Euler Equation Errors
Martin Lettau
New York University, CEPR, NBER
Sydney C. Ludvigson
New York University and NBER
Comments Welcome
First draft: September 1, 2004
This draft: August 9, 2005
Lettau: Department of Finance, Stern School of Business, New York University, 44 West Fourth Street,
New York, NY 10012-1126; Email: mlettau@stern.nyu.edu, Tel: (212) 998-0378; Fax: (212) 995-4233;
http://www.stern.nyu.edu/? mlettau. Ludvigson: Department of Economics, New York University, 269
Mercer Street, 7th Floor, New York, NY 10003; Email: sydney.ludvigson@nyu.edu, Tel: (212) 998-8927; Fax:
(212) 995-4186; http://www.econ.nyu.edu/user/ludvigsons/. Ludvigson acknowledges ?nancial support
from the Alfred P. Sloan Foundation and the CV Starr Center at NYU. We are grateful to Dave Backus,
John Y. Campbell, John Cochrane, George Constantinides, Wayne Ferson, Mark Gertler, Stephen Gordon,
Fatih Guvenen, Ravi Jagannathan, Narayana Kocherlakota, Stijn Van Nieuwerburgh, Martin Weitzman,
Motohiro Yogo, and seminar participants at the 2005 CIRANO/CIREQ (Montreal) Financial Econometrics
Conference, the 2005 Society for Economic Dynamics conference, Harvard University, New York University,
Unversit? Laval, West Virginia University and Wharton for helpful comments. We thank Massamiliano
Croce and Jack Favalukis for excellent research assistance. Any errors or omissions are the responsibility of
the authors.
Euler Equation Errors
Abstract
Among the most important pieces of empirical evidence against the standard represen-
tative agent, consumption-based asset pricing paradigm are the formidable unconditional
Euler equation errors the model produces for cross-sections of asset returns. Here we ask
whether calibrated leading asset pricing models? speci?cally developed to address empiri-
cal puzzles associated with the standard paradigm? explain the mispricing of the standard
consumption-based model when evaluated on cross-sections of asset returns. We ?nd that,
in many cases, they do not. We present several results. First, we show that if the true
pricing kernel that sets the unconditional Euler equation errors to zero is jointly lognormally
distributed with aggregate consumption and returns, such a kernel will not rationalize the
magnitude of the pricing errors generated by the standard model, particularly when the
curvature of utility is high. Second, we show that leading asset pricing models also do not
explain the signi?cant mispricing of the standard paradigm for plausibly calibrated sets of
asset returns, even though in those models the pricing kernel, returns, and consumption are
not jointly lognormally distributed. Third, in contrast to the above results, we provide one
example of a limited participation/incomplete markets model capable of explaining larger
pricing errors for the standard model; but we also ?nd many examples of such models, in
which the consumption of marginal assetholders behaves quite dicurrency1erently from per capita
aggregate consumption, that do not explain the large Euler equation errors of the standard
representative agent model.
JEL: G12, G10.
1 Introduction
Among the most important pieces of empirical evidence against the standard representative
agent, consumption-based asset pricing paradigm are the formidable unconditional Euler
equation errors the model produces for cross-sections of asset returns. Such Euler equation
errors, or pricing errors (terms we use interchangeably), are especially large for cross-sections
that include a broad stock market index return, a short term Treasury bill rate, and the
size and book-market sorted portfolio returns emphasized by Fama and French (1992,1993).
The large unconditional pricing equation errors of the standard model have been stressed
elsewhere as an indication of the model? empirical di? culties, e.g., Mankiw and Shapiro
(1986), Breeden, Gibbons, and Litzenberger (1989), Campbell (1996), Cochrane (1996), and
Lettau and Ludvigson (2001). We present further evidence on the size of these errors here.
The standard model, as we de?ne it, assumes that agents have unrestricted access to
?nancial markets, that assets can be priced using the Euler equations of a representative-
consumer maximizing the discounted value of power utility functions, and that the pricing
kernel M, or stochastic discount factor, is equal to the marginal rate of substitution in
consumption. This model takes the form
E [Mt+1Rt+1] = 1; Mt+1 = (Ct+1=Ct) ; (1)
where the ?rst equality is the Euler equation, Rt+1 denotes the gross return on any tradable
asset, Ct+1 is per capita aggregate consumption, is the coe? cient of relative risk-aversion
and is a subjective time-discount factor. For any model, the Euler equation error of a
tradable asset is the dicurrency1erence between E [Mt+1Rt+1] and unity.
In much of theoretical asset pricing, Euler equations give the equilibrium prices of tradable
assets, consequently the empirical errors in this equation are a fundamental measure of how
well any model explains asset returns. It is perhaps surprising, then, that little research has
been devoted to investigating whether leading asset pricing models can help us understand
the signi?cant mispricing of the standard model when confronted with cross-sections of stock
returns. After all, such models were developed with the express purpose of explaining the
empirical limitations of the standard model. Instead, theoretical research has proceeded by
focusing on well known ?puzzles?generated by the standard model, for example, the equity
premium puzzle, the risk-free rate puzzle, and the time-series predictability of excess stock
market returns.
In this paper, we argue that large empirical Euler equation errors constitute a puzzle for
the standard consumption-based asset pricing model that is at least as damning as these
other, more well known, conundrums. We employ the empirical facts on Euler equation
errors to evaluate some leading asset pricing models that were speci?cally developed to
1
address puzzles associated with the standard paradigm (1). This is of interest because the
underlying assumption in each of these leading models is that, by discarding the standard
pricing kernel in favor of the true kernel implied by the model, an econometrician would be
better able to model asset pricing data. In particular, if leading asset pricing models are
true, then in these models using (1) to price assets should generate large unconditional asset
pricing errors, as in the data.
We ?nd that this is not always the case. Often, in leading asset pricing models, parameters
of a standard representative agent ?pricing kernel?based on (1) can be found that imply the
standard model has virtually identical unconditional pricing implications as the true model
that prices assets correctly. Thus, an econometrician who observed data generated from any
of these leading models would fail to reject the standard consumption-based model in tests
of its unconditional moment restrictions, let alone replicate the sizable unconditional Euler
equation errors found when ?tting (1) to historical data.1
We note that the literature has already demonstrated a set of theoretical propositions
showing that any observed joint process of aggregate consumption and returns can be an
equilibrium outcome if the second moments of the cross-sectional distribution of consump-
tion growth and asset returns covary in the right way (Constantinides and Du? e (1996)).
Such existence proofs, important in their own right, are not the focus of this paper. Instead,
we ask whether particular calibrated economies of leading asset pricing models are quan-
titatively capable of matching the large pricing equation errors generated by the standard
consumption-based model when ?tted to historical data. This is important because it re-
mains unclear whether fully speci?ed models built on primitives of tastes, technology, and
underlying shocks, and calibrated to accord with the data in plausible ways, can in practice
generate the joint behavior of aggregate consumption and asset returns that we observe in
the data.
Our analysis uses simulated data from several leading asset pricing models: the represen-
tative agent external habit-persistence paradigms of (i) Campbell and Cochrane (1999) and
(ii) Menzly, Santos, and Veronesi (2004), (iii) the representative agent long-run risk model
based on recursive preferences of Bansal and Yaron (2004), and (iv) the limited participation
model of Guvenen (2003). Each is an explicitly parameterized economic model calibrated
to accord with the data, and each has proven remarkably successful in explaining a range of
1Note that these ?ndings are not a statement about the ?power? of empirical Euler equation errors to
reject alternative models, just as evidence that most business cycle models fail to explain a large equity
premium and a low and stable risk-free rate is not a statement about the power of key asset pricing moments
to reject alternative business cycle models. The interesting ?nding is not that the alternative models we
study have similar properties along this dimension, but rather that they all fail in the same way to explain
the signi?cant mispricing of the standard model.
2
asset pricing phenomena that the standard model is unable to explain.
Our focus on Euler equations is intentional, since they represent the set of theoretical
restrictions from which all asset pricing implications follow. Kocherlakota (1996) emphasizes
the importance of Euler equation errors for understanding the central empirical puzzles of
the standard consumption-based model, which he illustrates using annual data on aggregate
consumption and asset returns. Formal econometric tests of conditional Euler equations
using aggregate consumption data lead to rejections of the standard representative agent,
consumption-based asset pricing model, even when no bounds are placed on the coe? cient of
relative risk aversion or the rate of time preference (Hansen and Singleton (1982); Ferson and
Constantinides (1991); Hansen and Jagannathan (1991)). Similarly, we stress here that the
quarterly pricing errors for the unconditional Euler equations associated with cross-sections
of asset returns are large when ?tting aggregate data to (1), even when the parameters
and are left unrestricted and chosen to minimize those errors. Such Euler equation errors
place additional testable restrictions on asset pricing models: not only must such models
have zero pricing errors when the pricing kernel is correctly speci?ed according to the model,
they must also produce large pricing errors when the pricing kernel is incorrectly speci?ed
using power utility and aggregate consumption.
Our main ?ndings are as follows:
First, we consider the case in which consumption in (1) is mismeasured, perhaps because
per capita aggregate consumption is a poor measure of individual assetholder consump-
tion, or the consumption of stockholders as an aggregate. We show that if the true pricing
kernel based on assetholder consumption is jointly lognormally distributed with aggregate
consumption and returns, then estimation of (1) using per capita aggregate consumption
produces biased estimates of the assetholder? subjective discount factor and risk aversion
parameters, but does not rationalize the magnitude of the pricing errors generated by the
standard model, particularly when is large.
Second, we use simulated data from each of the leading asset pricing models mentioned
above to study the extent to which these models explain the mispricing of the standard
model. We show that some of these models can explain why an econometrician obtains
implausibly high estimates of and when freely ?tting aggregate data to (1). But, none
can explain the large unconditional Euler equation errors associated with such estimates
for plausibly calibrated sets of asset returns.2 Indeed, the asset pricing models we consider
2Campbell and Cochrane (2000) evaluate the pricing errors of the standard consumption-based model
implied by the habit model of Campbell and Cochrane (1999), by looking at the pricing errors for the most
mispriced portfolio. Their results suggest that there is scope for mispricing, but do not imply signi?cant
mispricing for the sets of stock portfolios we calibrate our models to match. Our approach dicurrency1ers from theirs
in that we do not analyze the most mispriced portfolio (which can look nothing like the stock portfolios
3
counterfactually imply that values of and can be found for which (1) satis?es the uncon-
ditional Euler equation restrictions just as well as the true pricing kernel, implying that the
standard model generates negligible pricing errors for cross-sections of asset returns.
Third, in contrast to the above results, we provide one example of an incomplete mar-
kets/limited participation model that can rationalize larger pricing errors for the standard
model, as long as the joint distribution of aggregate consumption, individual assetholder con-
sumption, and stock returns takes a particular non-normal form. But we also ?nd? within
the class of distributions we consider? that many models with non-normal distribution spec-
i?cations will not explain mispricing of the standard model, since in many cases the use of
(1) to price assets merely distorts the estimated preference parameters but not the pricing
errors.
We emphasize that this paper is not a criticism of existing asset pricing theory. In-
stead, we seek a diagnostic for understanding the directions in which future research may
be fruitfully applied by providing a dicurrency1erent perspective on whether leading paradigms fully
rationalize the joint behavior of asset prices and aggregate quantities that is central to the
empirical failure of the standard model. We also add to the literature by outlining the econo-
metric consequences, for estimation and testing of unconditional Euler equations, of ?tting
the standard pricing kernel (1) to data when the true pricing kernel that generated the data
is derived from some other model. Finally, we stress that our results do not imply that no
model can be made consistent with the testable restrictions we focus on here. Our point
is that many models written down today appear inconsistent with these restrictions and do
not explain the mispricing of the standard consumption-based model.
The rest of this paper is organized as follows. The next section lays out the empirical
Euler equation facts using post-war U.S. data on per capita aggregate consumption and
returns. Section 3 studies the implications of various economic theories for the same Euler
equation errors we measure in the data, beginning with a simple example in which the
true pricing kernel is jointly lognormally distributed with aggregate consumption growth
and asset returns. Next, we investigate the extent to which the four leading asset pricing
models mentioned above are capable of explaining the empirical facts. Our main ?ndings are
shown to be robust to time-aggregation of aggregate consumption data, to the introduction
of limited participation in the representative agent models, and to the use of small samples
to compute pricing errors. Finally, we explore the pricing implications of a number of
simple incomplete markets/limited participation models in which assetholder consumption
observed in historical data), but instead generate speci?c cross-sections of traded assets in the models to
match the properties of cross-sections in the data by directly relying on the models?own baseline calibrations
of asset returns, or by employing calibrations which deliver spreads in risk-premia commensurate with those
in our historical data set.
4
is permitted to behave quite dicurrency1erently from per capita aggregate consumption. Section 4
concludes.
2 Euler Equation Errors: Empirical Facts
The standard consumption based model, as de?ned above,assumes a representative-consumer
with constant relative risk aversion (CRRA) preferences over consumption given by
U = E
( 1X
t=0
t C
1
t 1
1
)
; > 0: (2)
At each date, agents maximize (2) subject to an accumulation equation for wealth. Agents
have unrestricted access to ?nancial markets and face no borrowing or short-sales constraints.
The asset pricing model comes from the ?rst-order conditions for optimal consumption choice,
which imply that for any traded asset indexed by j, with a gross return at time t+1 of Rjt+1;
the following Euler equation holds:
Et
"
C
t+1
Ct
Rjt+1
#
= 1: (3)
Here Et is the conditional expectation operator, conditional on time t information. The mar-
ginal rate of substitution in consumption, Mt+1 (Ct+1=Ct) , is the stochastic discount
factor, or pricing kernel in this model. By the law of iterated expectations, equation (3) also
implies a corresponding unconditional Euler equation taking the form (1), which we focus
on from here on.
We focus our attention on the unconditional Euler equation errors for cross-sections
of asset returns that include a broad stock market index return (measured as the CRSP
value-weighted price index return and denoted Rst), a short term Treasury bill rate (mea-
sured as the three-month Treasury bill rate and denoted Rft ), and six size and book-market
sorted portfolio returns available from Kenneth French? Dartmouth web site. These returns
are value-weighted portfolio returns of common stock sorted into two size (market equity)
quantiles and three book value-market value quantiles. We use equity returns on size and
book-to-market sorted portfolios because Fama and French (1992) show that these two char-
acteristics provide a ?simple and powerful characterization?of the cross-section of average
stock returns, and absorb the roles of leverage, earnings-to-price ratio and many other factors
governing cross-sectional variation in average stock returns. These returns are denoted as
a vector RFFt (R1t ;:::R6t )0. We analyze the pricing errors for the eight assets Rst;Rft ;RFFt
as a group, as well as for the set of two assets comprised of only Rst and Rft . The latter is
of interest because the standard model? inability to explain properties of these two returns
5
has been central to the development of a consensus that the model is ?awed. In addition,
almost all asset pricing models seek to match the empirical properties of these two returns,
whereas fewer generate implications for larger cross-sections of securities.
