777777777775¼

a1 � � � ak b1 � � � bk

1

0 � 0

1

c1 � � � ck d1 � � � dk

1

0 � 0

1

266666666666664

377777777777775

dpt�1

��

dpt�k

rt�1�Ddt�1

��

rt�k�Ddt�k

266666666666664

377777777777775þ

u1t

0

�0

u2t

0

�0

266666666666664

377777777777775

ð16Þ

13 We employ a dynamic least squares (DLS) technique proposed by Stock and Watson (1993) to generate theoptimal estimates of the cointegrating parameters in the spread, s, since the spread itself is endogenouslydetermined.


or

zt ¼ Azt�1 þ ut:

The first-order VAR representation is useful because we can obtain forecasts of futurezts as

E½ztþkjHt� ¼ Akzt; ð17Þwhere Ht includes current and past values of zt (i.e., dpt�j and rt�j � Ddt�j for all j P 0).

Define g1 0 and g2 0 as row vectors with 2k elements, all of which are zero except for thefirst element of g1 0 and the (k + 1)st element of g2 0, being unity. Then, these vectors canselect dpt and rt � Ddt as

dpt ¼ g10zt; and rt � Ddt ¼ g20zt:

Thus, by projecting the loglinear dividend yield model (4) onto the information set Ht, wecan characterize the loglinear dividend yield model as the following restriction:

dpt ¼ g10zt ¼X1j¼0

qjg20Ajþ1zt: ð18Þ

Assuming a non-singular variance–covariance matrix of u1t and u2t, we can rewrite Eq.(18) as

g10 ¼ g20AðI � qAÞ�1; or

g10 ¼ ðqg10 þ g20ÞA:ð19Þ

Specifically, the constraints imposed by Eq. (19) are

qai þ ci ¼ 1; for i ¼ 1;

qai þ ci ¼ 0; for i ¼ 2; 3; :::k; and

qbi þ di ¼ 0; for all i ¼ 1; 2; 3; . . . ; k:

ð20Þ

Thus, under the null hypothesis that the loglinear dividend yield model holds, the restric-tions in (19) (or (20)) should hold.

For the loglinear book-to-market model, we consider a BVAR representation of the logbook-to-market ht and (rt � ft) � (art � ft). For the loglinear cointegration model, we con-sider a BVAR representation of the spread st and rt � art. Then, by following a similarprocedure, we can summarize each model by a set of cross-equation restrictions on theBVAR coefficients.

In Table 2, we provide results of the cross-equation restriction tests for the loglineardividend yield model (model 1), the loglinear book-to-market model (model 2), and theLLCI model (model 3). For the convenience of comparison and to better illustrate the nat-ure of the loglinear cointegration model, we use both narrow dividends (cash dividends) inmodel 1a, and broad dividends (generated by the clean surplus relation) in model 1b formodel 1.

We observe in Table 2 that the restrictions derived from model 1 (including 1a and 1b)and model 2 are strongly rejected by the data summarized by a VAR for any lags consid-ered. In contrast, the restrictions from model 3 are not rejected by the data for any lagsconsidered. While the p-values for models 1 (including 1a and 1b) and 2 for any four lagsare substantially less than 1%, the p-values for model 3 with lags one to four are 0.525,

Table 2Cross-equation restriction tests for loglinear models

q Model 1a Model 1b Model 2 Model 3

v2 p-value v2 p-value v2 p-value v2 p-value

1 312.217 0.000 26.852 0.000 16.430 0.000 1.289 0.5252 591.033 0.000 37.947 0.000 19.271 0.001 3.936 0.4153 589.996 0.000 48.061 0.000 22.784 0.001 5.879 0.4374 598.750 0.000 56.055 0.000 26.167 0.001 14.289 0.075

We present the test results of cross-equation restrictions derived from loglinear models using the annual S&Pindustrial index data (from 1946 to 2004). The term ‘q’ represents the number of lag length in the VAR. Models1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividend yield model, book-to-market model,and loglinear cointegration model, respectively.


0.415, 0.437, and 0.075, respectively. In summary, the VAR-based cross-equation restric-tion tests show that model 3, the LLCI model, fits the data substantially better thanmodels 1 and 2.

