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A Note on the Economics and Statistics of

Predictability: A Long Run Risks Perspective∗

Ravi Bansal†

Dana Kiku‡

Amir Yaron§

November 14, 2007

Abstract

Asset return and cash flow predictability is of considerable interest in financial

economics. In this note, we show that the magnitude of this predictability in the

data is quite small and is consistent with the implications of the long-run risks model.

∗Yaron thanks the Rodney White Center for financial support.

†Fuqua School of Business, Duke University, and NBER, ravi.bansal@duke.edu.

‡The Wharton School, University of Pennsylvania, kiku@wharton.upenn.edu.

§The Wharton School, University of Pennsylvania and NBER, yaron@wharton.upenn.edu.

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1Introduction

Predictability of asset returns and cash flows is a topic of considerable interest for financial

economists. The source and magnitude of predictability in these components determine

asset price fluctuations and impose restrictions on economic models that help evaluate asset

pricing models. We use the long-run risks model of Bansal and Yaron (2004) to evaluate the

economic and statistical plausibility of predictability of returns and cash flows. That is, we

ask how much predictability is plausible in the data, both from a statistical and the long-run

risks model perspective.

The evidence on predictability is voluminous and contentious (see for example, Keim

and Stambaugh (1986), Campbell and Shiller (1988), Fama and French (1988), Hodrick

(1992), Stambaugh (1999), Goyal and Welch (2003), Valkanov (2003), Lewellen (2004), and

Boudoukh, Richardson, and Whitelaw (2006)). One view, (see Campbell and Cochrane

(1999) and Cochrane (2006)) is that returns are sharply predictable while consumption and

cash flow growth rates are not. This view, therefore, associates movements in asset prices to

discount rate variation rather than time varying cash flow growth. However, on statistical

grounds, Ang and Bekaert (2007), Boudoukh, Richardson, and Whitelaw (2006) question the

magnitude of return predictability in the data and argue that returns do not have significant

predictability. An alternative view is that cash flow growth rates are predictable in ways

that have important implications for asset prices (see Bansal and Yaron (2006), Lettau

and Ludvigson (2005), and Hansen, Heaton, and Li (2006)).Hence, the magnitude of

predictability of returns and cashflows in the data is a source of considerable debate and

discussion.

The main focus in this paper is about magnitudes: what is a plausible magnitude of

predictability from the statistical perspective and from the perspective of an economic model

– the long-run risks model. The economic model, which is broadly consistent with a wide-

range of asset market facts, provides a framework to evaluate the plausibility of predictability

in the data. We confine our attention to the standard excess return and consumption growth

rate predictability. Our evidence shows that based on dividend-price ratios returns are

modestly predictable, though this predictability is quite fragile. For example, when we use

dividend-price ratios adjusted by the risk-free rate, we get a more stationary and better

behaved predictor variable, however, the level of return predictability declines considerably

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and is close to zero.1The magnitude of predictability of consumption growth rate in the data

is also quite small. For both returns and consumption growth, the finite sample distribution

of the coefficients and adjusted R2’s are quite wide.

We calibrate a version of the long-run risks model of Bansal and Yaron (2004) and use

an improved model solution based on approximate analytical method from Bansal, Kiku,

and Yaron (2007) to show that the model can generate finite sample properties that are

consistent with the aforementioned empirical findings. Excess return predictability in the

model is due to the time variation of risk premia, induced by the presence of time varying

volatility of consumption and cash flows. Consumption growth in the model is driven by a

small, persistent component that, in equilibrium, governs the dynamics of asset prices. Thus,

current asset valuations should contain important information about future consumption

growth. However, price-dividend ratios in the model move not only on news about future

economic growth but also on news about future economic uncertainty (or discount-rate news).

Price fluctuations emanating from time-variation in discount rates may significantly diminish

the informational content of asset valuations about future growth and, consequently, limit

their ability to forecast future dynamics of consumption growth. Indeed, we show, that

consistent with the data evidence, the model-implied predictability of consumption growth

by the market dividend-price ratio is quite small.

