Jiang, X., & Lee, B. S. (2012). Do Decomposed Financial Ratios Predict Stock Returns and...

The Financial Review 47 (2012) 531–564

Do Decomposed Financial Ratios PredictStock Returns and Fundamentals Better?

Xiaoquan Jiang∗Florida International University

Bong Soo LeeFlorida State University

Abstract

We investigate the prediction of excess returns and fundamentals by financial ratios, whichinclude dividend-price ratios, earnings-price ratios, and book-to-market ratios, by decomposingfinancial ratios into a cyclical component and a stochastic trend component. We find bothcomponents predict excess returns and fundamentals. Cyclical components predict increasesin future stock returns, while stochastic trend components predict declines in future stockreturns in long horizons. This helps explain previous findings that financial ratios in theabsence of decomposition find weak predictive power in short horizons and some predictivepower in long horizons. We also find both components predict fundamentals.

Keywords: financial ratios, return predictability, decomposition, fundamentals

JEL Classifications: G12, G14

∗Corresponding author: Florida International University, 11200 SW 8th Street, RB 203B, Miami, FL33199; Phone: (305) 348-7910; Fax: (305) 348-4245; E-mail: [email protected].

We would like to thank Robert Van Ness (the editor) and two anonymous referees for constructiveand detailed suggestions. Helpful comments were received from the participants at seminar at FloridaInternational University, 2008 SUERF Colloquium (European Money and Finance Forum), Munich,Germany, 2009 SFA Conference, Captiva FL, and 2010 FMA Conference, New York, NY. All remainingshortcomings are our own responsibility.

C© 2012, The Eastern Finance Association 531

532 X. Jiang and B. S. Lee/The Financial Review 47 (2012) 531–564

1. Introduction

Whether financial ratios—such as dividend-price ratios, earnings-price ratios,and book-to-market ratios—predict future stock returns has drawn much attentionfrom both practitioners and academia. The prediction of stock returns using financialratios is theoretically rooted in present value models (e.g., the dividend discountmodel, earnings discount model, and residual income model). These models implythat stock return is determined by fundamentals embedded in financial ratios. Usinga log-linear present value model, Campbell and Shiller (1988a) demonstrate thatthe dividend-price ratio is either positively associated with future stock returns ornegatively associated with future dividend growth rates. Empirical evidence generallysuggests that financial ratios can predict future stock returns, especially in longhorizons.1 In contrast, short-horizon returns and the growth of fundamentals aremore difficult to predict.

A consensus view of a quarter century of empirical work has been summarizedin Cochrane’s book, “Asset Pricing” (2001, p. 388): “2. Returns are predictable. Inparticular, (a) Variables including the dividend/price ratio and term premium can infact predict substantial amounts of stock return variation. This phenomenon occursover business cycle and longer horizons. Daily, weekly, and monthly stock returnsare still close to unpredictable . . .”

Recently, this view has been questioned and challenged on several grounds. It iswell known that financial ratios are very persistent, and return shocks are negativelycorrelated with financial ratios. Because of the near-unit-root property of financialratios, the statistical inference leads to uninformative inference on the predictionrelation (e.g., Nelson and Kim, 1993; Stambaugh, 1999; Ferson, Sarkissian andSimin, 2003; Valkanov, 2003; Lewellen, 2004). Ang and Bekaert (2007) provideevidence that dividend yields do not significantly predict excess returns at longhorizons after carefully accounting for small-sample properties of standard tests.Goyal and Welch (2003, 2008) show that financial ratios have poor out-of-sample(OOS) forecast power.

The prediction relation between returns and financial ratios appears to sufferfrom structural instability over time. Particularly in the late 1990s, the predictionrelation does not seem robust (see Viceira, 1996; Lettau and Ludvigson, 2001; Goyaland Welch, 2003; Paye and Timmermann, 2006; Lettau and Van Nieuwerburgh, 2008).

We reexamine the predictive power of financial ratios in predicting stock returnsand fundamentals.2 For this purpose, we propose decomposing financial ratios into a

1 See Rozeff (1984), Fama and French (1988), Pontiff and Schall (1998), Campbell and Shiller (1988a,1988b, 2005), Cochrane (1992, 2008), Goetzman and Jorion (1993), Hodrick (1992), Lewellen (2004),Lettau and Ludvigson (2005), Hecht and Vuolteenaho (2006), Jiang and Lee (2007), Lettau and VanNieuwerburgh (2008), and others.

2 Jiang and Lee (2009) reexamine the intertemporal risk-return relation and find a positive risk-returnrelation by measuring expected returns and conditional variance in a consistent manner using firm funda-mentals. As measures of fundamentals, they use earnings and dividends.

X. Jiang and B. S. Lee/The Financial Review 47 (2012) 531–564 533

cyclical component and a stochastic trend component using Hodrick and Prescott’s(1997) Kalman filter procedure. To address a finite sample bias issue in the forecastingregressions, we use the bootstrap simulation procedure in testing the null hypothesisof no predictability. To circumvent the look-ahead bias in the OOS forecasts, weprovide an alternative decomposition.

Using the decomposed components of financial ratios, we find that both ex-cess returns and fundamentals are significantly predictable for the sample periodsfrom 1926:Q1 to 2008:Q4 and from 1952:Q1 to 2008:Q4. The cyclical componentsof financial ratios predict an increase in stock returns, while the stochastic trendcomponents of financial ratios predict a decrease in stock returns. Their predictivepower peaks at horizon of around 12 quarters. At the horizon of 12 quarters for thesample period from 1926:Q1 to 2008:Q4, the explanatory powers (R2s) of the twocomponents of dividend-price ratio, earnings-price ratio, and book-to-market ratioare 45%, 72%, and 77%, respectively. The cyclical components tend to dominate thestochastic trend component in the explanatory power. Consistent with present valuemodels, we also find that both components predict fundamentals. For example, fordividend-price ratio, we find the cyclical component tends to predict dividend growthwith a negative coefficient while the stochastic trend component tends to predictdividend growth with a positive coefficient.

We also aim to explore the channels through which returns are predicted sothat we can provide better understanding of previous findings that suggest financialratios are weak predictors of stock returns in short horizons, while they show strongerpredictive power in long horizons. For example, the log dividend ratio model antic-ipates that an increase in this financial ratio is associated with future increases instock returns and/or future declines in dividend growth rates. Return forecast studiesgenerally attribute the forecast power of financial ratios to (slow) mean reversion(see Campbell and Shiller, 1988a, 2005).3 We interpret the cyclical component asreflecting a local mean reversion effect, while the stochastic trend component reflectsa long-run persistence (slow mean reversion) effect. Ferson and Xie (2009) find thatthe long-run risk models better capture momentum effect, which is consistent withthis persistence effect.4 By decomposing the financial ratios into two components,we show that the two components of financial ratios can better predict both stockreturns and fundamentals than the financial ratios alone.

Therefore, we find returns and fundamentals are predictable through two chan-nels. In particular, we find that the stochastic trend components are persistent andhave higher explanation powers over long horizons. This indicates that the long-run persistence in the stochastic trend components of financial ratios is particularlyeffective over long horizons. Since the stochastic trend components tend to be more

3 In fixed income research, Fama (2006) also proposes local mean reverting and slow mean revertingcomponents to study the behavior of interest rates. Park (2010) shows the ratio of the short-term-movingto the long-term-moving average has significant predictive power for future returns.

4 Our model can be viewed as the cointegrated version of LRR model in Ferson and Xie (2009).


important in the financial ratios over long horizons, this helps explain previous find-ings that raw financial ratios tend to show some predictive power for stock returnsover long horizons. In addition, our finding that the two components predict stock re-turns in the opposite direction suggests that over short horizons, the two componentsof financial ratios tend to offset much of each other’s prediction. This helps explainthe failure of previous studies to find strong predictive power of raw financial ratiosover short horizons in the absence of decomposition. Our evidence also helps explainwhy the dividend-price ratio does not predict dividend growth. Lettau and Ludvigson(2005) argue that the failure of dividend-price ratio to predict dividend growth is dueto the positive correlation between the expected return and the expected dividendgrowth, suggesting that time-varying investment opportunities are poorly capturedby dividend-price ratio. Our research suggests that these investment opportunitiesare captured separately by each component of dividend-price ratio, and the failure ofdividend-price ratio to predict dividend growth is due to the offsetting effect betweenthe cyclical and stochastic trend components.

In their recent paper, Lettau and Van Nieuwerburgh (2008) suggest that thechanges in the steady-state mean of financial ratios cause the puzzling empirical pat-terns in return prediction. They estimate regime-switching models for the steady-statemean of financial ratios. In regime-switching or structural break models, however,the number of parameters to be estimated grows rapidly with the number of regimesor breaks. Generally, two or, at most, three regimes are allowed, and more regimesimply a lower power. The permanent change in the financial ratios can be attributedto such factors as technological innovations, changes in market participations, andtax codes. As such, two or three regimes may not suffice.

In our research, we employ Hodrick and Prescott’s (1997) filter procedure indecomposing financial ratios into stochastic trend and cyclical components, which ismore flexible than allowing for regime changes. To link Hodrick and Prescott’s (1997)Kalman filter decomposition to the regime-switching models, the cyclical compo-nent (stochastic long-run trend component) in Hodrick and Prescott’s Kalman (1997)filter is similar to the stochastic component (steady-state value) in regime-switchingsettings. The stochastic long-run trend component is, however, time-varying in lowfrequency and can be used to forecast stock returns and growth in fundamentalsas an additional predictor. Therefore, Hodrick and Prescott’s (1997) filter allowsus to generate the time-varying long-run stochastic trend component, which canbe viewed as a time-varying conditional mean. As a result, we do not directly ex-amine the structural break issue as in Lettau and Van Nieuwerburgh (2008). In-stead, we apply Hodrick and Prescott’s (1997) filter method to decompose financialratios.

Intuitively, the stochastic trend component is measured as a weighted averageof financial ratio over time, and cyclical component is measured as residuals. We useHodrick and Prescott’s (1997) filter for two reasons. Their decomposition reflectsnew classical economic theory (see Lucas, 1980, 1981). In terms of methodology,Hodrick and Prescott’s (1997) filter procedure is very flexible. It can be used to


Figure 1

Financial ratios (DP, EP, and BP) and their cyclical components and stochastic trend componentswith NBER business cycles

decompose highly persistent stationary series (e.g., value premium in Chen, Petkovaand Zhang, 2008), as well as potentially nonstationary series (e.g., gross domesticproduct [GDP]). This is particularly useful for financial ratios because it is stilldebated whether financial ratios contain a unit root. In theory, they are expected tobe stationary as we can see from various present value models. However, Figure 1shows that they are highly persistent with a downward trend.


