multifactor models and their consistency with the icapm · 2018. 8. 6. · icapm and nd that most...

Multifactor models and their consistency with the ICAPM

Paulo Maio1 Pedro Santa-Clara2

This version: February 20123

1Hanken School of Economics. E-mail: [email protected] School of Business and Economics, NBER, and CEPR. E-mail: [email protected] thank an anonymous referee, Antonio Antunes, Matti Keloharju, Timo Korkeamaki, Anders

Loflund, Peter Nyberg, Bill Schwert (the editor), and seminar participants at the Helsinki FinanceSeminar and the Bank of Portugal for helpful comments. We are grateful to Kenneth French, AmitGoyal, Lubos Pastor, and Robert Shiller for making data available on their webpages. A previousversion of this paper was entitled “The time-series and cross-sectional consistency of the ICAPM.” Maioacknowledges financial support from the Hanken Foundation. Santa-Clara is supported by a grant fromthe Fundacao para a Ciencia e Tecnologia (PTDC/EGE-GES/101414/2008). All errors are ours.

Abstract

Can any multifactor model be interpreted as a variant of the ICAPM? The ICAPM places restrictions on

time series and cross-sectional behavior of state variables and factors. If a state variable forecasts positive

(negative) changes in investment opportunities in time-series regressions, its innovation should earn a

positive (negative) risk price in the cross-sectional test of the respective multifactor model. Second, the

market (covariance) price of risk estimated from the cross-sectional tests must be economically plausible

as an estimate of the coefficient of relative risk aversion (RRA). We apply our ICAPM criteria to eight

popular multifactor models tested over 25 portfolios sorted on size and book-to-market (SBM25), and

25 portfolios sorted on size and momentum (SM25). Our results show that most factor models do not

satisfy the ICAPM restrictions. Specifically, the “hedging” risk prices have the wrong sign and the

estimates of RRA are not economically plausible. Overall, the Fama and French (1993), and Carhart

(1997) models perform the best in consistently meeting the ICAPM restrictions. The remaining models,

which represent some of the most relevant examples presented in the empirical asset pricing literature,

can still empirically explain the size, value, and momentum anomalies, but they are generally inconsistent

with the ICAPM.

Keywords: Asset pricing models; Intertemporal CAPM; Predictability of returns; Linear

multifactor models; Cross-section of stock returns; Size and value anomalies; Momentum; Time-

varying investment opportunities; Fama-French factors

JEL classification: G12; G14.

1 Introduction

Explaining the dispersion in average excess returns in the cross-section of stocks has been

one of the most important topics in the asset pricing literature. The inability of the Sharpe

(1964)–Lintner (1965) CAPM to price portfolios sorted on size, book-to-market, momentum,

and other stock characteristics has led to so-called size, value, and momentum anomalies [Fama

and French (1992, 1993, 1996), among others]. In response, several multifactor models seeking

to explain these various anomalies have emerged in the literature. Typically, these models

include factors in addition to the market return whose betas help match the dispersion in excess

portfolio returns observed in the cross-section. Many of these multifactor models have been

justified as empirical applications of the Intertemporal CAPM (ICAPM, Merton, 1973), leading

Fama (1991) to interpret the ICAPM as a “fishing license” to the extent that authors claim it

provides a theoretical background for relatively ad hoc risk factors in their models. However,

Cochrane (2005, Chapter 9) notes that although the ICAPM does not directly identify the “state

variables” underlying the risk factors, there are some restrictions that these state variables must

satisfy. According to Merton, the state variables relate to changes in the investment opportunity

set, which implies that they should forecast the distribution of future aggregate stock returns.

Moreover, the innovations in these state variables should be priced factors in the cross-section.

We examine the restrictions associated with the ICAPM that prevent it from being a “fishing

license” for any multifactor model that seeks to explain the cross-section of stock returns. We

identify three main conditions that a multifactor model must meet to be justifiable by the

ICAPM and find that most multifactor models in the literature do not satisfy these restrictions.