Overall our results support the view that there is a small time-varying component in

returns and in cash flows. The evidence in this paper shows that the long-run risks model

can quantitatively explain the level of predictability of returns and consumption growth

consistent with that observed in the data.

The paper continues as follows: Section 2 discusses the data and provides the results

of our empirical analysis. Section 3 presents the model and provides the corresponding

predictability results. Section 4 provides concluding comments.

1This difference in the magnitude of the R2between dividend-price and risk-free rate adjusted dividend-

price ratio is most likely due to the very high persistence in the dividend yield. For this issue also see Hodrick

(1992).

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2 Empirical Findings

We use annual data on consumption and asset prices for the time period from 1930 till 2006.

The annual data provides the longest available sample and is arguably the least susceptible

to measurement errors. Consumption data are based on seasonally adjusted per-capita series

on real consumption from the NIPA tables available on the Bureau of Economic Analysis

website. Aggregate consumption is defined as consumer expenditures on non-durables and

services. Growth rates are constructed by taking the first difference of the corresponding log

series. Our asset menu comprises the aggregate stock market portfolio on the value weighted

return of the NYSE/AMEX/NASDAQ from CRSP and a proxy of a risk-less asset. The

real interest rate is constructed by subtracting realized annual inflation from the annualized

yield on the 3-month Treasury bill taken from the CRSP treasury files.

Table I presents descriptive statistics for consumption growth, the return and dividend

yield of the aggregate stock market and the risk-free rate. All entries are expressed in real

percentage terms. Standard errors are based on the Newey and West (1987) estimator with 8

lags. This particular sample results in the standard and well known features of the data such

as a low risk free rate, a large equity premium and a relatively low consumption volatility.

Table II provides the results of consumption growth predictability using the log of the

dividend-price ratio as a regressor. The table presents estimates of slope coefficients (ˆβ),

robust t-statistics and R2s from projecting 1-, 3- and 5-year consumption growth onto lagged

log dividend-price ratio of the aggregate stock market portfolio. The point estimates are

insignificantly different from zero and the R2s are less than 2%. In addition, the right

columns display bootstrap distributions of the reported statistics. Empirical percentiles are

constructed by resampling the data 10,000 times in blocks of 8 years with replacement. At

the 5-year horizon, the median R2is 4 percent while the 90 percentile includes an R2as high

as 18%. This evidence suggests that the level of the consumption predictability in the data

includes a wide range of predictability estimates and R2s.

It is very important to note that the above predictability evidence is solely based on using

the dividend-price ratio as a predictive variable. Bansal, Kiku, and Yaron (2007) provide

evidence that when additional predictive variables are used, the consumption predictability is

considerably higher. For example, if the risk-free rate is included as an additional predictive

variable, the R2for the one-year horizon rises to 17% and at the two-year horizon is about

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12%. Clearly other forecasting variables, such as earnings to consumption ratio used in

Hansen, Heaton, and Li (2006), would further increase short- and long-run predictability

of consumption. Expanding the information set beyond financial ratios to forecast future

growth is motivated by economic considerations as discussed in Bansal, Kiku, and Yaron

(2007).

Table III provides evidence on predictability of multi-period excess returns. In panel

A the log of dividend-price ratio is used to forecast returns. Consistent with evidence is

earlier papers, the R2s rise with maturity from 4.5% at the 1-year to 29% at the 5-year

horizon. Note that the slope coefficient estimates are only marginally significant for all three

horizons. The bootstrap t-statistics and R2s have a wide distribution and range from 0.2

to 3 for the t-statistics and from 0 to 40% for the R2. This evidence of predictability is

highly fragile. Panel B of Table III runs the same regressions save for the fact the regressor

is now the log dividend-price ratio minus the risk free rate. We do so to ensure that the

predictive variable is well behaved — adjusting the dividend-price ratio for the risk free rate

lowers the high persistence in the predictive variable. The results of return predictability

are now much weaker. In particular, at all horizons, the slope coefficients are insignificant.