5. Forecasting expected returns

Empirical research finds that expected excess return has a positive relation with divi-dend yield and book-to-market ratio in both cross-section and time-series relations. Forexample, Litzenberger and Ramaswamy (1979), Kothari and Shanken (1992), and Bren-nan et al. (1998) find that dividend yield has some predictive power for cross-section excessstock return, while Shiller (1984), Fama and French (1988, 1989), Campbell and Shiller(1988, 2001), Kothari and Shanken (1997), and Lamont (1998) find that dividend yieldhas predictive power for time-series excess returns. Book-to-market ratio has also beenfound to have predictive power in both cross-section and time-series excess returns (e.g.,Fama and French, 1992, 1993; Kothari and Shanken, 1997; Lewellen, 1999). Guo(2006) provides evidence of out-of-sample forecast of stock returns. Similarly, we explorethe predictive ability of the loglinear spread in model 3 and compare the result with that ofthe log book-to-market ratio in model 2 and the dividend yield in model 1. Based on theabove theoretical analysis and empirical tests, we conjecture the predictive ability of thespread to perform at least as well as that of the log book-to-market ratio and the dividendyield.

Panel A of Table 3 presents the results of regressing annual log real return, rt, on thelagged log dividend yield (models 1a and 1b), lagged log book-to-market (model 2), andlagged log spread of the LLCI model (model 3), respectively. The model forecasts realreturns in 2004, using 2003 values of the regressors. Estimation error is the standard errorof the point estimate based on sampling error in the coefficients. Total forecast errorincludes both sampling error and residual error. The t-statistics computed usingNewey–West heteroskedastic-robust standard errors with two lags are in parentheses.The term R2 in the table is the adjusted R2.

The slope coefficients on dividend yield using both narrow and broad dividend, on logbook-to-market, and on LLCI model’s spread are 0.080, 0.042, 0.057 and 0.191, respec-tively. While the coefficients on dividend yield and book-to-market are not significantlydifferent from zero, the coefficient on LLCI model’s spread is significantly different fromzero. The adjusted R2 for the four models are 0.039, 0.016, 0.007 and 0.091, respectively.

Table 3OLS and VAR estimates

a b R2 Forecast return

Panel A: OLS

Model 1a rt+1 = a + bdpt + ut+1

�0.024 0.080 0.039 0.000 Forecast(�0.323) (1.480) 0.011 Estimate error[�0.435] [1.776] 0.168 Total forecast error

Model 1b rt+1 = a + bbdpt + ut+1

0.195** 0.042 0.016 0.057 Forecast(2.202) (1.287) 0.003 Estimate error[2.139] [1.411] 0.161 Total forecast error

Model 2 rt+1 = a + bbpt + ut+1

0.107*** 0.057 0.007 0.036 Forecast(2.903) (0.959) 0.007 Estimate error[2.834] [1.200] 0.166 Total forecast error

Model 3 rt+1 = a + bst + ut+1

0.070*** 0.191*** 0.091 0.070 Forecast(3.409) (2.648) 0.001 Estimate error[3.466] [2.602] 0.154 Total forecast error

Panel B: Vector autoregressive (VAR) model estimates

Model 1a Constant rt Ddt dpt R2 Forecast

�0.030 0.137 0.178 0.079 0.046 0.031 Forecast(�0.435) (1.241) (0.683) (1.624) 0.001 Estimate error

0.162 Total forecast error

Model 1b Constant rt Dbdt bdpt R2

0.178* 0.188 0.042 0.040 0.015 0.061 Forecast(1.779) (1.333) (0.728) (1.071) 0.001 Estimate error


Model 2 Constant rt Dbt bpt R2

0.114*** 0.197 �0.218 0.071 0.019 0.039 Forecast(3.828) (1.606) (�0.819) (1.407) 0.001 Estimate error


Model 3 Constant rt Dbpt Ddpt st R2

0.078*** �0.092 �0.346 0.064 0.229*** 0.141 0.031 Forecast(3.583) (�0.331) (�1.168) (1.350) (2.904) 0.001 Estimate error


We report regressions of stock returns using the S&P industrial index data of 1946–2004. The dependent variablert is log return. dpt, bdpt, bpt, and st are log (narrow) dividend yield, log broad dividend yield, log book-to-market, and loglinear cointegration model’s spread, respectively. All variables are in real values. Models 1a, 1b, 2,and 3 represent (narrow) dividend yield model, broad dividend yield model, book-to-market model, and loglinearcointegration model, respectively. Estimation error is the standard error of the point estimate, based on samplingerror in the coefficients. Total forecast error includes both sampling error and residual error. We compute the t-statistics in parentheses and brackets using the Newey–West heteroskedastic-robust standard errors with two lagsand using bootstrapped standard errors, respectively. R2 is the adjusted R2. *, **, and *** represent significance at10%, 5% and 1% levels, respectively.


In addition, the total forecast errors (estimation error) for the four models are 0.168(0.011), 0.161 (0.003), 0.166 (0.007) and 0.154 (0.001), respectively. This result shows that


the predictive ability of the LLCI model’s spread is better than that of either the dividendyield (narrow or broad) or the log book-to-market from the perspective of tracking stockreturn, goodness of fit and accuracy.