In terms of economic and finance intuition, Blanchard and Watson (1982) andFroot and Obstfeld (1991) show that, if stock price is allowed to deviate from itsintrinsic value (i.e., if bubbles are allowed), regardless of whether they are ratio-nal or irrational, financial ratios can be decomposed into two parts: the randomwalk and bubbles.5 Campbell and Vuolteenaho (2004) show that returns gener-ated by discount rate risk are offset by lower returns in the future (mean rever-sion), whereas returns generated by cash flow risk are never reversed subsequently(persistence). Motivated by this insight, we interpret the predictive power of cycli-cal components of financial ratios as due to a local mean reversion, and that ofstochastic trend components of financial ratios as due to a long-run persistenceeffect. Similarly, Fama and French (2007), in studying average returns on valueand growth portfolios, employ a decomposition of capital gains into three sources:growth in book value, convergence in market-to-book ratios, and drift in market-to-book. Another advantage is that besides the cyclical component, it also generatesthe long-run stochastic trend component, which is time-varying and can be used as apredictor.

Fama and French (2002) argue that average stock return of the last half century isa lot higher than expected. The observation that 1990s’ financial ratios are low, whilestock returns remain high makes the return prediction more difficult. The decomposedfinancial ratios help us better predict returns both in short and long horizons. Usingthe consumption-wealth ratio (CAY), Lettau and Ludvigson (2001, 2005) find thatexcess returns are predictable. Consistent with Lettau and Ludvigson (2001, 2005)and Ang and Bekaert (2007), we find both CAY and relative T-bill rates able topredict stock returns in the univariate regression. However, when we include the twocomponents of financial ratios in the prediction regression, the predictive power ofCAY and relative T-bill rates tends to disappear, while both the cyclical and stochastictrend components of financial ratios predict stock returns significantly in all horizonsconsidered. We find that the two components of financial ratios have more predictivepower than CAYs and relative T-bill rates.

We also find that dividend growth, earnings growth, and accounting returns arepredictable using the cyclical and stochastic trend components of financial ratios.Dividend growth is typically predictable by the cyclical component of dividend-price ratio, especially over short horizons. Accounting returns are predictable by thestochastic trend component of book-to-market ratio, in particular, over long horizons.Earnings growth is predictable by the cyclical component of the earning-price ratioover short horizons and by the stochastic trend component of the earning-price ratioover long horizons.

5 Wu (1997) uses a similar Kalman filter to decompose price-dividend ratio to account for U.S. stock pricevolatility.


2. Methodology

2.1. Decomposition procedure

In view of Figure 1, which shows highly persistent financial ratios, and consid-ering the insights provided by Fama (2006) and Lettau and Ludvigson (2005) andmethodologies developed by King, Plosser, Stock and Watson (1991) and Hodrickand Prescott (1997), we decompose financial ratios into a stochastic trend componentand a cyclical component. To implement the decompositions, we use the Hodrickand Prescott (1997) filter procedure. Hodrick and Prescott’s (1997) statistical ap-proach does not utilize standard time series analysis. Instead, Hodrick and Prescott(1997) proceed in a more cautious manner that requires only prior knowledge thatcan be supported by economic theory. Hodrick and Prescott (1997, p. 2) state, “Themaintained hypothesis, based upon growth theory considerations, is that the growthcomponent of aggregate economic time series varies smoothly over time.” Further-more, it does not require the underlying series to be necessarily nonstationary andis flexible enough to capture potential regime changes in a parsimonious manner.Therefore, it is widely adopted in existing studies.

Let FRt be a log financial ratio at time t. FRt can be log dividend-price ratio,log earnings-price ratio, or log book-to-market ratio. Following Hodrick and Prescott(1997), we model FRt as

FRt = Gt + Ct, (1)

where Gt is a stochastic trend component that varies smoothly over time, and Ct is acyclical component. Since the cyclical component is the residual, it should be closeto zero on average in the long run.

The stochastic trend component can be obtained by solving the following pro-gramming problem:

minGt

{T∑

t=1

C2t + λ

T∑t=1

[Gt − 2Gt−1 + Gt−2]2

}, (2)

where λ is a positive parameter that penalizes variability in the stochastic trendcomponent. The larger the value of λ, the smoother the stochastic trend component.

There are two attractive properties in this decomposition for predicting returns.First, the cyclical component, Ct, is stationary and not persistent. It is well knownthat highly persistent predictors tend to lead to a small-sample problem (Stambaugh,1986, 1999) and spurious predictions (Ferson, Sarkissian and Simin, 2003). A sta-tionary cyclical component is less subject to these criticisms.6 Second, the stochastictrend component is not a constant, instead it is time-varying and “smooth” (Hodrickand Prescott, 1997). Furthermore, it captures fundamental structural changes well.

6 In forecasting regressions, we use the first difference of the stochastic trend component.


Hodrick and Prescott (1997) recommend using the Kalman filter procedure to identifythe unobservable stochastic trend component. We employ the following state-spacerepresentation:

Measurement equation

FRt = [1, 0]

[Gt

Gt−1

]+ Ct . (3)

Transition equation

[Gt

Gt−1

]=

[2 −11 0

] [Gt−1

Gt−2

]. (4)

A nice property of the AR(2) process of Gt is that it allows the cyclical componentto be periodic with a peak in its spectral density function, which implies a positiveshort-run autocorrelation (momentum) and a negative long-run autocorrelation (meanreversion).7 Based on Equations (3) and (4), we employ the Kalman filter procedureto estimate the stochastic trend and cyclical components.

2.2. Models of financial ratios

Campbell and Shiller (1987) develop the following log-linear dividend-priceratio model:

δt = dt − pt = Et

⎡⎣ ∞∑

j=0

ρj[rt+j+1 − �dt+j+1

]⎤⎦ . (5)

Equation (5) states that the spread, the log dividend-price ratio, is an expecteddiscounted value of all future returns and dividend growth rates discounted at thediscount rate ρ. That is, the log dividend-price ratio is an expected discounted valueof all future one-period “growth-adjusted discount rate,” rt+j − �dt+j. As such, thelog dividend-price ratio provides the optimal forecast of the discounted value of allfuture returns or future dividend growth rates, or both.

Lintner (1956), Marsh and Merton (1986), and Lee (1996a, 1996b, 1998) suggestthat dividends are associated with permanent earnings (i.e., the permanent earningshypothesis of dividends).8 Similarly, Pindyck and Rotemberg (1993) express stockprice as a present value of future earnings. Combining Pindyck and Rotemberg’s(1993) present value model of earnings and Campbell and Shiller’s (1988a, 1988b)dynamic present value model, we obtain

7 In macroeconomics studies, Harvey (1985) and Clark (1987), among others, decompose real GDP asAR(2) process.

8 For the theoretical modeling of the idea of permanent earnings, see Lee (1996a, 1996b, 1998).


et − pt = Et

⎡⎣ ∞∑

j=0

ρj[rt+j+1 − �et+j+1

]⎤⎦ , (6)

where et denotes log earnings. Equation (6) states that the log earnings-price ratioprovides the optimal forecast of the discounted value of all future returns or futureearnings growth rates, or both.

Given unstable dividend policy and controversies about the appropriateness ofusing dividends as a proxy for cash flows, Vuolteenaho (2000, 2002) proposes analternative, accounting-based, approximate present value model, the log-linear book-to-market ratio model

bt − pt = Et

⎡⎣ ∞∑

j=0

ρj[rt+j+1 − art+j+1

]⎤⎦ + kt , (7)

where art denotes accounting returns, and kt is an approximation error. Equation (7)states that the log book-to-market ratio is an infinite discounted sum of expectedfuture stock returns and profitability. Analogous to the Campbell-Shiller model, thebook-to-market ratio provides the optimal forecast of the discounted value of allfuture returns or future accounting returns, or both.

In the following discussions, the choice of financial ratios (dividend-price ratio,earnings-price ratio, and book-to-market ratio) and the prediction of returns andfundamentals are based on Equations (5), (6), and (7).

2.3. Predictive regressions

To examine whether the decomposition of financial ratios helps predict stockreturns and growth in fundamentals, we consider the following regressions:

yt,h = α + βFRt + εt,t+h, (8a)

yt,h = α + δCt + εt,t+h, (8b)

yt,h = α + γGt + εt,t+h, (8c)

yt,h = α + δCt + γGt + εt,t+h, (8d)

here yt,h denotes∑h

j=1(rt+j − rf,t+j ) for return regressions or∑h

j=1(Ft+j ) for fun-damentals regressions. FRt denotes a financial ratio, which can be DPt (=dt – pt),EPt (=et – pt), or BPt (=bt – pt). Ct and Gt represent the cyclical and stochastictrend components of financial ratios, respectively. Ft denotes fundamentals such asdividend growth, earnings growth and accounting returns. We choose fundamentalsto match financial ratios based on the present value models (Equations [5], [6], and[7]). For example, dividend growth is used as fundamentals when the predictor isdividend-price ratio.


2.4. Econometric issues

Our forecast regression can be subject to a finite sample bias, although thepersistence of the cyclical and stochastic trend components is lower than that offinancial ratios. As documented by Kendall (1954), Stambaugh (1999), and Ferson,Sarkissian and Simin (2003), the finite sample bias in the forecast regressions can besubstantial and misleading. Long-horizon returns may have potential autocorrelationand heteroskedasticity. We address this issue by applying a wild bootstrap procedure.Our bootstrap follows Goncalves and Kilian (2004) and imposes the null of nopredictability in calculating the critical values. We posit that the data are generatedby the following system under the null hypothesis of no predictability:9

rt+1 = α + ε1t+1,

xt+1 = ρ0 + ρxt + ε2t+1.

We resample r̂t+1 = ∝̂ + u∗t+1 (under H0) where u∗

t+1 = ηtet+1 while e is theresidual from the above regression, η is any random distribution with zero meanand variance 1, and α̂ is the OLS estimator. Calculate the t-statistics T∗ from theregression r̂t+1 = α + βxt + ut+1.

Repeat this 10,000 times to have bootstrap distribution {T ∗}100001 . The bootstrap

critical values can be obtained as the proportion of {T ∗}100001 that is greater than T in

absolute value (in the two-tailed test).The stochastic trend component reflects the long-run relation and requires a

longer sample to estimate the component. However, there is a tradeoff. Similar to theconstruction of CAY in Lettau and Ludvigson (2001), our decomposition uses fullsample and the OOS forecast is subject to the look-ahead bias. We address this issueby using the moving average of the most recent 25 years’ financial ratios as a proxyfor the stochastic trend component.10 Then, we use the difference between financialratio and the stochastic trend component as a proxy for the cyclical component. Bydoing so, we try to circumvent the look-ahead bias as much as we can and showwhether stochastic trend and cyclical components are better able to predict stockreturns and fundamentals.

3. Data

We use stock prices (P), dividends (D, four-quarter moving sum of dividends),and earnings (E, four-quarter moving sum of earnings) of the Standard and Poor’s(S&P) 500 index from 1926:Q1 to 2008:Q4. Following suggestions of Graham andDodd (1934) and Campbell and Shiller (2005), we employ a ten-year moving aver-age to smooth earnings to circumvent the problems associated with seasonality and

9 We also consider AR(1) as the null hypothesis and the bootstrap p-values are similar.

10 Fama (2006) uses a similar approach in calculating the long-term expected spot rate.


earnings manipulation. Since the book value of the S&P 500 index for such a longperiod is not available, we rely on the clean surplus accounting relation to generatebook value (Bt) and assume that the book and market values are identical at thebeginning.11 The risk-free rates (Rf) are measured as one-month T-bill rates fromCRSP. The CAY is from Lettau and Ludvigson (2001). Inflation rates are based onthe Consumer Price Index from the Bureau of Labor Statistics.