The R2s are now below 4.5% for all horizons. The range for the bootstrap t-statistics and

R2s is now tighter and covers 0.23 to 2.8 for the t-statistic, and 0 to 21% for the R2. This

is consistent with a view that the actual magnitude for return predictability is quite small.

The difference in predictability between Panel A and Panel B also clearly suggests that

much of the ability of the dividend-yield to predict future returns might be spurious and

simply due to its very persistent nature for this particular sample. The fragility of the return

predictability evidence is one of the reasons for the ongoing debate about the presence and

magnitude of return predictability discussed in the introduction.

3Model

In this section we specify a model based on Bansal and Yaron (2004). The underlying

environment is one with complete markets and the representative agent has Epstein and Zin

(1989) type recursive preferences in which she maximizes her life-time utility,

Vt=

?

(1 − δ)C

1−γ

θ

t

+ δ

?

Et

?V1−γ

t+1

??1

θ?

θ

1−γ

,(1)

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where Ctis consumption at time t, 0 < δ < 1 reflects the agent’s time preferences, γ is the

coefficient of risk aversion, θ =

1−1

(IES). Utility maximization is subject to the budget constraint,

1−γ

ψ, and ψ is the elasticity of intertemporal substitution

Wt+1= (Wt− Ct)Rc,t+1,(2)

where Wtis the wealth of the agent, and Rc,tis the return on all invested wealth.

Consumption and dividends have the following joint dynamics:

∆ct+1 = µc+ xt+ σtηt+1

xt+1 = ρxt+ ϕeσtet+1

σ2

t+1

= ¯ σ2+ ν(σ2

t− ¯ σ2) + σwwt+1,(3)

∆dt+1 = µd+ φxt+ πσtηt+1+ ϕσtud,t+1

where ∆ct+1, and ∆dt+1are the growth rate of consumption and dividends respectively. In

addition, we assume that all shocks are i.i.d normal and are orthogonal to each other. As in

the long-run risks model of Bansal and Yaron (2004), µc+xtis the conditional expectation of

consumption growth, and xtis a small but persistent component that captures long-run risks

in consumption growth. For parsimony, as in Bansal and Yaron (2004), we have a common

time-varying volatility in consumption and dividends, which, as shown in their paper, leads

to time-varying risk premia. Dividends have a levered exposure to the persistent component

in consumption, xt, which is captured by the parameter φ. In addition, we allow the i.i.d

consumption shock ηt+1to influence the dividend process, and thus serve as an additional

source of risk premia. The magnitude of this influence is governed by the parameter π.2

Save for this addition, the dynamics are similar to those in Bansal and Yaron (2004).

As in Epstein and Zin (1989), it is easily shown that, for any asset j, the first order

condition yields the following asset pricing Euler condition,

Et[exp(mt+1+ rj,t+1)] = 1,(4)

where mt+1 is the log of the intertemporal marginal rate of substitution and rj,t+1 is the

2Note that equivalently we could have specified the correlation between ηt+1and ud,t+1to be non-zero,

and set π = 0.

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log of the gross return on asset j. Further, the log of the Intertemporal Marginal Rate of

Substitution (IMRS), mt+1, is

mt+1 = θlogδ −

θ

ψ∆ct+1 + (θ − 1)rc,t+1,(5)

where rc,t+1is the continuous return on the consumption asset. To solve for the return on

wealth (the return on the consumption asset), we use the log-linear approximation for the

continuous return on the wealth portfolio, namely,

rc,t+1= κ0+ κ1zt+1+ ∆ct+1− zt,(6)

where zt= log(Pt/Ct) is log price to consumption ratio (the valuation ratio corresponding

to a claim that pays consumption) and the κ’s are log linearization constants which are

discussed in more detail below.