Panel B of Table 3 reports the results of the VAR estimation and forecast based on thethree loglinear models. To save space, we only report the return forecast regression. Thestandard errors are corrected for heteroskedasticity (Newey and West, 1987). Consistentwith the OLS forecast, the LLCI model’s spread is the only regressor which can forecastthe stock returns. The coefficient on the spread is 0.229 and the t-statistic computed usingNewey–West heteroskedastic-robust standard errors with two lags is 2.904. The R2 statis-tics of expected real return equations in the loglinear dividend yield model (narrow andbroad), the loglinear book-to-market model, and the LLCI model are 0.046, 0.015,0.019, and 0.141 respectively. The total forecast errors in the four models are 0.162,0.161, 0.161 and 0.148 respectively. The total forecast error in the LLCI model is thesmallest. In summary, as in the OLS forecast, the VAR forecast shows that the LLCIspread is the best forecast variable among those considered.

6. Out-of-sample forecast

We now consider out-of-sample forecasts of each model to examine whether the aboveresults are materially affected by small-sample biases. We compare the root-mean-squared errors (RMSE) from a series of one-year-ahead out-of-sample forecasts obtainedfrom the LLCI model to that of the loglinear book-to-market model and the loglineardividend yield model (see Lettau and Ludvigson, 2001a,b). One possible concern maybe the potential for look-ahead bias because the spread st is estimated using the full sam-ple. To address this concern, we use recursive regressions, re-estimating both the spreadst and the forecast model for each period using only data available at the time of theforecast, adding one year observations at a time and calculating a series of one-step-ahead forecasts.

Since our forecast comparison is not purely regression-based but model-based, weimplement the Diebold and Mariano (1995) non-nested test. For the model-based com-parison, the stock returns are adjusted by the fundamentals according to the models’specifications. The ROE is used as a fundamental in both Vuolteenaho’s model andthe LLCI model while the dividend growth rate is used as a fundamental in the Camp-bell–Shiller model. To address whether the alternative models encompass the LLCImodel, we conduct the Diebold and Mariano (DM) test. The DM test provides a formalhypothesis testing procedure for the analysis of competing forecasts from non-nestedmodels. The null hypothesis of the DM test is that the competitor models – the loglineardividend yield model (narrow or broad dividends) and the loglinear book-to-marketmodel – and the preferred model, the LLCI model, have equal forecast accuracy. Thealternative hypothesis is that the preferred model provides superior forecasts to any ofthe competitor models.

In Table 4, we present the results using the RMSE of model 3 to that of model 1 (or 2),RMSE3/RMSE1,2, and the DM test. If the preferred model (e.g., the LLCI model) per-forms better than the competitor models (e.g., the loglinear dividend yield model andthe loglinear book-to-market model), the RMSE3/RMSE1,2, should be less than one.We observe from rows 1, 2, and 3 that the root-mean-squared error ratios, RMSE3/RMSE1,2, are 0.829, 0.910, and 0.863, respectively. They are all less than one and are sig-

Table 4Out-of-sample forecast

Row Model comparison RMSE3/RMSE1,2 DM statistic p-value

1 st vs. dpt 0.829 2.917*** 0.0022 st vs. bdpt 0.910 1.194 0.1163 st vs. bpt 0.863 2.526*** 0.006

RMSE1,2,3/RMSR0

4 dpt vs. constant 1.273 �1.225 0.8905 bdpt vs. constant 1.060 �0.690 0.7556 bpt vs. constant 1.176 �1.630 0.9497 st vs. constant 0.875 0.810 0.209

We report the results of the one-year-ahead, model-based out-of-sample forecast evaluation using the S&PIndustrial (1946–2004) index. dpt, bdpt, bpt, and st are log dividend yield, log broad dividend yield, log book-to-market, and loglinear cointegration model’s spread, respectively. All variables are in real values. In each row, twomodels are compared. Models 1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividend yieldmodel, book-to-market model, and loglinear cointegration model, respectively. The left-hand-side is the realreturn. Model 3 uses lagged s as a predictive variable, while models 1a, 1b and 2 use lagged dpt, bdpt and bpt aspredictive variables, respectively. The column labeled RMSE3/RMSE1,2 reports the ratio of the RMSE of model 3to that of model 1a, 1b, and 2. The rows 4, 5, 6, and 7 provide comparison of each model to a random walkmodel. RMSE1,2,3/RMSR0 reports the ratio of the RMSE of model 1a, 1b, 2, and 3 to that of the random walkmodel. The column labeled ‘‘DM Statistic’’ gives the Diebold and Mariano (1995) test statistic. In rows 1, 2, and3, the null hypothesis is that the model 1a, 1b, 2, and 3 have equal forecast accuracy. In rows 4, 5, 6, and 7, thenull hypothesis is that the model 1a, 1b, 2, 3, and the random walk model have equal forecast accuracy The initialestimation period begins in 1946 and ends in 1994. *, **, and *** represent significance at 10%, 5%, and 1% levels,respectively.


nificant in rows 1 and 3. Thus, the null hypothesis of equal forecast accuracy is rejected infavor of the LLCI model.

