MONEY AND THE (C)CAPM: THEORY AND EVALUATION

MARCH 2004

Ronald J. Balvers Dayong HuangDivision of Economics and Finance Division of Economics and FinanceP.O. Box 6025 P.O. Box 6025West Virginia University West Virginia UniversityMorgantown, WV 26506-6025 Morgantown, WV 26506-6025Phone: (304) 293-7880 Phone: (304) 293-7875Email: rbalvers@wvu.edu Email: dahuang@mail.wvu.edu

ABSTRACT

We consider asset pricing in a monetary economy where liquid assets are held to lower transactioncosts. The ensuing model extends the CAPM and the Consumption CAPM by deriving real moneygrowth as an additional factor determining returns. Empirically, the unconditional version of thismodel compares favorably to other theoretical asset pricing models. Allowing for conditionalvariation in factor sensitivities improves model performance so the model performs as well as thea-theoretical Fama-French three factor model. The paper further introduces a technique thatfacilitates derivation of dynamic asset pricing results in discrete time by generalizing Stein’s Lemmato multivariate cases.

JEL classification: G12

Keywords: Asset Pricing, Money Supply Growth, Consumption CAPM, Stein’s Lemma

Money and the (C)CAPM: Theory and Evaluation

ABSTRACT

We consider asset pricing in a monetary economy where liquid assets are held to lower transaction

costs. The ensuing model extends the CAPM and the Consumption CAPM by deriving real money

growth as an additional factor determining returns. Empirically, the unconditional version of this

model compares favorably to other theoretical asset pricing models. Allowing for conditional

variation in factor sensitivities improves model performance so the model performs as well as the

a-theoretical Fama-French three factor model. The paper further introduces a technique that

facilitates derivation of dynamic asset pricing results in discrete time by generalizing Stein’s Lemma

to multivariate cases.

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Money and the (C)CAPM: Theory and Evaluation

Introduction

The empirical performance of the basic asset pricing theories – the CAPM and the

Consumption CAPM – has been disappointing. Alternative theories such as Cochrane’s (1996)

investment-based model and Lettau and Ludvigson (2001a, 2001b) (C)CAPM conditional on the

consumption-to-wealth ratio perform considerably better, but the purely-empirical three factor model

of Fama and French (1996) still outperforms all theoretical asset pricing formulations. A major

challenge to the profession is to provide a set of pricing factors that is both reasonable from a

macroeconomic perspective and is competitive with the Fama-French model in pricing the cross

section of asset returns.

From a macroeconomic perspective, the monetary environment as related to money growth

and inflation appears to be a prime candidate for providing a systematic pricing factor. Early

empirical results starting with Chen, Roll, and Ross (1986) were suggestive of a role for money in

asset pricing by finding that both anticipated inflation and unanticipated inflation are significant

factors in pricing the cross-section of assets. Following these results, a large body of work has

investigated the link between equity prices and inflation or money growth.

Boyle and Young (1988), for instance, consider a static asset pricing model in which

individuals have direct utility over consumption as well as real money balances. In their framework,

both monetary shocks and real shocks affect the equity premium. Marshall (1992) obtains similar

results but employs a dynamic formulation in which the level of liquidity measured by real money

holdings affects the cost of doing transactions. Both of these papers focus primarily on the link

1 The money factor, even if it is theoretically equivalent to consumption, can outperform the consumption factorgiven that consumption services are difficult to measure. A similar motivation has led to the production-based assetpricing models of Balvers, Cosimano, and McDonald (1990) and Cochrane (1991) in which output growth andinvestment growth, respectively, are merely alternative proxy variables measuring the marginal rate of intertemporalsubstitution in consumption.

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between stock prices and inflation and do not consider cross-sectional pricing implications.

Similarly, Boyle and Peterson (1995), Thorbecke (1997), Jensen and Mercer (2002), and others

examine the monetary policy impact on asset pricing without considering cross-sectional asset

pricing implications.

Chan, Foresi and Lang (1996) examine the implications for cross-sectional asset pricing of

a liquidity constraint in a cash-in-advance economy in which certain goods can be purchased only

with money. In this environment real inside money holdings are equal to the consumption good that

requires cash-in-advance. Due to a separability in utility assumption, the stochastic discount factor

depends solely on the growth rate of this cash-in-advance consumption good. Hence, the growth

rate of real inside money balances becomes the asset pricing factor. They find that their model with

real inside money growth as the sole factor outperforms the C-CAPM, but performs marginally

worse than the market CAPM.1

From a different angle, Pástor and Stambaugh (2003) find that aggregate liquidity is a

systematic asset pricing factor that moreover can explain a large part of momentum profits. They

measure aggregate liquidity as the average price reversals induced by order flow. The concept of

liquidity used by Pástor and Stambaugh – related to whether financial assets can be traded quickly

and at low cost – is a micro liquidity concept and quite different from the macro liquidity concept

related to the monetary environment, which deals with whether consumption goods can be traded

easily (at low time and resource cost). Intriguingly, Chordia, Sarkar, and Subrahmanyam (2002) link

2 Marshall’s formulation originates with Brock (1974) and Feenstra (1986). See also the description in Walsh(2003, Chapter 3).

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the two liquidity concepts empirically by showing that improvements in stock market liquidity are

tied to (nominal) money growth.

This paper follows the formulation in Marshall (1992) to introduce money to the C-CAPM

in a very natural way. The monetary economy is grounded in a transactions and precautionary

demand for money that stems from the liquidity services of money holdings as means to lowering

the cost of doing transactions.2 In this environment, the marginal value of financial returns is

determined by the marginal utility of consumption as well as the marginal cost of doing transactions

which depends on the liquidity currently available. Accordingly, the stochastic discount factor

varies with real consumption growth as well as with real money growth, where the real money

growth factor combines the effects of nominal money growth and inflation. Using the envelope

condition, it follows that the marginal utility of wealth alternatively depends on real market returns

together with real money growth rates. Thus, a two-factor asset pricing model emerges with either

market return and real money growth, or consumption growth and real money growth, as the factors.

The model solution that yields the asset pricing implications is accomplished in discrete time

by introducing a new method based on extending Stein’s (1973, 1981) lemma to multivariate

applications. The main advantage of this discrete-time approach is that it allows a direct transfer

of the theoretical results into an empirical formulation, without worrying about time aggregation.

It also permits a setting that may be more intuitive than the continuous-time setting typically

employed for analyzing cross-sectional pricing in equilibrium dynamic asset pricing models, and,

moreover, facilitates dealing with lags and conditioning information.