We construct three financial ratios: log dividend-price ratio (DP), log earnings-price ratio (EP), and log book-to-market ratio (BP). All the ratios are measured as thedifference between the log of fundamentals and log of prices.12 Based on Equations(5), (6), and (7), we choose fundamentals DD, EE, and EB, which are constructedas the difference in log dividends, the difference in log earnings, and the differencebetween log earnings and log book value, respectively.

The decomposition of financial ratios into the two components requires theselection of the smoothing parameter λ. We follow the suggestion by Hodrick andPrescott (1997) in selecting λ. Assuming that the cyclical components and the seconddifferences of the stochastic trend components are identically and independentlydistributed normal variables with means zero and variances σ 2

1 and σ 22 , they suggest

λ = σ 21 /σ 2

2 . We set λ = 1,600 as an initial value to solve the programming problemin Equation (2). Then, we estimate variances σ 2

1 and σ 22 . We then choose λ as the

ratio of the variances σ 21 /σ 2

2 in Hodrick and Prescott (1997) decomposition.13

In Table 1, we report summary statistics of the three financial ratios (DP, EP, andBP), their cyclical components (CDP, CEP, and CBP), their stochastic trend components(GDP, GEP, and GBP), CAY, relative T-bill rates (RTB), and log excess returns (R).In Panel A, we report mean, standard deviation, autocorrelation, and unit-root testsbased on the augmented Dickey-Fuller test and the Philips-Perron test. We find thatR, CAY, and RTB are stationary, while all financial ratios are highly persistent. Thefirst-order autocorrelation of each financial ratio is greater than 0.96, and the nullhypothesis of unit root in DP and BP cannot be rejected by either the augmentedDickey-Fuller test or the Philips-Perron test.14 Furthermore, Panel A in Figure 1shows a downward trend for all three financial ratios.

In Panel B, we report a correlation matrix between variables. It is interestingto note that all financial ratios are positively and closely correlated, but they are

11 The clean surplus accounting relation requires that all gains and losses affecting book value also areincluded in earnings; that is, the change in book value is equal to earnings minus dividends. This approachhas been used in both accounting and finance research (see, Vuolteenaho, 2002; Jiang and Lee, 2006,2007).

12 See Campbell and Shiller (2005).

13 For the robustness of our results, we also use λ = 1,600 to decompose the financial ratios. The resultsare very similar. We do not report the result to save space.

14 This evidence is consistent with the finding of Campbell and Shiller (1987), Froot and Obstfeld (1991),Jiang and Lee (2007), among others. The null hypothesis of unit root in EP is marginally rejected.


Table 1

Summary statistics

This table reports summary statistics for the financial ratios (log dividend-price ratio (DP), log earnings-price ratio (EP), and log book-to-market ratio (BP)) and their cyclical components (CDP, CEP, and CBP)and stochastic trend component (GDP, GEP, and GBP), consumption-wealth ratio (CAY), relative T-billrates (RTB), and log excess returns (R). The ρ is the first-order autocorrelation. ADF and PP denote theaugmented Dickey-Fuller test and the Philips-Perron test with four lags. The critical values for ADF andPP are: 1%, –3.453; 5%, –2.871; and 10%, –2.572, respectively.

Panel A: Summary statistics and unit-root tests for return and ratios

Series Sample period Mean Std ρ ADF PP

R 1926:Q1–2008:Q4 0.014 0.099 −0.059 −8.161∗∗∗ −19.018∗∗∗DP 1926:Q1–2008:Q4 −3.325 0.455 0.974 −2.558 −2.418EP 1926:Q1–2008:Q4 −2.912 0.379 0.965 −3.000∗∗ −2.751∗BP 1926:Q1–2008:Q4 −0.349 0.468 0.977 −2.335 −2.147CAY 1951:Q2–2008:Q4 0.000 0.014 0.882 −2.732∗ −3.513∗∗∗RTB 1926:Q1–2008:Q4 0.000 0.002 0.714 −7.221∗∗∗ −7.475∗∗∗

Panel B: Correlation matrix for returns and ratios

R DP EP BP CAY RTB

R 1.00 −0.044 −0.097 −0.094 −0.006 −0.175DP 1.00 0.952 0.963 0.162 0.033EP 1.00 0.992 0.086 −0.005BP 1.00 0.064 0.006CAY 1.00 −0.161RTB 1.00

Panel C: Summary statistics and unit-root tests for cyclical and trend components

Series Sample period Mean Std ρ ADF PP

CDP 1926:Q1–2008:Q4 0.000 0.174 0.827 −7.438∗∗∗ −6.146∗∗∗CEP 1926:Q1–2008:Q4 0.000 0.169 0.827 −7.236∗∗∗ −6.198∗∗∗CBP 1926:Q1–2008:Q4 0.000 0.164 0.818 −7.198∗∗∗ −6.231∗∗∗GDP 1926:Q1–2008:Q4 −3.324 0.408 1.000 −2.030 0.027GEP 1926:Q1–2008:Q4 −2.912 0.326 0.999 −3.097∗∗ −0.455GBP 1926:Q1–2008:Q4 −0.349 0.429 1.000 −3.204∗∗ −0.337

Panel D: Correlation matrix for returns and cyclical components

R CDP CEP CBP CAY RTB

R 1.00 −0.356 −0.353 −0.355 −0.006 −0.175CDP 1.000 0.943 0.952 0.381 −0.134CEP 1.000 0.995 0.371 −0.189CBP 1.000 0.387 −0.176CAY 1.000 −0.161RTB 1.000

Panel E: Correlation matrix for returns and trend components

R GDP GEP GBP CAY RTB

R 1.00 −0.234 −0.227 −0.241 −0.006 −0.175GDP 1.000 0.922 0.944 −0.587 0.059GEP 1.000 0.970 −0.675 0.059GBP 1.000 −0.608 0.042CAY 1.000 −0.161RTB 1.000∗∗∗, ∗∗, ∗ indicate statistical significance at the 0.01, 0.05 and 0.10 level, respectively.


weakly and negatively correlated with excess returns.15 In Panel C, we report sum-mary statistics and unit-root tests for the cyclical and stochastic trend components.Not surprisingly, autocorrelations of the cyclical components of financial ratios arelower than those of financial ratios. The null hypothesis of unit root in the cyclicalcomponents is rejected at conventional levels by both the augmented Dickey-Fullertest and the Philips-Perron test. The low autocorrelation in the cyclical componentsof financial ratios is very attractive in prediction regressions, since small-sample biasin predictive regression is primarily due to highly persistent predictors. In contrast,autocorrelations of the stochastic trend components of financial ratios are close toone and the null hypothesis of unit root in the trend components is not rejected atconventional levels by the Philips-Perron test.

In Panel D, we report a correlation matrix between excess returns (or CAY, RTB)and cyclical components of financial ratios. As in Panel B, the cyclical components offinancial ratios are closely and positively correlated with each other. In contrast, thecyclical components also are highly and negatively correlated with excess returns.The correlations between the cyclical components of DP, EP, and BP and excessreturns are –0.36, –0.35, and –0.36, respectively. They are much stronger in theirabsolute value than the correlations between excess returns and both CAY and RTB(–0.006 and –0.175, respectively). Panel B in Figure 1 shows that there is no trendfor all three cyclical components of financial ratios. In sum, the cyclical componentsof financial ratios are less persistent and more closely correlated with excess returns.In Panel E, we find that the stochastic trend components of financial ratios alsoare associated with excess returns: the correlation between excess returns and thestochastic trend components of dividend-price ratio, earning-price ratio and book-to-market are –0.23, –0.23, and –0.24, respectively. The absolute values of thesecorrelations also are much higher than the correlations between excess returns andboth CAY and RTB.

4. Long-horizon prediction

We examine three issues in this section: (1) Are excess returns or fundamentalspredictable? (2) Between the cyclical component and the stochastic trend component,which component predicts excess returns and fundamentals growth? (3) What is themechanism (channel) of the prediction, if there is any?

4.1. Prediction of returns

Theoretical models in Equations (5), (6), and (7) imply that financial ratios (DP,EP, and BP) predict either stock returns or fundamentals. Here, we investigate em-

15 This is consistent with the findings in the return prediction research that CAY and RTB have morepredictive power than financial ratios (see Lettau and Ludvigson, 2001, 2005; Ang and Bekaert, 2007).


pirically whether financial ratios predict excess returns in various horizons. First, wedecompose financial ratios (DP, EP, and BP) into a cyclical component and a stochas-tic trend component using the Kalman filter method recommended by Hodrick andPrescott (1997). In Table 2, we report the estimation results of univariate predictionregressions using each component separately. Since the stochastic trend is highlypersistent, we use the first difference (dG) of the stochastic trend in the regression.We report forecast regression coefficient, Newey-West corrected standard errors inparentheses, bootstrap p-value in squared brackets, and adjusted R2 statistics in curlybrackets. The sample period is from the first quarter of 1926 to the fourth quarter of2008.

We find that each component of financial ratios predicts future excess returnssignificantly.16 The cyclical components predict future excess returns with a posi-tive coefficient. The adjusted R2s increase monotonically as the prediction horizonincreases and reach the highest level at the prediction horizon of 16 quarters for CDP,and 12 quarters for CEP and CBP, respectively. The adjusted R2s are 30%, 53%, and52% for CDP, CEP, and CBP, respectively, at the peak. The coefficients and adjustedR2s in regressions of future returns on the cyclical components increase in proportionto horizons of eight quarters (two years). This implies that the variation in the cyclicalcomponents contains a similar variation of one-period expected returns up to eightquarters. The coefficients and adjusted R2s in the regressions increase at a decayingrate for longer return horizons, which suggests that the cyclical component containsless variation of one-period expected return in more distant horizons. This patternsuggests that the cyclical component is associated with the expected return with amean-reverting component around a business-cycle-length.17

The stochastic trend components predict future excess returns with a negativecoefficient. The bootstrap p-values and adjusted R2s show that the stochastic trendcomponents are able to forecast future stock returns, particularly in long horizons.It suggests that the stochastic trend components tend to forecast long-horizon futurestock returns. To our knowledge, this is new evidence. Previous research (e.g., Camp-bell and Shiller, 1988a, 1988b; Lettau and Van Nieuwerburgh, 2008) assumes thatthe long-run stochastic trend is constant and tends to ignore its predictive power. TheHodrick and Prescott (1997) filter allows us to generate the time-varying long-runstochastic trend component. The adjusted R2s are 16%, 17%, and 25% at the 16-quarter horizon for the stochastic trend components of DP, EP, and BP, respectively.

A similar observation is made when we run multivariate forecast regressionusing both cyclical and stochastic trend components as predictors in Table 3. Theadjusted R2s of two components for DP, EP, and BP at the one-quarter horizon are5%, 13%, and 13%, respectively, which are much higher than those of raw financial

16 The exception is the one-quarter-ahead forecast for CDP.

17 Fama and French (1988) and Summers (1986) show that the expected return is highly autocorrelatedwith a long-run slow mean reversion.


