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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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