First, the candidates for ICAPM state variables must forecast the first or second moments

of aggregate stock returns. We assess the forecasting power of each variable by conducting

time-series long-horizon regressions.

Second, if a given state variable forecasts positive expected aggregate returns, its innovation

(the risk factor) should earn a positive risk price in cross-sectional tests, while state variables

that forecast negatively expected aggregate returns should earn a negative risk price. Risk

premiums with opposite signs should accrue to innovations to state variables that forecast

market volatility. Thus, it is not enough that the candidate state variables forecast future

aggregate expected returns or the volatility of returns, the corresponding factors should also

be priced in the cross-section. The intuition for this result is simple. An asset that covaries

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positively with innovations to the state variable also covaries positively with future expected

returns. It does not provide a hedge for reinvestment risk because it offers lower returns when

aggregate returns are expected to be lower. Hence, a risk-averse rational investor will require a

positive risk premium to invest in such an asset, implying a positive price of risk for the factor.

A similar argument applies to assets that covary with innovations to market volatility.

The third restriction associated with the ICAPM is that the market (covariance) price of

risk estimated from the cross-sectional tests must be economically plausible as an estimate of

the coefficient of relative risk aversion (RRA) of the representative investor.

Most of the empirical literature on the ICAPM uses state variables from the predictability

literature (short-term interest rates, bond yields, and aggregate financial ratios) in order to meet

the first ICAPM restriction that the state variables should forecast expected market returns.

Yet authors largely neglect the other constraints of the ICAPM: that the market price of risk

corresponds to the risk aversion of the representative investor and especially that there must

be consistency between the “hedging” factor risk prices and the corresponding slopes from the

predictive regressions.

In Campbell (1996), the risk prices associated with the VAR state variables that forecast

market returns are constrained in the sense that they are linked with the estimated slopes from

the VAR. However, Campbell only tests a specific parametrization with Epstein-Zin preferences

and a VAR to estimate market discount rate news. This paper extends this work, focusing

on whether commonly used empirical factor models satisfy the consistency between time series

slopes and cross-sectional risk prices to be justifiable as ICAPM applications. Our work is also

related to Lewellen, Nagel, and Shanken (2010), and Lewellen and Nagel (2006), who advocate

that cross-sectional tests of asset pricing models in general, and the conditional CAPM in

particular, should impose the models’ theoretical restrictions on the factor risk prices.

We apply our ICAPM criteria to eight multifactor models, tested over 25 portfolios sorted

on size and book-to-market (SBM25) and 25 portfolios sorted on size and momentum (SM25).

We include the market return in the set of testing assets, which enables us to merge the cross-

sectional literature on the ICAPM with the literature on the time-series aggregate risk-return

trade-off. Hence, we have a total of 16 empirical tests in the cross-section: eight models and

two sets of portfolios.

Table 1 summarizes the main results regarding the multifactor models satisfying the ICAPM

criteria. When investment opportunities are driven by changing expected market returns, our

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results show that only two models – the Fama and French (1993) three-factor model tested

over SBM25, and the Carhart (1997) model tested over SBM25 and SM25 – meet the ICAPM

consistency criteria.

When we consider changes in the investment opportunity set driven by the second moment

of aggregate returns, only the Fama and French (1993) model satisfies the ICAPM criteria when

tested with the SBM25 portfolios.

In most other models the “hedging” risk prices have the wrong sign and the estimates of

RRA are not economically plausible. The Koijen, Lustig, and Van Nieuwerburgh (2010) and

Pastor and Stambaugh (2003) models in tests with SBM25 meet the sign restriction on the

hedging risk prices but do not produce a reasonable estimate for the risk aversion coefficient.

The Hahn and Lee (2006) model in the test with SM25 produces a plausible estimate for RRA,

but fails the sign restriction on the risk prices for the state variable factors. These rejections

show that the ICAPM is not really a “fishing license” after all.