Table 2

Univariate forecast of quarterly stock returns: Decomposition regressions (1926:Q1–2008:Q4)

This table reports univariate estimates from regressions of excess stock returns on the lagged decomposedfinancial ratios. The dependent variable is the sum of H-period log excess returns on the S&P 500 indexover a three-month T-bill rate. Financial ratios are log dividend-price ratio (DP), log earnings-price ratio(EP), and log book-to-market (BP), respectively. They all are from the S&P 500 index. C and dG are thecyclical and stochastic growth rate (the first difference of trend component) components of financial ratios.The table reports OLS estimates of regressors, Newey-West corrected standard errors (in parentheses),bootstrapping p-value (in brackets) and adjusted R2 (in curly brackets). The sample period is from 1926:Q1to 2008:Q4.

H CDP CEP CBP dGDP dGEP dGBP

1 0.103 0.183 0.188 −1.424 −1.595 −1.500(0.073) (0.070) (0.070) (0.403) (0.626) (0.569)[0.154] [0.014] [0.010] [0.000] [0.014] [0.008]{0.029} {0.093} {0.092} {0.025} {0.033} {0.036}

2 0.188 0.333 0.340 −2.708 −3.055 −2.888(0.069) (0.061) (0.062) (0.558) (0.762) (0.688)[0.007] [0.000] [0.000] [0.000] [0.000] [0.000]{0.055} {0.168} {0.164} {0.051} {0.068} {0.074}

3 0.312 0.513 0.524 −3.921 −4.411 −4.194(0.069) (0.060) (0.060) (0.702) (0.984) (0.876)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.097} {0.254} {0.248} {0.068} {0.090} {0.099}

4 0.464 0.713 0.729 −5.106 −5.686 −5.441(0.085) (0.062) (0.062) (0.840) (1.182) (1.060)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.152} {0.345} {0.338} {0.080} {0.104} {0.117}

8 0.850 1.223 1.253 −9.057 −9.649 −9.568(0.082) (0.110) (0.107) (1.311) (1.693) (1.508)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.262} {0.517} {0.509} {0.125} {0.152} {0.183}

12 1.053 1.455 1.495 −11.956 −12.137 −12.584(0.131) (0.135) (0.136) (1.589) (1.803) (1.656)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.292} {0.530} {0.524} {0.153} {0.172} {0.226}

16 1.169 1.470 1.525 −13.558 −13.537 −14.686(0.113) (0.097) (0.099) (1.782) (1.950) (1.829)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.295} {0.443} {0.447} {0.156} {0.173} {0.249}

20 1.229 1.334 1.406 −13.702 −14.257 −16.041(0.102) (0.112) (0.112) (2.033) (2.097) (1.962)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]{0.285} {0.318} {0.331} {0.135} {0.165} {0.255}


Table 3

Multivariate forecast of quarterly stock returns: Decomposition regressions (1926:Q1–2008:Q4)

This table reports multivariate estimates from regressions of excess stock returns on the lagged decomposedfinancial ratios. The dependent variable is the sum of H-period log excess returns on the S&P 500 indexover a three-month T-bill rate. Financial ratios are log dividend-price ratio (DP), log earnings-price ratio(EP), and log book-to-market (BP), respectively. They are all from the S&P 500 index. C and dG are thecyclical and stochastic growth rate (the first difference of trend component) components of financial ratios.The table reports OLS estimates of regressors, Newey-West corrected standard errors (in parentheses),and bootstrapping p-value (in brackets). Adjusted R2 also is reported. The sample period is from 1926:Q1to 2008:Q4.

H CDP dGCP R2 CEP dGEP R2 CBP dGBP R2

1 0.103 −1.433 0.054 0.185 −1.629 0.128 0.189 −1.521 0.129(0.073) (0.397) (0.071) (0.599) (0.072) (0.539)[0.151] [0.000] [0.013] [0.005] [0.013] [0.005]

2 0.189 −2.714 0.106 0.335 −3.111 0.239 0.341 −2.918 0.241(0.068) (0.542) (0.061) (0.723) (0.062) (0.641)[0.009] [0.000] [0.000] [0.000] [0.000] [0.000]

3 0.313 −3.927 0.166 0.517 −4.502 0.349 0.527 −4.243 0.351(0.066) (0.668) (0.056) (0.884) (0.056) (0.771)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

4 0.464 −5.106 0.233 0.718 −5.812 0.455 0.732 −5.509 0.459(0.083) (0.780) (0.062) (0.986) (0.062) (0.859)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

8 0.858 −9.227 0.394 1.239 −10.074 0.686 1.268 −9.879 0.706(0.071) (1.073) (0.077) (0.955) (0.072) (0.833)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

12 1.066 −12.224 0.453 1.480 −12.750 0.722 1.519 −13.053 0.770(0.113) (1.212) (0.086) (0.840) (0.085) (0.772)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

16 1.183 −13.861 0.460 1.498 −14.184 0.636 1.555 −15.195 0.716(0.098) (1.423) (0.070) (0.921) (0.076) (0.911)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

20 1.242 −14.005 0.427 1.361 −14.804 0.498 1.436 −16.480 0.603(0.088) (1.661) (0.084) (1.287) (0.081) (1.262)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000]

ratios.18 The adjusted R2s increase monotonically as the prediction horizon increasesand reach the highest level at the horizon of around 12 quarters. The adjusted R2sfall as the prediction horizon increases beyond 12 or 16 quarters. These adjusted R2s

18 In an unreported table, we forecast regression market excess return on lagged financial ratios sepa-rately. For one-quarter-ahead forecast regression, the adjusted R2s are 1%, 2%, and 2% DP, EP, and BP,respectively.


are substantially high relative to the findings in previous studies.19 We believe thehigher adjusted R2s are due to the decomposition of financial ratios; each componentmay capture different component of expected returns, local mean reversion and long-run slow mean reversion. For financial ratio alone (without decomposition), the twoeffects may cancel out partially, resulting in a relative low adjusted R2. This patternof the adjusted R2 is interesting. It is worthwhile to note that the predictive powerof the cyclical and stochastic trend components of financial ratios is at its best overthe horizon of three to four years, which is similar to the duration of business cycles.Our evidence suggests that the expected returns are highly autocorrelated with a slowmean reversion.

Lettau and Ludvigson (2001, 2005) demonstrate that CAY has a strong predictivepower for excess returns at business cycle frequencies, showing direct linkage thatrisk premia vary countercyclically.20 Ang and Bekaert (2007) provide evidence thatthe dividend-price ratio has little predictive power for future excess returns at longhorizons. At short horizons, the dividend-price ratio is able to predict future excessreturns only with short rates. They claim that the strongest predictive power comesfrom the short rate rather than from the dividend yield.

A natural question is whether each variable has independent information aboutfuture excess returns. To address this question, in Table 4 we report the results of thecomparisons using the two components of financial ratios, CAY and the relative T-billrate for the sample period from 1952:Q1 to 2008:Q4 (CAY is available from 1952:Q1).The cyclical component (C) and the stochastic trend component (dG) exhibit a similarpredictive pattern as in Table 2. For example, at the one-quarter horizon, the cyclicalcomponents of DP, EP, and BP predict excess returns significantly. Both Newey-West corrected standard errors and bootstrap p-values are significant at the 1% level.The adjusted R2s are about 8% for the cyclical components and around 5% for thestochastic trend components in one-quarter horizon. The predictive power reaches itspeak at eight (16-)-quarter horizon for the cyclical (stochastic trend) components offinancial ratios.

We also confirm the findings of Lettau and Ludvigson (2001, 2005) and Angand Bekaert (2007) that CAY predicts an increase in excess returns, and RTB predictsa decline in excess returns. The adjusted R2s of CAY (RTB) at prediction horizons of1, 2, 3, 4, 8, 12, 16, and 20 quarters are 4% (2%), 8% (2%), 11% (3%), 14% (4%),23% (1%), 30% (1%), 32% (1%), and 31% (1%), respectively. It is interesting to notethat the adjusted R2s at the one-quarter horizon are 4% and 2% for CAY and RTB,respectively, which are much higher than those in the raw financial ratios but much

19 Lettau and Van Nieuwerburgh (2008) report the adjusted R2 of 5% (13%) for prediction regression ofexcess returns on dividend-price ratio (with one break) using annual data from 1927 to 2004.

20 Idiosyncratic volatility also has been found associated with expected returns (see Goyal and Santa-Clara,2003; Ang, Hodrick, Xing and Zhang, 2006; Jiang and Lee, 2006).


Table 4

Univariate forecast of quarterly stock returns: Decomposition regressions (1952:Q1–2008:Q4)

This table reports univariate estimates from regressions of excess stock returns on the lagged decomposedfinancial ratios, lagged CAY and lagged RTB. The dependent variable is the sum of H-period log excessreturns on the S&P 500 index over a three-month T-bill rate. Financial ratios are log dividend-price ratio(DP), log earnings-price ratio (EP), and log book-to-market (BP), respectively. They all are from the S&P500 index. C and dG are the cyclical and growth rate (the first difference of trend component) componentsof financial ratios. CAY is the consumption-wealth ratio, RTB is the relative T-bill rates. The table reportsOLS estimates of regressors, Newey-West corrected standard errors (in parentheses), bootstrapping p-value(in brackets) and adjusted R2 (in curly brackets). The sample period is from 1952:Q1 to 2008:Q4.

H CDP CEP CBP dGDP dGEP dGBP CAY RTB

1 0.194 0.193 0.192 −1.301 −1.397 −1.392 1.027 −3.770(0.049) (0.048) (0.048) (0.374) (0.449) (0.419) (0.266) (1.834)[0.000] [0.000] [0.000] [0.001] [0.002] [0.001] [0.000] [0.044]{0.077} {0.077} {0.076} {0.051} {0.046} {0.051} {0.035} {0.016}

2 0.444 0.432 0.430 −2.404 −2.622 −2.614 2.217 −5.485(0.056) (0.058) (0.058) (0.545) (0.652) (0.615) (0.414) (2.949)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.067]{0.190} {0.183} {0.181} {0.080} {0.076} {0.085} {0.081} {0.016}

3 0.654 0.634 0.630 −3.421 −3.769 −3.758 3.110 −9.359(0.058) (0.063) (0.064) (0.636) (0.765) (0.730) (0.513) (3.724)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.012]{0.278} {0.264} {0.262} {0.108} {0.105} {0.118} {0.107} {0.034}

4 0.865 0.842 0.837 −4.440 −4.908 −4.905 4.201 −12.321(0.070) (0.077) (0.078) (0.748) (0.876) (0.848) (0.634) (4.504)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.009]{0.357} {0.343} {0.340} {0.132} {0.130} {0.147} {0.144} {0.043}

8 1.314 1.304 1.304 −8.093 −8.967 −9.123 7.423 −9.456(0.112) (0.112) (0.113) (1.012) (1.157) (1.140) (0.861) (4.576)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.039]{0.440} {0.437} {0.438} {0.226} {0.230} {0.267} {0.235} {0.011}

12 1.378 1.394 1.402 −10.938 −12.291 −12.524 9.967 −9.151(0.136) (0.136) (0.136) (1.157) (1.193) (1.181) (0.982) (4.491)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.041]{0.360} {0.370} {0.376} {0.292} {0.315} {0.363} {0.297} {0.006}

16 1.272 1.333 1.353 −12.821 −14.837 −15.031 11.697 −10.205(0.157) (0.148) (0.149) (1.363) (1.236) (1.200) (1.035) (5.720)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.071]{0.258} {0.285} {0.295} {0.322} {0.377} {0.426} {0.324} {0.007}

20 1.502 1.510 1.532 −14.246 −16.987 −17.235 13.080 −14.100(0.171) (0.168) (0.169) (1.658) (1.437) (1.379) (1.150) (6.803)[0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.042]{0.289} {0.292} {0.303} {0.304} {0.382} {0.431} {0.314} {0.012}


lower than (similar to) those of cyclical (stochastic trend) components of financialratios.