There are two ways to present the pricing errors implied by the standard consumption-
based model. One is to focus on the Euler equations of raw returns:
E
"
C
t+1
Ct
Rjt+1
#
1 = 0 j = s;f;1;:::;6: (4)
Another is to focus on the Euler equation errors for excess returns:
E
"
Ct+1
Ct
Rjt+1 Rft+1
#
= 0 j = s;1;:::;6: (5)
For both Euler equations above, we refer to the dicurrency1erence between an estimate of the left-
hand-side and zero as the unconditional Euler equation error, or alternatively the pricing
error, for the jth asset return. If the standard model is true then these errors should be zero
for any traded asset, given some values of the parameters and .
Regardless of whether the Euler equations are stated in terms of excess or raw returns,
we choose the parameters and to minimize a weighted sum of squared pricing errors, an
application of Generalized Method of Moments (GMM, Hansen (1982)):
min
;
gT ( ; ) w0T ( ; ) WwT ( ; ) ; (6)
where Wis the identity matrix and wT ( ; ) is the vector of average pricing errors for each
asset, with jth element wjT ( ; ) given either by
wjT ( ) = wjT ( ) = 1T
TX
t=1
C
t+1
Ct
Rjt+1 Rft+1
; j = s;1;:::;6:
in the case of excess returns, or
wjT ( ; ) = 1T
TX
t=1
C
t+1
Ct
Rjt+1 1; j = s;f;1;:::;6
in the case of raw returns. Let b and b denote the arg min gT ( ; ).
We use the identity weighting matrix because these it preserves the structure of the
original test assets, which are based on economically interesting characteristics and deliver
a wide spread in cross-sectional average returns. Use of alternate matrixes that re-weight
the Euler equations amount to minimizing the pricing errors for re-weighted portfolios of
the original test assets and destroy this structure. It should be noted, however, that other
weighting matrixes such as the optimal weighting matrix of Hansen (1982) and the second
6
moment matrix of Hansen and Jagannathan (1997) produce similar results although they
are not reported in what follows.
The estimation uses quarterly, per capita data on nondurables and services expenditures
measured in 1996 dollars as a measure of consumption Ct, in addition to the return data
mentioned above.3 Returns are de?ated by the implicit price de?ator corresponding to the
measure of consumption Ct. The data span the period from the fourth quarter of 1951 to
the fourth quarter of 2002. A detailed description of the data is provided in the Appendix.
Figure 1 displays the square root of the average squared Euler equation errors (RMSE)
for the excess returns in (5) over a range of values of . The solid line plots the case where
the single excess return Rst+1 Rft+1 is priced; the dotted line plots the case for the seven
returns returns Rst+1 Rft+1 and RFFt Rft+1. To give a sense of how large pricing errors are
relative to the returns being priced, we plot RMSE/RMSR, where RMSR is the square root
of the cross-sectional average of the squared mean returns of the assets under consideration.
Two aspects of Figure 1 warrant emphasis. First, notice that in the case of the single
excess return on the aggregate stock market, Rst+1 Rft+1 (solid line), the RMSE is itself just
the pricing error (5), where this error is computed as the sample mean of the expression in
square brackets in (5), scaled by the value of that minimizes an equally weighted average
of Euler equation errors for Rst and Rft . The solid line shows that the pricing error (5) for the
excess return on the aggregate stock market cannot be driven to zero, or indeed even to a
small number, for any value of . The lowest pricing error is 5.2% per annum, which occurs
at = 117. The ?gure displays this error as a fraction of the average excess stock market
return, and is shown to be almost 60 percent of the average annual CRSP excess return. At
other values of this error rises precipitously and reaches several times the average annual
stock market return when is outside the ranges displayed in Figure 1. Thus, there is no
value of that sets the pricing error (5) to zero.4
Second, the dashed line in Figure 1 shows that the root mean-squared pricing errors
for the seven asset case Rst+1 Rft+1;RFFt Rft+1 is also large. As a fraction of the square
root of the average squared excess returns being priced, the minimum RMSE is about 60%,
about the same as that for the single excess return Rst+1 Rft+1; and this occurs at = 118:
At other values of the RMSE rises precipitously, just as it does for the single asset case.
Therefore, the degree of mispricing in the standard model is about the same regardless of
whether we consider the single excess return on the market or a larger cross-section of excess
stock market returns.5
3We exclude shoes and clothing expenditure from this series since they are partly durable and should not
be included in a measure of the service ?ow of consumption.
4Note that (5) is a nonlinear function of : Thus, there is not necessarily a solution.
5In computing the pricing errors above, we use the standard timing convention that end-of-period returns
7
Next we report the Euler equation errors in (4) for raw returns. Table 1 shows that when
and are chosen to minimize (6) for Rst+1 and Rft+1 alone, the RMSE is 2.7% per annum, a
magnitude that is 48% of the square root of the average squared returns on these two assets.
Since there are just two moments in this case, this again means that there are no values of
and that set the two pricing errors to zero. When and are chosen to minimize (6)
for the eight asset returns Rst+1;Rft+1;RFFt , the RMSE is 3.05% per annum, a magnitude
that is 33% of the square root of the average squared returns on the eight assets. Notice
that the estimates b and b (which are left unrestricted) are close to 1.4 and 90, respectively,
regardless of which set of test assets are used. The ?nal two columns of Table 1 report the
results of statistical tests of the model, discussed below.
Why are the pricing errors so large? The lower panel of Table 1 provides a partial answer:
a signi?cant part of the unconditional Euler equation errors generated by the standard model
are associated with recessions, periods in which per capita aggregate consumption growth
is steeply negative. For example, if we remove the data points associated with the smallest
six observations on consumption growth, the RMSE is 0.73% per annum or 13% of the root
mean squared returns for Rst+1 and Rft+1; and 1.94% per annum or 21 percent of the root
mean squared returns on the eight asset returns Rst+1;Rft+1;RFFt . Table 2 identi?es these
six observations as they are located throughout the sample. Each occur in the depths of
recessions in the 1950s, 1970s, early 1960s, 1980s and 1990s, as identi?ed by the National
Bureau of Economic Research. In these periods, aggregate per capita consumption growth
is steeply negative but the aggregate stock return and Treasury-bill rate is, more often than
not, steeply positive. This result echoes the ?ndings in Ferson and Merrick (1987) who report
less evidence against the standard consumption-based model in non-recession periods. Since
the product of the marginal rate of substitution and the gross asset return must be unity
on average, such negative comovement (positive comovement between Mt+1 and returns)
contributes to large pricing errors.6 One can also reduce the pricing errors by using annual
dated in quarter t should be paired with consumption growth measured from t 1 to t. If, instead, returns
at t are paired with consumption growth from t to t + 1; a value for can be found that sets the pricing
error to zero for the single excess return Rs Rf . By contrast, the choice of timing convention has very
little acurrency1ect on the RMSE for the set of seven excess returns Rs Rf ; RFFt Rf . We use the former timing
convention as it is standard empirical practice in estimation of Euler equations. We stress, however, that
the timing convention itself is not important for the comparisons with theoretical models that follow, since
those models always produce zero pricing errors regardless of which timing convention is used.
6Eliminating the recession periods, however, results in preference parameter estimates that are even more
extreme than they are in the full sample; for example c > 300: Therefore, if one? criterion for success is
reasonable preference parameter estimates, then the standard model does worse when the recession periods
are removed than when they are included. If is restricted to be less than 100 in the sample without
recessions, the pricing errors move up considerably. For example, in the two asset case the RMSE moves up
to 1.94% from 0.73%.
8
returns and year-over-year consumption growth.7 This procedure averages out the worst
quarters for consumption growth instead of removing them. Either procedure eliminates a
substantial proportion of the cyclical variation in consumption. For example, on a quarterly
basis the largest declines in consumption are about six times as large at an annual rate as
those on a year-over-year basis. This explains why Kocherlakota (1996), who focuses on
annual data, is able to locate parameter values for and that exactly satisfy the Euler
equations (4) for a stock return and Treasury-bill rate.
Of course, these quarterly recession episodes are not outliers to be ignored, but signi?cant
economic events to be explained. Indeed, we argue that such Euler equation errors?driven
by periods of important economic change?are among the most damning pieces of evidence
against the standard model. An important question is why the standard model performs so
poorly in recessions relative to other times.
Although not reported above, we note that the pricing error of the Euler equation asso-
ciated with the CRSP stock market return is always positive, implying a positive ?alpha?
in the expected return-beta representation of the model.8 This says that unconditional risk
premia are too high to be explained by the stock market? covariance with the marginal
rate of substitution of aggregate consumption, a familiar result from the equity premium
literature. The high alphas generated by the standard consumption-based model constitute
one of the most remarked-upon failures in the history of asset pricing theory.
2.1 Sampling Error and Tests for Joint Normality
We can use GMM distribution theory to ask whether the estimated pricing errors wT ( ; )
are jointly dicurrency1erent from zero, that is larger than what would be implied by sampling error
alone. When there are more moments than parameters to be estimated, such an assessment
can be interpreted as a test of overidentifying restrictions. The last two columns of Table
1 report p values from chi-squared tests of the model? overidentifying restrictions for es-
timation of the eight Euler equations in (4). Although the results presented so far have
used the identity weighting matrix, the last column in Table 1 presents the p values from
the same statistical test using an estimate of the optimal GMM weighting matrix (Hansen
(1982)). The results from either weighting matrix are the same: we may strongly reject the
hypothesis that the Euler equation errors are jointly statistically indistinguishable from zero;
7Jagannathan and Wang (2004) study the ability of a linearized version of the standard model to explain
a large cross-section of asset returns using forth quarter over fourth quarter consumption growth and annual
asset returns. They ?nd more support for the model when year-over-year growth rates are restricted to the
fourth quarter.
8The alpha in the expected return-beta representation is equal to the pricing error, scaled by 1=E [Mt];
see Cochrane (2005) for an exposition.
9
the p values for this test are less than 0.0001.
For the two-asset case, the model is just-identi?ed, so the overidentifying tests above are
not applicable. But note that the expectation in (5) is estimated using the sample means of
et+1 =
C
t+1
Ct
Rst+1 Rft+1
;
which are excess returns discounted by the pricing kernel
Ct+1
Ct
: Fixing and , it is
possible to compute the sampling variation in the sample mean of et+1, given as 2 = 2e=T;
where e is the sample standard deviation of et+1 and T is the sample size.9 Not surprisingly,
the sampling error of the mean of et+1 is quite large when evaluated at the optimal values
of = 1:4 and = 117: a con?dence interval formed by plus and minus two standard errors
is ( 0:55%;11%) as a percent per annum. This large range, which includes zero, arises
partly for the same reason that it is di? cult to estimate the equity premium accurately:
excess returns are highly volatile. But it is also because the data force a very high value
for in an attempt to ?t the equity premium. Such a high value of generates extreme
volatility in the pricing kernel, making discounted returns even harder to estimate precisely
than nondiscounted returns. Unless one views = 117 as plausible, however, such wide
standard error bands for mean discounted returns merely provide further evidence of the
model? empirical limitations, which even at = 117 leaves a pricing error that is more than
half of the average annual stock return. If instead we restrict the value of risk aversion to
lie in the range 0 89, the pricing errors are always statistically dicurrency1erent from zero
at the ?ve percent level of signi?cance. In short, when is as high as 117, the sample
mean of et+1 is statistically insigni?cant not because the pricing errors are small?indeed they
are economically large?but rather because discounted returns are so extremely noisy when
= 117.
For the case of raw returns and only two assets Rst+1 and Rft+1; we ask? given sampling
error? how likely is it that we would observe the pricing errors we observe under the null
hypothesis that the standard model is true and the Euler equations are exactly satis?ed in
population?10 Models that postulate joint lognormality for consumption and asset returns
9We also calculated standard errors for the mean of et+1 using a nonparameteric correction for serial
correlation. Since et+1 is close to serially uncorrelated, this correction has little acurrency1ect on the error bands.
10For the case of raw returns and only two assets Rst+1 and Rft+1; we have an exactly identi?ed GMM
system, so sampling error could in principle be assessed by conducting a block bootstrap simulation of the
raw data. This approach is inappropriate for the application here, however, because such a procedure would
ecurrency1ectively treat the low consumption growth periods in our sample as outliers, in the sense that a nontrivial
fraction of the simulated samples would exclude those observations. But as we have argued above, these
episodes of low or negative consumption growth? the hallmark of recessions? are not outliers to be ignored,
but signi?cant economic events to be explained.
10
are null models of this form, since in this case values for and always exist for which
the population Euler equations of any two asset returns are exactly satis?ed. Consequently,
only sampling error in the estimated Euler equations could cause non-zero pricing errors for
two asset returns. To address the question just raised, we suppose the data were generated
by the standard CRRA representative agent model, with returns and consumption jointly
lognormally distributed, and ask how likely is it that we would ?nd results like those reported
in Table 1, in a sample of the size we have.
Consider a simple model where ln Ct+1 i:i:d:N( ; 2), and preferences are of the
CRRA form with (for example) = 0:99 and = 2. Since the log dicurrency1erence in consumption
is i.i.d. and normally distributed, the return to a risky asset that pays consumption, Ct, as
its dividend is also normally distributed, as is risk-free rate. The equilibrium returns have an
analytical solution in this case, and can be solved from the (exactly satis?ed) Euler equations.
Using this model, we simulate 1000 arti?cial samples of consumption data equal to the size
our quarterly data set (204), with and set to match their respective sample estimates.
Using the analytical solutions for returns we use the simulated data for consumption growth
to obtain corresponding simulated data for returns. Finally, we use these simulated data to
solve for the values of and that minimize the empirical Euler equation errors for the risky
and risk-free asset return and store the absolute value of those errors. The 95% centered
con?dence for these errors, in percent annum, is found to be (9.5 10 11, 7.0 10 9) for the
risky return and (1.3 10 10, 6.5 10 9) for the risk-free return. These ?ndings suggest that it
is extremely unlikely that we would ?nd results like those reported in Table 1, in a sample
of the size we have, if this simple version of the standard CRRA representative agent model,
where consumption and returns are jointly lognormally distributed, were true.
Given these results, it is natural to assess whether joint lognormality is a plausible de-
scription of our consumption and return data, once we take into account sampling error. We
do so by performing formal statistical tests of the data based on multivariate skewness and
kurtosis for the vector Yt
h
log (Ct+1=Ct) ;log Rst+1 ;log
Rft
i0
: We also perform joint
normality tests for the larger set of variables
Xt
h
log (Ct+1=Ct) ;log Rst+1 ;log
Rft
;log R1t ;:::;log R6t
i
:
Normality tests for the larger cross-section will help inform the results in the next section in
which models that assume joint lognormality are studied.11
Statistical tests based on multivariate skewness and kurtosis provide strong evidence
against joint normality. For Yt multivariate skewness is estimated to be 1.54 and multivariate
excess kurtosis is 4.64, with p values for the null hypothesis that these statistics are equal to
11Multivariate skewness and kurtosis statistics are computed following Mardia (1970). Let xt be a p-
dimensional random variable with mean and variance-covariance matrix V of sample size T. Multivariate
11
those of a multivariate normal distribution less than 0.0001. Similarly for Xt, multivariate
skewness is 4.65 and multivariate kurtosis is 35.93, and the statistical rejections of normality
are even stronger. The same conclusion arises from examining quantile-quantile plots (QQ
plots) for the vector time-series Yt and Xt, given in Figure 3. This ?gure plots the sample
quantiles for the data against those that would arise under the null of joint lognormality,
along with pointwise standard errors bands.12 The QQ plots show substantial departures
from normality: a large number of quantiles lie far outside the standard error bands for joint
normality.