Following Lettau and Ludvigson (2001b) we employ the Fama-MacBeth (1973) method

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(with Shanken and GMM-based corrections for measurement error in the betas) to price the cross-

section of the 25 Fama and French portfolios ranked by size and book-to-market ratio. We find that

our real money-growth-augmented CAPM explains 65%, and the real money-growth-augmented C-

CAPM explains 60% of the cross-sectional returns variation, in either case with insignificant Jensen

alphas. Both compare favorably with the CAPM, the C-CAPM, and with the theories of Cochrane

(1996), Campbell and Cochrane (1999), and Lettau and Ludvigson (2001a, 2001b). If we allow for

conditional variation in the betas, using the standard instrument of the lagged term premium as the

conditioning factor, then our model performance improves further. Explaining 75% of the cross-

sectional returns variation this conditional model performs as well as the a-theoretical Fama-French

(1992, 1993, 1996) three-factor model.

In Section 1 we derive a discrete-time factor model based on the assumption of conditional

normality of returns and factors. We derive both a money CAPM and a money C-CAPM and

consider unconditional and conditional specifications of each. Section 2 empirically evaluates each

version and compares to alternative asset pricing models. For the sake of robustness and to facilitate

comparisons we provide results for alternative conditioning variables, and also conduct alternative

evaluations and obtain comparable estimates based on a GMM-stochastic discount factor approach.

Section 3 concludes.

1. Asset Pricing in a Monetary Economy

We add money to the standard C-CAPM model of Breeden (1979). Money is held because

it lowers the transactions cost of purchasing consumption goods, as modeled in Marshall (1992)

who, however, did not consider cross-sectional asset pricing issues. The addition of money is best

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modeled in discrete time since money must be acquired first before it can be used to facilitate

transactions. Analogously to the Brownian motion assumption in continuous time, we assume

conditional normality and are able to obtain a specific solution for the asset pricing equation by

providing and applying an extension to Stein’s lemma. The resulting model has substantially

different implications compared to Chan, Foresi, and Lang (1996) – the only previous paper to have

considered the theoretical cross-sectional asset pricing implications of money – by generating real

money balances as an additional factor to consumption instead of an alternative proxy.

(a) The Model

A representative consumer-investor in an endowment economy maximizes expected lifetime

utility subject to a budget constraint and given the fact that transactions costs to purchase

consumption goods are mitigated by money holdings.

The maximized value of life time utility in real terms is given by the Bellman equation of

dynamic programming as:

, (1)

where and represent real financial wealth (excluding money holdings) and real money

holdings, respectively; and indicates a set of state variables, exogenous to the consumer-investor,

that is sufficient to represent changes in the investment opportunity set. The agent in period t

chooses current consumption , real money holdings (for use in the upcoming period), and the

portfolio shares of a risk free asset (asset 0) and n risky assets. Life time utility is time separable,

3 See Feenstra (1986, p. 278), Marshall (1992, pp. 1319-1321), and Walsh (2003, pp.119-120) for morediscussion on the conditions on the transactions cost function. Feenstra (1986) shows that the transactions costformulation used here can be derived from standard theories of transactions and precautionary demand for money. Lowliquidity implies, for instance more trips to the bank and more costs of having to buy goods on credit. He also showsthat the transactions cost approach is equivalent to the Sidrauski assumption of putting money directly in the utilityfunction. Further, the popular “cash-in-advance” approach is just a limiting case of this approach when becomes infinite for . Thus, the model employs a quite general approach to incorporating money.

4 To assure that second-order conditions for the maximization problem are met, we need to have a concaveobjective and a convex constraint set. This is guaranteed by the concavity of the utility function and the conditions onthe transaction cost function – see for instance Lucas and Stokey (1989, Chapter 4).

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future utility is discounted with factor , and felicity u(.) in each period is a concave function of

consumption. Optimization is subject to the budget constraint for each period:

, (2)

with

, , (3)

. (4)

represents the real transactions cost of purchasing the current level of consumption.

Indicating partial derivatives by subscripts, the conditions on the transactions cost are that

, , , , , . These

conditions are intuitive. Transaction costs are positive, increase in consumption and decrease in

real balances. The second derivative conditions assure that increases in consumption and decreases

in real money holdings have a constant effect or become increasingly costly. The cross derivative

condition implies that the marginal cost of consumption stays constant or falls as more real money

balances are held.3 4

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Investable wealth in equation (2) equals minus current expenditure on consumption

and future money holdings . The gross real portfolio return is a weighted

average of returns on the individual assets as given by equation (3). Changes in the distribution over

time of the asset returns are captured by the state variables (the Merton, 1973, factors). The

choice of real money balances at the current price level to be used in the next period differs from

the real balances actually used in the next period at the then relevant price level, due to the gross

inflation rate and due to an assumed transfer of government revenues from nominal

money creation, , to the representative consumer-investor, yielding equation

of motion (4).

The first-order conditions derived from (1) - (4) are:

, (5)

, (6)

. (7)

The envelope conditions for the two state variables are:

, (8)

. (9)

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From aggregating equation (7) multiplied by the portfolio shares over all i and using equation (8)

it follows that:

. (10)

Defining excess returns as , equation (7) becomes

, (7')

yielding the typical result that we have the marginal value of wealth serve as

the stochastic discount factor pricing all assets.

However, it follows from equations (5) and (8) that

, (11)

which is non-standard in that the marginal utility of consumption is not sufficient to capture the

marginal value of wealth. The marginal value of wealth depends on consumption but also on the

real value of money holdings, determining the cost of purchasing consumption goods at the margin.

Given consumption, if money holdings turn out to be low after the price level and market returns

are realized (a liquidity squeeze), then the marginal cost of consumption is higher at this given

consumption level so that the marginal value of wealth is lower according to equation (11).

We continue in the spirit of Merton (1973) and Breeden (1979) by developing an equilibrium

5 A general equilibrium version of the model embedded in a Lucas (1978) endowment economy is availablefrom the authors. Here dividend processes and nominal money growth are the exogenous endowment processes andMerton factors can be ruled out if dividends and money growth are i.i.d. Even if the endowment processes followconditionally normal processes, multivariate conditional normality of asset returns and the factors, as we assume, is notguaranteed in general but holds for some examples in general equilibrium.

6 If the distributions of returns are not predictable then investment opportunities do not change, but the realizedlevel of real money balances still affects return realizations in our model. Thus, even if equilibrium real money balancesare not predictable (as in many newclassical macro models), they may still be a priced risk factor.

7 In fact the solution technique only requires that the factors are functions of conditionally normal randomvariables but for simplicity we avoid this slight generalization. Conditional normality can imply a large variety ofunconditional return distributions, analogously to the continuous-time formulation where the Brownian motion buildingblocks can imply other return distributions when aggregated over time.

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asset pricing model without formally specifying a full general equilibrium environment.5 To isolate

the contribution of the money factor, we rule out Merton factors by assuming that there are no

changes over time in the exogenous dividend processes, ruling out shifts in the investment

opportunities set (except, possibly, through changes in the real money supply).6 Then we have

. (12)

We further assume that all asset returns and the factors are conditionally multivariate

normal.7 This is an assumption made in many equilibrium asset pricing models. Even if normality

does not hold exactly, the analysis is appropriate if one is willing to accept the assumption,

bolstered by various Central Limit theorems, that a variety of different shocks adds up to an

approximate normal distribution. Working in discrete time with conditionally normal distributions

requires the following lemma.