It is also interesting to note that when we run the multivariate regression ofexcess returns on cyclical component, stochastic trend component, CAY, and RTBin Table 5, the coefficient of CAY at the one-quarter horizon decreases from 1.03 to–0.49, and it becomes no longer significant, while the coefficients of both cyclicaland stochastic trend components of financial ratios change little, and their t-statisticsremain highly significant. The coefficient of RTB also decreases in its absolute value,and becomes insignificant. At long horizons, we find the coefficients of CAY to beunstable. This implies that the predictive power of CAY and RTB is partially capturedby the cyclical and stochastic trend components of financial ratios.

In sum, in the subsample period from 1952:Q1 to 2008:Q4, the univariateforecast regression shows that at horizons from one to 12 quarters, the adjusted R2s arehigher using cyclical components as predictors. At horizons beyond 12 quarters, theadjusted R2s are higher using stochastic trend components as predictors. This evidenceindicates that the cyclical components of financial ratios dominate the predictivepower in relative short horizons while the stochastic trend components of financialratios dominate the predictive power in the relative long horizons. In multivariateforecast regressions, we compare the ratio of total variance of return explained byeach component, which also confirms that the cyclical (stochastic trend) componentshave more predictive power in relative short (long) horizons.21 Furthermore, thecomponents of financial ratios outperform CAY and RTB in all horizons.

4.2. Prediction of fundamentals

The present value models in Equations (5), (6), and (7) also imply that financialratios (DP, EP, and BP) may predict fundamentals. However, empirical evidence sug-gests that financial ratios have little predictive power for fundaments (see Campbell,1991; Lettau and Ludvigson, 2001, 2005; Campbell and Shiller, 2005).22 Cochrane(2001, p. 398) points out: “It is nonetheless an uncomfortable fact that almost allvariation in price/dividend ratios is due to variation in expected excess returns. Hownice it would be if high prices reflected expectations of higher future cash flows.Alas, that seems not to be the case.”

Here, we investigate whether the decomposed financial ratios predict such fun-damentals as log dividend growth (DD), log earnings growth (EE), and log accountingreturns (EB) in long horizons. Again, we choose these fundamentals based on the

21 We thank the referee’s suggestion in this direction.

22 Ang and Bekaert (2007) find the marginal predictability of dividend growth using dividend-price ratio.Using annual data, Lettau and Ludvigson (2005) find that dividend-price ratio significantly predictsdividend growth but with the wrong (positive) sign. Ang and Bekaert (2007) find that earnings-price ratiopredicts future cash flows.


Tabl

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Mul

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Q1–

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estim

ates

from

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ons

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4.

HC

DP

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RT

BR

2C

EP

dGE

PC

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RT

BR

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dGB

PC

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BR

2

10.

207

−1.5

54−0

.493

−2.7

280.

131

0.21

8−1

.957

−0.7

61−2

.380

0.12

90.

213

−1.8

33−0

.678

−2.5

760.

134

(0.0

56)

(0.5

62)

(0.4

84)

(1.6

86)

(0.0

54)

(0.7

49)

(0.5

73)

(1.6

74)

(0.0

55)

(0.6

44)

(0.5

20)

(1.6

65)

[0.0

00]

[0.0

05]

[0.3

09]

[0.1

10]

[0.0

00]

[0.0

09]

[0.1

86]

[0.1

60]

[0.0

00]

[0.0

05]

[0.1

92]

[0.1

20]

20.

465

−2.8

02−0

.740

−3.0

330.

273

0.47

4−3

.523

−1.1

86−2

.222

0.26

30.

468

−3.3

72−1

.094

−2.6

270.

273

(0.0

64)

(0.6

49)

(0.5

40)

(2.5

88)

(0.0

66)

(0.8

21)

(0.6

05)

(2.5

76)

(0.0

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(0.7

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(0.5

71)

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693

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−5.5

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405

0.70

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0.38

90.

695

−5.0

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.862

−4.8

980.

405

(0.0

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(0.7

60)

(0.6

43)

(3.0

46)

(0.0

71)

(0.9

48)

(0.7

18)

(3.0

24)

(0.0

69)

(0.8

42)

(0.6

59)

(2.9

91)

[0.0

00]

[0.0

00]

[0.0

47]

[0.0

71]

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[0.0

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[0.0

07]

[0.1

58]

[0.0

00]

[0.0

00]

[0.0

07]

[0.1

00]

40.

901

−5.1

57−1

.365

−6.7

870.

507

0.91

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0.49

10.

911

−6.4

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.154

−5.8

110.

513

(0.0

61)

(0.7

97)

(0.6

81)

(3.4

57)

(0.0

70)

(0.9

59)

(0.7

28)

(3.4

85)

(0.0

67)

(0.8

49)

(0.6

55)

(3.4

67)

[0.0

00]

[0.0

00]

[0.0

49]

[0.0

55]

[0.0

00]

[0.0

00]

[0.0

02]

[0.1

45]

[0.0

00]

[0.0

00]

[0.0

01]

[0.0

97]

81.

349

−8.5

22−0

.739

1.04

50.

662

1.46

9−1

1.44

5−2

.788

3.88

00.

697

1.45

1−1

1.11

0−2

.524

2.72

80.

727

(0.1

06)

(0.8

82)

(0.8

42)

(2.8

23)

(0.0

96)

(1.1

32)

(0.9

61)

(2.8

14)

(0.0

87)

(0.8

69)

(0.7

89)

(2.8

14)

[0.0

00]

[0.0

00]

[0.3

92]

[0.7

10]

[0.0

00]

[0.0

00]

[0.0

05]

[0.1

61]

[0.0

00]

[0.0

00]

[0.0

01]

[0.3

35]

121.

315

−9.7

431.

403

2.48

70.

656

1.54

3−1

3.81

6−1

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5.70

20.

738

1.51

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3.41

4−1

.282

4.30

90.

771

(0.1

08)

(1.0

73)

(0.9

65)

(2.7

29)

(0.0

98)

(1.1

31)

(1.0

11)

(2.7

19)

(0.0

85)

(0.8

42)

(0.7

99)

(2.6

99)

[0.0

00]

[0.0

00]

[0.1

55]

[0.3

62]

[0.0

00]

[0.0

00]

[0.0

87]

[0.0

39]

[0.0

00]

[0.0

00]

[0.1

13]

[0.1

18]

161.

104

−10.

302

3.50

30.

656

0.59

21.

444

−15.

540

−0.8

264.

168

0.70

51.

417

−15.

106

−0.2

002.

579

0.74

6(0

.125

)(1

.384

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.124

)(3

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)(0

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)(1

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)(2

.834

)(0

.097

)(1

.017

)(0

.900

)(2

.702

)[0

.000

][0

.000

][0

.002

][0

.837

][0

.000

][0

.000

][0

.476

][0

.143

][0

.000

][0

.000

][0

.829

][0

.342

]

201.

335

−12.

045

3.29

2−2

.105

0.59

71.

649

−18.

489

−1.9

101.

702

0.70

21.

621

−17.

981

−1.0

39−0

.144

0.75

1(0

.134

)(1

.538

)(1

.152

)(3

.816

)(0

.105

)(1

.311

)(1

.222

)(3

.285

)(0

.095

)(0

.993

)(0

.990

)(3

.184

)[0

.000

][0

.000

][0

.005

][0

.589

][0

.000

][0

.000

][0

.121

][0

.600

][0

.000

][0

.000

][0

.292

][0

.963

]


present value models in Equations (5), (6), and (7). The other reason to examine theprediction of fundamentals is that it helps to understand the mechanism and drivingforces of the return prediction, which we will discuss later.

We present the univariate estimation results in Table 6. CDP significantly predictsdividend growth, DD, with a negative coefficient up to 12 quarters. From the one-quarter horizon to the 20-quarter horizon, the adjusted R2s for CDP are 25%, 32%,33%, 30%, 11%, 4%, 1%, and 0%, respectively. The stochastic trend component of DPdoes not seem to have significant predictive power for dividend growth. The cyclicalcomponent of EP (BP) also predicts decreasing earnings growth (accounting returns)in short horizon while the stochastic trends of EP (BP) tend to predict decreasingearnings growth (accounting returns) in long horizons.

In a multivariate forecast regression in Table 7, we find that the cyclical (stochas-tic trend) component of DP predicts decreasing (increasing) dividend growth inrelative short horizons. The adjusted R2 is highest at three quarters for DD. Thepredictability is not stable for the cyclical components of EP and BP although theytend to predict decreasing fundamental growth in short horizons. Both stochastictrends of EP and BP tend to predict decreasing fundamental growth, particularly inlong horizons. Both univariate and multivariate regressions show that cyclical andstochastic trend have some predictability of fundamental growths.

Lettau and Ludvigson (2001, 2005) document that there is a common variationbetween expected returns and expected dividend growth, which offsets the predictivepower of dividend yield, but not of the CAY. As a robustness check, we investigate theprediction of fundamentals using the data from 1952:Q1 to 2008:Q4. We also comparethe predictive power using multivariate regressions incorporating the two componentsof financial ratios, CAY, and RTB. In Table 8, we find that the cyclical componentof the dividend-price ratio predicts dividend growth with a significantly negativecoefficient up to four quarters. However, the stochastic trend component of thedividend-price ratio predicts dividend growth with a positive coefficient. The adjustedR2s of the stochastic trend component is higher than that of cyclical component andincreases as the horizon increases. Our evidence helps explain why the dividend-price ratio does not predict dividend growth. Lettau and Ludvigson (2005) argue thatthe failure of dividend-price ratio to predict dividend growth is due to the positivecorrelation between expected return and expected dividend growth, suggesting thattime-varying investment opportunities are poorly captured by dividend-price ratio.Our evidence suggests that these investment opportunities are possibly capturedseparately by each component of dividend-price ratio, and the failure of dividend-price ratio to predict dividend growth is due to the offsetting effect between thecyclical and stochastic trend components. CAY predicts dividend growth weakly inunivariate regression, while RTB seems to predict dividend growth at short horizons.