3 Euler Equation Errors: The Theories
How capable are asset pricing theories of explaining the large pricing errors of the standard
model? In this section, we address this question by considering a number of distinct asset
pricing models. We begin with a simple model of limited participation/incomplete markets
model in which the true pricing kernel based on assetholder consumption is jointly lognor-
mally distributed with aggregate consumption and returns. Although the empirical results
reported above suggest that any model that implies aggregate consumption and returns
are jointly lognormally distributed will be unable to match the data, studying a lognormal
model is instructive for considering how the use of a mismeasured pricing kernel (for exam-
ple because per capita aggregate consumption is used in place of stockholder consumption)
might distort parameters compared to pricing errors. Next we evaluate the Euler equa-
tion errors generated by leading asset pricing models in which the log pricing kernel and
returns are not generally lognormally distributed. As mentioned, these include the external
habit-formation models of Campbell and Cochrane (1999) and Menzly, Santos, and Veronesi
(2004), the long-run risk model of Bansal and Yaron (2004), and the limited participation
model of Guvenen (2003). Finally, we present a number of additional results for simple lim-
skewness S and (excess) kurtosis K and asymptotic distributions are given by
S =
1
T2
TX
t=1
TX
s=1
g3ts
!1=2
TS2
6
2
p(p+1)(p+2)=6
K = 1T
TX
t=1
g2tt p (p +2)
pTK
p8p (p +2) N (0; 1) ;
where gts = (xt ^)0 ^ 1(xs ^) and ^ and ^ are sample estimates of and V. S and K are zero if x is
jointly normally distributed. If x is univariate S and K are equivalent to the standard univariate de?nitions
of skewness and kurtosis.
12Pointwise standard error bands are computed by simulating from the multivariate normal distribution
with length equal to the size of our data set.
12
ited participation/incomplete markets models in which assetholder consumption, aggregate
consumption and asset returns are not jointly lognormally distributed.
3.1 A Limited Participation/Incomplete Markets Model With Joint
Lognormality
We investigate the acurrency1ect on parameter estimates and pricing errors of estimating (1) on
aggregate consumption data when the return data were generated from a model with limited
stock market participation or incomplete markets. For this purpose, a model of limited
stock market participation is isomorphic to that of incomplete markets since what matters
is the common implication that the consumption of the marginal assetholder may behave
dicurrency1erently from per capita aggregate consumption.13 Thus, one can interpret the example in
this section as an illustration of the in?uence of measurement error on empirically observed
pricing errors. In this case, stockholder consumption corresponds to correctly measured
consumption for which the model holds exactly, and aggregate consumption is an error-
ridden empirical measure of true consumption.
As a benchmark case in this section, we assume aggregate consumption, stockholder or
individual consumption, and asset returns are jointly lognormally distributed. Later we
consider asset pricing models in which the joint distribution is permitted to deviate from
lognormality. For the rest of the paper, we use lowercase letters to denote log variables, e.g.,
ct+1 log (Ct+1=Ct) :
Denote the marginal rate of substitution (MRS) of an individual asset-holder as
Mit+1
Ci
t+1
Cit
; (7)
where Cit is the consumption of assetholder i, is the subjective time discount factor of
this assetholder, and is the coe? cient of relative risk aversion. If agents have unrestricted
access to ?nancial markets, then Mit+1 correctly prices any traded asset return, implying that
E Mit+1Rjt+1 = 1; j = 1;:::;N (8)
for N asset returns.
We can interpret the MRS, Mit+1; either as that of a representative stockholder in a
limited participation setting (Cit is then the consumption of a representative assetholder), or
13With limited stock market participation, the set of Euler equations of stockholder consumption imply that
a representative stockholder? marginal rate of substitution is a valid stochastic discount factor. Similarly,
with incomplete consumption insurance the set of Euler equations of household consumption imply that any
household? marginal rate of substitution is a valid stochastic discount factor.
13
as that of an individual assetholder in an incomplete markets setting (Cit is the consumption
of any marginal assetholder, e.g., Constantinides and Du? e (1996)). It functions as the
stochastic discount factor in this model. The risk-free rate is de?ned as a one-period riskless
bond, Rft+1 = 1=Et Mit+1 :
Now denote the misspeci?ed ?MRS,? for some parameters c and c, that would be
computed if an econometrician erroneously used per capita aggregate consumption, Ct in
place of Cit
Mct+1 c
C
t+1
Ct
c
: (9)
For any asset return indexed by j, the pricing error associated with the true MRS, Mit+1, is
by construction zero, but the pricing error associated with the erroneous MRS, Mct+1, is not
necessarily zero and is denoted PEj, where (dropping the time subscripts for brevity)
PEj = E McRj 1: (10)
Throughout this paper, when we refer to pricing errors, we mean the pricing error generated
for any asset by erroneously using the ?pricing kernel?Mc in place of the true pricing kernel,
since only the former are potentially nonzero if the model is true.
Under joint lognormality of Ct+1=Ct and returns, the pricing error may be written
PEj = E Rj E [Mc] exp Cov mc;rj 1 (11)
Use the fact that the pricing error is identically zero under Mi to write
E Rj E Mi exp Cov mi;rj = 1;
implying under joint lognormality,
PEj = E [M
c]
E [Mi] exp
Cov mc;rj Cov mi;rj 1 (12)
= E [M
c]
E [Mi] exp
cCov
c;rj + Cov ci;rj 1: (13)
How are the parameters and pricing errors distorted by using Mct+1 to price assets in
place of the true pricing kernel Mit+1? For N > 2 asset returns, it is not possible to give a
intuitively appealing analytical expression for this distortion, although values can be obtained
numerically. It is, however, possible to illustrate analytically the distortion in c to a very
close approximation, by focusing on log pricing errors and assuming that the risk-free rate
is constant. In this case we can choose c so that E [Mi] = E [Mc] ; which insures that the
14
pricing error for the risk-free rate is zero.14 While this is an approximation, it turns out to
be very well satis?ed in the data, since the Treasury-bill rate is extremely stable.15 As a
result, ?ndings based on the full numerical solution are almost identical to those based on
this approximation when the returns being priced are calibrated to match the means and
volatilities of those in the data. We maintain this approximation purely for expositional
purposes; the reader should be aware that exact results are very close.16
With this approximation in hand, the pricing error of the jth asset is now
PEj = exp cCov c;rj + Cov ci;rj 1:
With c set as just described and c is chosen to minimize the sum of squared log pricing
errors, pej log (1 + PEj), as in GMM estimation using the identity weighting matrix, the
resulting value of c is given by
bc =
P
j cj ijP
j
2
cj
!
; (14)
where cj Cov( c;rj) ; ij Cov( ci;rj) ; and ?Pj? indicates summation over all j =
s;f;1;:::;N asset returns being priced. In the two-asset case, when only Rf and Rs are
priced, this collapses to
bc =
is
cs
: (15)
Notice that, in the two asset case, this value of c, along with the value of c discussed above,
insures that the log pricing errors for Rst+1 and constant Rf are identically zero.17 This follows
14Note that this does not imply that the risk-free rate puzzle is trivial, since c is unrestricted and in
particular can be chosen to be greater than unity if required to set the pricing error to zero.
15If Mit is the true pricing kernel, then E Mit = E
h
1=Rft
i
: Since we assume E Mit = E [Mct ], our
assumption implies E [Mct ] = E
h
1=Rft
i
, which prices the risk-free rate exactly if Rft is constant. It follows
that the approximation error in pricing the risk-free rate is E
h
1=Rft
i
1=E
h
Rft
i
, which is -0.01 percent per
annum. Therefore, the approximation implies we pick so that E Mit = E [Mct ], with the resulting pricing
error in the risk-free rate of -0.01%.
16The calculations below are similar in spirit to those in Vissing-Jorgensen (1999), who shows how limited
stock market participation biases estimates of relative risk aversion based on aggregate consumption. Vissing-
Jorgensen? calculations presume heterogenous households rather than a representative-stockholder, as below.
17When there are only two asset returns, simple analytical expressions for the values of c and c that
insure the pricing errors are identically zero can be obtained without assuming that the risk-free rate is
constant. For a single risky asset return Rst+1and the risk-free return Rft+1, these values are given by
bc =
is if
cs cf
;
b c = exp
c c
2c 2c
2 i +
2 2i
2 + c cs is
;
15
because, under lognormality, the log model is linear and the problem collapses to solving two
linear equations in two unknowns. Therefore an erroneous pricing kernel based on aggregate
consumption can always be found that unconditionally prices any two assets just as well
as the true pricing kernel based on assetholder consumption. The estimates of c and c
that result from ?tting (9) to data will not correspond to any marginal investor? true risk-
aversion or time discount factor. But a representative agent pricing kernel based on per capita
aggregate consumption can nevertheless be found that has the same unconditional asset-
pricing implications as the true pricing kernel based on individual assetholder consumption.
Equation (14) shows that c will be higher the higher is assetholder risk aversion, but
that this relation depends on the statistical properties of aggregate consumption growth,
individual consumption growth, and returns. Again, a more intuitively appealing expression
can be obtained by focusing on the two-asset case. Consider an orthogonal decomposition of
aggregate consumption growth into a part that is correlated with asset-holder consumption
and a part, "it, orthogonal to asset-holder consumption:
ct = cit + "it; (16)
where = Cov( ct; cit)Var( ci
t)
ci c i : Here ci denotes the correlation between ct and cit. Using
this decomposition, (15) can be re-written as
bc = + "is
is
; (17)
where "is Cov "it;Rst+1 . Now consider assets that are uncorrelated with "it, the component
of aggregate consumption that is orthogonal to stockholder consumption. Any risky asset
that is on the log mean-variance e? cient frontier will be included in this category. In this
case "is = 0 and therefore
bc = = i
ci c
: (18)
The formula tells us that limited participation and/or incomplete consumption insurance
can in principal account for implausibly high estimated values of c and c obtained when ?t-
ting data to (9), but to do so, assetholder consumption must be more volatile than aggregate
consumption and/or very weakly correlated correlated with it. Notice, however, that even
where if Cov ci; rf ; cf Cov c; rf ; c is the mean growth rate of aggregate consumption, and i
is the mean growth rate of the consumption of asset-holder i: Notice that, in equilibrium, bc and bc will take
the same value regardless of the identity of the assetholder. This follows because any two households must
in equilibrium agree on asset prices, so that the Euler equation holds for each individual household. Thus,
c = i
is if
cs cf
= k
ks kf
cs cf
for any two asset-holders i and k:
16
if assetholder consumption behaves very dicurrency1erently from per capita aggregate consumption,
this is not enough to explain the large unconditional Euler equation errors that arise from
?tting (9) to data in the two-asset case. In that case, the only consequence of using aggregate
per capita consumption in this setting is a bias in the estimated parameters bc and bc; there
is no consequence for the Euler equation errors, which remain zero.
How do the pricing errors under lognormality compare with those estimated in the data
when there are more asset returns? Figure 2 provides an answer for both the two- and
eight-asset cases using actual historical return data. The ?data?line plots RMSE/RMSR
over a range of values for c, that arise from choosing c to minimize the sum of squared
pricing errors in (10), which do not impose lognormality. The top panel plots for the two-
asset case (these lines reproduce those in Figure 1), the bottom panel for the eight-asset
case. The line labeled ?lognormality?plots the RMSE/RMSR that arise from choosing c
to minimize the sum of squared pricing errors in (11), which impose lognormality. One
way to interpret the ?lognormal? line is to note that we can always ?nd a pricing kernel
Mit+1 = expflog ( ) cit+1g, for some , , and normally distributed cit+1, which along
with (16) and a statistical model for log returns, such as
rjt = j cit + jt;
generates a set of asset returns with the same means, variances and covariances with ct
as those in the historical data, and prices those asset exactly.18 The dashed line labeled
?lognormality? then gives the pricing errors that would arise from ?tting Mct+1 to data
generated from this model.
Figure 2 shows that a lognormal model cannot explain the pricing errors in the data,
especially when c is large. When only two assets are priced (top panel), values for c
and c can be found for which the pricing errors of the CRSP stock market return and the
Treasury-bill rate are exactly zero, whereas this is not true in the data when no distributional
assumptions are imposed. Similarly, the bottom panel shows that the lognormal model
cannot match the magnitude of the Euler equation errors for the eight-asset case, increasingly
so as c rises.
We note that the results above hold for any pricing kernel Mit+1 that is jointly lognormally
distributed with returns and aggregate consumption growth. It is not necessary that the
pricing kernel take the form given in (7). Referring to (12) it is evident that the resulting
solutions for c and c would be a function of the means, variances and covariances of ct,
returns and mit, whatever form the latter may take. As long as the true kernel Mit+1 is
jointly lognormally distributed with aggregate consumption and returns, then values for the
18This is done by choosing j to match the mean excess return for each asset, choosing var j to match
the volatility of each return, and choosing cov ; "i to match the cov rj; c from the data.
17
discount factor relative risk aversion can be always be found such that the standard model
generates identical (zero) unconditional asset pricing implications for two asset returns, and
for which the root mean-squared pricing errors are much smaller than in the data (especially
for large values of c) for larger cross-sections of asset returns. These results suggest that
models that are approximately lognormal will also have di? culty explaining the large pricing
errors of the standard model.
3.2 Leading Asset Pricing Models
We now turn our attention to investigating how well leading asset pricing models explain
the pricing errors of the standard model by examining the properties of model-generated
data. All of the models generate predictions endogenously for a stock market return and a
risk-free rate, and none imply that the pricing kernel is unconditionally jointly lognormally
distributed with aggregate consumption growth and returns.19 In addition, the Menzly,
Santos, and Veronesi (2004) (MSV) model is multi-asset extension of the Campbell and
Cochrane (1999) (CC) habit model; thus we can extend our analysis of nonlinear habit-
based models to study multiple risky asset returns by applying the MSV framework. It is
also straightforward to study the multi-asset properties of the long-run risk model of Bansal
and Yaron (2004) (BY), since the cash-?ows in BY are exogenously modeled. By contrast,
the limited participation model of Guvenen (2003) (GUV) generates implications for only
two asset returns, a single risky (stock market) return, and a risk-free return. Since cash-
?ows are endogenously determined by the properties of a general equilibrium setting in that
model, the extension to multiple-assets is not trivial and would have to be developed. Thus,
we focus only on the implications of the Guvenen model for Rs and Rf below.