It is interesting to compare each model with a constant expected return model (model0). We report the comparison results in rows 4–7 of Table 4. It is not surprising to find thatnone of the models considered outperforms the constant expected return model in the out-of-sample forecast.14 However, it is noted that the RMSE1,2,3/RMSR0 is greater than onefor models 1a, 1b, and 2, while it is less than one for model 3. This is consistent with theresults in rows 1–3 that model 3 provides a better out-of-sample forecast than models 1a,1b, and 2.

In summary, the results in Table 4 show that the LLCI spread st has statistically signif-icant out-of-sample predictive power for the real returns and contains information that isnot included in either log book-to-market or dividend yield. The DM test results suggestthat the LLCI model outperforms both the loglinear dividend model and the loglinearbook-to-market model. The superior performance may also suggest that the LLCI spreadis a better indicator for intrinsic fundamentals than the dividend yield or the book-to-mar-ket ratio separately.

14 Our finding that dividend yield and book-to-market do not outperform the constant model in the out-of-sample forecast is consistent with previous finding; see Bossaerts and Pierre (1999) and Goyal and Welch(forthcoming).


7. Bootstrapping forecast

Stambaugh (1986, 1999) shows that the OLS estimator of the regressor coefficient maybe biased in small samples if the regressor is highly persistent. Consider the model ofreturns analyzed by Stambaugh (1986, 1999), Mankiw and Shapiro (1986), and Nelsonand Kim (1993):

rt ¼ aþ bX t�1 þ ut; ð21ÞX t ¼ dþ cX t�1 þ vt; ð22Þ

where rt is the stock return, Xt�1 is a candidate forecastor variable, and {(ut,vt)0} is an

independently and identically distributed vector sequence. Eq. (21) is the forecast equationand Eq. (22) specifies the evolution of the forecaster. Since Xt follows an AR(1) process,the residuals ut and vt are correlated. The OLS estimator of c in a sample of T observationsis biased toward zero with the bias given by

Eðc� cÞ � � 1þ 3cT

� �: ð23Þ

Stambaugh (1986, 1999) shows that the size of bias in the OLS of b in the forecast equa-tion is proportional to the bias of c in AR(1) process:

Eðb� bÞ ¼ ruv

r2v

Eðc� cÞ: ð24Þ

It is noted from Eq. (23) that the bias in estimates c and b can be large for small T. Sev-eral recent studies discuss alternative econometric methods for correcting the Stambaughbias and conducting valid inference (Cavangh et al., 1995; Ang and Bekaert, 2003; Janssonand Moreira, 2003; Polk et al., 2006; Lewellen, 2004; Torous et al., 2004; Campbell andYogo, 2006).

To deal with the small sample issue, we evaluate the forecasting models by using a boot-strap. In this bootstrap, the observed distribution of the random variables is the best esti-mate of the actual distribution. We implement the bootstrap-based statistical inference asfollows:

1. Estimate each forecasting model and calculate residuals for each model.2. Generate 1000 bootstrap error samples of size 59 from each forecasting model.3. Use the bootstrap errors to compute 1000 series of bootstrap returns for each

model.4. Run a forecasting regression for each model to obtain the root-mean-squared error

ratio (RMSE3/RSME1,2) for in-sample and out-of-sample cases.

We choose the RMSE ratio for three reasons. First, it is simple. Second, RMSE is argu-ably the most commonly used measure of forecasting ability (West, 2005). Third, we cancompare the results for in-sample and out-of-sample.

Table 5 presents the bootstrap forecasting results. We observe that the RMSE3/RMSE1,2 for the in-sample (out-of-sample) forecasts are 0.861 (0.898), 0.860 (0.952),and 0.856 (0.914), respectively. They are all less than one. The significance level is alwaysless than 10%. The bootstrap results show that the LLCI outperforms other loglinermodels in forecasting returns, which is consistent with the forecasting results in the

Table 5Bootstrapping for in-sample forecast and out-of-sample forecast evaluation

Row Model comparison In sample forecast Out-of-sample forecast

RMSE3/RMSE1,2 p-value RMSE3/RMSE1,2 p-value

1 st vs. dpt 0.861 0.034** 0.898 0.030**2 st vs. bdpt 0.860 0.029** 0.952 0.053*3 st vs. bpt 0.856 0.031** 0.914 0.036**