When the dependent variable is earnings growth (EE), both the cyclical andstochastic trend components of the earnings-price ratio predict earnings growth witha positive coefficient at long horizons. When the dependent variable is accountingreturns (EB), we find that the cyclical component of book-to-market seems to predict


Table 6

Univariate forecast of quarterly fundamentals: Decomposition regressions (1926:Q1–2008:Q4)

This table reports estimates from regressions of fundamentals on the lagged decomposed financial ratios.The dependent variable is the sum of H-period log dividend growth (DD), log earnings growth (EE), andlog accounting return (EB). Financial ratios are log dividend-price ratio (DP), log earnings-price ratio(EP), and log book-to-market (BP), respectively. They all are from the S&P 500 index. C and dG are thecyclical and stochastic growth rate (the first difference of trend component) components of financial ratios.The table reports OLS estimates of regressors, Newey-West corrected standard errors (in parentheses),bootstrapping p-value (in brackets) and adjusted R2 (in curly brackets). The sample period is from 1926:Q1to 2008:Q4.

Dependent variable: DD Dependent variable: EE Dependent variable: EB

H CDP dGDP CEP dGEP CBP dGBP

1 −0.098 0.169 −0.052 −1.310 −0.035 −0.474(0.013) (0.105) (0.025) (0.493) (0.010) (0.107)[0.000] [0.112] [0.041] [0.007] [0.001] [0.000]{0.245} {0.000} {0.006} {0.025} {0.043} {0.051}

2 −0.198 0.307 −0.048 −2.225 −0.068 −0.969(0.027) (0.189) (0.048) (0.726) (0.020) (0.215)[0.000] [0.104] [0.300] [0.002] [0.001] [0.000]{0.320} {0.000} {0.000} {0.032} {0.042} {0.055}

3 −0.282 0.404 −0.006 −3.067 −0.098 −1.496(0.038) (0.275) (0.069) (0.973) (0.031) (0.323)[0.000] [0.144] [0.930] [0.001] [0.002] [0.000]{0.327} {0.000} {−0.003} {0.034} {0.039} {0.059}

4 −0.341 0.458 0.094 −3.791 −0.120 −2.056(0.047) (0.363) (0.092) (1.197) (0.042) (0.433)[0.000] [0.214] [0.291] [0.001] [0.006] [0.000]{0.302} {−0.001} {0.002} {0.036} {0.033} {0.064}

8 −0.344 0.272 0.607 −5.459 −0.129 −4.595(0.074) (0.746) (0.154) (1.929) (0.096) (0.873)[0.000] [0.719] [0.000] [0.004] [0.182] [0.000]{0.114} {−0.003} {0.094} {0.035} {0.008} {0.087}

12 −0.241 −0.327 0.946 −5.788 −0.033 −7.545(0.100) (1.118) (0.165) (2.417) (0.152) (1.299)[0.021] [0.775] [0.000] [0.015] [0.825] [0.000]{0.035} {−0.003} {0.167} {0.027} {−0.003} {0.113}

16 −0.157 −1.134 0.924 −5.465 0.061 −10.748(0.117) (1.414) (0.181) (2.643) (0.201) (1.692)[0.178] [0.426] [0.000] [0.040] [0.761] [0.000]{0.009} {−0.001} {0.140} {0.020} {−0.002} {0.140}

20 −0.069 −1.965 0.810 −4.774 0.119 −14.073(0.115) (1.607) (0.174) (2.631) (0.242) (2.053)[0.540] [0.227] [0.000] [0.070] [0.610] [0.000]

{−0.001} {0.003} {0.104} {0.014} {−0.001} {0.165}


Table 7

Multivariate forecast of quarterly fundamentals: Decomposition regressions (1926:Q1–2008:Q4)

This table reports estimates from regressions of fundamentals on the lagged decomposed financial ratios.The dependent variable is the sum of H-period log dividend growth (DD), log earnings growth (EE), andlog accounting return (EB). Financial ratios are log dividend-price ratio (DP), log earnings-price ratio(EP), and log book-to-market (BP), respectively. They all are from the S&P 500 index. C and dG are thecyclical and stochastic growth rate (the first difference of trend component) components of financial ratios.The table reports OLS estimates of regressors, Newey-West corrected standard errors (in parentheses),and bootstrapping p-value (in brackets). Adjusted R2 also is reported. The sample period is from 1926:Q1to 2008:Q4.

Dependent variable: DD Dependent variable: EE Dependent variable: EB

H CDP dGCP R2 CEP dGEP R2 CBP dGBP R2

1 −0.098 0.178 0.247 −0.051 −1.300 0.031 −0.035 −0.470 0.094(0.014) (0.100) (0.023) (0.492) (0.009) (0.100)[0.000] [0.077] [0.031] [0.008] [0.000] [0.000]

2 −0.198 0.313 0.322 −0.047 −2.218 0.032 −0.068 −0.963 0.096(0.027) (0.182) (0.043) (0.730) (0.018) (0.201)[0.000] [0.083] [0.267] [0.002] [0.000] [0.000]

3 −0.282 0.410 0.329 −0.004 −3.066 0.031 −0.097 −1.487 0.098(0.038) (0.266) (0.062) (0.975) (0.027) (0.305)[0.000] [0.126] [0.954] [0.001] [0.001] [0.000]

4 −0.341 0.458 0.303 0.097 −3.808 0.039 −0.119 −2.044 0.097(0.047) (0.356) (0.082) (1.187) (0.036) (0.412)[0.000] [0.196] [0.220] [0.001] [0.001] [0.000]

8 −0.345 0.340 0.112 0.616 −5.671 0.132 −0.122 −4.565 0.094(0.074) (0.758) (0.136) (1.751) (0.078) (0.856)[0.000] [0.655] [0.000] [0.002] [0.116] [0.000]

12 −0.241 −0.267 0.032 0.958 −6.184 0.199 −0.019 −7.539 0.110(0.099) (1.148) (0.143) (1.965) (0.120) (1.291)[0.019] [0.816] [0.000] [0.001] [0.879] [0.000]

16 −0.156 −1.094 0.009 0.935 −5.869 0.164 0.082 −10.775 0.139(0.116) (1.445) (0.164) (2.127) (0.156) (1.682)[0.175] [0.449] [0.000] [0.006] [0.601] [0.000]

20 −0.067 −1.949 0.002 0.820 −5.104 0.121 0.145 −14.118 0.165(0.113) (1.627) (0.163) (2.186) (0.182) (2.037)[0.551] [0.241] [0.000] [0.020] [0.411] [0.000]

accounting returns negatively in short horizons while positively in long horizons.The stochastic trend component of book-to-market predicts accounting returns witha positive coefficient, especially in long horizons beyond 16 quarters. In Table 9 withmultivariate forecast regression, we find that cyclical component tends to forecastdecreasing dividend growth (DD) and accounting returns (EB), particularly in shorthorizons. The stochastic trend component tends to forecast increasing fundamental

554 X. Jiang and B. S. Lee/The Financial Review 47 (2012) 531–564Ta

ble

8

Uni

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ate

fore

cast

ofqu

arte

rly

fund

amen

tals

:D

ecom

posi

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regr

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(195

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Thi

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estim

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regr

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ons

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).Fi

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PC

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1−0

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0.23

3−0

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1.64

4−0

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−0.7

77−0

.437

7.39

2−0

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0.00

9−0

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3.66

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)(0

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)(0

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)(0

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.684

)(0

.346

)(2

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)(0

.010

)(0

.105

)(0

.085

)(0

.417

)[0

.010

][0

.001

][0

.072

][0

.000

][0

.201

][0

.276

][0

.209

][0

.004

][0

.000

][0

.933

][0

.008

][0

.000

]{0.

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{0.04

1}{0.

008 }

{0.10

3}{0.

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{0.00

5}{0.

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{0.04

6}{0.

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{−0.

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{0.02

8}{0.

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2−0

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4−0

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3.01

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−0.9

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)(4

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)(0

.019

)(0

.197

)(0

.169

)(0

.808

)[0

. 018

][0

.000

][0

.123

][0

.000

][0

.842

][0

.303

][0

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][0

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][0

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][0

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][0

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][0

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]{0.

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{0.06

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{0.11

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{−0.

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{0.04

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{−0.

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{0.02

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10.2

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)(0

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)(1

.141

)(0

.691

)(5

.420

)(0

.027

)(0

.285

)(0

.248

)(1

.172

)[0

.034

][0

.000

][0

.073

][0

.000

][0

.697

][0

.382

][0

.772

][0

.209

][0

.000

][0

.642

][0

.020

][0

.000

]{0.

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{0.07

9}{0.

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{0.11

6}{−

0.00

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{−0.

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{0.00

8}{0.

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{−0.

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{0.02

3}{ 0.

318}

4−0

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1.03

1−0

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4.80

50.

213

−0.9

781.

038

−2.2

48−0

.120

0.22

6−0

.677

12.3

70(0

.029

)(0

.202

)(0

.186

)(0

.740

)(0

.163

)(1

.329

)(0

.996

)(4

.134

)(0

.034

)(0

.366

)(0

.329

)(1

.512

)[0

.080

][0

.000

][0

.094

][0

.000

][0

.196

][0

.464

][0

.309

][0

.590

][0

.001

][0

.547

][0

.043

][0

.000

]{0.

012}

{0.09

3}{0 .

006}

{0.09

5}{0.

009}

{−0.

001}

{0.00

1}{−

0.00

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036}

{−0.

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{0.01

8}{0.

273}

(Con

tinu

ed)


Tabl

e8

(con

tinu

ed)

Uni

vari

ate

fore

cast

ofqu

arte

rly

fund

amen

tals

:D

ecom

posi

tion

regr

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ons

(195

2:Q

1–20

08:Q

4)

Dep

ende

ntva

riab

le:D

DD

epen

dent

vari

able

:EE

Dep

ende

ntva

riab

le:E

B

HC

DP

dGD

PC

AY

RT

BC

EP

dGE

PC

AY

RT

BC

BP

dGB

PC

AY

RT

B

80.

018

2.17

5−0

.332

3.22

00.

761

0.19

12.

763

−31.

323

−0.0

800.

720

−0.8

4414

.657

(0.0

52)

(0.3

46)

(0.3

32)

(1.3

04)

(0.2

09)

(1.7

43)

(1.6

56)

(5.4

88)

(0.0

62)

(0.6

43)

(0.6

24)

(2.6

57)

[0.7

25]

[0.0

00]

[0.3

12]

[0.0

14]

[0.0

00]

[0.9

09]

[0.1

01]

[0.0

00]

[0.1

96]

[0.2

66]

[0.1

76]

[0.0

00]

{−0.

004}

{0.13

6}{−

0.00

1}{0.

010}

{0 .08

7}{−

0.00

5}{0.

016}

{0.09

7}{0.

001}

{0.00

1}{0.

006}

{0.12

1}12

0.09

03.

345

−0.4

100.

084

0.93

61.

976

3.99

7−3

9.60

10.

128

1.32

3−0

.163

11.1

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.070

)(0

.458

)(0

.449

)(1

.679

)(0

.189

)(1

.882

)(1

.633

)(7

.544

)(0

.086

)(0

.861

)(0

.888

)(3

.119

)[0

.205

][0

.000

][0

.361

][0

.963

][0

.000

][0

.302

][0

. 015

][0

.000

][0

.135

][0

.127

][0

.856

][0

.000

]{0.

006}

{0.17

9}{−

0.00

1}{−

0.00

5}{0.

115}

{0.00

1}{0.

030}

{0.13

4}{0.