Our findings are robust to estimating the multifactor models with an intercept, estimating

each model by second-stage GMM, using alternative measures of the innovations in the state

variables, using alternative test equity portfolios, adding bond risk premia to the menu of test

assets, estimating the models in expected return-beta form, conducting a bootstrap simulation

for the slopes in the predictive regressions, and using alternative proxies for the expected market

return.

Our paper is related to the growing empirical literature on the ICAPM. An incomplete list

of empirical tests of the ICAPM over the cross-section of stock returns includes Shanken (1990),

Brennan, Wang, and Xia (2004), Ang, Hodrick, Xing, and Zhang (2006), Gerard and Wu (2006),

Hahn and Lee (2006), Lo and Wang (2006), Petkova (2006), Bali (2008), Guo and Savickas

(2008), Ozoguz (2009), and Bali and Engle (2010). In related work, Campbell (1993) develops a

theoretical model based on a representative agent with Epstein and Zin (1991) preferences that

leads to two risk factors – the excess market return (as in the standard CAPM) and expectations

about future market returns (discount rate news). Campbell (1996), Chen (2003), Guo (2006a),

Campbell and Vuolteenaho (2004), Chen and Zhao (2009), and Maio (2012b) represent variants

or extensions of the two-factor model developed in Campbell (1993). A related literature focuses

on the time-series aggregate risk-return trade-off. An incomplete list of recent papers includes

Scruggs (1998), Whitelaw (2000), Brandt and Kang (2004), Ghysels, Santa-Clara, and Valkanov

(2005), Guo and Whitelaw (2006), Lundblad (2007), Pastor, Sinha, and Swaminathan (2008),

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Bali, Demirtas, and Levy (2009), and Guo, Savickas, Wang, and Yang (2009).

This paper is organized as follows. In Section 2 we discuss the theoretical restrictions

associated with the ICAPM. In Section 3 we analyze the forecasting power of ICAPM state

variable candidates with regards to the expected market return. In Section 4 we analyze whether

the factor risk prices from cross-sectional asset pricing tests are consistent with the time-series

predictability of market returns. In Section 5 we conduct a sensitivity analysis. In Section 6

we evaluate the predictive ability of the ICAPM state variables with regards to the volatility

of market return. In Section 7, we conduct a Monte Carlo simulation experiment to assess the

plausibility of our results.

[Table 1 about here.]

2 Time-series and cross-sectional implications of the ICAPM

A simplified version of the Merton (1973) Intertemporal CAPM (ICAPM) is based on the

consumption/portfolio choice of a representative investor in continuous time.1 We discuss the

restrictions this model imposes on multifactor asset pricing models.

There are N risky assets, and asset i has an instantaneous rate of return given by

dSi

Si= µi(z, t)dt+ σi(z, t)dξi, i = 1, ..., N, (1)

where Si denotes the price of asset i; dξi is a Wiener process; and the covariance between two

arbitrary risky assets is equal to σijdt.

In this model, investment opportunities are time-varying since both the mean (µi) and

volatility (σi) of asset returns are functions of a single state variable, z, which also evolves as a

diffusion process:

dz = a(z, t)dt+ b(z, t)dζ, (2)

where dζ denotes another Wiener process, and the covariance with the return on risky asset i

is equal to σizdt.2

The N + 1th asset is a risk-free asset with instantaneous rate of return equal to r:

dB

B= rdt. (3)

1For a textbook treatment see Pennacchi (2008), Chapter 13.2The Merton (1973) ICAPM does not directly identify the state variables. The main restriction in this simple

model is that the state variables forecast the first two moments of stock returns.

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To simplify the exposition, we assume that the risk-free rate is constant.

The dynamics of wealth (W ) are given by

dW =N∑

i=1

ωi(µi − r)Wdt+ (rW − C)dt+N∑

i=1

ωiWσidξi, (4)

where ωi denotes the portfolio weight for asset i, and C stands for consumption. The investor

maximizes lifetime utility:

J(W, z, t) = maxC,ωi

Et

[∫ ∞s=t

U(C, s)ds], (5)

subject to the intertemporal budget constraint (4), where J(W, z, t) denotes the value function.