(b) A Multivariate Version of Stein’s Lemma

Based on equation (7') we derive a two factor model under the assumption of conditional

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normality. We assume that the returns of all assets as well as real money growth are conditionally

multivariate normal. To be able to derive a multi-factor model we must first provide a

generalization of Stein’s Lemma which has not appeared elsewhere in the finance or statistics

literature.

Multivariate Extension of Stein’s Lemma: If X, Y, and Z are multivariate

normal and h( ) is differentiable, with for , then∞<|),(| ZYhE i

. (13)

Proof: See Appendix.

Note that the statement of the lemma here is specialized to a trivariate normal case, which is the case

we need in this paper, but that the general multivariate case and its proof are provided in the

appendix.

(c) Derivation of the Money CAPM

The derivation of the asset pricing equations extends a standard approach for single-factor

models (see for instance Huang and Litzenberger, 1988) to multi-factor models. Use the definition

of covariance in equation (7') and employ equation (12) to dispense with the Merton variables:

. (14)

Apply the multivariate version of Stein’s Lemma in equation (13) to equation (14), noting from

8 From equation (10) and the definition of return, this asset would have a price of: . Similarly, we have the price of a riskless asset (with unit payoff) given

as . Accordingly, an asset position m exists with excess return for which .

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equation (2) and our normality assumptions that is conditionally normal, to produce

, (15)

with

. (16)

Here double subscripts indicate second derivatives.

Assume that an asset exists with a payoff that moves perfectly with real money growth. Such

an asset, has excess return

, (17)

where and is positive and known at time t. 8

Relating the covariances in equation (15) to the market excess return and to the real

money supply growth rate gives

, (18)

where, from equation (2) and the definition of ,

9 The sign of can be obtained by differentiating in equation (11) with respect to m, usingthe fact that optimal consumption is determined by the state variables, . The effect of higher moneyholdings is to raise the marginal value of wealth as the marginal transactions cost falls (given ). However,higher money holdings also have a price effect in that consumption becomes cheaper due to lower transactions cost. Thislowers the marginal value of wealth as the higher optimal consumption level reduces the marginal utility of consumptionand increases the marginal transactions cost.

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, , (19)

Based on equations (5) - (9) it can be shown that (confirming risk aversion) and that

the sign of is ambiguous.9 Thus, from equation (16), and 0.

Applying equation (18) to the market asset and asset m implies:

, (20)

. (21)

If the sign of is positive, then < 0 (from equations 16 and 19). In this case, an asset

with return positively correlated with money growth has a lower expected return as follows from

equation (18). The money growth factor then is a “hedging” factor in the sense that positive

exposure of some portfolio to money growth mitigates risk since this portfolio tends to pay out well

when the marginal value of wealth is high (when > 0). Even if < 0, from equation

(21) the expected excess return on the asset whose return is perfectly correlated with money growth,

, may be positive or negative. The reason is that a positive hedging benefit may be

offset by positive exposure to market risk if the covariance between market return and money

10 In our data, the is in fact positive, equal to 0.244. Similarly, and.

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growth is positive.10 Clearly, if # 0 then the money growth risk premium must be

positive given that the covariance between market return and money growth is positive.

Solving for and from equations (20) and (21):

, (22)

and plugging into equation (18) yields

, (23)

where

, (24)

. (25)

Here the coefficients are analogous to those obtained in a multivariate regression.

Equation (23) thus provides a two-factor model of asset returns. The two factors are the observable

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market excess return and the money growth rate minus an unobservable variable that is constant

across assets. There is a standard market beta and a real money growth “gamma”. For a positive

money factor risk premium, , if asset i has positive exposure to money growth

risk, , then asset i’s expected return is higher since the money factor exacerbates risk.

Intuitively, when liquidity is high so that transactions are cheap and accordingly optimal

consumption is high (lower marginal utility of consumption), the extra wealth from a higher return

on asset i positively correlated with liquidity tends to be less useful. Asset i is thus less attractive

so that a higher expected return is required.

(d) Derivation of the Money C-CAPM

Instead of using the marginal value of wealth as the stochastic discount factor, based on the

envelope condition provided in equation (11), we can directly employ the transaction-cost adjusted

marginal utility of consumption condition instead to derive a C-CAPM with money growth as an

additional factor. An analogous derivation to that of the Money CAPM in equation (23) yields

, (26)

with

, (27)

. (28)

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Here represents the consumption growth rate and a positive constant known at time t. The

signs of both risk premia are ambiguous a priori. The money growth factor matters to deal with the

risk of changes in transactions cost. These transactions cost have both an “income” and a

“substitution” effect. The substitution effect occurs because a lower cost of doing transactions

implies higher optimal consumption (lower marginal utility). So an asset that pays out mostly when

liquidity is high is less valuable (and requires a higher return). The confounding income effect

occurs for given planned consumption: an asset that pays out most when liquidity is high is worth

more (and requires a lower return) because it allows more transactions exactly when transactions

cost are low. The risk premium on the money growth factor is positive if the substitution effect

dominates the income effect, but this cannot be guaranteed a priori.

(e) From Theory to Measurement

Unconditional models

Under the same general equilibrium assumptions that rule out Merton factors, that nominal

money growth and dividend processes are i.i.d., the betas are constant over time. Taking

unconditional expectations, equation (23) then becomes

. (29)

Similarly, equation (26) becomes:

(30)

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These represent our unconditional versions of the models to be estimated. The standard Fama-

MacBeth (1973) two-pass “beta” approach under the assumption that the betas are stationary

provides consistent estimates of the coefficients of each asset in the first pass. In the second

pass, then, the specifications in equations (29) and (30) make clear that the constant should be zero

(asymptotically only due to measurement error in the generated regressors) if the estimates of the

coefficients are used as regressors. Standard errors in tests of significance use the Shanken

correction to adjust for measurement error. As discussed in the previous section, the signs of the

risk premia to be obtained in the second pass are ambiguous, except for the sign of which

should be positive because average market excess returns are positive in the data. One other

restriction is that the risk premium for the money factor, , should be identical in both

the Monetary CAPM and the Monetary C-CAPM versions.

Equation (29) can be used to derive straightforwardly a stochastic discount factor (see

Cochrane, 2001, section 6.3):

. (31)

This factor is the unique stochastic discount factor that can be mimicked by a portfolio of existing

assets. Although it should price all assets just as well as the stochastic discount factor in equation

(10), , the latter is generally much different since it need not be normally distributed

and then, given our normality assumption, cannot be generated by a portfolio of existing assets.

Conditional models

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If investment opportunities vary over time, when the endowment shocks are not i.i.d., the

factor loadings generally covary with the factor premia. To deal with this issue, we derive the

mimicking stochastic discount factor directly from the conditional formulation in equation (23).