3.2.1 Misspeci?ed Preferences
We begin with the representative-agent models and consider three prominent representative
agent models: the external habit-persistence models of Campbell and Cochrane (1999) and
Menzly, Santos, and Veronesi (2004), and the long-run risk model of Bansal and Yaron
(2004). All three of these models display a striking ability to match a range of asset pricing
phenomena, including a high equity premium, low and stable risk-free rate, long-horizon
predictability of excess stock returns, and countercyclical variation in the Sharpe ratio. In
what follows, we describe only the main features of each model, and refer the reader to the
original article and the Appendix for details. Except where noted, our simulations use the
baseline parameter values of each paper.
19Joint lognormality of consumption growth, the risky asset return, and the risk-free return can be statis-
tically rejected in simulated data from the models discussed in this section.
18
The utility function in the CC and MSV models take the form
U = E
( 1X
t=0
t (Ct Xt)
1 1
1
)
; > 0 (19)
where Ct is individual consumption and Xt is habit level, which they assume to be a function
of aggregate consumption, and is the subjective discount factor. In equilibrium, identical
agents choose the same level of consumption, so Ct is equal to aggregate consumption. The
key innovation in each of these models concerns the speci?cation of the habit process Xt,
which in both cases evolves according to heteroskedastic autoregressive processes. (The
Appendix provides a detailed description of the models in this section.) The stochastic
discount factors in both models take the form
Mt+1 =
C
t+1 Xt+1
Ct Xt
but dicurrency1er in their speci?cation of Xt (see the Appendix). We denote as MCCt+1 the speci?cation
of the stochastic discount factor corresponding to the Campbell-Cochrane model of Xt, and as
MMSVt+1 the speci?cation of the stochastic discount factor corresponding to the MSV model of
Xt. Both CC and MSV assume that the log dicurrency1erence in consumption, ct log (Ct=Ct 1),
follows an i.i.d. process:
ct = + vt;
where vt is a normally distributed, i.i.d. shock. Both models derive equilibrium returns for a
risk-free asset and a risky equity claim (stock market claim) that pays aggregate consumption
as its dividend. As above, the returns to these assets are denoted Rft+1, and Rst+1, respectively.
Campbell and Cochrane set = 2 and = 0:89 at an annual rate. Menzly, Santos and
Veronesi choose = 1 and = 0:96: Notice that the curvature parameter , is no longer
equal to relative risk-aversion.
The MSV model is a multi-asset extension of the CC model that generates implications
for multiple risky securities, each distinguished by a distinct dividend process with dynamics
characterized by ?uctuations in the share it represents in aggregate consumption:
sjt = D
j
t
Ct for j = 1;:::;N;
where n represents the total number of risky ?nancial assets indexed by j, each paying a
dividend Dj. Cross-sectional variation in unconditional mean returns across risky securities
is governed by cross-sectional variation in the covariance between shares sjt and aggregate
consumption growth ct. In analogy to the empirical exercise, we create a model-generated
cross-section of asset returns comprised of six risky securities plus the aggregate wealth
19
portfolio (stock market) return and the risk-free rate, for a total of 8 asset returns. The
Appendix gives a detailed description of the stochastic process for the shares.
Bansal and Yaron (2004) consider a representative agent who maximizes utility given
by recursive preferences of Epstein and Zin (1989, 1991) and Weil (1989). The stochastic
discount factor under Epstein-Zin-Weil utility used in BY takes the form
MBYt+1 =
C
t+1
Ct
1
!
R 1w;t+1; (20)
where Rw;t+1 is the simple gross return on the aggregate wealth portfolio, which pays a
dividend equal to aggregate consumption, Ct, (1 ) =(1 1= ) ; is the intertemporal
elasticity of substitution in consumption (IES), is the coe? cient of relative risk aversion,
and is the subjective discount factor. Bansal and Yaron assume that both aggregate
consumption growth and aggregate dividend growth have a small predictable component
that is highly persistent. They also incorporate stochastic volatility into the exogenous
processes for consumption and dividends to capture evidence of time-varying risk premia.
Taken together, the dynamics of consumption growth and stock market dividend growth,
dt, take the form
ct+1 = + xt + t t+1 (21)
dt+1 = d + xt + d tut+1; (22)
xt+1 = xt + c tet+1
2t+1 = 2 + 1 2t 2 + wwt+1;
where 2t+1 represents the time-varying stochastic volatility, 2 is its unconditional mean, and
; d, , d; , c, 1 and w are parameters, calibrated as in BY. Here, the stock market
asset is the dividend claim, given by (22), rather than a claim to aggregate consumption,
given by (21). We denote the return to this dividend claim Rst+1, since it corresponds the
model? stock market return. BY calibrate the model so that xt is very persistent, with a
small unconditional variance. Thus, xt captures long-run risk, since a small but persistent
component in the aggregate endowment can lead to large ?uctuations in the present dis-
counted value of future dividends. Their favored speci?cation sets = 0:998, = 10 and
= 1:5.
As for the MSV model, we can analyze the multi-asset implications of the BY model by
considering risky securities, indexed by j; that are distinguished by their cash-?ow processes:
djt+1 = jd + jxt + jd tut+1: (23)
20
By considering a grid of values for j, we create risky securities with dicurrency1erent risk-premia,
since this parameter governs the correlation of equilibrium returns with the stochastic dis-
count factor. By altering jd, we control the variance in the risky security returns, and jd
controls the mean price-dividend ratio across risky assets.
For both the MSV and BY models, we choose parameters of the cash-?ow processes
to create a cross-section of asset returns that include a risk-free rate, an aggregate equity
return, and six additional risky securities, or eight securities in total. For each model, we
exactly replicate the original calibration of the risk-free return and aggregate equity return,
themselves chosen to match the properties of these assets in U.S. data. For the six additional
risky securities, we choose parameters of the cash-?ow processes that allow us to come as close
as possible to matching the spread spread in risk-premia found in the six size/book-market
sorted portfolio returns in the data. For the BY model, we are able to generate a cross-
section of returns that come very close to matching the historical spread. For example, the
largest spread in average annualized returns is given by the dicurrency1erence between the portfolio
in the smallest size and highest book-market category and the portfolio in the largest size
and lowest book-market category, equal to about seven percent. Thus, we create six arti?cial
returns for which the largest spread is 6.7 percent per annum. Constructing such returns for
the MSV framework is more complicated, since the solutions for the multi-asset model hold
only as an approximation (see the Appendix for the approximate relation). Unfortunately,
we ?nd that the approximation error in this model can be substantial under parameter values
required to make the maximal spread as large as seven percent.20 As a result, we restrict
the parameter values to ranges that limit approximation error to reasonably small degrees.
This still leaves us with a maximal spread of 4.5 percent per annum in the returns of the six
arti?cial securities created.
To study the implications of these representative-agent models, we simulate a large time-
series (e.g., 20,000 periods) from each model and compute the pricing errors that would arise
in equilibrium if Mct+1 = c
Ct+1
Ct
c
were ?tted to data generated by these models. As
in the historical data, the parameters c and c are chosen by GMM to minimize the Euler
equation errors E[Mct+1Rjt+1] 1; j = 1;:::;N. We denote the parameters that minimize the
GMM criterion as bc and bc. As in the historical data, we focus on the case of N = 2 asset
returns (Rst+1 and Rft+1), and the case of N = 8 asset returns, (Rst+1;Rft+1;R1t+1;:::;R6t+1).
The results are presented in Table 3. First consider the CC and MSV habit models.
For each model, we ?nd the pricing errors that arise from ?tting Mct+1 to model-generated
data are numerically zero, just as they are when the true habit pricing kernel is used. This
20Menzly, Santos, and Veronesi (2004) state that the approximation error is small for the parameters they
employ, but it is not small for our parameters, which were chosen to mimic returns of the Fama-French
portfolios.
21
result does not depend on the number of assets being priced; it is the same for the two-asset
case and eight-asset case. Values of c and c can in each case be found that allow the
standard consumption-based model to unconditionally price assets just as well as the true
pricing kernel, as measured by the root mean-squared pricing error. In the CC model, the
values of c and c that set these pricing errors for Rst+1 and Rft+1 to zero are 1.28 and 57.48,
respectively. In the MSV model, the corresponding values are 1.71 and 30.64, respectively.
Thus, the habit models can explain what many would consider the implausible estimates of
time preference and risk aversion obtained when freely ?tting aggregate data to (1). (Recall
that the true preference parameters are = 2 and = 0:89 in CC and = 1 and = 0:96 in
MSV.) But, it is in those parameters that all of the distortion from erroneously using Mct+1
to price assets arises. No distortion appears in the Euler equation errors themselves.
The conclusions for the Bansal-Yaron long-run risk model, also displayed in Table 3, are
the same. Here we follow BY and simulate the model at monthly frequency, aggregate to
annual frequency, and report the model? implications for pricing errors and parameter values.
The monthly consumption data are time-aggregated to arrive at annual consumption, and
monthly returns are continuously compounded to annual returns.21 We ?nd that c is close
to the true value, but c is estimated to be about ?ve times as high as true risk aversion. As
for the habit models, this framework explains why an econometrician obtains high estimates
of risk aversion when ?tting data to the standard consumption-based model. But, also like
the habit models, if an econometrician ?t Mct+1 to data generated by MBYt+1, the resulting
Euler equation errors would be ecurrency1ectively zero, in contrast to what is found using historical
data.22 In the two-asset case the RMSE is zero to numerical accuracy, and it is 0.01% per
annum in the eight-asset case.
3.2.2 Misspeci?ed Consumption
We now consider the limited participation model of Guvenen (2003) (GUV). Like the rep-
resentative agent models considered above, this model has remarkable success in explaining
many of the empirical puzzles associated the standard representative agent consumption-
based model. It can account for a high equity premium and low and stable risk-free rate,
predictable stock market returns, and countercyclical Sharpe ratio. Here we suppose (1)
21The resulting Euler equation errors are unchanged if they are computed for quarterly time-aggregate
consumption and quarterly returns rather than annual time-aggregated consumption and annual returns.
22For models based on recursive preferences, Kocherlakota (1990) shows that there is an observational
equivalence to the standard model with power utility preferences, if the aggregate endowment growth is i.i.d.
However, the endowment growth process in the BY model is not i.i.d., but instead serially correlated with
stochastic volatility. Moreover, the annual consumption data are time-aggregated, which further distorts the
time-series properties from those of the monthly endowment process.
22
is ?tted to data generated by this non-representative agent model in which asset prices
are determined not by per capita aggregate consumption but rather by the consumption of
stockholders.
The Guvenen model has two types of consumers, stockholders and nonstockholders. The
latter are exogenously prevented from participating in the stock market. The discount factor
in this model is denoted
MGUVt+1
Ci
t+1
Cit
;
where Cit is stockholder consumption, which by assumption is not the same as aggregate
per capita consumption, is the subjective discount factor of the stockholder, and is
the stockholder? relative risk aversion. In other respects, the model is a standard one-
sector real business cycle model with adjustment costs in capital. Both stockholders and
nonstockholders receive labor income with wages determined competitively by the marginal
product of labor, and ?rms choose output by maximizing the present discounted value of
expected future pro?ts. Both agents have access to a riskless bond.
We follow the same procedure discussed above to quantify pricing errors in this model. We
simulate a large time series of arti?cial data (20,000 observations) for the equilibrium values
of the variables in this model, and use these data to quantify the magnitude of unconditional
pricing errors that an econometrician would ?nd from ?tting (1) to data generated by MGUVt+1 .
In Guvenen? baseline model, stockholders have risk aversion = 2 and subjective dis-
count factor = 0:99. Table 4, panel A shows that stockholder consumption growth is about
two and a half times as volatile as aggregate consumption growth in the baseline model, and
perfectly correlated with it. The model also implies that stockholder consumption is over
four times as volatile as nonstockholder consumption growth, but the two are almost per-
fectly correlated , with correlation 0.99. This is not surprising since both types of consumers
participate in the same labor market and bond markets; the agents dicurrency1er only in their ability
to hold equities, and in their risk-aversion (nonstockholders have higher risk-aversion). As a
consequence, stockholder? marginal rate of substitution, MGUVt+1 , is highly correlated with an
aggregate consumption pricing kernel Mct+1 c(Ct+1=Ct) c, for a variety of values of c and
c. Panel B of Table 4 shows this correlation for two such cases, ?rst with c = = 0:99 and
c = = 2, and second with c and c set to the values that minimize the equally-weighted
sum of squared pricing errors for the stock return and riskless bond. These latter values are
b c = 0:99 and b c = 4:49; thus, unlike the representative-agent models discussed above, this
model does not explain the high estimated values of and obtained when ?tting data to
the standard consumption-based model. In both cases, the correlation between MGUVt+1 and
Mct+1 is extremely high, 0.99.
Panel C of Table 4 shows the pricing errors in Guvenen? model that would arise if
23
aggregate consumption were erroneously used in the pricing kernel in place of stockholder
consumption. For comparison, the table also reports the pricing errors using the true kernel
MGUVt+1 based on stockholder consumption, which are quite small (0.02% on an annual basis)
but not exactly zero due to the rarely-binding borrowing constraints that apply to both
stockholders and nonstockholders. To compute the pricing errors this model produces when
Mct+1 is employed to explain asset returns, we ?rst set the parameters c and c equal to
the stockholder? discount factor, , and risk aversion, , in the baseline parameterization.
When c = = 0:99 and c = = 2, the pricing errors using aggregate consumption are
larger than that using stockholder consumption, equal to about 0.4% at an annual rate for
the stock return and -0.34% for the risk-free rate, but still small in magnitude compared
to the data. By contrast, when c and c are chosen to minimize the sum of squared
pricing errors for these two asset returns, as in empirical practice, the pricing errors are,
to numerical accuracy, zero for the stock return and risk-free return. By simply increasing
by a factor of 2.5, to 4.5 from 2, the Guvenen model delivers a power utility pricing
kernel using aggregate consumption that explains the historical mean return on the stock
market and risk-free (Treasury bill) return just as well as the true pricing kernel based on
stockholder consumption. Thus, the model does not explain the equity premium puzzle of
Mehra and Prescott (1985), which is the puzzle that a high value of (greater than 10
according to Mehra and Prescott) is required to explain the equity premium when the power
utility model is calibrated to aggregate consumption data. These results echo the empirical
?ndings in Brav, Constantinides, and Geczy (2002), which suggest that limited stock market
participation plays a minimal role in explaining the historical equity premium.
3.2.3 Misspeci?ed Preferences and Misspeci?ed Consumption
One possible reaction to the results above, is that we should take the representative agent
nature of the CC, MSV and BY models less literally and assume that they apply only to a
representative stockholder, rather than to a representative household of all consumers. Would
the results for these models be better reconciled with the data if we accounted for limited
participation? Not necessarily. As an illustration, we consider a limited-participation version
of the MSV model and show that the conclusions are unchanged from the representative agent
setup.