We report ratios of root-mean-squared errors for model 3 to models 1a, 1b, and 2 for both in sample and out-of-sample forecasts using the S&P Industrial (1946–2004) index. dpt, bdpt, bpt, and st are log dividend yield, logbroad dividend yield, log book-to-market, and loglinear cointegration model’s spread, respectively. All variablesare in real values. The rows 1, 2, and 3 provide the model-based forecast comparisons. The column labeledRMSE3/RMSE1,2 reports the ratio of bootstrapping mean of the root-mean-squared forecasting error of model 3to that of models 1a, 1b, and 2. Models 1a, 1b, 2, and 3 represent (narrow) dividend yield model, (broad) dividendyield model, book-to-market model, and loglinear cointegration model, respectively. P-value is the bootstrappingsignificance level. *, **, and *** represent significance at 10%, 5% and 1% levels, respectively.


previous section. They imply that the superior performance of the LLCI model is not dueto a small sample bias.

8. Cointegration among dividends, book value, and market value

Our finding of the bivariate cointegrations between log dividend yield and log book-to-market ratio suggests that there may be a trivariate cointegration relation among log div-idends (dt), log book values (bt), and log market values (pt). Thus, it is worth examining thedynamic relations among these variables in a unified framework.

For this purpose, we test for possible cointegration among the three variables by usingthe procedure of Johansen (1988, 1991). In Panel A of Table 6, we present the results ofthree variable cointegration tests based on the maximal eigenvalue test and the trace testof Johansen. The term r denotes the number of linearly independent cointegrating vec-tors. The null of zero cointegration vector (r = 0) is rejected, whereas the null of eitherless than one cointegration vector (r 6 1) or less than two cointegration vectors (r 6 2)is not rejected at the conventional significance level of 10%. To confirm this, we followJohansen and Juselius (1992) and further examine the determination of the number ofcointegrating vectors based on a formal testing. Using three eigenvalues, we computethree possible cointegration terms: S1, S2, and S3. Unit root tests and stationary testsin Panel B of Table 6 show that S1 is stationary, but S2 and S3 are non-stationary. Over-all, the tests indicate that there is one cointegration vector. Further unit root testsfor possibly as many as three cointegration terms confirm that there is indeed one coin-tegration term, which we call S1. This indicates that dividends, book values, and marketvalues tend to commove over time sharing a common stochastic trend.15 We reportthe estimation result of the trivariate VECM in Panel C of Table 6. We find that thecointegration term is significant in the dividend change and the book value changeequations.

15 We thank a referee for suggesting cointegration test for dividends, book value, and market value.

Table 6Trivariate cointegration test

Eigenvalue k-max Trace H0: r p � r k-max90 Trace90

Panel A: Johansen cointegration test

0.344 24.00 36.24 r = 0 3 13.39 26.700.159 9.85 12.24 r 6 1 2 10.60 13.310.041 2.39 2.39 r6 2 1 2.71 2.71

Variables Lag q ADF PP KPSS (mu) KPSS (tau)

Panel B: unit root and KPSS tests

S1 1 �3.940*** �4.492*** 0.661** 0.371***2 �3.632*** �4.488*** 0.505** 0.285***3 �3.372*** �4.498*** 0.431* 0.245***

S2 1 �2.263 �2.195 0.485** 0.319***2 �1.710 �2.128 0.347** 0.230***3 �2.042 �2.150 0.277 0.185**

S3 1 �0.530 �0.319 1.606*** 0.285***2 �0.718 �0.346 1.121*** 0.202**3 �0.749 �0.345 0.878*** 0.160**

Constant Dpt�1 Ddt�1 Dbt�1 S1t�1 R2

Panel C: Vector error correction model estimates

Dpt �0.017 0.122 0.310 �0.273 �0.017 �0.012t-stat �0.357 1.218 1.076 �0.861 �1.319Ddt 0.118*** 0.089** 0.338** �0.168 0.033*** 0.274t-stat 5.196 2.185 2.420 �0.904 5.198Dbt 0.072*** 0.063 0.244** �0.075 0.017*** 0.088t-stat 4.038 1.587 2.494 �0.574 2.605

We report the results of the Johansen cointegration tests for dt (log dividend), bt (log book value), and pt (logprice) using the S&P Industrial (1946–2004) index. We also report the cointegration terms’ unit root tests usingthe augmented Dickey–Fuller (ADF), Phillips–Perron (PP), and KPSS test. We consider the lag lengths (q) of oneto three for each variable for robustness checks. In panel A, r is the number of linearly independent cointegratingvectors. Tracestatistic ¼ �T