This yields

. (32)

We allow the factor loadings to vary linearly with a single conditioning variable, again as in Lettau

and Ludvigson (2001b), and follow Cochrane (1996) and others in utilizing the term premium

at each time as a conditioning variable. Then:

. (33)

This stochastic discount rate formulation can be transformed back into a beta model, which becomes

after taking unconditional expectations:

. (34)

Similarly, from equation (26) we can derive the conditional Money C-CAPM as:

. (35)

11 As Jagannathan and Wang (1998) point out concerning the measurement error issue, the Fama-MacBethmethod need not overestimate the precision of the standard errors in the presence of heteroskedasticity. We thus providethe Fama-MacBeth t-values as well as the Shanken t-values. Cochrane (2001) provides a convenient GMM approachfor calculating the t-statistics, which we apply here with five lags (see John Cochrane’s website for details). These GMMt-statistics impove on the Shanken adjusted t-values in the presence of heteroskedastic or autocorrelated errors.

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Equations (34) and (35) are our formulation of the conditional versions of the Money CAPM and

the Money C-CAPM. In the following we focus on estimating stochastic versions of equations (29),

(30), (34), and (35) and comparing the results to alternative asset pricing models.

2. Empirical Results and Comparison to Alternative Asset Pricing Models

In this section we discuss our methodology, the data, the results for both unconditional and

conditional models, and alternative evaluation methods.

(a) Methodology

We employ the Fama-MacBeth (1973) two-pass “beta” approach to estimate and evaluate

the various asset pricing models. Under the assumption of beta stationarity it is optimal to estimate

betas using the full sample. In this case, Cochrane (2001) shows that coefficient estimates from

running a cross-sectional second pass are identical to those from averaging the coefficient estimates

of the cross-sectional regressions for each time period. We use the latter approach but making

Shanken (1992) and GMM corrections to the t-statistics to account for measurement error in the

generated regressors (the betas) and to provide results robust to heteroskedasticity and serial

correlation in the errors.11

The beta approach under beta stationarity is also equivalent to a stochastic discount factor

approach with identity weighting matrix in the GMM estimation. We follow Lettau and Ludvigson

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in not conducting second-stage GMM estimation with an optimal weighting matrix. The reasons

are that estimation of the weights is poor when the time series is short relative to the cross section.

This is the case here since we have quarterly data while working with 25 test assets. Further, since

the 25 Fama-French portfolios are chosen as test assets because of their economic relevance,

focusing on extreme positions on these assets, as the optimal weighting is likely to do, may not be

desirable. Lastly, as the optimal weights on the different test assets differ from model to model,

comparison across models is more difficult. We also consider the Hansen-Jagannathan weights for

model evaluation (although for our purposes these share some of the shortcomings of the optimal

weights) because they are constant across the models and therefore allow comparison. For further

discussion on these issues see Lettau and Ludvigson (2001b, p. 1276) and Cochrane (2001, Ch. 11).

The beta approach allows evaluation of the results relying on two general criteria: First, the

significance of the coefficient estimates (the risk premia) relative to their theoretical values. The

theory implies (a) an intercept of zero. As our derivations show, this holds even if the factors are

not returns. Additionally, the theory may suggest (b) the sign of a risk premium or the difference

between the actual value of the risk premium and the average return on the factor (if the factor is a

return). In a one-factor model such as the CAPM, implications (a) and (b) are perfectly correlated

in that a deviation from (a) implies a deviation from (b) and vice versa. Since, most of the signs of

the risk premia are ambiguous in our model, we focus on (a), checking the intercept. Note that this

requires that we consider returns net of the return on a risk free asset. In this we differ from Lettau

and Ludvigson (2001b) since they assume that no risk free asset exists.

The second evaluation criterion deals with the “fit” of the model in explaining the returns

of the set of test assets. The theory implies that the stochastic discount factor should price all assets.

12 Monthly consumption growth data are available and used by Hodrick and Zhang (2001). However, thesedata are unreliable as a measure of the stream of consumption services. In particular, they exhibit significantmeasurement error as evidenced by a significant first-order correlation coefficient of -0.35. Accordingly, we follow theprevious literature (e.g., Lettau and Ludvigson, 2001a, 2001b) in employing quarterly consumption data.

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In principle, considering equations (29) and (30) for instance, the average returns should be

explained perfectly by the factor sensitivities. Thus, (a) a higher R-square suggests a better model.

Further, (b) additional factors should have no significant impact in explaining the cross-sectional

returns.

Accordingly, in evaluating our models and comparing against the standard alternative models

we focus on the significance of the intercept, Jensen’s alpha; the adjusted R-squared measure of the

goodness of fit as well as the average pricing error; and the influence of additional factors.

(b) The Data

As the test assets we employ the time series of the returns on the 25 portfolios sorted by size

and book-to-market ratio which are provided by Fama and French (available from Kenneth French’s

website). These are the assets that are most commonly used to test asset pricing models and present

the greatest challenge to asset pricing theory. Since we consider excess returns, we cannot use the

risk free asset as a 26th test asset. The three month t-bill rate is used as the risk free asset. Nominal

money growth is based on the time series of M2 and is deflated by the Consumer Price Index,

consumption is proxied by quarterly non-durables consumption. The term premium is constructed

as the difference between ten-year government bonds and the three-month T-bill rate. These series

are available from the Federal Reserve Bank of St. Louis. Market returns are the value-weighted

returns index from CRSP. Due to the quarterly availability of the consumption data, all series are

compounded from monthly to quarterly.12 The availability of M2 data further limits our sample to

13 Other data used for evaluating competing models are the cay ratio from Sydney Ludvigson’s web site, thebook-to-market and size variables (average of logs over time) from Kenneth French’s website, the investment growthdata (non residential and residential) from the St. Louis Fed website, the dividend yield from CRSP.

14 Where there are definite predictions, the signs of the risk premia are consistent with the theory. First, asshown in Table 1, the risk premia for the money factor are not significantly different in the two models; and, second, themarket return risk premium is positive, although not significant.

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start in 1959Q1. The sample ends with 2003Q1.13

(c) Results for Unconditional Model Specifications

Table 1 provides the results for our unconditional models: (1) the Money CAPM and (2) the

Money C-CAPM as well as unconditional alternatives: (3) the CAPM, (4) the C-CAPM, (5) the

Fama-French (1996) three factor model, and (6) Cochrane’s (1996) investment-based model.