Since the MSV model is a representative agent model, we modify it in order to study
the role of limited participation. Assume that asset prices are determined by the framework
above, where a valid stochastic discount factor is a function of any stockholder? consumption
Cit and stockholder? habit Xit. The process for stockholder consumption is the same as in
24
MSV, described above, but now with i subscripts:
cit = i + ivit;
where vit is a normally distributed i.i.d. shock. Aggregate consumption is assumed to follow
a separate process given by
ct = c + cvct;
with vct a normally distributed i.i.d. shock. We analyze the results over a range of cases for
the correlation between vit and vct, and their relative volatilities i= c.
Asset prices are determined by the stochastic discount factor of individual assetholders,
denoted
MMSV it+1
Ci
t+1 X
i
t+1
Cit Xit
;
where Xit+1 is the habit modeled as in MSV, not a function of Cit. We then compute two
types of pricing errors. First, we assume that the data are generated by MMSV it+1 and compute
the pricing errors that arise from using
Mcht+1 c
C
t+1 Xt+1
Ct+1 Xt+1
c
to price assets, where Xt is now computed from the MSV habit speci?cation using aggregate
consumption. Second, we assume the data are generated by MMSV it+1 and compute the pricing
errors that arise from using
Mct+1 c (Ct+1=Ct) c
to price assets. This latter case is what we refer to as misspeci?ed preferences and misspec-
i?ed consumption; an econometrician who tried to ?t Mct+1 to asset return data would be
employing both the wrong preferences and the wrong consumption measure. In both cases,
c and c are chosen to minimize an equally-weighted sum of squared pricing errors of the
assets under consideration, as with the historical data.
The results are presented in Table 5 for the exercise using Mcht+1 and Table 6 for the
exercise using Mct+1. We report the pricing errors for a range of parameter speci?cations.
The standard deviation of asset-holder consumption growth is allowed to range from one
times to ?ve times as volatile as that of aggregate consumption growth, the correlation
from -1.0 to 1.0. The pricing errors (as measured by RMSE/RMSR) are reported in the
bottom subpanels. The top panel reports these errors for the two-asset case where only Rst+1
and, Rft+1 are priced; the bottom panel reports for the eight-asset case with six additional
risky securities. For each parameter con?guration, we also report the values bc and bc that
minimize the quadratic form gT ( c; c), as above.
25
Several aspects of Tables 5 and 6 are of interest. First, consider the two-asset case. Table
5 shows that the pricing errors that arise from using Mcht+1 to price assets are always zero, even
if assetholder consumption growth has very dicurrency1erent properties from aggregate consumption
growth. For example, Table 5 shows that aggregate consumption growth can be perfectly
negatively correlated with assetholder consumption growth and ?ve times as volatile, yet the
pricing errors that arise from using Ct in place of Cit are still zero. Notice, however, that
the parameters c and c can deviate substantially from the true preference parameters of
stockholders. This is similar to the lognormal example, in which the use of mismeasured
consumption distorts preference parameters, but does not explain the large pricing errors
generated by the standard consumption-based model.
Second, Table 6 shows that the same result holds if one uses Mct+1 in place of the true
pricing kernel Mit+1. Here the model used to explain asset returns is based both the wrong
consumption measure and the wrong preferences. Nevertheless, values of c and c exist such
that Mct+1explains the Euler equations just as well as Mit+1. The values for c and c are more
distorted from their true values than is the case in Table 5 where we have merely substituted
the wrong consumption measure into the class of habit preferences, but the pricing errors
are still zero.23 These ?ndings reinforce the conclusion that changing the pricing kernel does
not necessarily change the pricing implications.
Third, results for the multi-asset case are qualitatively the same as those for two-asset
case. For example, Table 6, bottom panel shows that the root mean-squared pricing error
that arises from erroneously using Mct+1 to price assets is a tiny fraction of the square-root
of the average squared returns of the assets under consideration. The highest is 4% per
annum. These numbers should be contrasted with the 33% ?gure obtained for a cross-
section of 8 asset returns in U.S. data (Table 1). Moreover, the numbers in Table 6 actually
overstate the true pricing errors. This is because there are two sources of error that result
in nonzero pricing errors even using the true pricing kernel MMSV it+1 . The ?rst is the discrete-
time approximation to the continuous-time model of MSV. We eliminate much of this error
by shrinking the time-interval over which we simulate the model and reporting annualized
values in the table. The second source of error is the approximation in (26). Taken together,
these errors mean that the true kernel generates pricing errors that are often of the same
order of magnitude as those reported in Table 6.
The results reveal a striking implication of leading asset pricing models: the unconditional
23Variation in i= c has little acurrency1ect on the estimated value of the risk-aversion parameter c. This happens
because we adjust the parameter in the MSV habit speci?cation (see the Appendix) at the same time as we
adjust i= c so that the mean excess return Rs Rf remains roughly what it is in MSV. Since the volatility
of aggregate consumption is kept the same and is adjusted to keep the returns of the same magnitude, c
doesn? change much.
26
pricing errors of the standard consumption-based model can be virtually identical to those
using the true pricing kernel, even when (i) the true kernel has preferences dicurrency1erent from the
CRRA form of the standard model, (ii) the consumption of marginal assetholders behaves
dicurrency1erently from per capita aggregate consumption, and (iii) the number of assets exceeds
the number of free parameters to be estimated. This implies that the explanation for the
high average pricing errors produced by the standard model has to be something more than
limited participation and/or nonstandard preferences per se, since in many models parameter
values can be found that allow the standard model to price cross-sections of assets almost
as well as the true pricing kernel that generated the data.
3.2.4 Time Aggregated Consumption
What if the decision interval of households may be shorter than the data sampling interval,
leading to time-aggregated consumption observations? We have repeated the same exercise
for all the models above using time-aggregated consumption data, assuming that agents?de-
cision intervals are shorter than the data sampling interval, for a variety of decision intervals.
For example, we assume that agents make decisions quarterly but that the data sampling
interval is annual. As above, we also allow for the possibility that aggregate consumption is
a misspeci?ed measure of assetholder consumption. For all models the essential results for
the Euler equation errors remain small: values of c and c can always be found such that
the unconditional pricing errors associated with using Mct+1 to price assets are very small
relative to the data, even when using time-averaged data. As an example, Table 7 shows
results for the MSV model with limited participation. (To conserve space, we report only
the results for this model, since the conclusion is unchanged for the other models, although
note that the results above for the BY model are already based on time-aggregate data.)24
Most values of RMSE/RMSR are close to zero. The largest occurs for the eight asset case
and is equal to 0.07, which occurs only if we assume stockholder consumption growth is
negatively correlated with aggregate consumption growth. This is far smaller than the value
of 0.33 found in the data. Since time-averaging changes both the serial dependence of the
consumption data and its unconditional correlation with returns, this suggests that the exact
time-series properties of consumption growth are not crucial for explaining the large pricing
errors of the standard model.
24Again, because of numerical error, these ?gures actually overstate the true relative pricing errors, since
the RMSE for the true MSV stochastic discount factor is of the same order of magnitude as that for the
CRRA model.
27
3.2.5 Finite Sample Pricing Errors
The results above are based on long samples of model-generated data, providing estimates
of the population Euler equation errors. The estimates using historical data are based on
a ?nite sample of 204 observations. In Table 8 we show that our main conclusions are
robust to using samples equal in size to that of our historical dataset. The table reports the
maximum RMSE/RMSR over 1,000 samples of size 204 that arises from ?tting Mct+1 to data
generated from the relevant model. We do not report small-sample results for the eight-
asset MSV model. The small sample behavior of the MSV model is problematic because
the model is solved in continuous time and moreover holds only as an approximation for
multiple risky securities. As a result, we ?nd that small amounts of approximation error are
compounded by discretization error in small samples and it is not possible to reduce these
errors to reasonable levels unless the number of decisions within the period is almost in?nite.
Nevertheless, we are able to report the results for the two-asset case, since the solutions for
the aggregate consumption claim and risk-free rate are not approximate. Table 8 shows that
for the three representative agent models, CC, MSV, and BY, the maximum Euler equation
errors are numerically zero, in both the two-asset and eight-asset cases. The Guvenen model
produces a slightly higher maximum RMSE/RMSR in ?nite samples, equal to about 0.87%
at an annual rate, but this is still well below the value of almost 50% found in historical data
(Table 1). In short, the large empirical Euler equation errors of the standard model are not
explained by small sample biases.
To summarize, the results above suggest that if the data on asset returns and consumption
were generated by any of the leading models considered above, we would ?nd zero Euler
equation errors and the consequence of using the wrong pricing kernel would simply be
incorrect estimates of and . We now move on to consider an alternative way to explain
the large historical pricing errors in frictionless models, by further studying the potential
roles of limited participation/incomplete markets. We saw above that when all variables
are jointly lognormally distributed, the standard model does not in general generate the
magnitude of pricing errors found in the data. Thus the next section considers models in
which these variables are allowed to depart from joint lognormality.
3.3 Perturbations from Normality: Limited Participation/Incomplete
Markets
How do the unconditional pricing implications of models with limited participation/incomplete
markets change when variables are not jointly lognormal? We approach this question by al-
lowing for ?rst-order Hermite expansions around the multivariate normal distribution. Since
28
many economic models are close to, if not exactly lognormal, this is advantageous because
the leading term in the expansion is Gaussian, while higher-order terms accommodate devia-
tions from normality. One caveat is that the distributions we consider cannot accommodate
conditional heteroskedasticity or other forms of conditional temporal dependence. Allowing
for such dependence along with arbitrary non-normalities would require the calibration of
an infeasible number of Hermite parameters about which we have no information. We begin
this section by considering the Euler equation errors associated with a stock market return
and a risk-free rate and later move on to consider a larger cross-section of asset returns.
Let yt = ( ct; cit; dt)0 (y1;t;y2;t;y3;t)0, ct is aggregate consumption growth, cit
is individual asset-holder consumption growth, and dt is dividend growth of an aggregate
stock market claim. We will consider asset pricing models in which these variables are i.i.d.,
but not necessarily jointly lognormally distributed.
Let the joint density of yt be denoted h(y). A Hermite expansion is a polynomial in
y times the standard Gaussian density. Gallant and Tauchen (1989) show that such an
expansion can be put in tractable form by specifying the density as
h(y) = a(y)
2 f (y)
R R R a(u)2 f (u) du
1du2du3
:
Here, f (y) is the multivariate Gaussian density with variance-covariance matrix and mean
= ( 1; 2; 3)0, and a(y) is the sum of polynomial basis functions of the variables in y;
it is squared to insure positivity and divided by the integral over R3 to insure the density
integrates to unity.
In our calibrated examples, we set a(y)2 = (a0 + a1y1;t + a2y2;t + a3y3;t)2, a ?rst-order
expansion but one that can nonetheless accommodate quite signi?cant departures from nor-
mality. We investigate a large number of possible joint distributions by varying the para-
meters a0,...,a3: When a0 = 1 and a1 = a2 = a3 = 0, h(y) collapses to the Gaussian joint
distribution, f (y). It is important to keep the degree of the Hermite expansion manageable
since, lacking a su? ciently long times series on asset-holder consumption, we cannot estimate
the parameters of f (y) and a(y).
For the equity claim, in equilibrium we must have
Et
Mit+1
P
t+1
Dt+1 + 1
D
t+1
Dt
= PtD
t
; (24)
where Mit+1 Cit+1=Cit is the true pricing kernel based on individual assetholder con-
sumption, Dt denotes dividends paid at time t, and Pt is the end-of-period stock price
at time t: For the risk-free rate, an analogous equation holds using the de?nition Rft+1
Et Mit+1 1, but notice that since all variables are i.i.d., conditional expectations are just
29
the same as unconditional expectations and h(y) can be used to compute (24) and the equi-
librium risk-free rate. Also, the equilibrium price-dividend ratio is a constant, P=D, that
satis?es P=D
P=D + 1 =
Z Z
i exp iy2 exp (y3) h(y2;y3) dy2dy3:
Given a distribution h(y) and the equilibrium value for P=D, it is straightforward to compute
the pricing errors associated with erroneously using Mct+1 c (Ct+1=Ct) c to price assets.
We assume the asset return data are generated by Mit+1 and solve numerically for the values
of c and c that minimize an equally-weighted sum of squared pricing errors that arise from
using Mct+1 to price assets.
For our numerical computations, parameters of the leading normal density f (y) are
calibrated to match data on aggregate consumption growth and dividend growth for the
CRSP value-weighted stock market index, on an annual basis. We take the the mean of
c to be 2% annually and the mean of d to be 4% annually from annual post-war data
used in Lettau and Ludvigson (2005). From the same annual data, the standard deviation
of aggregate consumption growth is c = 1:14% and the standard deviation of dividend
growth is d = 12:2%. The covariance between c and d, denoted cd, is notoriously
hard to measure accurately and appears to depend on the horizon, sampling frequency, and
sample size. It is estimated to be negative, equal to -0.000177 in the annual post-war data
used by Lettau and Ludvigson (2005), but others have estimated a weak positive correlation
(e.g., Campbell (2003)). We therefore consider both small negative values for this covariance
(equal to the point estimate from Lettau and Ludvigson (2005)), and small positive values
of the same order of magnitude, e.g., 0.000177. Finally, the parameters for asset-holder
consumption and assetholder preferences are somewhat arbitrary since there is insu? cient
data available to measure these empirically. We therefore consider a range for ; , i= c,
i= c; ci; and id, where id is the correlation between asset-holder consumption growth and
dividend growth. Because our calibration corresponds to an annual frequency, the Euler
equation errors are comparable to the annualized errors from U.S. data reported in Table 1.
We begin with an example of a joint distribution that can roughly replicate the large Euler
equation errors that arise from ?tting the data to (1). We stress that this is only one example,
but within the class of distributions we investigate here, it seems to be representative of
what is required. Clearly distributions outside this class could provide other examples. The
marginal distributions for c, ci, and d for this example are presented in Figure 4. The
parameters in the leading normal are set as follows: i= c = 4, i= c = 1:5, ci = 0:1, and
id = 0:9. Assetholder risk aversion is set to a moderate value of = 5 and the time discount
factor is set to = 0:99. The Hermite parameters ao;:::;a3, are set to obtain the density
shapes displayed in Figure 4.
30
For this particular joint distribution model, the RMSE that arises from erroneously using
Mct+1 c (Ct+1=Ct) c to price the two assets is 2.81%, close to the value in the historical
data for the CRSP stock return and 3-month Treasury bill rate (Table 1). The average
stock return in this example is about 11%, and the average risk-free rate 4% annually. The
latter is a bit higher than in the historical data, but the pricing errors as a fraction of the
average returns are reasonably close to the data. The corresponding standard deviation of
d is a somewhat higher and its mean somewhat lower than the corresponding ?gures for
the CRSP-VW return.