004}

{0.00

6}{−

0.00

4}{0.

037}

160.

055

4.48

9−0

.730

−2.2

500.

857

3.91

44.

540

−38.

343

0.33

82.

264

1.24

06.

490

(0.0

78)

(0.5

45)

(0.5

29)

(2.0

65)

(0. 1

79)

(1.9

82)

(1.5

64)

(7.4

42)

(0.0

96)

(1.0

35)

(1.0

63)

(3.4

69)

[0.4

80]

[0.0

00]

[0.1

73]

[0.2

71]

[0.0

00]

[0.0

46]

[0.0

04]

[0.0

00]

[0.0

01]

[0.0

27]

[0.2

49]

[0.0

62]

{−0.

002}

{0.22

7}{0.

003}

{−0.

002}

{0.09

3}{0.

017}

{0.03

7}{0.

120}

{0.04

0}{0.

019}

{0.00

4}{0.

006}

20−0

.046

5.43

6−1

.103

− 3.8

180.

535

5.37

83.

739

−31.

684

0.46

13.

619

2.74

34.

046

(0.0

86)

(0.6

02)

(0.5

86)

(2.5

69)

(0.1

58)

(1.9

37)

(1.6

53)

(7.2

77)

(0.1

01)

(1.1

67)

(1.1

32)

(4.1

10)

[0.5

93]

[0.0

00]

[0.0

61]

[0.1

42]

[0.0

02]

[0.0

05]

[0.0

26]

[0.0

00]

[0.0

00]

[0.0

02]

[0.0

15]

[0.3

20]

{−0.

003}

{0.26

2}{0.

009}

{0.00

2}{0.

036}

{0.03

8}{0.

024}

{0 .08

6}{0.

066}

{0.04

4}{0.

031}

{−0.

001}


Tabl

e9

Mul

tiva

riat

efo

reca

stof

quar

terl

yfu

ndam

enta

ls:

Dec

ompo

siti

onre

gres

sion

s(1

952:

Q1–

2008

:Q4)

Thi

sta

ble

repo

rts

estim

ates

from

regr

essi

ons

offu

ndam

enta

lson

the

lagg

edde

com

pose

dfi

nanc

ial

ratio

s.T

hede

pend

ent

vari

able

isth

esu

mof

H-p

erio

dlo

gdi

vide

ndgr

owth

(DD

),lo

gea

rnin

gsgr

owth

(EE

),an

dlo

gac

coun

ting

retu

rns

(EB

).Fi

nanc

ialr

atio

sar

elo

gdi

vide

nd-p

rice

ratio

(DP)

,log

earn

ings

-pri

cera

tio(E

P),

and

log

book

-to-

mar

ketr

atio

(BP)

,res

pect

ivel

y.T

hey

alla

refr

omth

eS&

P50

0in

dex.

Can

ddG

are

the

cycl

ical

and

stoc

hast

icgr

owth

rate

(the

firs

tdif

fere

nce

oftr

end

com

pone

nt)

com

pone

nts

offi

nanc

ialr

atio

s.T

heta

ble

repo

rts

OL

Ses

timat

esof

regr

esso

rs,N

ewey

-Wes

tcor

rect

edst

anda

rder

rors

(in

pare

nthe

ses)

,and

boot

stra

ppin

gp-

valu

e(i

nbr

acke

ts).

Adj

uste

dR

2is

repo

rted

sepa

rate

ly.T

hesa

mpl

epe

riod

isfr

om19

52:Q

1to

2008

:Q4.

Dep

ende

ntva

riab

le:D

DD

epen

dent

vari

able

:EE

Dep

ende

ntva

riab

le:E

B

HC

DP

dGD

PC

AY

RT

BR

2C

EP

dGE

PC

AY

RT

BR

2C

BP

dGB

PC

AY

RT

BR

2

1−0

.025

0.28

50.

167

1.56

90.

151

0.04

4−2

.221

−1.5

897.

089

0.07

2−0

.019

−0.1

68−0

.155

3.47

20.

355

(0.0

10)

(0.1

07)

(0.1

02)

(0.2

36)

(0.0

58)

(1.5

68)

(1.1

64)

(2.1

69)

(0.0

10)

(0.1

19)

(0.1

08)

(0.4

15)

[0.0

12]

[0.0

07]

[0.1

06]

[0.0

00]

[0.4

42]

[0.1

65]

[0.1

75]

[0.0

02]

[0.0

70]

[0.1

59]

[0.1

56]

[0.0

00]

2−0

.047

0.65

90.

421

2.89

10.

196

0.06

5−1

.797

−0.7

7610

.598

0.04

5−0

.038

−0.2

30−0

.201

6.91

60.

362

(0.0

17)

(0.1

70)

(0.1

63)

(0.4

28)

(0.0

95)

(1.2

55)

(0.8

33)

(4.8

75)

(0.0

20)

(0.1

90)

(0.1

86)

(0.8

29)

[0.0

06]

[0.0

00]

[0.0

11]

[0.0

00]

[0.4

87]

[0.1

57]

[0.3

62]

[0.0

32]

[0.0

62]

[0.2

35]

[0.2

84]

[0.0

00]

3−0

.061

0.96

20.

556

3.92

70.

200

0.12

5−2

.096

−1.0

947.

546

0.00

7−0

.052

−0.3

18−0

.292

9.77

40.

333

(0.0

23)

(0.2

30)

(0.2

13)

(0.6

10)

(0.1

47)

(2.0

47)

(1.4

34)

(5.6

45)

(0.0

28)

(0.2

93)

(0.2

82)

(1.2

03)

[0.0

09]

[0.0

00]

[0.0

08]

[0.0

00]

[0.3

86]

[0.3

16]

[0.4

42]

[0.1

79]

[0.0

62]

[0.2

77]

[0.3

06]

[0.0

00]

4−0

.071

1.34

60.

764

4.38

70.

192

0.22

2−1

.141

−0.2

50−0

.457

−0.0

02−0

.060

−0.3

23−0

.292

11.8

950.

280

(0.0

29)

(0.2

84)

(0.2

59)

(0.7

76)

(0.1

95)

(2.0

03)

(1.4

80)

(4.6

82)

(0.0

37)

(0.3

99)

(0.3

78)

(1.5

72)

[0.0

15]

[0.0

00]

[0.0

04]

[0.0

00]

[0.2

52]

[0.5

74]

[0.8

67]

[0.9

24]

[0.1

14]

[0.4

14]

[0.4

37]

[0.0

00]

8−0

.039

2.95

11.

393

2.43

10.

161

0.57

02.

172

1.49

0−2

7.16

40.

152

−0.0

070.

029

−0.3

1414

.324

0.11

0(0

.053

)(0

.455

)(0

.436

)(1

.244

)(0

.217

)(1

.948

)(2

.005

)(5

.181

)(0

.072

)(0

.791

)(0

.776

)(2

.753

)[0

.447

][0

.000

][0

.002

][0

.052

][0

.010

][0

.267

][0

.463

][0

.000

][0

.930

][0

.971

][0

.689

][0

.000

]

120.

002

4.48

71.

768

−0.8

320.

212

0.52

97.

426

5.58

4−3

4.87

90.

248

0.15

21.

277

0.43

311

.950

0.04

6(0

.072

)(0

.597

)(0

.610

)(1

.465

)(0

.199

)(1

.811

)(1

.887

)(7

.119

)(0

.102

)(1

.066

)(1

.106

)(3

.197

)[0

.979

][0

.000

][0

.005

][0

.576

][0

.009

][0

.000

][0

.003

][0

.000

][0

.137

][0

.237

][0

.699

][0

.000

]

16−0

.063

6.01

22.

193

−3.4

240.

274

0.26

312

.233

9.61

9−3

4.56

80.

287

0.23

24.

006

2.80

98.

593

0.08

8(0

.072

)(0

.608

)(0

.640

)(1

.777

)(0

.191

)(1

.819

)(1

.941

)(6

.621

)(0

.112

)(1

.213

)(1

.318

)(3

.501

)[0

.376

][0

.000

][0

.000

][0

.058

][0

.165

][0

.000

][0

.000

][0

.000

][0

.033

][0

.001

][0

.033

][0

.016

]

20−0

.183

7.04

72.

519

−5.5

930.

311

−0.1

2614

.721

11.3

21−3

0.37

90.

267

0.19

17.

514

6.05

06.

937

0.20

0(0

.070

)(0

.639

)(0

.657

)(2

.100

)(0

.168

)(1

.634

)(1

.980

)(5

.991

)(0

.105

)(1

.259

)(1

.334

)(3

.910

)[0

.010

][0

.000

][0

.000

][0

.008

][0

.447

][0

.000

][0

.000

][0

.000

][0

.071

][0

.000

][0

.000

][0

.082

]


growth, particularly in long horizons. CAY tends to forecast fundamental growth inlong horizons, too. RTB tends to forecast DD and EE with a positive sign in shorthorizons and a negative sign in long horizons.23

In sum, cyclical components tend to predict decreasing fundamental growths inshort horizons while stochastic trend components tend to predict increasing funda-mental growths in long horizons.

4.3. Behavior of financial ratios

So far, we observe that the decomposed financial ratios (DP, EP, and BP) predictstock excess returns. This prompts us to ask the following: How do financial ratiospredict excess returns? What does this say about the channels of the prediction?Summers (1986), Fama and French (1988), and Campbell and Shiller (1988b, 2005)suggest a simple theory of slow mean reversion to explain the predictability; thatis, stock prices cannot drift too far from their fundamentals (dividend, earnings, orbook value). The theory of slow mean reversion requires that financial ratios have tobe stationary in the long run. The predictive power comes from the long-run meanreversion in financial ratios. Previous findings also support the finding that financialratios have more predictive power in long horizons.

We observe that the cyclical components of the financial ratios predict increasesin stock returns and decreases in fundamentals, particularly in short horizons, whilethe stochastic trend components of the financial ratios predict decreases in stockreturns, particularly in long horizons. Based on the adjusted R2s in univariate forecastregressions and the relative importance of each component in multivariate forecast re-gressions, we show that the cyclical components tend to capture more predictability inrelatively shorter horizons while the stochastic trend components tend to capture thepredictability in relatively longer horizons. By construction, the cyclical componentsare less persistent than the stochastic trend components. Therefore, the cyclical com-ponent reflects a local mean reversion effect, while the stochastic trend componentreflects a slow mean reversion effect.

Our conjecture helps explain previous findings that raw financial ratios tendto show some predictive power for stock returns in long horizons. In addition, ourfinding that the two components predict stock returns in the opposite directionssuggests that, in short horizons, the two components of financial ratios tend to offsetmuch of each other’s prediction. This helps explain the failure of previous studies tofind strong predictive power of raw financial ratios in short horizons in the absenceof decomposition.

23 In a further robust check, we implement Lewellen’s (2004) bias-corrected method, since the firstdifferenced stochastic trend component is still quite persistent. In an unreported table, we confirm thepredictive power of the decomposed financial ratios in Tables 2–5.