To solve for asset prices we provide a simpler and more efficient way to solve the Bansal

and Yaron (2004) long-run risks model. We use approximate analytical solutions (instead

of the polynomial-based numerical approximation in the original paper), which, we find,

provide a more accurate solution to the model.3This easier-to-implement solution and a

refined configuration leads to similar economic magnitudes but allows us to better address

certain predictability dimensions. Specifically, we conjecture the price to consumption ratio

follows,

zt= A0+ A1xt+ A2σ2

t

(7)

and solve for the A’s using the Euler equation (4), the return equation (6) and the conjectured

equation (7) for the price-consumption ratio. The solution for the A’s depends on all the

preference and technology parameters and is derived in Bansal and Yaron (2004) and Bansal,

Kiku, and Yaron (2007), and for completeness is reproduced in the Appendix. In solving

for the price-consumption ratio we impose model consistency between the average price

consumption ratio ¯ z and the approximation κ’s, which themselves depend on the average

price-consumption ratio. This is important to impose, as any change in the model parameters

will alter ¯ z and hence the approximation κ’s. The model-based endogenous solution to ¯ z can

3Bansal, Kiku, and Yaron (2007) evaluate the various approaches and find the approximate-analytical

solution to be the most accurate and easy to implement.

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be obtained by solving the equation,

¯ z = A0(¯ z) + A2(¯ z)¯ σ2, (8)

recognizing that κ0= log(1 + exp(¯ z)) − κ1¯ z and κ1=

to solve for ¯ z is quite easy in practice. The endogeneity of ¯ z has also been emphasized in

exp(¯ z)

1+exp(¯ z). Implementing equation (8)

Campbell and Koo (1997).

Given the solution for zt, the innovation to the return to wealth can be derived, which

in turn allows us to specify the innovations to the IMRS and thus facilitate computing risk

premia for various assets. In particular, it immediately follows that the risk premium on the

market portfolio (that is, the return on the dividend paying asset) carries three sources of

risks. That is

Et[rm,t+1− rf,t+ 0.5σ2

t,rm] = βη,mλησ2

t+ βe,mλeσ2

t+ βw,mλwσ2

w

(9)

where βm,j, j = {η,e,w} are respectively the betas of the market return with respect to the

“short run” risk, ηt, the long-run risk innovation, et, and the economic uncertainty (volatility)

risk, wt. The λ’s represent the corresponding market prices of risks. The appendix and

Bansal, Kiku, and Yaron (2007) provide the solution for the market price of risks and the

market return’s betas in terms of the underlying preference and technology parameters.

Table IV provides the parameter configuration we use to calibrate the model — these are

chosen to match several key statistics of consumption data, dividend data, and asset returns.

Table V presents moments of simulated annualized consumption and dividend growth rates

along with asset pricing implications of the model. Reported statistics are based on 10,000

simulated samples with 77×12 monthly observations that match the length of the actual data.

The entries represent the median, 5thand 95thpercentiles of the monte-carlo distributions of

the corresponding statistics. These results show that the model distribution for the mean,

standard deviation and first autocorrelation of consumption and dividend growth and their

correlation are consistent with the data. Moreover, the model generates a distribution of

asset returns that captures the key features of the data. In particular, the model’s median

equity premium is just under 7% and the volatility of the market return is just about that

of the data at 19%. The model further generates a low risk free rate level and volatility and

a plausible level and volatility of the price-dividend ratio.

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Table VI provides the model-implied consumption growth predictability by the log of

the dividend-price ratio. The model’s median estimate indicates a significant negative slope

coefficient for predicting the one- to five-year ahead consumption growth. Although the

median R2s are somewhat large relative to their data counterpart, the data t-statistics and

R2s are within the 95% confidence interval generated by the model. It is important to

note that, to maintain parsimony and keep the number of calibrated parameters small,

we have assumed that all consumption shocks are orthogonal to each other.We have,

however, explored the sensitivity of the model implications to the innovation correlation

structure. We find that relaxing a zero-correlation restriction between long-run and volatility

risks allows the model to even better capture the low ability of the dividend-price ratio to

forecast future consumption growth, as observed in the data. Under reasonable correlation

parameterizations, the model is able to diminish short- and long-horizon consumption growth

predictability to about 5-6% without altering other asset pricing predictions (this evidence

is available upon request).4

Panel A of Table VII provides the analogous results for model-implied statistics for return

predictability by the log dividend-price ratio. As in the data, the regression coefficients and