What features distinguish this example? First, notice from Figure 4 that assetholder con-
sumption growth and the risky return are highly correlated with one another, but neither is
highly correlated with aggregate consumption growth. Second, both assetholder consump-
tion growth and dividend growth are much more volatile than aggregate consumption growth,
with the former six times as volatile as that of aggregate consumption growth. Third, the
density of aggregate consumption is almost identical to the leading normal. By contrast,
stockholder consumption and dividend growth have distributions that dicurrency1er signi?cantly
from normality, with both displaying bimodal densities. Assetholder consumption and divi-
dend growth have about equal mass points at steeply negative and positive growth rates not
present in the density of aggregate consumption. With probability 0.25, assetholder con-
sumption can decline by 5%, while such a steep decline receives no weight in the density of
aggregate consumption growth. Similarly, with probability 0.2, assetholder consumption can
by 10% while dividend growth on the risky asset can grow 25%, again zero-probability events
for aggregate consumption growth. It follows that simulations from such a distribution would
deliver periods in which the joint behavior of Mct+1 and returns would be quite dicurrency1erent from
the joint behavior of Mit+1 and returns. Notice that assetholder consumption growth and
returns are quite non-normal in this example, similar to ?ndings in Brav, Constantinides,
and Geczy (2002) that suggest higher-order moments of assetholder consumption growth
have an important role in the pricing kernel.
Within the class of models we consider, how common is this example? To address this
question, we evaluated pricing errors obtained from a wide grid (over 20,000 parameter
combinations) for the Hermite parameters a0 through a3. Since it is infeasible to report the
output from tens of thousands of distributional assumptions, we report a limited number of
results. Two restrictions place limitations on the number of valid parameter combinations
that can be considered. First, must be positive semi-de?nite.25 Second, the price-dividend
ratio must be ?nite. Thus, risk-aversion cannot be too low if dividend growth is too high.
25Since the data suggest a weak correlation between aggregate consumption growth and dividend growth,
this requires that the correlation between ct and cit be relatively small.
31
The table reports results for which is set to 5, is set to 0.99, i= c = 1, 2, 4, i= c = 0:85,
1:5, ci = 0:1, id = 0:9.
Table 9 shows a range of cases in which the joint distribution deviates considerably
from normality and yet the pricing errors associated with erroneously using Mct+1 to price
assets in place of Mit+1 are, to numerical accuracy, zero. For example, the kurtosis of the
marginal distribution of ct is often greater than 11, and the skewness greater than 4. The
values of c and c that drive the pricing errors to zero vary, but are typically not close
to the true preference parameters for asset-holder i. The parameter c is much larger than
the true when asset-holder consumption growth is much more volatile than aggregate
consumption growth or when it is not highly correlated with it, as suggested by (18). Also,
when Cov( c; d) = cd is parameterize to be negative, c takes on negative values. This
is similar to the normal case (15), where the expression in (15) collapses to c = id= cd in
this model, so that c is negative when cd is negative.
Figure 5 provides a graphical description of two of the perturbed densities that created
the output in Table 9. Notice that the shapes can dicurrency1er considerably from Gaussian and yet
values for c and c can still be found for which Mct+1 prices assets just as well as the true
kernel based on assetholder consumption. The densities in the left-hand column are bimodal
for ci and d, while aggregate consumption is close to normal. This is similar to the
example above (Figure 4), which does deliver large pricing errors, but unlike that case the
negative mass points are much smaller relative to the positive mass points. Also, in Figure
4 assetholder consumption growth has a higher mean than aggregate consumption growth,
whereas in Figure 5 it has about the same mean. By contrast, the densities in the right-
hand column of Figure 5 are close to normal for ci and d; while the density of aggregate
consumption has skewness of about 4 and kurtosis around 11, strongly non-normal.
To evaluate the pricing errors for a larger cross-section of returns, we consider simple
models of N assets, indexed by j, whose dividend processes take the form
dj = j cit + "jt; j = 1;:::N;
where "jt is an i.i.d. shock uncorrelated with cit. In analogy to the two-asset case above,
the vector of variables yt = ct; cit; d1t ;:::; dNt 0 is assumed to be i.i.d. The ?leverage?
parameter j controls the covariance of each asset return with the stochastic discount factor,
and "jt controls the variance of individual risky returns. Assets on the log mean-variance
e? cient frontier (i.e., those that are perfectly correlated with mi) have shocks "jt with zero
variance. By varying j across assets, we create a spread in the covariance of returns with
stockholder consumption growth, and therefore a spread in risk premia.
We calibrate the leading normal for N = 8 arti?cial assets, including a risk-free return,
with j = "jt = 0, and a mean-variance e? cient return that is perfectly correlated with the
32
log stochastic discount factor, j = 1 and "jt = 0: The six other asset returns are generated
by a grid of values of j and Var "jt . Equilibrium returns Rjt+1 are computed as described in
the previous subsection for the two-asset case. As above, we assume the data are generated
by Mit+1 Cit+1=Cit and search numerically for values for c and c that minimize the
Euler equation errors associated with using Mct+1 c (Ct+1=Ct) c to price assets. The
pricing errors are summarized by reporting the root mean-squared pricing error as a fraction
of the root mean squared returns of the assets under consideration (RMSE/RMSR).
Table 10 presents both the maximum and average values of RMSE/RMSR obtained over
a large grid search of distributional parameter values, including the special case of joint
lognormality. The average pricing errors are often very small, indeed close to zero, even for
signi?cant perturbations from joint lognormality. We ?nd a small number of cases in which
the RMSE/RMSR is as large as 10 percent. Nevertheless, the 10 percent magnitude is still
signi?cantly smaller than in the data, and these cases are relatively rare, occurring in less
than 0.2% of the parameter permutations. Most non-normal models we considered imply
that the wrong pricing kernel based on aggregate consumption delivers tiny pricing errors
even when the joint distribution of ct, cit, and returns are signi?cantly non-normal. This
suggests that the explanation for the large pricing errors of the standard representative agent
model must be more than limited participation per se. The joint distribution of assetholder,
aggregate consumption and returns has to be of a particular form, and it is that form that
must be the central part of the story.
4 Conclusion
We view the evidence presented above as a convenient diagnostic for what remains missing
in modern-day asset pricing theories designed to remedy shortcomings of the standard rep-
resentative agent, consumption-based asset pricing model. In this paper we emphasize one
shortcoming of the standard model that provides a margin upon which it fails overwhelm-
ingly: its inability to explain the average returns on cross-sections of risky assets. This failure
is quantitatively large and present even when the range of parameters for risk aversion and
time preference is left unrestricted and chosen to maximize the model? chance of success.
We argue that these empirical facts constitute a puzzle that is at least as damning as other,
more well known, puzzles commonly emphasized when studying calibrated models.
Are prominent modi?cations to the standard model capable of explaining its mispricing?
If so, then an econometrician who ?t the standard model to data generated from leading asset
pricing models should ?nd large unconditional asset pricing errors, as in the historical data.
Alas, we ?nd that new pricing kernels do not necessarily generate new pricing implications.
33
Instead, we ?nd that parameter values can often be found that imply the standard model
has virtually the same explanatory power in tests of unconditional asset pricing restrictions
as those models currently at the forefront of theoretical asset pricing. This is true both
for explaining the behavior of one risky and one risk-free asset, and for explaining larger
cross-sections of risky returns. Moreover, some leading models imply that the standard
consumption-based model is equally capable of explaining asset returns even when it is
based both on the wrong consumption measure (aggregate consumption instead of individual
assetholder consumption) and on the wrong model of underlying preferences (CRRA instead
of habit or recursive preferences). The asset pricing models we explore can, in many cases,
explain why an econometrician obtains implausibly high estimates of and when ?tting
the standard consumption-based model to historical data. But they cannot explain why
the standard model fails so resoundingly to satisfy the most basic unconditional moment
restrictions implied by theory. A complete explanation of aggregate stock market behavior
should account for these empirical regularities.
Inability to account for these empirical regularities cannot be uncovered by studying
calibrated models or by procedures that rely solely on a model? ?rst-order conditions for
estimation and testing. That is because the ?rst-order conditions of any model are not a
complete description of the joint distribution of asset returns and aggregate quantities. But
an econometrician who observes this joint distribution in the data can assess whether its key
properties are matched by the simulated data of theoretical models.
Intuitively, how is it that asset pricing models capable of explaining a host consumption-
based asset pricing puzzles are incapable of explaining the large unconditional Euler equation
errors of the standard model? In thinking about this, it is helpful to consider the equity
premium puzzle as an example. We know that the equity premium puzzle can be ?solved?
by taking the standard consumption-based model and applying su? ciently high risk-aversion
(Mehra and Prescott (1985)).26 The di? culty with this resolution of the puzzle is that, in
order to show that high risk-aversion delivers the right equity premium as an equilibrium
outcome, the resulting equilibrium returns must be derived from theoretical Euler equations
that are exactly satis?ed. To the extent that these Euler equations are not satis?ed in
historical data, such a resolution would seem to rest on a fundamental misspeci?cation of
the joint behavior of asset returns and aggregate quantities.
What types of changes might bring asset pricing models more in line with the data along
these lines? We considered examples of limited participation/incomplete markets models in
which non-normalities are important, a ?nding also hinted at by the work of Brav, Constanti-
nides, and Geczy (2002). But we also found that it is insu? cient for assetholder consumption
26For the sake of argument, ignore the risk-free rate puzzle and other problems this resolution would leave.
34
to merely behave dicurrency1erently from aggregate consumption. This suggests that careful atten-
tion to the joint properties of the pricing kernel, aggregate consumption, and returns is
crucial for explaining the mispricing of the standard paradigm in frictionless models. Alter-
natively, classes of economic models with endogenously distorted beliefs, as surveyed in the
work of Hansen and Sargent (2000) or illustrated in the learning model of Cogley and Sargent
(2004), may present interesting possibilities for explaining these phenomena. In such models,
beliefs are distorted away from what a model of rational expectations would impose, so asset
return volatility can be driven by ?uctuations in beliefs not necessarily highly correlated
with consumption. Other candidates include any modi?cations to the standard model that
would make unconditional Euler equations more di? cult to satisfy, especially in recessions.
Possibilities include binding restrictions on the ability to trade and smooth consumption,
such as borrowing constraints, short-sales constraints, and transactions costs (e.g., Luttmer
(1996); He and Modest (1995); Heaton and Lucas (1996, 1997); Ludvigson (1999)). An im-
portant area for future research will be to determine whether such modi?cations are capable
of delivering the empirical facts, once introduced into plausibly calibrated economic models
with empirically credible frictions.
35
5 Appendix
1. Data Description
This appendix describes the data. The sources and description of each data series we use
are listed below.
CONSUMPTION
Consumption is measured as expenditures on nondurables and services, excluding shoes and
clothing. The quarterly data are seasonally adjusted at annual rates, in billions of chain-
weighted 1996 dollars. The components are chain-weighted together, and this series is scaled
up so that the sample mean matches the sample mean of total personal consumption expen-
ditures. Our source is the U.S. Department of Commerce, Bureau of Economic Analysis.
POPULATION
A measure of population is created by dividing real total disposable income by real per
capita disposable income. Consumption, is in per capita terms. Our source is the Bureau of
Economic Analysis.
PRICE DEFLATOR
Real asset returns are de?ated by the implicit chain-type price de?ator (1996=100) given for
the consumption measure described above. Our source is the U.S. Department of Commerce,
Bureau of Economic Analysis.
ASSET RETURNS
Three-Month Treasury Bill Rate: secondary market, averages of business days, discount
basis%; Source: H.15 Release ?Federal Reserve Board of Governors.
Six size/book-market returns: Six portfolios, monthly returns from July 1926-December
2003. The portfolios, which are constructed at the end of each June, are the inter-
sections of 2 portfolios formed on size (market equity, ME) and 3 portfolios formed
on the ratio of book equity to market equity (BE/ME). The size breakpoint for
year t is the median NYSE market equity at the end of June of year t. BE/ME
for June of year t is the book equity for the last ?scal year end in t-1 divided by
ME for December of t-1. The BE/ME breakpoints are the 30th and 70th NYSE
percentiles. Source: Kenneth French? homepage, http://mba.tuck.dartmouth.edu/
pages/faculty/ken.french/data_library.html.
The stock market return is the Center for Research and Security Prices (CRSP) value-
weighted stock market return. Our source is the Center for Research in Security Prices.
2. Detailed Description of Models
The utility function in the CC and MSV models take the form
U = E
( 1X
t=0
t (C
i
t X
i
t)
1 1
1
)
; > 0 (25)
where Cit is individual consumption and Xt is habit level which they assume to be a function
of aggregate consumption, and is the subjective discount factor. In equilibrium, identical
agents choose the same level of consumption, so Cit is equal to aggregate consumption, Ct.
CC de?ne the surplus consumption ratio
St Ct XtC
t
< 1;
and model its log process as evolving according to a heteroskedastic ?rst-order autoregressive
process (where as before lowercase letters denote log variables):
st+1 = (1 ) s + st + (st) (ct+1 ct g) ;
where , g, and s are parameters. (st) is the so-called sensitivity function that CC choose
to satisfy three conditions: (1) the risk-free rate is constant, (2) habit is predetermined at
steady state, and (3) habit moves nonnegatively with consumption everywhere. We refer the
reader to the CC paper for the speci?c functional form of (st) : The stochastic discount
factor in the CC model is given by
MCCt+1 =
C
t+1
Ct
St+1
St
:
In all of the models considered here, the return on a risk-free asset whose value is known
with certainty at time t is given by
Rft+1 (Et [Mt+1]) 1 ;
where Mt+1 is the pricing kernel of whichever model we are considering.
MSV model the behavior of Yt; the inverse surplus consumption ratio:
Yt = 11 (X
t=Ct)
> 1:
Following Campbell and Cochrane (1999), MSV assume that Yt follows a mean-reverting
process, perfectly negatively correlated with innovations in consumption growth:
Yt = k Y Y (Yt ) ( ct Et 1 ct) ;
where Y is the long-run mean of Y and k, ; and are parameters, calibrated as in MSV.
Here ct log (Ct+1=Ct), which they assume it follows an i.i.d. process
ct = + vt;
where vt is a normally distributed i.i.d. shock. The stochastic discount factor in the MSV
model is
MMSVt+1 =
C
t+1
Ct
Yt
Yt+1
:
Since the MSV model is a representative agent model, we modify it in order to study
the role of limited participation. Assume that asset prices are determined by the framework
above, where a valid stochastic discount factor is a function of any stockholder? consumption
Cit and stockholder? habit Xit. The process for stockholder consumption is the same as in
MSV, described above, but now with i subscripts:
cit = i + ivit;
where vit is a normally distributed i.i.d. shock. Aggregate consumption is assumed to follow
a separate process given by
ct = c + cvct;
with vct a normally distributed i.i.d. shock. We analyze the results over a range of cases for
the correlation between vit and vct, and their relative volatilities i= c.
For the representative stockholder, we model the ?rst dicurrency1erence of Y it as in MSV:
Y it = k
Y i Y i
Y it cit Et 1 cit ;
and compute equilibrium asset returns based on the stochastic discount factor MMSV it+1 =
Cit+1=Cit Y it =Y it+1 : As before, this is straightforward to do using the analytical
solutions provided in MSV.