The Money CAPM yields a good fit with adjusted R-squared of 65%, positive risk premia

for market return and the real money growth rate – with the latter being strongly significant even

using the Shanken t-statistic. The intercept implies a risk-adjusted return – Jensen’s alpha – that is

insignificant and equals 0.90 % per quarter. The Money C-CAPM yields an R-squared of 60% and

an insignificant intercept of 1.07 % per quarter. Thus, results for both versions of the model are

similar as would be expected (given the model which implies that both equations 29 and 30 are

correct) if consumption growth is measured about as accurately as wealth growth/the market

return.14

As is well known the regular CAPM and C-CAPM both perform poorly with adjusted R-

squared around 6%, a significant intercept implying a 3% risk-adjusted return per quarter, and wrong

signs for the factors. The adjusted R-squared for Cochrane’s investment-based model is 21% with

a (statistically insignificant) risk-adjusted return of 1.92% per quarter. The Fama-French three

15 Since the previous literature (e.g., Chen, Roll, and Ross, 1986, and Flannery and Protopapadakis, 2002) hasillustrated the importance of a variety of macro variables, we also considered various macro variables as factors suchas real GNP, the risk free rate, inflation, and nominal money growth. Adding any of these factors to consumption growthor the market risk premium yields results that are significantly inferior to our benchmark results, with R-squared valuesvarying from 15% to 48%. When we add both nominal money growth and inflation separately to the consumptiongrowth or the market risk premium model, the risk premia on these factors are of opposite sign and similar in absolutevalue, as predicted by our model, but a formal test narrowly rejects the restriction that the absolute values are equal (p-value of 0.04 in both cases). These results are available from the authors.

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factor model has adjusted R-squared of 75% but has a risk-adjusted return of 3.17% per quarter

which is statistically significant.15

Our Money CAPM and Money C-CAPM both seem to perform quite well unconditionally;

we do not observe the puzzling performance that the CAPM and C-CAPM present. Figure 1

illustrates the fit for each of the six unconditional models by showing the deviations of each test

asset’s return from that predicted by the model. For our models, the pricing errors are largest for

the “11” and “41” portfolios (smallest size quintile and smallest book-to-market ratio, and smallest

size quintile and 4th smallest book-to-market ratio, respectively) which is similar for the Fama-

French three factor model.

(d) Results for Conditional Model Specifications

Table 2 provides the results for six conditional models: (1) the Money CAPM and (2) the

Money C-CAPM with the term premium as the conditioning factor, (3) the C-CAPM with the term

premium as the conditioning factor, (4) the Market Cay and (5) the Consumption Cay models of

Lettau and Ludvigson (2001a, 2001b) , and (6) the Habit-Based Consumption CAPM of Campbell

and Cochrane (1999) and Li (2001).

Each of the conditional models have insignificant intercepts. The conditional Money CAPM

and Money C-CAPM formulations generate adjusted R-squared values of 75% and 70%,

16 Although Table 3 does not display the results for all the models, each of the 12 models that we consider isrejected based on either the size or the book-to-market variable.

24

respectively. The C-CAPM using the term premium as a conditioning factor and the Habit-Based

Consumption model, which to our best knowledge has not been subjected previously to cross-

sectional testing, each yield an R-squared of 62%. Both Cay models have adjusted R-squared of

around 54%. These results are depicted in Figure 2.

(e) Further Evaluation

Size and book-to-market variables

In order to test for irrelevant factors, we include the size and book-to-market variables as

additional “factors” in the second pass as suggested by Jagannathan and Wang (1998). Table 3,

panel A provides the results for the size factor and panel B provides the results for the book-to-

market factor added individually to our four specifications, the two Cay models, and the Fama-

French three-factor model. Each of the models is rejected based on one or both of these added

variables.16 This includes also the Fama-French three factor model. The blanket rejection of all

models along this dimension suggests a further search for a specification that more fully absorbs the

size and value effects, and implies that our money-modified CAPM is not the full story (as, in the

end, no model can ever be). In particular, the fact that our real money factor and the Fama-French

factors independently add explanatory power suggests that the view of the value and size factors as

proxies for liquidity risk may be only a part of the explanation for the importance of these factors.

Beta-approach pricing errors

Table 4 presents the squared pricing errors for the book-to-market and size quintiles for all

25

models. Ranking of each model by its average pricing error is equivalent to that implied by ranking

the adjusted R-squared values as pointed out by Lettau and Ludvigson (2001b). Evaluation based

on the P2-statistic is somewhat different since besides the pricing errors both the covariance matrix

of the factors and of the pricing errors are involved. Cochrane’s investment growth model

outperforms all other unconditional models from this perspective. However, none of our four

models can be rejected based on this test.

GMM distance tests

An alternative model evaluation method measures the pricing errors when the expected

returns are based on each model’s stochastic discount factor (the pricing kernel). The distance

between each model’s stochastic discount factor and a true stochastic discount factor is given by the

Hansen-Jagannathan (HJ) distance. This measure is obtained as the square root of the minimized

Generalized Method of Moments (GMM) objective when we use the second moment matrix of asset

returns to weight the moment conditions; it is appropriate for comparing models as the weights do

not depend on model characteristics. The HJ-distance is reported in Table 5 for each of the 12

models together with p-values for the null-hypothesis that the distance is zero. None of these

models survive the HJ test. Ahn and Gadarowski (2004) point out however that the HJ distance

causes the correct model to be rejected too often for commonly used sample sizes.

As recommended by Altonji and Segal (1996), Cochrane (1996, 2001), and Lettau and

Ludvigson (2001b), we compute an alternative distance measure using the identity matrix instead

of the HJ weighting matrix. In this case, the distance is given by the square root of the minimized

GMM objective using the identity matrix as the weighting matrix. The distance results for the

17 To check for consistency we also obtain via one-stage GMM the stochastic discount factor coefficients foreach unconditional model and calculate the implied risk premia and p-values via the delta method (following Cochrane,2001, and Hodrick and Zhang, 2001). The results are very close to those in Table 1 and are not displayed.

26

identity weighting matrix are also reported in Table 5. The six models with the smallest distance

are our four models plus the Fama-French model and the Campbell-Cochrane Habit C-CAPM.

While the distance measures are not significantly different from zero for all of the conditional

models as well as our two unconditional models, the Fama-French model is rejected based on the

distance with identity weighting matrix. Lettau and Ludvigson (2001b) also report rejecting the

Fama-French model based on this measure and point out that this may be in part because models

with returns as factors have smaller sampling errors in the estimated betas. Note that, as expected

from the equivalence of the Fama MacBeth method and first-stage GMM method with identity

matrix weights, the p-values reported here are consistent with the joint test of pricing errors results

reported in Table 4.17

Other conditioning variables

We do not consider other conditioning variables jointly with the term premium, because each

conditioning variable adds additional factors equal to the number of unconditional factors to the

model. To check robustness we instead try alternative proxies for changes in investment opportunity

sets individually: the risk free rate (suggested originally by Merton, 1970) and the dividend yield

(typically used as a conditioning variable in the asset pricing context – see for instance Cochrane,

1996). Table 6 shows that the results for both our MCAPM and the MC-CAPM are at least as good

(with R-square ranging from 78% to 89%) when the risk free rate or the divided yield is used as the

conditioning variable instead of the term premium.