Pni¼rþ1 lnð1� kiÞ; k-max statistic = �T ln(1 � ki), where T is the number of obser-

vations, n is the dimension of the vector (here n = 3), and ki is the ith smallest squared canonical correlations inJohansen (1988, 1991) or Johansen and Juselius (1992). Various spreads Si for i = 1, 2, and 3 are calculated asfollows. s1t = �10.083 * dt + 1.967 * bt + 2.120 * pt, s2t = 3.911 * dt � 8.538 * bt + 2.496 * pt, and s3t = 4.182 *dt � 4.135 * bt � 1.243 * pt.For the ADF and PP unit root tests of the spreads Si for i = 1, 2, and 3, critical values with 100 (200) observationsare 10%, �3.03(�3.02); 5%, �3.37(�3.37); and 1%, �4.07(�4.00), respectively (see Engle and Yoo, 1987, Table,p. 157). For KPSS tests, critical values are 10%, 0.347 (0.119); 5%, 0.463 (0.146); and 1%, 0.739 (0.216), for mu(tau), respectively. *, **, and *** represent 10%, 5% and 1% significant levels, respectively.


Campbell and Shiller (1987) show that a present value model implies a cointegrationbetween two variables in the model. Based on the dividend discount model, they show thatdividends and stock prices are cointegrated, although the cointegration vector is not[1,�1]. That is, dividends and stock prices tend to move together over time. Similarly,the RIM (residual income model), which is a present value model, suggests that there isa cointegration relation among market value, book value, and earnings. Some dividendmodels (e.g., the permanent earning hypothesis and the partial adjustment hypothesis),which are represented as present value models, suggest that earnings and dividendsare cointegrated. Considered together, it is quite possible that dividends, book value,


and market value are cointegrated, sharing a common trend that may reflect a permanentcomponent of fundamentals.16 Intuitively, market value and book value tend to movetogether over time around their fundamental variables such as earnings and dividends.

9. Concluding remarks

Campbell–Shiller’s dividend yield model has been widely used in previous research thatrelates stock price to cash flow fundamentals. Given unstable corporate dividend policy, aloglinear book-to-market model has been proposed recently by Vuolteenaho based on theaccounting clean surplus relation. However, these models rely on the assumption that div-idend yield and book-to-market ratio are stationary. Empirical evidence for this is, at best,mixed.

By extending Campbell and Shiller’s and Vuolteenaho’s models, we have proposed aloglinear cointegration model, which accounts for potential cointegration and explainsfuture profitability and excess stock returns in terms of a linear combination of logbook-to-market and log dividend yield. The loglinear cointegration model takes intoaccount possible non-stationarity of these variables. We have shown that the loglinearcointegration model performs better than either the log dividend yield model or the logbook-to-market model in terms of cross-equation restrictions tests, excess return forecast-ing performance, and out-of-sample forecasting performance comparisons. The superiorperformance of the loglinear cointegration model suggests that the spread may be a betterindicator for intrinsic fundamentals than dividend yield or book-to-market ratio. Since theloglinear cointegration model can be viewed as a combination of the log dividend yieldmodel and the log book-to-market model, its superior performance also suggests thatthe spread may contain useful information that is not contained in either the dividendyield or the book-to-market ratio separately.

Acknowledgements

We would like to thank seminar participants at the University of Northern Iowa, 2003Midwest Finance Association, St. Louis, MO and 2004 Financial Management Associa-tion, New Orleans, LA for their useful comments. We especially thank Giorgio Szego(the Editor), and two anonymous referees for detailed and valuable comments.

References

Ali, Ashiq, Hwang, Lee-seok, Trombley, Mark A., 2003a. Arbitrage risk and the book-to market anomaly.Journal of Financial Economics 69 (2), 355–373.

Ali, Ashiq, Hwang, Lee-seok, Trombley, Mark A., 2003b. Residual-income-based valuation predicts future stockreturns: Evidence on mispricing vs. risk explanations. Accounting Review 78 (2), 377–396.

Ang, Andrew, Bekaert, Geert, 2003. Stock return predictability: Is it there? Unpublished paper, ColumbiaUniversity.

Bali, Turan G., Wu, Liuren, 2005. Estimating the intertemporal capital asset pricing model using industry andsize/book-to-market portfolios. Working paper, Baruch College, CUNY.

16 For a similar cointegration among earnings, dividends, and stock prices, see Lee (1996, 1998). See alsoLamont (1998).


Bossaerts, Peter, Pierre, Hillion, 1999. Implementing statistical criteria to select return forecasting models: Whatdo we learn? Review of Financial Studies 12, 405–428.

Breen, William, Korajczyk, Robert A., 1993. On selection biases in book-to-market based tests of asset pricingmodels. Working paper, Northwestern University.