It can be shown that the ICAPM equilibrium relation between expected return and risk is

given by

µi − r = γσim + γzσiz, (6)

where γ ≡ −WJWW (W,z,t)JW (W,z,t) denotes the parameter of relative risk aversion; σim and σiz denote the

covariances between the return on asset i and the market return and state variable, respectively;

and γz denotes the (covariance) risk price associated with the state variable, which is given by

γz ≡ −JWz(W, z, t)JW (W, z, t)

, (7)

where JW (·) denotes the marginal value of wealth; JWW (·) is the growth in the marginal value

of wealth; and JWz(·) represents a second-order cross-derivative relative to wealth and the state

variable.

As in Cochrane (2005), Chapter 9, we can approximate Eq. (6) in discrete time, leading to

the pricing equation:

Et(Ri,t+1)−Rf,t+1 = γ Covt(Ri,t+1, Rm,t+1) + γz Covt(Ri,t+1,∆zt+1), (8)

where Ri,t+1 is the return on asset i between t and t + 1; Rf,t+1 denotes the risk-free rate,

which is known at t; Rm,t+1 is the market return; and ∆zt+1 denotes the innovation or change

in the state variable. In Eq. (8), the novelty relative to the standard CAPM (Sharpe, 1964;

and Lintner, 1965) is the second term on the right-hand side, γz Covt(Ri,t+1,∆zt+1). This

means that if the risk price associated with the state variable is zero, γz = JWz(·) = 0, we

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are back to the standard “static” CAPM. This pricing equation represents the theory behind

many multifactor models in the empirical asset pricing literature, leading Fama (1991) to call

the ICAPM a fishing license. Yet, as should be clear from the derivation of (8), the non-market

factors in such models should proxy for the innovation in some state variable, ∆zt+1, so they

cannot be just anything.

By using the law of iterated expectations, we can obtain the ICAPM in unconditional form:

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1, Rm,t+1) + γz Cov(Ri,t+1,∆zt+1). (9)

In Eq. (9) there are two sources of risk that explain average risk premiums.3 The first is captured

by the “static” market risk premium associated with the CAPM, γ Cov(Ri,t+1, Rm,t+1), which

postulates that an asset that covaries positively with the market return earns a positive risk

premium over the risk-free rate. The intuition is that such an asset does not provide a hedge

against changes in current aggregate wealth, as it pays in good times (periods with high returns

on wealth), so a risk averse investor is willing to hold such an asset only if it offers a premium

over the risk-free rate. The estimate for the relative risk aversion (RRA) coefficient should be

between one and ten (see Mehra and Prescott, 1985, for example).

To understand that the second source of risk in Eq. (9) is captured by the term, γz Cov(Ri,t+1,∆zt+1),

consider a state variable that predicts future market returns. If the risk price for intertemporal

risk (γz) is positive, an asset that covaries positively with changes in the state variable (and

is thus positively correlated with future market expected return) earns a risk premium. The

intuition is that the asset does not provide a hedge against future negative shocks in the returns

of aggregate wealth (reinvestment risk), as it offers low returns when future aggregate returns

are also expected to be low. Therefore, a rational investor is willing to hold such asset only if

it offers an expected return in excess of the risk-free rate.

This central economic intuition of the ICAPM puts a constraint on the sign of γz, when the

model is forced to price a set of assets in the cross-section. Specifically, if the state variable is

3If the factor risk prices were time-varying (as a function of state variables), there would be additional (scaled)risk factors in the pricing equation from the interaction between the original factor and the state variables (seeJagannathan and Wang, 1996; Lettau and Ludvigson, 2001; and Cochrane, 2005, Chapter 8, among others).In our case, and following most of the empirical literature on the ICAPM, we assume that the risk prices areconstant through time. In the next sections, we test only the unconditional versions of the empirical multifactormodels that are candidates to ICAPM applications.