5. Out-of-sample prediction

Our in-sample prediction results are striking. However, since both componentsare estimated using the full sample, there is a concern that the results are subjectto a “look-ahead” bias. We address this concern by implementing OOS predictions.The Hodrick-Prescott (1997) decomposition is based on the relative long-run equi-librium relation. A meaningful and consistent estimation requires a large number ofobservations. On the other hand, the OOS prediction has to rely on the real-timedata that may induce sampling errors in forming the cyclical and stochastic trendcomponents. Therefore, we consider two types of OOS prediction. First, we predictOOS using the ex post decomposition (components are estimated by using the fullsample). Second, we predict OOS using alternative ex ante decomposition. In thisapproach, we estimate the cyclical and stochastic trend components using real-timedata (available at the time of prediction). We call the first approach pseudo OOS, andthe second pure OOS.24

We choose the random walk model as a benchmark model (restricted model).We implement nested prediction comparisons. We run the prediction regression usingdata from 1926:Q1 to 1989:Q4, and thus the first period for prediction is 1990:Q1.We report the ratio of the root mean squared errors (MSE) for the unrestrictedand restricted model forecasts, Theil’s U, MSE-F (H0: restricted and unrestrictedpredictions are equal) by McCracken (2004), ENC-F (H0: restricted prediction en-compasses the unrestricted prediction) by Clark and McCracken (2001), and pseudoR2 measured as one minus the ratio of mean square error of unrestricted to restrictedmodels.

Table 10 reports the results for the pseudo OSS prediction with bootstrappingerrors. When the unrestricted models include raw financial ratios (DP, EP, and BP),we find that the prediction errors in the restricted model are smaller than those in theunrestricted models. Theil’s U is greater than one, and pseudo R2 are negative. BothMSE-F and ENC-F show that financial ratios do not perform well in OOS predictions.Our evidence is consistent with evidence provided by Goyal and Welch (2008) andLettau and Van Nieuwerburgh (2008) in that the random walk model outperformsthe models with raw financial ratios. We find a similar pattern for the model withRTB as a predictor. We find that CAY with fixed cointegrating parameters for thefull sample outperforms the random walk model in the Clark and McCracken (2001)test.

We are particularly interested in the OOS prediction with the cyclical or stochas-tic trend component of financial ratios as a predictor. When we compare the perfor-mance of the random walk model with that of the model with the ex post cyclicalcomponent of financial ratios, we find that the Theil’s U is less than one, and pseudoR2 is positive for CDP, CEP, and CBP. Both MSE-F and ENC-F are statistically

24 We use the terms following Lettau and Van Nieuwerburgh (2008).


Table 10

Pseudo out-of-sample (OOS) forecast

This table reports the results of one-quarter-ahead nested prediction comparison of excess returns on theS&P 500 index over a three-month T-bill rate and fundamentals. Financial ratios are log dividend-priceratio (DP), log earnings-price ratio (EP), and log book-to-market (BP), respectively. They all are fromthe S&P 500 index. C and dG are the cyclical and stochastic growth rate (the first difference of trendcomponent) components of the financial ratios. The table reports pseudo OOS prediction for CAY, RTB,C and dG. The benchmark (restricted) model is the random walk model. We report the Theil’s U (theratio of the root mean squared errors for the unrestricted and restricted model forecasts), MSE-F (H0:restricted and unrestricted predictions are equal) by McCracken (2004), ENC-F (H0: restricted predictionencompasses the unrestricted prediction) by Clark and McCracken (2001), and pseudo R2 measured asone minus the ratio of mean square error of unrestricted to restricted models.

Return regression Fundamentals regression

Variable Theil’s U MSE-F ENC-F Pseudo R2 Theil’s U MSE-F ENC-F Pseudo R2

DP 1.014 −2.117 1.218 −0.029 1.958 −47.309 −2.545 −2.834[0.010] [1.456] [0.660] [0.020] [0.010] [1.230] [0.639] [0.020]

EP 1.050 −7.016 1.198 −0.102 1.016 −1.984 0.183 −0.032[0.010] [1.399] [0.637] [0.020] [0.008] [1.031] [0.542] [0.017]

BP 1.027 −3.895 1.059 −0.054 0.867 21.212 51.344 0.249[0.011] [1.595] [0.694] [0.023] [0.008] [1.032] [0.573] [0.016]

CAY 1.029 −4.211 2.550 −0.059[0.007] [1.079] [0.571] [0.014]

RTB 1.058 −8.118 −2.670 −0.120[0.002] [0.357] [0.178] [0.005]

CDP 0.977 3.618 2.631 0.045 1.319 −32.348 −2.918 −0.741[0.007] [1.052] [0.545] [0.014] [0.007] [1.112] [0.587] [0.015]

CEP 0.987 1.969 4.477 0.025 1.005 −0.695 −0.257 −0.009[0.007] [1.047] [0.526] [0.014] [0.009] [1.292] [0.641] [0.018]

CBP 0.993 1.148 4.372 0.015 0.962 6.062 3.675 0.074[0.007] [1.012] [0.517] [0.014] [0.006] [0.976] [0.524] [0.013]

dGDP 0.971 4.669 6.217 0.058 0.968 5.165 3.335 0.064[0.011] [1.596] [0.953] [0.022] [0.013] [1.881] [1.063] [0.026]

dGEP 0.964 5.765 6.119 0.071 0.990 1.472 1.027 0.019[0.009] [1.353] [0.736] [0.018] [0.013] [1.873] [1.043] [0.027]

dGBP 0.959 6.643 6.301 0.080 0.977 3.554 4.610 0.045[0.009] [1.396] [0.813] [0.019] [0.009] [1.398] [0.753] [0.019]

significant with bootstrap. When we compare the performance of the random walkmodel with that of the model with the ex post stochastic trend component of finan-cial ratios, we find a similar pattern. Our evidence shows that the models includingthe cyclical or stochastic trend components of financial ratios (estimated in the fullsample) in the restricted model have superior OOS predictions.

Table 11 reports the results for the pure OOS prediction. Here, we use the 25years’ moving average of financial ratio as a measure of stochastic trend componentwhile using the difference between the financial ratio and the stochastic trend as a


Table 11

Pure out-of-sample (OOS) forecast

This table reports the results of one-quarter-ahead nested prediction comparison of excess returns on theS&P 500 index over a three-month T-bill rate and fundamentals. Financial ratios are log dividend-priceratio (DP), log earnings-price ratio (EP), and log book-to-market (BP), respectively. They all are fromthe S&P 500 index. C and dG are the cyclical and stochastic growth rate (the first difference of trendcomponent) components of the financial ratios. Here, to avoid the look-ahead bias, we simply use 25 years’moving average as a proxy for the long-run growth component, the difference of the financial ratio seriesand the moving average as a proxy for the cyclical component. The table reports pure OOS predictionfor C and dG. The benchmark (restricted) model is the random walk model. We report the Theil’s U (theratio of the root mean squared errors for the unrestricted and restricted model forecasts), MSE-F (H0:restricted and unrestricted predictions are equal) by McCracken (2004), ENC-F (H0: restricted predictionencompasses the unrestricted prediction) by Clark and McCracken (2001), and pseudo R2 measured asone minus the ratio of mean square error of unrestricted to restricted models.

Return regression Fundamentals regression

Variable Theil’s U MSE-F ENC-F Pseudo R2 Theil’s U MSE-F ENC-F Pseudo R2

CDP 0.986 2.249 2.056 0.029 1.495 −42.017 −3.481 −1.236[0.015] [2.116] [1.393] [0.031] [0.007] [1.060] [0.631] [0.014]

CEP 0.980 3.190 2.488 0.040 0.999 0.086 0.190 0.001[0.006] [0.955] [0.477] [0.013] [0.008] [1.249] [0.749] [0.017]

CBP 0.989 1.638 1.740 0.021 0.845 30.500 19.782 0.286[0.014] [2.002] [1.444] [0.027] [0.007] [1.103] [0.591] [0.015]

dGDP 0.991 1.442 1.704 0.019 1.255 −27.709 3.650 −0.574[0.007] [1.038] [0.596] [0.014] [0.007] [1.005] [0.564] [0.014]

dGEP 1.001 −0.095 −0.046 −0.001 1.006 −0.937 −0.262 −0.012[0.006] [0.957] [0.494] [0.013] [0.008] [1.164] [0.579] [0.016]

dGBP 1.000 −0.068 −0.033 −0.001 0.843 30.924 22.768 0.289[0.007] [0.994] [0.543] [0.013] [0.007] [0.999] [0.555] [0.013]

measure of the cyclical component. In this way, we use the cyclical and stochastictrend components estimated using real-time data in OOS prediction tests. We findthat the Theil’s U is less than one for all cyclical components of three financialratios, suggesting that the MSE for the unrestricted model is less than the MSE forthe restricted model. MSE-F rejects the null of equal forecast accuracy between theconstant expected returns benchmark model and the cyclical component-augmentedmodel. ENC-F also rejects the null hypothesis that cyclical components have nopredictive power for excess returns. However, bootstrap errors show that only theMSE-F and ENC-F for the cyclical component in earnings-price ratio are significant.For the stochastic trend components, the Theil’s U in dividend-price ratio is lessthan one while the others are greater than one. The bootstrap errors show that onlythe ENC-F for the stochastic component of dividend-price ratio is significant. Insum, the result shows some of the cyclical components of financial ratios displaystatistically significant OOS predictive power for excess returns. The stochastic trend


components display little OOS predictive power for excess returns. For the growth infundamentals, both the cyclical and stochastic trend components of book-to-marketratio show statistically significant predictive power for accounting returns. The Theil’sU of the cyclical and stochastic trend components are 0.845 and 0.843, respectively.MES-F and ENC-F are 30.500 and 19.782 (30.924 and 22.768), respectively, forthe cyclical (stochastic trend) component. Bootstrap errors also suggest the OOSpredictability.

6. Conclusions

Cochrane (2008, p. 1535) points out “If both returns and dividend growth areunforecastable, then present value logic implies that the price/dividend ratio is con-stant, which is obvious not.” Some explanation should be explored. Motivated by thefinding that financial ratios are highly persistent and that the mean of financial ratiosis time-varying, we have examined the prediction of excess returns and fundamentalsby financial ratios in-sample and OOS. We examine the prediction by decomposingthe financial ratios into a cyclical component and a stochastic trend component usingHodrick and Prescott’s (1997) Kalman filter procedure. Previous studies find thatfinancial ratios predict long-horizon returns but scant short-horizon returns and fun-damentals. In contrast, we find that decomposed financial ratios significantly predictexcess returns and fundamentals in both long and short horizons. The OOS forecastresult shows that the cyclical component of earnings-price ratio is superior in fore-casting market return, and the stochastic trend component of book-to-market ratio issuperior in forecasting fundamentals.

We also find that the cyclical components of financial ratios predict increasesin returns, while the stochastic trend components predict decreases in returns. Weinterpret these findings as evidence that the cyclical components reflect a local meanreversion effect while the stochastic trend components reflect a long-run persistenceeffect. This helps explain previous findings that, in the absence of decomposition,financial ratios find little predictive power in short horizons due to offsetting effects.It also helps explain the failure of dividend-price ratio to predict dividend growth.Our result shows that the decomposed financial ratios based on the Hodrick andPrescott (1997) method predict stock returns and fundamentals better than financialratios alone.

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