Brennan, Michael J., Chordia, Tarun, Subrahmanyam, A., 1998. Alternative factor specifications, securitycharacteristics, and the cross-section of expected stock returns. Journal of Financial Economics 4, 345–373.

Campbell, John Y., Yogo, Motohiro, 2006. Efficient tests of stock return predictability. Journal of FinancialEconomics. 81, 27–60.

Campbell, John Y., Shiller, Robert J., 1987. Cointegration and tests of present value model. Journal of PoliticalEconomy 95, 1088–1602.

Campbell, John Y., Shiller, Robert J., 1988. The dividend–price ratio and expectations of future dividends anddiscount factors. Review of Financial Studies 1, 195–228.

Campbell, John Y., Shiller, Robert J., 2001. Valuation ratios and the long-run stock market outlook: An update.NBER Working Paper 8221.

Campbell, John Y., Vuolteenaho, Tuomo, 2004. Bad beta, good beta. American Economic Review 94 (5), 1249–1275.

Cavangh, Christopher L., Graham, Elliott, Stock, James H., 1995. Inference in models with nearly integratedregressors. Econometric Theory 11, 1131–1147.

Dechow, M. Patricia, Hutton, Amy P., Sloan, Richard G., 1999. An empirical assessment of the residual incomevaluation model. Journal of Accounting and Economics 26, 1–34.

Diebold, F.X., Mariano, R.S., 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics13, 134–144.

Engle, R., Yoo, B., 1987. Forecasting and testing in cointegrated systems. Journal of Econometrics 35, 143–159.Fama, E.F., French, K.R., 1988. Dividend yields and expected stock returns. Journal of Financial Economics 22,

3–25.Fama, E.F., French, K.R., 1989. Business conditions and expected returns of stocks and bonds, dividend yields

and expected stock returns. Journal of Financial Economics 22, 3–25.Fama, E.F., French, K.R., 1992. The cross-section of expected returns. Journal of Finance 47, 427–465.Fama, E.F., French, K.R., 1993. Common risk factors in the returns on bonds and stocks. Journal of Financial

Economics 33, 3–56.Fama, E.F., French, K.R., 1995. Size and book-to-market factors in earnings and returns. Journal of Finance 50,

131–155.Fama, E.F., French, K.R., 1996. Multifactor explanations of asset pricing anomalies. Journal of Finance 51, 55–

84.Fama, E.F., French, K.R., 2001. Disappearing dividends: Changing firm characteristics or increased reluctance to

pay? Journal of Financial Economics 60, 3–43.Feltham, A. Gerald, Ohlson, James A., 1995. Valuation and clean surplus accounting for operating and financial

activities. Contemporary Accounting Research 11, 689–731.Ferson, Wayne E., Harvey, Campbell R., 1999. Conditioning variables and the cross section of stock returns.

Journal of Finance 54 (4), 1325–1360.Frankel, Richard, Lee, Charles M.C., 1998. Accounting valuation, market expectation, and cross-sectional stock

returns. Journal of Accounting and Economics 25, 283–319.Goyal, Amit, Welch, Ivo, forthcoming. A comprehensive look at the empirical performance of equity premium

prediction. Review of Financial Studies.Griffin, John M., Lemmon, Michael L., 2002. Book-to-market equity, distress risk, and stock returns. Journal of

Finance 57 (5), 2317–2335.Guo, Hui, 2006. On the out-of-sample predictability of stock market returns. Journal of Business 79 (2), 645–670.Harvey, Campbell R., Siddique, Akhtar, 2000. Conditional Skewness in asset pricing tests. Journal of Finance 55

(3), 1263–1295.Hodrick, Robert J., 1992. Dividend yield and expected stock returns: Alternative procedures for inference and

measurements. Review of Financial Studies 5, 357–386.Jansson, Michael, Moreira, Marcelo J., 2003. Optimal inference in regression models with nearly integrated

regressors. Unpublished paper, Harvard University.Jiang, Xiaoquan, Lee, Bong Soo, 2005. An empirical test of the accounting-based residual income model and

traditional dividend discount model. Journal of Business 78 (4), 1465–1504.


Johansen, Soren, 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control12, 231–254.

Johansen, Soren, 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vectorautoregressive models. Econometrica 59, 1551–1580.

Johansen, Soren, Juselius, Katarina, 1992. Testing structural hypotheses in a multivariate cointegration analysisof the PPP and the UIP for UK. Journal of Econometrics 53, 211–244.

Kim, Dongcheol, 1997. A reexamination of firm size, book-to-market, earnings price in the cross-section ofexpected stock returns. Journal of Financial and Quantitative Analysis 32 (4), 389–463.