27

3. Conclusion

The promise of the consumption-based asset pricing model of Breeden (1979) that a

consumption growth factor alone is adequate for pricing any asset is based on the fact that the

consumption level is a sufficient statistic for capturing marginal utility of the representative

consumer, or any consumer in complete financial markets. However, monetary theory suggests that

a further factor is relevant. Specifically, liquidity as determined by the real quantity of money

balances, affects the cost of acquiring consumption goods and thus affects the transactions-cost-

adjusted marginal utility of consumption. A generalized version of Stein’s Lemma permits the

derivation of a two-factor model, the Money CAPM, or, using the envelope condition, an alternative

two-factor model, the Money C-CAPM. This two-factor model, in both versions, is subjected to the

data.

Both model versions perform about equally well and outperform existing models in

explaining the cross-section of the 25 portfolio returns of firms sorted by size and book-to-market

ratio. Conditional versions of these models perform even better, explaining as much of the cross-

sectional variation as even the a-theoretical Fama-French three factor model. Thus, a logical

extension to Breeden’s model appears to provide results that avoid most of the puzzles of the

original C-CAPM. That such an extension was not attempted previously may be somewhat

surprising but is at least in part due to the difficulty of dealing with the cash-in-advance lag and

conditioning information in a continuous time framework. As such, the extension of Stein’s lemma

derived here was an important ingredient in obtaining a theoretical solution.

In spite of the logic of the theory and the robustly strong explanatory power of the different

versions of the model, a specification test that includes size or book-to-market ratios in addition to

28

our theoretical factors suggests shortcomings in the models, as well as in all other models considered

here – including the Fama-French three-factor model itself. Additional research is called for to

propose an alternative specification that better clarifies the role of the size and book-to-market

variables.

29

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32

Appendix: A MULTIVARIATE EXTENSION OF STEIN’S LEMMA

In the finance literature Stein’s lemma is taken to be the statement that

when X and Y are bivariate normal. See for instance Huang andXYX XhEXhYCov σ])([)](,[ =

Litzenberger (1988, pp. 101, 116) and Cochrane (2001, pp. 164-165). The lemma was first used and

proven indirectly in the finance literature by Rubinstein (1973) using exact Taylor expansions.

Ingersoll (1987, pp. 13-14) proves this result directly without naming it Stein’s lemma. The result

is generally attributed to Stein (1973, 1981).

Below we prove a multivariate generalization that to our knowledge has not been proven or

used elsewhere: , where X represents a vector of multivariateYXX

n

i

i

ihEhYCov σ])([)](,[

1XX ∑

=

=

random variables. This form of Stein’s lemma would appear to be of great use in multi-factor asset

pricing models. We first state the existing result:

Stein’s Lemma [Stein (1981), Lemma 2]. Assume: (1) a random variable Y and

random n-vector such that Y and are independent multivariate normal, and (2)ε ε

the partial derivative exists and has . Then: ),( εYgY ∞<|),(| εYgE Y

. (A1)2)(]),([)],(,[ YY YgEYgYCov σεε =

Proof: See Stein (1981), or the following proof provided for completeness. By definition:

(A2)eeeεε

dydyfyygYgYCovY

YY∫∫ −= ),()(),()],(,[ εµ

33

where y and e are realizations of the random variables Y and g with joined multivariate

density . Note that superscripts indicate particular random variables and subscripts indicate),( eyf Yε

partial derivatives. Using independence and the fact that for the)(])/()[()( 2 xfxxf XXXXX σµ−−=

normal distribution , we obtain

(A3).)()(),()()],(,[ 2 eeeε ε

ε

dfydyfygYgYCovY

YY

Y ∫∫−= σ

Integration by parts of the inner integral yields:

, (A4)eeeeε ε

ε

dfyfygydyfygYgYCov Y

Y

YY

Y )(|)](),([)(),()()],(,[ 2 ∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∞

∞−σ

where vanishes for any e if . This is guaranteed∞∞−|)()(),( ee εfyfyg Y )]/[exp()(),( 22

yyofyg σ=ee ε

if .∞<|),(| εYgE Y

Equation (A4) then becomes

, (A5)eeeε ε

ε

ddyfyfygYgYCovY

YY

Y )()(),()()],(,[ 2 ∫ ∫= σ

which is Stein’s lemma, equation (A1). G

Next extend the lemma to allow for dependence among the random variables.

34

Stein’s Extended Lemma. Assume: (1) a random variable Y and random n-vector X

such that Y and are multivariate normal, and (2) the n-vector of partialX )(XXh

derivatives exists with . Then:∞<||)(|| XXhE

. (A6)XYhEhYCov σ])([)](,[ ′= XX X

Proof: It is always possible to find coefficients b such that:

, with and . (A7)εbX += Y 0),( =εYCov 2)/(),( YYCov σXb =

Then:

(A8),),()()()](,[ xxxXX

dydyfyhhYCovY

Y∫∫ −= µ

(A9),),()()()](,[ εebebXε

dydyyfyyhhYCovY

Y∫∫ +−+= µ

Set:

, so that . (A10))(),( εbε +≡ YhYg bεbε X )(),( ′+= YhYgY

Using equation (A10) we can apply Stein’s lemma to equation (A9) to yield

, (A11) YYY hEYgEhYCov X

X XεX σσ ])([)(]),([)](,[ 2 ′==

where the second equality follows from equations (A7) and (A10) . To complete the proof note that

implies and so that ,∞<||)(|| XXhE ∞<|)(| XXhE ∞<|)'(| bXXhE ∞<|),(| εYgE Y

which was needed to apply Stein’s lemma to (A9). G

35

Table 1. Fama-Macbeth estimation for 6 unconditional linear factor models. The factors in eachmodel (1 to 6: MMKT, MCCAPM, CCAPM, FF3, MKT, COCH) are market return and moneysupply growth (Mkt, ∆Μ2); consumption growth and money supply growth (∆C, ∆Μ2);consumption growth (∆C); the Fama-French 3 factors (Mkt, Smb, Hml); Market Return (Mkt); andthe Cochrane investment growth measures for nonresidential and residential sectors (∆Nriq, ∆Riq).For each model, we report the risk premiums, their t-statistic, Shanken’s (1992) adjusted t-statistic,GMM t-statistics (based on five lags) and the Adjusted R-square (Adj. R2). Data are from 1959Q1to 2003Q1.