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positively correlated with future aggregate returns,

Covt(Rm,t+2, zt+1) = Covt[Et+1(Rm,t+2), zt+1] = Covt[Et+1(Rm,t+2),∆zt+1] > 0, (10)

then the intertemporal risk price must be positive.

To see this point, assume without loss of generality that the return on asset i is positively

correlated with the (innovation in the) state variable:

Covt(Ri,t+1, zt+1) = Covt(Ri,t+1,∆zt+1) > 0. (11)

These two conditions imply that the return on asset i is also positively correlated with the

future expected market return:

Covt(Ri,t+1, Rm,t+2) = Covt[Ri,t+1,Et+1(Rm,t+2)] > 0. (12)

This last condition has an important economic content: This asset does not provide a hedge for

reinvestment risk, and should earn a higher risk premium than an asset with Cov(Ri,t+1, Rm,t+2) =

0, that is, γz Covt(Ri,t+1,∆zt+1) > 0. This in turn implies that γz > 0, given the assumption

that Covt(Ri,t+1,∆zt+1) > 0. If we assume instead that the state variable forecasts negative

expected market returns, then the intertemporal risk price must be negative, and the argument

is just symmetric.4

The main practical implication of this result is that if a state variable positively forecasts

expected returns, the asset’s covariance with its innovation should earn a risk premium in the

cross-section. Thus, it is not enough that the candidate state variables forecast future aggregate

returns. It must be the case that the (covariance) risk prices with (the innovations in) those

state variables have the correct sign. Otherwise, the risk based explanation associated with

those factors is inconsistent with the rational explanation underlying the ICAPM.

4This positive correlation between the factor risk price and the forecasting slope is also valid under the (morerestrictive) Campbell (1993) version of the ICAPM, as long as the coefficient of relative risk aversion is greaterthan one, γ > 1. In the (implausible) case of an investor less risk averse than the log investor the risk priceof “discount rate news” would be negative, that is, an asset that is positively correlated with good news aboutfuture market returns (or alternatively, an asset that is positively correlated with a state variable that forecastspositive market returns) would earn a lower risk premium than an asset that is uncorrelated with future marketdiscount rates. The intuition is that for an investor with very low risk aversion, the upside effect of a positivecorrelation between a given asset and future market returns (in the sense that it allows the investor to profitfrom the improvement in future investment opportunities) outweighs the downside effect (a reduced ability tohedge changes in future investment opportunities). However, we follow the extant literature and assume that therepresentative investor is more risk averse than the log investor, thus ruling out this perverse effect.

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In the predictability literature, most predictive variables forecast positive expected equity

market returns. This is the case for aggregate financial ratios (dividend-to-price ratio; earnings-

to-price ratio; book-to-market ratio), or bond yield spreads like the slope of the yield curve or

the default spread.

Next, we consider a state variable that forecasts the future variance of market return, and

reexamine the corresponding implications for the sign of γz in the cross-sectional asset pricing

tests. The main result in this case is the following: If the state variable is positively correlated

with the future volatility of aggregate returns,

Covt(R2m,t+2, zt+1) = Covt[Et+1(R2

m,t+2), zt+1] = Covt[Et+1(R2m,t+2),∆zt+1] > 0, (13)

then the risk price of intertemporal risk must be negative.5

To see this point, assume again without loss of generality that the return on asset i is

positively correlated with the (innovation in the) state variable, which implies that the return

on asset i is also positively correlated with the future volatility of market return:

Covt(Ri,t+1, R2m,t+2) = Covt[Ri,t+1,Et+1(R2

m,t+2)] > 0. (14)

The economic implication of this last condition is that such an asset provides a hedge for

reinvestment risk, as it pays high returns when future aggregate volatility is also high. Thus,

such an asset should earn a lower risk premium than an asset with Covt(Ri,t+1, R2m,t+2) = 0,

that is, γz Covt(Ri,t+1,∆zt+1) < 0, which implies that γz < 0, given the assumption that

Covt(Ri,t+1,∆zt+1) > 0. If we assume instead that the state variable forecasts negative market

volatility, then the intertemporal risk price must be positive, and the argument is just symmetric.