Kothari, S.P., Shanken, Jay., 1992. Stock return variation and expected dividends: A time series and crosssectional analysis. Journal of Financial Economics 31, 177–210.

Kothari, S.P., Shanken, Jay., 1997. Book-to-market, dividend yield, and expected market returns: A time seriesanalysis. Journal of Financial Economics 44, 169–204.

Kothari, S.P., Shanken, Jay., Sloan, Richard G., 1995. Another look at the cross-section of expected stockreturns. Journal of Financial Economics 50 (1), 185–224.

Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y., 1992. Testing the null hypothesis of stationarity againstthe alternative of unit root. Journal of Econometrics 54, 159–178.

Lamont, Owen., 1998. Earnings and expected returns. Journal of Finance 53, 1563–1587.Lee, Bong Soo, 1996. Comovements of earnings, dividends, and stock prices. Journal of Empirical Finance 3,

327–346.Lee, Bong Soo, 1998. Permanent, temporary, and nonfundamental components of stock prices. Journal of

Financial and Quantitative Analysis 33, 1–32.Lettau, Martin., Ludvigson, Sydney, 2001a. Consumption, aggregate wealth, and expected stock returns. Journal

of Finance 56, 812–849.Lettau, Martin, Ludvigson, Sydney, 2001b. Resurrecting the (C)CAPM: A cross-sectional test when risk premia

are time-varying. Journal of Political Economy 109, 1238–1287.Lewellen, Jonathan, 1999. The time-series relations among expected returns, risk, and book-to-market. Journal of

Financial Economics 54, 5–43.Lewellen, Jonathan, 2004. Predicting returns with financial ratios. Journal of Financial Economics 74, 209–235.Litzenberger, R.H., Ramaswamy, K., 1979. The effect of personal taxes dividends on capital asset prices: Theory

and empirical evidence. Journal of Financial Economics 7, 163–195.Lo, A.W., Mackinlay, A.C., 1990. Data-snooping biases in tests of financial asset pricing models. Review of

Financial Studies 3, 431–467.Mankiw, N. Gregory, Shapiro, Matthew, 1986. Do we reject too often? Small sample properties of tests of

rational expectations models. Economics Letters 20, 139–145.Nelson, Charles, Kim, M., 1993. Predictable stock returns: The role of small-sample bias. Journal of Finance 48,

641–661.Newey, W.K., West, K.D., 1987. A simple, positive semidefinite, heteroskedasticity and autocorrelation

consistent covariance matrix. Econometrica 55, 703–708.Ohlson, James A., 1990. A synthesis of security valuation theory and the role of dividends, cash flows, and

earnings. Contemporary Accounting Research 6 (2), 648–676.Ohlson, James A., 1991. The theory of value and earnings, and an introduction to the Ball-Brown analysis.

Contemporary Accounting Research 8, 1–19.Ohlson, James A., 1995. Earnings, book values, and dividends in equity valuation. Contemporary Accounting

Research 11, 661–687.Ohlson, James A., 2005. A simple model relating the expected return (risk) to the book-to-market and the

forward earnings-to-price ratios. Working paper, Arizona State University.Polk, Christopher, Thompson, Samuel, Vuolteenaho, Tuomo, 2006. Cross-sectional forecasts of the equity

premium. Journal of Financial Economics 81, 101–141.Pontiff, Jeffrey, Schall, Lawrence D., 1998. Book-to-market ratios as predictors of market returns. Journal of

Financial Economics 49, 141–160.Shiller, Robert J., 1984. Stock prices and social dynamics. Brookings Papers on Economic Activity, 457–498.Stambaugh, Robert F., 1986. Bias in regression with lagged stochastic regression. Working paper, University of

Chicago.Stambaugh, Robert F., 1999. Predictive regressions. Journal of Financial Economics 54, 375–421.Stock, James, Watson, Mark, 1993. A simple estimator of cointegrating vectors in higher order integrated system.

Econometrica 6, 783–820.


Torous, Walter, Rossen, Valkanov, Shu, Yan, 2004. On predicting stock returns with nearly integratedexplanatory variables. Journal of Business 77, 937–966.

Vuolteenaho, Vuomo, 2000. Understanding the aggregate book-to-market ratio and its implications to currentequity-premium expectations. Working paper, Harvard University.

Vuolteenaho, Vuomo, 2002. What drives firm-level stock returns? Journal of Finance 57, 233–264.West, Kenneth D., 2005. Forecast evaluation. Working Paper, University of Wisconsin.Zhang, Lu, 2005. Value premium. Journal of Finance 60 (1), 67–103.

Stock Returns, Dividend Yield, And Book-To-market Ratio

Documents

Transcript of Stock Returns, Dividend Yield, And Book-To-market Ratio