R2s rise with the horizon. The median return predictability coefficients are not significant at

conventional levels and the median estimate for R2is 5% at the 5-year horizon. The data’s

estimates across horizons are all well within their corresponding 90% model-based confidence

intervals. Panel B provides the analogous projections when the log dividend price ratio is

adjusted for the risk free rate. The model-based results are very close to those in the data

reported in Panel B of Table III — both the level of the slope coefficient and the R2are a

close match. In all this evidence implies that the model can match the return predictability

observed in the data. Note that in the model the level of predictability is not sensitive

to using the dividend-price ratio or the adjusted dividend-price ratio; this is because the

information in the predictive variable, consistent with theory, should not change when one

adjusts the price-dividend ratio for the risk free rate. This model-based evidence, along with

the sharp differences in the data across Panels A and B of Table III, indicates that the actual

return predictability is close to what one finds using the risk-free adjusted dividend-price

4Using the parameter configuration of Bansal and Yaron (2004) may lead to somewhat higher

predictability of consumption growth as pointed out in Bui (2007).

configuration we rely on does equally well in reproducing key asset pricing features as well as replicating low

predictability of consumption as shown in Table VI. As discussed above, the model-implied predictability of

consumption growth is even lower when one allows for a non-zero correlation in shocks in the consumption

growth dynamics.

However, the somewhat different

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ratio. The model, as documented above, can completely match this data feature.

4 Conclusions

The debate regarding return and cash flow predictability has been at center stage in finance

for several decades. From the statistical point of view, both returns and consumption growth

are predictable by the dividend-price ratios only to a limited extent. We show that the

implications of the long-run risks model are consistent with the view that the data contains

a small predictive component in both returns and consumption growth.

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References

Ang, Andrew, and Geert Bekaert, 2007, Stock Return Predictability: Is it There?, Review

of Financial Studies 20,3, 651–707.

Bansal, Ravi, Dana Kiku, and Amir Yaron, 2007, Risks for the long run: Estimation and

Inference, Working paper, The Wharton School, University of Pennsylvania.

Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset

pricing puzzles, Journal of Finance 59, 1481–1509.

Bansal, Ravi, and Amir Yaron, 2006, The asset pricing-macro nexus and return-cash flow

predictability, Working paper, The Wharton School, University of Pennsylvania.

Boudoukh, Jacob, Matthew Richardson, and Robert Whitelaw, 2006, The Myth of Long-

Horizon Predictability, forthcoming, Review of Financial Studies.

Bui, Ming P., 2007, Long-Run Risks and Long-Run Predictability: A Comment, Working

paper, Harvard University.

Campbell, John, and Hyeng Keun Koo, 1997, A comparison of a numerical and approximate

analytical solutions to an intertemporal consumption choice problem, Journal of Economic

Dynamics and Control 21, 273–295.

Campbell, John, and Robert Shiller, 1988, Stock Prices, Earnings, and Expected Dividends,

Journal of Finance 43, 661–676.

Campbell, John Y., and John H. Cochrane, 1999, By Force of Habit: A Consumption-Based

Explanation of Aggregate Stock Market Behavior, Journal of Political Economy 107, 205–

255.

Cochrane, John, 2006, The Dog That Did Not Bark: A Defense of Return Predictability,

forthcoming, Review of Financial Studies.

Epstein, Larry G., and Stanley E. Zin, 1989, Substitution, risk aversion, and the

intertemporal behavior of consumption and asset returns:A theoretical framework,

Econometrica 57, 937–969.

Fama, Eugene, and Kenneth French, 1988, Dividend yields and expected stock returns,

Journal of Financial Economics 22, 3–27.

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Goyal, Amit, and Ivo Welch, 2003, Predicting the equity premium with dividend ratios,

Managment Science 49, 639654.

Hansen, Lars, John Heaton, and Nan Li, 2006, Consumption strikes back?, Working paper,

University of Chicago.