Thus we have an opposite result relative to the case of expected returns: If a state variable

negatively forecasts the volatility of returns, the asset’s covariance with its innovation should

earn a risk premium in the cross-section. The intuition is that an asset that covaries negatively

with future aggregate volatility does not provide an hedge for risk averse investors (who dislike

increased uncertainty in their future wealth), who in turn are willing to invest in such an asset

if it offers a premium.

Because Rf,t+1 is known at the beginning of the period, we have Covt(Rf,t+1, Rm,t+1) =

5This proposition is consistent with the model developed by Chen (2003).

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Covt(Rf,t+1,∆zt+1) = 0, leading to the pricing equation that we test on our data:

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, Rm,t+1) + γz Cov(Ri,t+1 −Rf,t+1,∆zt+1). (15)

3 Do state variables forecast future investment opportunities?

In this section, we test the first restriction associated with the Intertemporal CAPM (ICAPM),

that is, do the candidates for state variables forecast future investment opportunities? Second,

and most important, we want to assess the sign of the correlation between the state variables and

future aggregate returns, for comparison with the factor risk prices we estimate, which represent

our main criteria to evaluate the ICAPM. Our proxy for the investment opportunity set is the

aggregate equity market, which is captured by the monthly return on the value-weighted stock

market index available from the Chicago Center for Research in Security Prices (CRSP).

We conduct long-horizon predictive regressions, which are commonly used in the predictabil-

ity literature to assess the forecasting power of the state variables over future expected market

returns [Keim and Stambaugh, 1986; Campbell, 1987; Fama and French (1988, 1989), among

others]:

rt,t+q = aq + bqzt + ut,t+q, (16)

where rt,t+q ≡ rt+1 + ... + rt+q is the continuously compounded return over q periods (from

t+ 1 to t+ q), and ut,t+q denotes a forecasting error with zero conditional mean, Et(ut,t+q) = 0.

It follows that the conditional expected return at time t is given by Et(rt,t+q) = aq + bqzt.

The sign of the slope coefficient, bq, indicates whether a given state variable forecasts positive

or negative changes in future expected aggregate stock returns, and the associated t-statistic

indicates whether this effect is statistically significant. We use forecasting horizons of 1, 3, 12,

24, 36, 48, and 60 months ahead.6 The original sample is 1963:07-2008:12, which corresponds

to the time span used in most empirical asset pricing studies of the cross-section. We evaluate

the statistical significance of the regression coefficients by using both Newey and West (1987),

and Hansen and Hodrick (1980) asymptotic standard errors with q lags.7

The first set of state variables we use in our empirical test are variables from the predictability

6There is a recent debate between in-sample versus out-of-sample predictability of stock market returns (seeCampbell and Thompson, 2008; Cochrane, 2008; and Goyal and Welch, 2008, among others). In our case, itmakes sense to use in-sample regressions since we are interested in the long run predictive power of the statevariables, and also to use the same time series that is used in the estimation of the factor covariances (betas) andaverage excess returns, which are employed in the cross-sectional tests.

7We use q lags to correct for the serial correlation in the residuals caused by the overlapping returns.

9

literature. The first two variables are bond yield spreads: The slope of the Treasury yield curve

(TERM , Campbell, 1987; Fama and French, 1989) and the corporate bond default spread

(DEF , Keim and Stambaugh, 1986; Fama and French, 1989). We measure TERM as the yield

spread between the ten-year and the one-year Treasury bond, while DEF represents the yield

spread between BAA and AAA corporate bonds from Moody’s. The yield data are available

from the FRED (St. Louis Fed) database.