Next, we compute two types of unconditional pricing errors. First, we compute the
pricing errors generated from erroneously using aggregate consumption in the pricing kernel
in place of assetholder consumption. That is, we compute the pricing errors that arise from
using Mcht+1 c (Ct+1=Ct) c Y ct =Y ct+1 c in place of MMSV it+1 to price assets, where c and
c are chosen freely to ?t the data, and where Y ct follows the process
Y ct = k Y c Y c (Y ct ) ( ct Et 1 ct) :
With the exception of , all parameters are set as in MSV. The parameter is set to keep
the mean return on the aggregate wealth portfolio the same as in MSV. Thus, if i= c = 2,
the value of in MSV is divided by two.
To model multiple risky securities, MSV model the share of aggregate consumption that
each asset produces,
sjt = D
j
t
Ct for j = 1;:::;n;
where n represents the total number of risky ?nancial assets paying a dividend D. MSV
assume that these shares are bounded, mean-reverting and evolve according to
sjt = j sj sjt + sjt (si) t;
where (sj) is an N dimensional row vector of volatilities and t is an N dimensional
column vector of standard normal random variables, and j and sj are parameters. (N
n + 1 because MSV allow for other sources of income, e.g., labor income, that support
consumption.) Cross-sectional variation in unconditional mean returns across risky securities
in this model is governed by cross-sectional variation in the covariance between shares and
aggregate consumption growth: Cov
sjt
sjt ;
ct
ct
, for j = 1;:::;n. This in turn is determined
by cross-sectional variation in j, sj and (sj) : We create n arti?cial risky securities using
an evenly spaced grid of values for these parameters. The values of j lie on a grid between
0 and 1, and the values of sj 2 [0;1) lie on a grid such that the sum over all j is unity.
The parametric process for (sj) follows the speci?cation in MSV in which the volatilities
depend on a N-dimensional vector of parameters vj as well as the individual share processes
(sj) = vj
nX
k=0
skt vk:
We choose the parameters j, sj, and vj, to generate a spread in average returns across
assets. In analogy to the empirical exercise (Panel B of Table 1), we do this for n = 6 risky
securities plus the aggregate wealth portfolio return and the risk-free for a total of 8 asset
returns.
Closed-form solutions are not available for the individual risky securities, but MSV show
that equilibrium price-dividend ratios on the risky assets are given by the approximate
relation
Pjt
Djt a
j
0 + a
j
1St + a
j
2
sj
sjt + a
j
3
sj
sit St; (26)
where St 1=Y it and where Y it again denotes the inverse surplus ratio of an individual
assetholder indexed by i, which should not be confused with the indexation by j, which
denotes a security. The parameters aj0, aj1; aj2; and aj3 are all de?ned in terms of the other
parameters above. Using these solutions for individual price-dividend ratios, we create a
cross-section of equilibrium risky securities using
Rit+1 =
Pjt+1=Djt+1 + 1
Pjt =Djt
!
exp djt+1 : (27)
Bansal and Yaron (2004) consider a representative agent who maximizes utility given by
recursive preferences of Epstein and Zin (1989, 1991) and Weil (1989). The utility function
to be maximized takes the form
U = E
( 1X
t=0
t
n
(1 ) C
1
t +
E
tU
1
t+1
1 o 1 ); (28)
where (1 ) =(1 1= ) ; is the intertemporal elasticity of substitution in consump-
tion (IES), is the coe? cient of relative risk aversion, and is the subjective discount factor.
The stochastic discount factor under Epstein-Zin-Weil utility used in BY takes the form
MBYt+1 =
C
t+1
Ct
1
!
R 1w;t+1; (29)
where Rw;t+1 is the simple gross return on the aggregate wealth portfolio, which pays a
dividend equal to aggregate consumption, Ct.
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Table 1: Pricing Errors with CRRA Preferences
Assets hatwide hatwide RMSE (in %) RMSE/RMSR p (W = I) p (W = S-1)
Rs,Rf 1.41 89.78 2.71 0.48 N/A N/A
Rs,Rf, 6 FF 1.39 87.18 3.05 0.33 0.00 0.00
Excluding Periods with low Consumption Growth
Rs,Rf 2.55 326.11 0.73 0.13 N/A N/A
Rs,Rf, 6 FF 2.58 356.07 1.94 0.21 0.00 0.00
Notes: This table reports the minimized annualized postwar data pricing error for CRRA preferences. The pref-
erence parameters hatwidec and hatwidec are chosen to minimize the mean square pricing error for different sets of returns:
mindeltac,gammacbracketleftbigg(deltac,gammac)primeWg(deltac,gammac)bracketrightbig where g(deltac,gammac) = E[deltac(Ct/Ct-1)-gammacRt - 1]. Rs is the CRSP-VW stock returns, Rf is
the 3-month T-bill rate and Ct is real per-capita consumption of nondurables and services excluding shoes and clothing.
The table also reports results when the periods with the lowest six consumption growth rates are eliminated. The table
reports estimated hatwide, hatwide and the minimized value of RMSR/RMSRR where RMSE is the square root of the average squared
pricing error and RMSR is the square root of the averaged squared returns of the assets under consideration for W = I.
The last two colums report chi2 p-values for tests for the null hypothesis that pricing errors are jointly zero for W = I and
W = S-1 where S is the spectral density matrix at frequency zero. The data span the period 1951Q4 to 2002Q4.
Table 2: Low Consumption Growth Periods
Quarter NBER Recession Dates Ct/Ct-1 - 1 Rst Rft
1980Q02 80Q1-80Q3 -1.28 16.08 3.59
1990Q04 90Q3-91Q1 -0.87 8.75 2.16
1974Q01 73Q4-75Q1 -0.85 -1.26 2.37
1958Q01 57Q3-58Q2 -0.84 7.03 0.65
1960Q03 60Q2-61Q1 -0.64 -4.93 0.67
1953Q04 53Q1-54Q2 -0.60 7.87 0.47
Notes: This table reports consumption growth, the return of the CRSP-VW stock returns Rs and the 3-month T-bill
rate Rf (all in in percent per quarter) in the six quarters of our sample with the lowest consumption growth rates. The
consumption measure is real per-capita expenditures on nondurables and services excluding shoes and clothing. The data
span the period 1951Q4 to 2002Q4.
Table 3: Pricing Errors
Model hatwidec hatwidec RMSE/RMSR (Rs,Rf) RMSE/RMSR (8 assets)
CC Habit 1.28 57.48 0.00 N/A
MSV Habit 1.71 30.64 0.00 0.00
BY LR Risk 0.93 48.97 0.00 0.00
Notes: This table reports the annualized pricing errors for stock returns Rs and the riskfree rate Rf from simulated data
from Campbell and Cochrane?s habit model (CC Habit), Menzly, Santos and Veronesi?s habit model (MSV Habit) and
Bansal and Yaron?s long run risk model (BY LR Risk) for CRRA preferences. The preference parameters hatwidec and hatwidec are
chosen to minimize the mean square pricing error mindeltac,gammacbracketleftbigg(deltac,gammac)primeg(deltac,gammac)bracketrightbig where g(deltac,gammac) = E[deltac(Ct/Ct-1)-gammacRt -1].
RMSR is the square root of the averaged squared returns of the assets under consideration. RMSE is the square root of
the average squared pricing error. Pricing errors are computed from simulations with 10,000 observations.
Table 4: Properties of Guvenen?s Model
Panel A: Consumption Growth
Ct/Ct-1 - 1 Cit/Cit-1 - 1 Cnt /Cnt-1 - 1 Rst Rft
Mean 0.01 0.02 0.00 1.31 0.64
Std. Dev. 2.04 4.53 0.83 7.30 1.69
Correlation 1.00 1.00 0.99 1.00 0.17
1.00 1.00 0.98 0.99 0.17
0.99 0.98 1.00 0.99 0.16
1.00 0.99 0.99 1.00 0.19
0.17 0.17 0.16 0.19 1.00
Panel B: Stochastic Discount Factors
Mit(0.99,2.00) Mct (0.99,2.00) Mct (0.99,4.49)
Mean 0.99 0.99 0.99
Std. Dev. 0.09 0.04 0.09
Correlation 1.00 1.00 1.00
1.00 1.00 1.00
1.00 1.00 1.00
Panel C: Pricing Errors
Consumption (delta,gamma) E[Mt(delta,gamma) Rst - 1] E[Mt(delta,gamma) Rft - 1]
SH (0.99, 2.00) 0.02% 0.02%
AC (0.99, 2.00) 0.39% -0.34%
AC (0.99, 4.49) 0.00% 0.01%
Notes: This table reports properties of Guvenen?s model. Panel A reports the properties of consumption growth rates
of aggregate consumption Ct/Ct-1, stockholders consumption Cit/Cit-1, nonstockholders consumption Cnt /Cnt-1, stock
returns Rst and the riskfree rate Rft in Guvenen?s model. Panel B reports properties of stochastic discount factors. The
first row reports properties of the SDF for stockholders consumption. The remaining rows report SDF properties for
total consumption and different preference parameters. The stochastic discount factors are of the CRRA form Mt =
delta(Ct/Ct-1)-gamma. The first parameter in parenthesis is delta, the second one is gamma. Panel C reports the annual pricing error
Guvenen?s model. The preference parameters delta and gamma are chosen to minimize the equally weighted sum of pricing errors
for the stock returns Rs and the riskfree rate Rf. The first row labelled ?SHquotedleft reports the pricing errors for stockholders
consumption. The remaining rows labelled ?ACquotedleft report pricing errors for aggregate consumption and different preference
parameters. All statistics are quarterly.
Table 5: Limited Participation Habit Model Estimated with Aggregate Consumption Habit SDF
sigmai/sigmac rho(Cit/Cit-1,Ct/Ct-1)
-1.0 -0.5 -0.25 0.25 0.5 1.0
2 Assets: Rs,Rf
hatwidedeltac
1 0.84 0.66 0.25 0.39 0.84 0.96
2 0.85 0.65 0.22 0.46 0.83 0.96
5 0.84 0.66 0.23 0.46 0.82 0.96
hatwidec
1 -1.69 -2.91 -4.89 5.11 2.35 0.84
2 -1.59 -3.09 -5.46 5.08 2.34 1.14
5 -1.75 -2.99 -5.96 4.88 2.26 1.11
RMSE/RMSR
1 0.00 0.00 0.00 0.00 0.00 0.00
2 0.00 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00 0.00
8 Assets
hatwidedeltac
1 0.84 0.65 0.27 0.42 0.84 0.96
2 0.84 0.65 0.27 0.40 0.83 0.96
5 0.85 0.63 0.23 0.36 0.81 0.96
hatwidec
1 -1.71 -2.93 -5.21 5.08 2.28 1.02
2 -1.73 -2.96 -5.17 5.24 2.37 1.03
5 -1.76 -3.07 -5.46 5.39 2.51 1.18
RMSE/RMSR
1 0.04 0.03 0.03 0.03 0.03 0.02
2 0.05 0.03 0.04 0.03 0.02 0.02
5 0.05 0.04 0.04 0.03 0.02 0.02
Notes: This table reports preference parameters and pricing errors in Menzly, Santos and Veronesi?s (2004) habit model.
Consumption growth of stockholders is assumed to follow a random walk with a mean of 2% and standard deviation of
1%. All parameters are as in Menzly, Santos and Veronesi except alpha, which is set obtain the same average stock return as
in Menzly-Santos-Veronesi. The preference parameters hatwidec and hatwidec are chosen to minimize the mean square pricing error
mindeltac,gammacbracketleftbigg(deltac,gammac)primeWg(deltac,gammac)bracketrightbig where g(deltac,gammac) = E[Mcht Rt - 1],Mcht = deltac( CtCt-1 Yt-1Yt )-gammac. Ct is aggregate consumption,
Yt is the inverse of the consumption surplus ratio computed from aggregate consumption, Rs is the return of equity, Rf
is the riskfree rate, and W = I. In the top panel, R = [Rs,Rf]prime, in the bottom panel R includes the return of the market
Rs, the riskfree rate Rf and the returns of six individual assets. RMSR is the square root of the averaged squared returns
of the assets under consideration. RMSE is the square root of the average squared pricing error. The weighting matrix
W is the identity matrix.
Table 6: Limited Participation Habit Model Estimated with Aggregate Consumption CRRA SDF
sigmai/sigmac rho(Cit/Cit-1,Ct/Ct-1)
-1.0 -0.5 -0.25 0.25 0.5 1.0
2 Assets: Rs,Rf
hatwidedeltac
1 0.51 0.24 0.03 5.27 2.69 1.61
2 0.52 0.24 0.03 5.20 2.75 1.83
5 0.48 0.23 0.03 4.94 2.81 1.79
hatwidec
1 -30.71 -60.15 -128.80 127.03 58.59 27.93
2 -29.22 -61.24 -132.02 117.99 61.69 33.28
5 -33.48 -64.30 -131.01 117.94 64.43 32.56
RMSE/RMSR
1 0.00 0.00 0.00 0.00 0.00 0.00
2 0.00 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00 0.00
8 Assets
hatwidedeltac
1 0.50 0.24 0.04 5.44 2.76 1.74
2 0.50 0.23 0.04 5.60 2.80 1.74
5 0.48 0.21 0.03 5.74 2.94 1.85
hatwidec
1 -30.83 -61.99 -123.23 124.24 61.51 31.21
2 -31.69 -62.76 -124.21 126.92 62.34 31.22
5 -33.73 -67.43 -134.53 133.41 65.50 34.11
RMSE/RMSR
1 0.03 0.03 0.03 0.03 0.04 0.03
2 0.04 0.03 0.03 0.03 0.03 0.03
5 0.03 0.03 0.04 0.03 0.03 0.04
Notes: This table reports preference parameters and pricing errors in Menzly, Santos and Veronesi?s (2004) habit model.
Consumption growth of stockholders is assumed to follow a random walk with a mean of 2% and standard deviation of
1%. All parameters are as in Menzly, Santos and Veronesi except alpha, which is set obtain the same average stock return as
in Menzly-Santos-Veronesi. The preference parameters hatwidec and hatwidec are chosen to minimize the mean square pricing error
mindeltac,gammacbracketleftbigg(deltac,gammac)primeWg(deltac,gammac)bracketrightbig where g(deltac,gammac) = E[Mct Rt - 1],Mct = deltac( CtCt-1 )-gammac. Ct is aggregate consumption, Rs is
the return of equity, Rf is the riskfree rate, and W = I. R includes the return of the market Rs, the riskfree rate Rf
and the returns of six individual assets. RMSR is the square root of the averaged squared returns of the assets under
consideration. RMSE is the square root of the average squared pricing error. The weighting matrix W is the identity
matrix.