Hodrick, Robert, 1992, Dividend Yields and Expected Stock Returns:Alternative

Procedures for Inference and Measurement, Review of Financial Studies 5-3, 357–386.

Keim, Donald, and Robert Stambaugh, 1986, Predicting returns in the stock and bond

markets, Journal of Financial Economics 17, 357390.

Lettau, Martin, and Sydney Ludvigson, 2005, Expected Returns and Expected Dividend

Growth, Journal of Financial Economics 76, 583–626.

Lewellen, Jonathan, 2004, Predicting returns with financial ratios, Journal of Financial

Economics 74, 209235.

Stambaugh, Robert, 1999, Predictive Regressions, Journal of Financial Economics 54,

375421.

Valkanov, Rosen, 2003, Long-horizon regressions:theoretical results and applications,

Journal of Financial Economics 68, 201232.

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5 Appendix

The solutions for As are given by,

A0 =

1

1 − κ1

?

logδ + κ0+

?

1 −1

ψ

?

µc+ κ1A2(1 − ν)¯ σ2+θ

2

?

κ1A2σw

?2?

A1 =

1 −1

1 − κ1ρ

ψ

(10)

A2 = −

(γ − 1)(1 −1

2 (1 − κ1ν)

ψ)

?

1 +

?κ1ϕe

1 − κ1ρ

?2?

As shown in Bansal and Yaron (2004) the solution for the market price of risks,

λη

λe = (1 − θ)κ1A1ϕe= (γ −1

= (1 − θ)κ1A2= −(γ − 1)(γ −1

= γ

ψ)

κ1ϕe

1 − κ1ρ

ψ)

(11)

λw

κ1

2 (1 − κ1ν)

?1 + (

κ1ϕe

1 − κ1ρ)2?

where these respectively represent the market prices of transient (ηt+1), long-run (et+1) and

volatility (wt+1) risks respectively.

The price-dividend ratio for the market claim to dividends, zm,t= A0,m+A1,mxt+A2,mσ2

where

?

φ −1

1 − κ1,mρ

1

1 − κ1,mν

?

t,

A0,m =

1

1 − κ1,m

Γ0+ κ0,m+ µd+ κ1,mA2,m(1 − ν)¯ σ2+1

2

?

κ1,mA2,m− λw

?2σ2

w

?

A1,m =

ψ

(12)

A2,m =

?

Γ2+1

2

?

ϕ2+ (π − λη)2+ (κ1,mA1,mϕe− λe)2??

?

where Γ0= logδ −1

ψµc− (θ − 1)A2(1 − ν)¯ σ2+θ

2

κ1A2σw

?2?

and Γ2= (θ − 1)(κ1ν − 1)A2

The risk premium is determined by the covariation of the return innovation with the

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innovation into the pricing kernel. Thus, the risk premium for rm,t+1is equal to the asset’s

exposures to systematic risks multiplied by the corresponding risk prices,

Et(rm,t+1− rf,t) + 0.5σ2

t,rm

= −Covt

?

mt+1− Et(mt+1),rm,t+1− Et(rm,t+1)

?

= λησ2

tβη,m+ λeσ2

tβe,m+ λwσ2

wβw,m

where the asset’s βs are defined as,

βη,m = π

βe,m = κ1,mA1,mϕe

βw,m = κ1,mA2,m

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Table I

Summary Statistics

Mean Volatility

EstimateEstimate SESE

Cons. Growth (∆c)

Market Return (R)

Div. Yield (D/P)

Risk-free Rate (Rf)

1.95

8.51

3.97

0.95

0.31

1.69

0.43

0.91

2.13

19.65

1.52

3.95

0.44

2.14

0.23

0.75

Table I presents descriptive statistics for consumption growth, return and dividend yield of the aggregate

stock market, and the risk-free rate. All entries are expressed in percentage terms. Standard errors are based

on the Newey and West (1987) estimator with 8 lags. The data are real, sampled on an annual frequency

and cover the period from 1930 to 2006.