Model-1 Const. Mkt ∆Μ2 Adj. R2Risk Premium % 0.896 0.352 1.256 0.653

t-statistics 0.815 0.284 3.079Shanken- t 0.553 0.209 2.117

GMM-t 0.574 0.239 2.249

Model-2 Const. ∆C ∆Μ2 Adj. R2Risk Premium % 1.072 -0.255 0.916 0.597

t-statistics 0.946 -0.841 2.084Shanken- t 0.656 -0.588 1.461

GMM-t 0.607 -0.607 1.727

Model-3 Const. ∆C Adj. R2Risk Premium % 2.961 -0.231 0.059

t-statistics 3.134 -0.759Shanken- t 2.976 -0.722

GMM-t 2.496 -0.729

Model-4 Const. Mkt Smb Hml Adj. R2Risk Premium % 3.168 -1.840 0.544 1.379 0.745

t-statistics 2.504 -1.286 1.168 2.773Shanken- t 2.376 -1.233 1.165 2.751

Model-5 Const. Mkt Adj. R2Risk Premium % 2.914 -0.821 0.066

t-statistics 3.126 -0.728 Shanken- t 3.112 -0.726

Model-6 Const. ∆Nriq ∆Riq Adj. R2Risk Premium % 1.920 0.980 2.662 0.211

t-statistics 2.220 0.731 3.033Shanken- t 1.437 0.483 1.986

GMM-t 1.577 0.590 1.728

36

Table 2. Fama-Macbeth estimation for 6 conditional linear factor models. The factors in each model(7 to 12) are MMKT conditioned on the lagged term premium; the MC-CAPM conditioned on thelagged term premium; the C-CAPM conditioned on the lagged term premium; the MKT conditionedon CAY; the CCAPM conditioned on CAY; the CCAPM conditioned on lagged consumption. Foreach model, we report the risk premiums, their t-statistic, Shanken’s (1992) adjusted t-statistic,GMM t-statistics (based on five lags) and Adj. R2. See Table 1 for descriptions of the model labels.

Model-7 Const. TP Mkt )M2 TP.Mkt TP.)M2 Adj. R2

Risk Premium % -1.314 0.374 2.420.49 1.496 0.013 -0.001 0.747

t-statistics -1.027 4.080 1.732 3.418 1.638 -0.304

Shanken-t -0.532 2.163 0.978 1.798 0.872 -0.162

GMM-t -0.525 2.668 1.088 1.740 0.737 -0.156

Model-8 Const. TP )C )92 TP.)C IP.)92 Adj. R2

Risk Premium % - -0.882 0.346 0.193 1.281 0.000 -0.001 0.702

t-statistics -0.901 3.324 0.898 2.832 0.145 -0.946

Shanken- t -0.499 1.871 0.508 1.589 0.082 -0.536

GMM-t -0.390 1.948 0.520 1.499 0.080 -0.463

Model-9 Const. TP ∆C TP.∆C Adj. R2

Risk Premium % 1.977 0.630 -0.272 0.000 0.623

t-statistics 1.938 2.504 -0.927 0.228

Shanken- t 1.280 1.676 -0.618 0.153

GMM-t 1.226 1.712 -0.630 0.159

Model-10 Const. Cay Mkt Cay.Mkt Adj. R2

Risk Premium % 0.930 -0.638 0.367 0.056 0.531

t -statistics 0.919 -1.891 0.309 2.220

Shanken-t 0.704 -1.474 0.253 1.746

GMM-t 0.671 -1.586 0.296 1.490

Model-11 Const. Cay ∆C Cay.∆C Adj. R2

Risk Premium % 1.766 -0.323 0.246 0.008 0.535

t -statistics 1.727 -0.900 0.806 2.413

Shanken-t 1.075 -0.573 0.506 1.520

GMM-t 1.269 -0.506 0.512 1.124

Model-12 Const. ∆C(t−1) ∆C ∆C(t−1).∆C Adj. R2

Risk Premium % 1.977 0.630 -0.272 0.000 0.623

t-statistics 1.938 2.504 -0.927 0.228

Shanken-t 1.280 1.676 -0.618 0.153

GMM-t 1.226 1.712 -0.630 0.159

37

Table 3. Model misspecification test incorporating the average log size or average log book-to-market ratio. The models considered are the MMKT and MC-CAPM, the MMKT and MCCAPMconditioned on the term premium and the CAY and FF3 models. See Table 1 and Table 2 for factordetails. Panel A provides the results for the size variable and Panel B provides the results for thebook-to-market ratio.

Panel A: Size

Const. Mkt ∆Μ2 Size Adj. R2Risk Premium % 6.727 -2.710 0.067 -0.411 0.758

t-statistics 4.149 -2.362 0.234 -2.918

Const. ∆C ∆Μ2 Size Adj. R2Risk Premium % 3.522 -0.464 0.342 -0.173 0.622

t-statistics 2.421 -2.137 1.097 -1.147

Const. TP Mkt ∆Μ2 TP.Mkt ΤP.∆Μ2 Size Adj. R2Risk Premium % 3.927 0.229 -0.501 0.395 0.010 -0.001 -0.348 0.814

t-statistics 2.280 2.537 -0.400 1.217 1.295 -0.841 -2.475

Const. TP ∆C ∆Μ2 TP.∆C ΤP.∆Μ2 Size Adj. R2Risk Premium % 1.786 0.282 0.040 0.596 0.001 -0.002 -0.210 0.731

t-statistics 1.055 2.559 0.189 1.651 0.819 -1.585 -1.476

Const. Cay Mkt Cay.Mkt Size Adj. R2Risk Premium % 6.518 -0.295 -2.620 0.014 -0.399 0.766

t -statistics 4.857 -0.990 -2.491 0.575 -3.214

Const. Cay ∆C Cay.∆C Size Adj. R2Risk Premium % 4.995 -0.608 -0.220 0.003 -0.260 0.700

t -statistics 4.143 -1.804 -1.147 1.170 -2.051

Const. Mkt Smb Hml Size Adj. R2Risk Premium % 6.454 0.365 -2.163 0.923 -0.848 0.809

t-statistics 3.991 0.220 -2.033 1.850 -3.002

38

Table 3 (continued)

Panel B: Book-to-market ratio

Const. Mkt ∆Μ2 BM Adj. R2Risk Premium % 0.753 0.839 0.690 0.578 0.756

t-statistics 0.679 0.656 2.154 2.723

Const. ∆C ∆Μ2 BM Adj. R2Risk Premium % 0.614 0.143 0.561 0.695 0.752

t-statistics 0.553 0.529 1.283 3.049

Const. TP Mkt ∆Μ2 TP.Mkt ΤP.∆Μ2 BM Adj. R2Risk Premium % -0.033 0.262 1.596 0.748 -0.008 -0.002 0.694 0.827

t-statistics -0.028 2.968 1.202 1.969 -1.182 -1.338 3.283

Const. TP ∆C ∆Μ2 TP.∆C ΤP.∆Μ2 BM Adj. R2Risk Premium % 0.388 0.176 0.073 0.450 -0.002 -0.002 0.789 0.846

t-statistics 0.441 1.796 0.352 1.209 -2.292 -1.374 3.574

Const. Cay Mkt Cay.Mkt BM Adj. R2Risk Premium % 1.035 0.023 0.915 0.005 0.887 0.683

t -statistics 1.035 0.080 0.736 0.230 4.096

Const. Cay ∆C Cay.∆C BM Adj. R2Risk Premium % 0.875 -0.107 0.336 -0.003 1.035 0.704

t -statistics 0.806 -0.297 1.067 -1.057 3.857

Const. Mkt Smb Hml BM Adj. R2Risk Premium % 3.879 -2.183 0.425 0.339 0.567 0.769

t-statistics 2.961 -1.511 0.920 0.440 1.901

39

Table 4. Average squared pricing errors (in percentages) for size and book-to-market quintiles. S1refers to the smallest size group and B1 refers to the smallest book-to-market ratio group. Thedefinitions for S1 and B1 extend to S2-S5 and B2-B5. The last 3 rows are the average squaredpricing error, the joint test statistic for significance of the pricing errors and its p-value for the 25size and book-to-market sorted portfolios. The pricing errors are provided for each of the sixunconditional and six conditional models. See Tables 1 and 2 for model details.

