A Note on the Economics and Statistics of
Predictability: A Long Run Risks Perspective∗
November 14, 2007
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, email@example.com.
‡The Wharton School, University of Pennsylvania, firstname.lastname@example.org.
§The Wharton School, University of Pennsylvania and NBER, email@example.com.
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
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
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
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
Model-Implied Dynamics of Growth Rates and Prices
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.
Model-Implied Predictability of Consumption Growth
5%10% 90% 95%
-0.09 -0.08-0.02 -0.01
-4.22 -3.96 -1.40-0.83
-0.29 -0.250.01 0.05
-3.89 -3.50 0.17 0.85
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.
Model-Implied Predictability of Excess Returns
Panel A: Predictability by Dividend-Price Ratio
-0.11-0.06 0.27 0.33
-0.95 -0.57 1.982.30
0.00 0.00 0.060.08
-0.24 -0.14 0.690.82
-0.83 -0.47 2.082.40
Panel B: Predictability by Dividend-Price Ratio Adjusted for Risk-free Rate
5% 10% 90% 95%
-0.02 -0.020.04 0.05
-1.17 -0.811.74 2.08
-1.15 -0.801.77 2.10
0.00 0.00 0.120.17
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.