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Table II

Predictability of Consumption Growth

Horizon (yr)

Estimate

5%10% 50% 90%95%

ˆβ

t-stat

R2

-0.007

-1.286

0.022

-0.02-0.01 -0.000.00 0.00

1

-1.97-1.64 -0.590.72 1.12

0.00 0.000.010.05 0.06

ˆβ

t-stat

R2

-0.004

-0.363

0.001

-0.02-0.010.010.030.04

3

-1.31 -0.860.70 1.852.12

0.00 0.00 0.02 0.080.11

ˆβ

t-stat

R2

0.005

0.294

0.002

-0.03-0.02 0.020.070.09

5

-1.29-0.811.042.38 2.71

0.000.000.04 0.180.23

Table II presents estimates of slope coefficients (ˆβ), robust t-statistics and R2s from projecting 1-, 3- and

5-year consumption growth onto lagged dividend-price ratio of the aggregate stock market portfolio. Robust

t-statistics are computed using Hodrick (1992)-adjusted standard errors. The right columns display bootstrap

distributions of the reported statistics. Empirical percentiles are constructed by resampling the data 10,000

times in blocks of 8 years with replacement. The data employed in estimation are real, compounded

continuously, sampled on an annual frequency and cover the period from 1930 to 2006.

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Table III

Predictability of Excess Returns

Panel A: Predictability by Dividend-Price Ratio

Horizon (yr)

Estimate

5% 10%50%90%95%

ˆβ

t-stat

R2

0.094

1.779

0.045

0.030.05 0.100.200.25

1

0.580.901.912.83 3.10

0.00 0.010.05 0.130.15

ˆβ

t-stat

R2

0.276

1.755

0.175

0.080.13 0.29 0.56 0.67

3

0.540.90 1.962.873.10

0.010.030.150.32 0.38

ˆβ

t-stat

R2

0.455

1.857

0.294

0.050.120.38 0.76 0.87

5

0.210.561.67 2.58 2.81

0.01 0.020.17 0.400.46

Panel B: Predictability by Dividend-Price Ratio Adjusted for Risk-free Rate

Horizon (yr)

Estimate

5% 10%50% 90%95%

ˆβ

t-stat

R2

0.009

1.418

0.034

0.000.000.01 0.010.02

1

0.230.48 1.402.48 2.80

0.000.000.03 0.080.10

ˆβ

t-stat

R2

0.012

0.839

0.033

-0.02-0.010.010.03 0.04

3

-0.83-0.46 0.932.24 2.54

0.000.000.03 0.150.20

ˆβ

t-stat

R2

0.017

1.025

0.045

-0.02 -0.01 0.010.040.05

5

-1.08 -0.700.762.032.33

0.000.00 0.030.160.21

Panel A of Table III presents estimates of slope coefficients (ˆβ), robust t-statistics and R2s from projecting

1-, 3- and 5-year excess returns onto lagged dividend-price ratio of the aggregate stock market portfolio.

Evidence on predictability of multi-period excess returns by the dividend-price ratio adjusted for the risk-

free rate is reported in Panel B. Robust t-statistics are computed using Hodrick (1992)-adjusted standard

errors. The right columns display bootstrap distributions of the reported statistics. Empirical percentiles are

constructed by resampling the data 10,000 times in blocks of 8 years with replacement. The data employed

in estimation are real, compounded continuously, sampled on an annual frequency and cover the period from

1930 to 2006.

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Table IV

Configuration of Model Parameters

Preferences

δγ

10

ψ

1.50.9989

Consumption

µρφx

¯ σνσw

0.00150.9750.038 0.00720.999 0.0000028

Dividends

µd

φϕd

5.96

π

0.0015 2.52.6

Table IV reports configuration of investors’ preferences and time-series parameters that describe dynamics

of consumption and dividend growth rates. The model is calibrated on a monthly decision interval.