MMKT MCCAPM CCAPM FF3 MKT COCHS1 0.591 0.618 0.906 0.490 0.922 0.925S2 0.262 0.357 0.621 0.207 0.602 0.543S3 0.377 0.450 0.470 0.165 0.425 0.396S4 0.325 0.267 0.383 0.339 0.380 0.358S5 0.356 0.369 0.757 0.395 0.771 0.603B1 0.560 0.573 0.652 0.556 0.666 0.955B2 0.262 0.315 0.546 0.187 0.530 0.246B3 0.262 0.235 0.454 0.203 0.450 0.298B4 0.320 0.304 0.680 0.314 0.690 0.606B5 0.489 0.585 0.869 0.311 0.854 0.608

Average 0.398 0.429 0.655 0.341 0.653 0.600Chi-2 31.866 33.804 64.269 54.934 69.498 28.076

p-value 0.080 0.051 0.000 0.000 0.000 0.173

MMKTTP MCCAPMTP CCAPMTP MKTCAY CCAPMCAY CCAPMCS1 0.449 0.398 0.452 0.765 0.758 0.452S2 0.179 0.277 0.328 0.440 0.419 0.328S3 0.295 0.449 0.358 0.159 0.285 0.358S4 0.417 0.396 0.450 0.360 0.298 0.450S5 0.289 0.295 0.467 0.368 0.374 0.467B1 0.470 0.400 0.398 0.570 0.519 0.398B2 0.312 0.354 0.398 0.326 0.307 0.398B3 0.259 0.296 0.259 0.241 0.337 0.259B4 0.257 0.218 0.479 0.561 0.633 0.479B5 0.355 0.510 0.498 0.515 0.427 0.498

Average 0.340 0.369 0.415 0.462 0.460 0.415Chi-2 16.589 18.459 26.148 39.106 27.229 26.148

p-value 0.618 0.492 0.201 0.010 0.163 0.201

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Table 5. The distance test for linear factor pricing models using the HJ weighting matrix (Hansenand Jagannathan, 1997) and Identity matrix (Lettau and Ludvigson, 2001b). The test assets are the25 portfolios sorted by size and book-to-market ratio. The distances and their statistical significanceare provided for each of the six unconditional and six conditional models. See Tables 1 and 2 formodel details. The p-values for the weighted P2 statistic of Jagannathan and Wang (1996) to testfor the significance of the distance measure (for both weighting matrices) is based on 10,000simulations.

MMKT MCCAPM CCAPM FF3 MKT COCHHJ-Matrix 0.6140 0.6222 0.6248 0.5697 0.6167 0.6052

p-value 0.0002 0.0001 0.0003 0.0031 0.0005 0.0043

Identity Matrix 0.0197 0.0212 0.0318 0.0165 0.0317 0.0294p-value 0.1151 0.0903 0.0041 0.0049 0.0021 0.1498

MMKTTP MCCAPMTP CCAPMTP MKTCAY CCAPMCAY CCAPMCHJ-Matrix 0.5846 0.5776 0.5775 0.5988 0.6211 0.5775

p-value 0.0062 0.0052 0.0124 0.0013 0.0003 0.0124

Identity Matrix 0.0172 0.0186 0.0203 0.0229 0.0226 0.0203p-value 0.3816 0.2985 0.1549 0.0089 0.1804 0.1549

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Table 6. Fama-MacBeth regressions using different conditioning variables for the MCAPM and theMC-CAPM. DP is the lagged dividend-price ratio and Rf is the lagged risk free rate. Note that theconditioning variables are demeaned. See Table 1 for details on the models and variables.

Const. DP Mkt ∆Μ2 DP.Mkt DP.∆Μ2 Adj. R2Risk Premium % 1.014 0.050 0.301 0.500 0.019 0.003 0.780

t-statistics 0.900 0.629 0.246 1.333 2.294 2.106Shanken-t 0.608 0.433 0.181 0.916 1.590 1.445GMM-t 0.561 0.352 0.183 0.947 1.406 1.156

Const. DP ∆C ∆Μ2 DP.∆C DP.∆Μ2 Adj. R2Risk Premium % 0.925 -0.055 -0.159 0.095 0.003 0.004 0.890

t-statistics 1.057 -0.792 -0.589 0.347 3.250 2.933Shanken- t 0.562 -0.434 -0.317 0.192 1.761 1.590

GMM-t 0.651 -0.380 -0.436 0.221 1.866 1.489

Const. Rf Mkt ∆Μ2 Rf.Mkt Rf.∆Μ2 Adj. R2Risk Premium % -0.211 -0.414 1.214 1.301 -0.002 0.009 0.782

t-statistics -0.190 -2.207 0.996 3.060 -0.169 4.440Shanken- t -0.106 -1.265 0.620 1.737 -0.098 2.575

GMM-t -0.149 -1.910 1.004 2.069 -0.117 1.894

Const. Rf ∆C ∆Μ2 Rf.∆C Rf.∆Μ2 Adj. R2Risk Premium % 0.199 -0.328 -0.214 0.975 0.002 0.011 0.785

t-statistics 0.192 -1.982 -0.812 2.265 1.199 4.995Shanken- t 0.097 -1.036 -0.415 1.161 0.614 2.603

GMM-t 0.122 -1.377 -0.494 1.524 0.768 1.834

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Figure 1. Actual return and predicted return for 6 unconditional linear factor models. Models 1 to6 are MMKT, MCCAPM, CCAPM, FF3, MKT and COCH The two-digit numbers denote the Fama-French 25 portfolios. The first digit refers to the size quintile and the second digit refers to the book-to-market quintile. See Table 1 for factor details.

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Figure 2. Actual return and predicted return for 6 conditional linear factor models. Models areMMKT-TP, MCCAPM-TP, CCAPM_TP, MKT-CAY, CCAPM-CAY and CCAPM-C The two-digitnumbers denote the Fama-French 25 portfolios. The first digit refers to the size quintile and thesecond digit refers to the book-to-market quintile. See Table 2 for factor details.