Table 7: Limited Participation Habit Model Estimated with Aggregate Consumption CRRA SDF:
Time Aggregated Data
sigmai/sigmac rho(Cit/Cit-1,Ct/Ct-1)
-1.0 -0.5 -0.25 0.25 0.5 1.0
2 Assets: Rs,Rf
hatwidedeltac
1 0.53 0.22 0.07 4.30 2.48 1.56
2 0.55 0.29 0.06 6.05 2.64 1.77
5 0.49 0.27 0.06 5.45 2.75 1.77
hatwidec
1 -28.74 -64.81 -104.82 104.40 53.96 26.56
2 -26.32 -53.79 -110.70 129.91 59.36 31.78
5 -32.16 -58.47 -108.09 129.49 63.45 32.38
RMSE/RMSR
1 0.00 0.00 0.00 0.00 0.00 0.00
2 0.00 0.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 0.00 0.00
8 Assets
hatwidedeltac
1 0.49 0.24 0.04 5.51 2.78 1.75
2 0.49 0.23 0.04 5.57 2.86 1.76
5 0.48 0.21 0.03 5.63 2.96 1.83
hatwidec
1 -32.13 -61.76 -122.15 126.00 61.95 31.39
2 -32.02 -62.97 -125.44 127.60 63.43 31.83
5 -33.53 -66.57 -135.09 131.36 65.84 33.66
RMSE/RMSR
1 0.03 0.03 0.06 0.03 0.04 0.04
2 0.04 0.03 0.06 0.04 0.03 0.04
5 0.04 0.03 0.07 0.03 0.04 0.04
Notes: This table reports preference parameters and pricing errors in Menzly, Santos and Veronesi?s (2004) habit model.
Consumption growth of stockholders is assumed to follow a random walk with a mean of 2% and standard deviation of
1%. All parameters are as in Menzly, Santos and Veronesi except alpha, which is set obtain the same average stock return as
in Menzly-Santos-Veronesi. The preference parameters hatwidec and hatwidec are chosen to minimize the mean square pricing error
mindeltac,gammacbracketleftbigg(deltac,gammac)primeWg(deltac,gammac)bracketrightbig where g(deltac,gammac) = E[Mct Rt - 1],Mct = deltac( CtCt-1 )-gammac. R includes the return of the market
Rs, the riskfree rate Rf and the returns of six individual assets. RMSR is the square root of the averaged squared returns
of the assets under consideration. RMSE is the square root of the average squared pricing error. The weighting matrix W
is the identity matrix. Ct is aggregate consumption, Rs is the return of equity, Rf is the riskfree rate, and W = I. The
model is simulated on a weekly frequency. The pricing errors are computed using the growth rate of annual consumption
and compounded annual returns.
Table 8: Small Sample Pricing Errors
Model max. RMSE/RMSR (2 Assets) max. RMSE/RMSR (8 Assets)
CC Habit 0.01 N/A
MSV Habit 0.00 N/A
BY LR Risk 0.00 0.01
Guvenen Lim. Part. 0.87 N/A
Notes: This table reports the annualized pricing errors for stock returns Rs and the riskfree rate Rf from simulated
data from Campbell and Cochrane?s habit model (CC Habit), Menzly, Santos and Veronesi?s habit model (MSV Habit),
Bansal and Yaron?s long run risk model (BY LR Risk) and Guvenen?s limited participation model. The preference
parameters hatwidec and hatwidec are chosen to minimize the mean square pricing error mindeltac,gammacbracketleftbigg(deltac,gammac)primeWg(deltac,gammac)bracketrightbig where g(deltac,gammac) =
E[deltac(Ct/Ct-1)-gammacRt - 1]. R = [Rs,Rf]prime and W = I. ?max. RMSE/RMSRquotedleft report the maximum absolute value of
RMSE/RMSR from 5,000 simulations with 204 observations.
???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.8536.2112.56134.95E-104.90E-104.091711.195-0.004233730.0361113.000950.990.10.911.536.2172.39882.12E-102.10E-104.089911.181-0.00422830.0360633.000950.990.10.920.8571.4956.06751.14E-091.12E-094.095211.2070.007850930.046993.001550.990.10.921.571.535.68691.50E-091.49E-094.093411.1930.007840330.0469273.001550.990.10.940.85129.0814.2359.75E-089.67E-084.101811.2290.0320213.00070.0687513.003250.990.10.941.5129.2213.395-9.01E-08-8.50E-084.111.2150.0319773.00070.0686583.0031???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.85-64.060.1052-1.41E-14-1.40E-144.103711.313-0.00415963-0.664373.289950.990.10.911.5-64.050.0987-1.27E-08-1.27E-084.103411.303-0.00415433-0.663563.289150.990.10.920.85-118.70.0121-6.34E-14-6.22E-144.107111.3240.00771343-0.653043.280250.990.10.921.5-118.60.0113-8.05E-11-8.07E-114.106711.3140.00770363-0.652253.279550.990.10.940.85-210.40.0002-6.58E-13-6.60E-134.113411.3460.0314593.0007-0.630433.261450.990.10.941.5-210.20.0001-2.31E-13-2.31E-134.113111.3360.031423.0007-0.629663.2607???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.8535.4881.7114-7.93E-09-8.00E-090.21753.03070.496922.48040.461532.294650.990.10.911.535.4881.6032-7.94E-09-7.99E-090.21753.03070.496912.48040.461532.294650.990.10.920.8570.9782.74459.82E-099.64E-090.21753.03070.496922.48040.461542.294650.990.10.921.570.9782.5719.82E-099.64E-090.21753.03070.496912.48040.461532.294650.990.10.940.85141.964.36122.26E-072.25E-070.21753.03070.496922.48040.461542.294650.990.10.941.5141.964.08552.26E-072.25E-070.21753.03070.496922.48040.461532.2946
Table 9: Lim. Partic./Inc. Markets Pricing Errors for Stock Return and Risk-Free Rate: Hermite Densities
Cov(?c,?d)=0.00017Cov(?c,?d)=0.00017Cov(?c,?d)=-0.00017
???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.85-35.460.4115-4.87E-08-4.88E-08-0.2183.03080.496912.48040.461522.294650.990.10.911.5-35.460.3855-4.88E-08-4.88E-08-0.2183.03080.49692.48040.461512.294650.990.10.920.85-70.920.15874.66E-154.66E-15-0.2183.03080.496912.48040.461522.294650.990.10.921.5-70.920.14874.22E-155.33E-15-0.2183.03080.49692.48040.461512.294650.990.10.940.85-141.80.01461.21E-131.22E-13-0.2183.03080.496922.48050.461532.294650.990.10.941.5-141.80.01371.15E-131.18E-13-0.2183.03080.496912.48040.461522.2946???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.8534.6141.6892-3.03E-10-2.94E-100.23893.03730.541472.57130.504962.3950.990.10.911.534.6191.5825-3.03E-10-2.96E-100.24253.03850.555432.60080.518532.420450.990.10.920.8569.2412.69481.24E-111.20E-110.23773.03690.537292.56140.501282.381850.990.10.921.569.2492.52471.26E-111.24E-110.24133.03810.551132.59050.514722.411850.990.10.940.85138.644.3376-2.37E-09-1.87E-090.23533.03610.529112.5420.494092.365950.990.10.941.5138.644.0636-2.40E-09-1.89E-090.23893.03730.542692.57050.507272.3952???(?c,?c
i
)?(?c
i
,?d)?(i)/?(c)?(?c
i
)/?(?c)?
c
?
c
PrErrR(s)PrErrR(f)Sk[c]Ku[c]Sk[i]Ku[i]Sk[d]Ku[d]50.990.10.910.85-36.690.3989-2.05E-11-2.08E-11-0.2183.03110.542072.57120.5062.392350.990.10.911.5-36.680.3737-1.81E-11-1.83E-11-0.2213.03210.556072.60070.519612.422850.990.10.920.85-73.970.1449-3.45E-10-3.50E-10-0.2153.03030.537882.56130.502312.384150.990.10.921.5-73.950.1358-3.37E-10-3.42E-10-0.2183.03130.551742.59050.515782.414250.990.10.940.85-150.40.010401.33E-15-0.2093.02870.529672.54180.495082.368150.990.10.941.5-150.40.00986.66E-164.44E-16-0.2123.02960.543282.57040.508292.3975Cov(?c,?d)=-0.00017Cov(?c,?d)=0.00017Cov(?c,?d)=-0.00017
Notes: This table reports output on the pricing error associated with erroneously using aggregate consumption in place of asset-holder consumption, for a range of parameter values and joint distributions. ?
i
is the persumed value of asset-holder risk-aversion; ?
i
is the presumed value of the asset-holder's subjective discount rate; ?(?c,?c
i
) denotes the correlation between aggregate consumption growth and asset-holder consumption growth in the leading normal; ?(?c
i
,?d) denotes the correlation between asset-holder consumption growth and dividend growth in the leading normal; ?(?c
i
)/?(?c) denotes the standard deviation of asset-holder consumption growth divided by the standard deviation of aggregate consumption growth in the leading normal; ?(?c
i
)/?(?c) denotes the mean of asset-holder consumption growth divided by the mean of aggregate consumption growth in the leading normal; ?
c
and ?
c
are the values of ? and ? that minimize the pricing errors using aggregate consumption; PrErrR(s) is the pricing error for the Euler equation associated with the stock return; PrErrR(f) is the pricing error of the Euler equation associated with the risk-free rate, and Sk[ ], Ku[ ] refer to the skewness and kurtosis of aggregate consumption (c), asset-holder consumption (i), and dividends (d).
Distribution???(?c,?c
i
)?(?c
i
)/?(?c)?(?c
i
)/?(?c)Cov(?c,?d)?
c
?
c
Max RMSEAvg RMSEJ. Log N.50.990.1321.50.0001778.242.770.02%0.02%Non-Normal50.990.1321.50.000176.831.0910.10%0.25%J. Log N.50.990.1321.5-0.00017-78.240.120.02%0.02%Non-Normal50.990.1321.5-0.00017-82.80.2090.58%0.03%Table 10: Lim. Partic./Inc. Markets Pricing Errors in a Larger Cross-Section: Hermite Densities
Notes: This table reports the average pricing errors for models with 8 asset returns. The column labled "Distribution" denotes whether the joint distribution of ?c, ?c
i
and dividend growth for each of the 8 assets is modeled as lognormal or not. "J. Log N." reports results for the jointly lognormal case; "Non-Normal" reports the results for cases in which a perturbation from the lognormal was used to describe the joint distribution of aggregate consumption, asset-holder consumption, and the 8 asset returns. The numbers in the column labled "Max RMSE" give the square root of the average squared pricing error, as a fraction of the cross-sectional average mean return, that is the maximum over all Non-Normal perturbations (over 100) considered. The numbers in the column labled "Avg RMSE" give the square root of the average squared pricing error, as a fraction of the cross-sectional average mean return, that is the average of over Non-Normal perturbations (over 100) considered. ?
i
is the persumed value of asset-holder risk-aversion; ?
i
is the presumed value of the asset-holder's subjective discount rate; ?(?c,?c
i
) denotes the correlation between aggregate consumption growth and asset-holder consumption growth in the leading normal; ?(?c
i
,?d) denotes the correlation between asset-holder consumption growth and dividend growth in the leading normal; ?(?c
i
)/?(?c) denotes the standard deviation of asset-holder consumption growth relative to the standard deviation of aggregate consumption growth in the leading normal; ?(?c
i
)/?(?c) denotes the mean of asset-holder consumption growth divided by the mean of aggregate consumption growth in the leading normal. ?
c
and ?
c
are the values of ? and ? that minimize the equally weighted sum of squared pricing errors when aggregate consumption is used in place of stockholder consumption, for the hermite distribution that delivers the maximumRMSE, as a percentage of the cross-sectional mean return. For the jointly lognormal case, the average is the maximum since there is only one distribution to average over.
Figure 1: Pricing Errors for CRRA Preferences: Excess Returns
Notes: The figure plots RMSE/RMSR as a function of gamma for excess returns. The pricing errors
are PE = E
bracketleftBig
(Ct+1/Ct)-gamma(Rt+1 - Rft+1)
bracketrightBig
. The solid line shows RMSE/RMSR for R = Rs, the
dotted line shows RMSE/RMSR for R = (Rs,6 FF).
Figure 2: Pricing Errors: With and Without Lognormality
Notes: This figure plots RMSE/RMSR with and without the assumption of joint log-
normality as a function of gammac. deltac is chosen to minimize the RMSE for each value
of gammac. The top panel shows the case for R = (Rs,Rf), in the bottom panel
R = (Rs,Rf,6 FF). The pricing error for asset j without assuming lognormality is
PEj = deltacE [exp {-gammacdeltac + rj}] - 1. Under the assumption of joint lognormality, the pricing
error is PEj = deltac exp {-gammacEdeltac + gamma2c sigma2c/2 + Erj + sigma2r/2 - gammacCov(deltac,rj)} - 1.
Figure 3: QQ Plots
Notes: This figure shows multivariate quantile-quantile (QQ) plots of log consumption growth
and asset returns. Each panel plots the sample quantiles (on the y-axis) against the quantiles
of a given distribution (on the x-axis) as well pointwise 5% and 95% bands. The top panel
shows the QQ plot for the joint distribution of deltac,rs and rf, i.e. the quantiles of the squared
Mahalanobis distances against those of a chi23 distribution. The bottom panel shows the QQ
plot for the joint distribution of deltac,rs,rf and 6 FF portfolios, i.e. the quantiles of the squared
Mahalanobis distances against those of a chi29 distribution. The squared Mahalanobis distance Mt
for a p-dimensional multivariate distribution xt with mean ?x and variance-covariance matrix
V is defined as Mt = (xt - ?x)primeV -1(xs - ?x). Under the null hypothesis that deltac,rs and rf
are jointly normally distributed, Mt has a chi2p distribution.
-0.04 -0.02 0 0.02 0.04 0.06 0.080
0.1
0.2
0.3
0.4
Density of Aggregate Consumption Growth
Hermite
Normal
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.30
0.1
0.2
0.3
0.4
Density of Assetholder Consumption Growth
Hermite
Normal
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80
0.1
0.2
0.3
0.4
Density of Dividend Growth
Hermite
Normal
Figure 4
Notes: An example of distributions that produce large pricing errors when aggregate consumption
and returns are fitted to a power utility model.
-0.05 0 0.05 0.1 0.150
0.1
0.2
0.3
0.4
Density of Aggregate Consumption Growth
Hermite
Normal
-0.2 -0.1 0 0.1 0.2 0.30
0.1
0.2
0.3
0.4
Density of Assetholder Consumption Growth
Hermite
Normal
-1 -0.5 0 0.5 10
0.2
0.4
0.6
0.8
Density of Dividend Growth
Hermite
Normal
-0.05 0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
Density of Aggregate Consumption Growth
Hermite
Normal
-0.2 -0.1 0 0.1 0.2 0.30
0.1
0.2
0.3
0.4
Density of Assetholder Consumption Growth
Hermite
Normal
-1 -0.5 0 0.5 10
0.1
0.2
0.3
0.4
Density of Dividend Growth
Hermite
Normal
Figure 5
Notes: Plots of marginal densities for two Hermite parameter configurations that do not explain large
pricing errors of the standard model.