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Table V

Model-Implied Dynamics of Growth Rates and Prices

Moments

Median5%95%

Consumption:

E[∆c]

σ(∆c)

AC(1)

1.80

2.46

0.39

0.88

1.58

0.18

2.69

3.62

0.57

Dividends:

E[∆d]

σ(∆d)

Corr(∆d,∆c)

1.81

13.87

0.46

-2.02

8.95

0.25

5.63

20.37

0.62

Market:

E[R]

σ(R)

E[D/P]

σ[D/P]

8.16

20.39

4.42

0.83

4.38

13.28

3.71

0.43

13.73

30.62

6.07

1.73

Risk-free Rate:

E[Rf]

σ(Rf)

1.23

0.95

0.28

0.58

1.81

1.48

Table V presents moments of simulated annualized consumption and dividend growth rates along with asset

pricing implications of the model. Reported statistics are based on 10,000 simulated samples with 77 × 12

monthly observations that match the length of the actual data. The entries represent the median, 5thand

95thpercentiles of the monte-carlo distributions of the corresponding statistics.

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Table VI

Model-Implied Predictability of Consumption Growth

Horizon (yr)

Median

5%10% 90% 95%

ˆβ

t-stat

R2

-0.05

-2.88

0.15

-0.09 -0.08-0.02 -0.01

1

-4.22 -3.96 -1.40-0.83

0.010.030.31 0.36

ˆβ

t-stat

R2

-0.10

-2.35

0.11

-0.20 -0.18-0.020.01

3

-4.02 -3.72-0.440.22

0.000.010.31 0.37

ˆβ

t-stat

R2

-0.12

-1.94

0.09

-0.29 -0.250.01 0.05

5

-3.89 -3.50 0.17 0.85

0.00 0.000.310.38

Table VI reports implications of the Long-Run Risks model for consumption growth predictability. The

entries represent estimates of slope coefficients (ˆβ), robust t-statistics and R2s from projecting 1-, 3- and

5-year consumption growth onto lagged dividend-price ratio of the aggregate stock market portfolio. Robust

t-statistics are computed using Hodrick (1992)-adjusted standard errors. The entries present distributions

of the corresponding moments across 10,000 simulated samples.

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Table VII

Model-Implied Predictability of Excess Returns

Panel A: Predictability by Dividend-Price Ratio

Horizon (yr)

Median

5% 10%90%95%

ˆβ

t-stat

R2

0.09

0.76

0.01

-0.11-0.06 0.27 0.33

1

-0.95 -0.57 1.982.30

0.00 0.00 0.060.08

ˆβ

t-stat

R2

0.27

0.87

0.03

-0.24 -0.14 0.690.82

3

-0.83 -0.47 2.082.40

0.000.000.14 0.18

ˆβ

t-stat

R2

0.42

0.90

0.05

-0.38-0.201.04 1.21

5

-0.86-0.45 2.112.44

0.000.000.20 0.25

Panel B: Predictability by Dividend-Price Ratio Adjusted for Risk-free Rate

Horizon (yr)

Median

5% 10% 90% 95%

ˆβ

t-stat

R2

0.01

0.52

0.01

-0.02 -0.020.04 0.05

1

-1.17 -0.811.74 2.08

0.000.00 0.050.07

ˆβ

t-stat

R2

0.03

0.52

0.02

-0.06 -0.040.110.13

3

-1.15 -0.801.77 2.10

0.00 0.00 0.120.17

ˆβ

t-stat

R2

0.04

0.54

0.04

-0.10-0.07 0.160.19

5

-1.19-0.821.822.14

0.00 0.000.18 0.23

Table VII reports predictability evidence for excess returns implied by the Long-Run Risks model. Panel A

presents estimates of slope coefficients (ˆβ), robust t-statistics and R2s from projecting 1-, 3- and 5-year excess

returns onto lagged dividend-price ratio of the aggregate stock market portfolio. Evidence on predictability

of multi-period excess returns by the dividend-price ratio adjusted for the risk-free rate is reported in Panel

B. Robust t-statistics are computed using Hodrick (1992)-adjusted standard errors. The entries present

distributions of the corresponding moments across 10,000 simulated samples.

20