We also use the market dividend-to-price ratio [DY , Fama and French (1988, 1989); Camp-

bell and Shiller (1988a)], and the aggregate price-earnings ratio [PE, Campbell and Shiller

(1988b); Campbell and Vuolteenaho, 2004], both for the S&P 500 index. DY is computed

as the log ratio of the sum of annual dividends to the level of the S&P 500 index, and PE

corresponds to the log ratio of the price of the S&P 500 index to a ten-year moving average

of earnings. The price, dividend, and earnings data for the S&P 500 index are available from

Robert Shiller’s website.

The next two state variables are the one-month Treasury bill rate (RF , Fama and Schwert,

1977; Campbell, 1991; Hodrick, 1992), which is available from Kenneth French’s website, and

the value spread (V S, Campbell and Vuolteenaho, 2004). The value spread is computed as

the difference between the monthly log book-to-market ratios of small-value and small-growth

stocks.8 The last state variable is the Cochrane and Piazzesi (2005) factor (CP ), which is

related to bond risk premia.9

Hahn and Lee (2006) use TERM and DEF in their ICAPM application. Petkova (2006)

uses DY and RF , in addition to TERM and DEF , to proxy for changes in future investment

opportunities. Campbell and Vuolteenaho (2004) use PE rather than DY and the value spread

in addition to TERM to forecast aggregate returns in their ICAPM application. In Koijen,

Lustig, and Van Nieuwerburgh (2010), the two state variables are TERM and CP .10 Therefore,

in addition to single-variable forecasting regressions, we conduct multiple-variable regressions

8The value spread is calculated from six portfolios sorted on both size and book-to-market, available fromKenneth French’s website. For details on the construction of V S, see the appendix in Campbell and Vuolteenaho(2004).

9CP is the fitted value from a regression of an average of excess bond returns on forward rates. For details onthe construction of CP , see Cochrane and Piazzesi (2005). We thank an anonymous referee for suggesting theinclusion of CP in the empirical analysis conducted in the paper.

10The proxy for the level of the yield curve in Koijen, Lustig, and Van Nieuwerburgh (2010) is constructeddifferently than TERM , however, both proxies are highly correlated.

10

to assess the joint forecasting power of the state variables in four ICAPM applications:11

rt,t+q = aq + bqTERMt + cqDEFt + ut,t+q, (17)

rt,t+q = aq + bqTERMt + cqDEFt + dqDYt + eqRFt + ut,t+q, (18)

rt,t+q = aq + bqTERMt + cqPEt + dqV St + ut,t+q, (19)

rt,t+q = aq + bqTERMt + cqCPt + ut,t+q. (20)

The second set of state variables is based on risk factors widely used in the empirical asset

pricing literature. Although several of these multifactor models are justified as applications of

the ICAPM, it is not clear whether the associated state variables do actually forecast future

stock returns. The first two variables are the size (SMB) and value (HML) factors used by

Fama and French (1993, 1996).12 We also use the momentum factor (UMD) from Carhart

(1997). These three factors are obtained from Kenneth French’s website. The fourth empirical

factor we use is the liquidity factor from Pastor and Stambaugh (2003) (L), which is obtained

from Lubos Pastor’s website.13

To obtain the associated state variables we use the cumulative sums on the factors for UMD

and L. For example, in the case of L, the cumulative sum is obtained as:

CLt =t∑

s=t−59

Ls. (21)

We use the cumulative sum over the last 60 months since the total cumulative sum is close to

being non-stationary (auto-regressive coefficients around one).

In the case of SMB, the corresponding state variable, SMB∗, is constructed as the difference

between the monthly market-to-book ratios of small and big stocks, using the six portfolios

sorted on both size and book-to-market, available from Kenneth French’s website:

SMB∗ =MBSL +MBSM +MBSH

3− MBBL +MBBM +MBBH

3, (22)

where MBSL, MBSM , MBSH , MBBL, MBBM , and MBBH denote the monthly market-to-

11When there are multiple state variables, we should focus on the marginal predictive role of each variable forchanges in future investment opportunities, conditional on all other variables. This is why we use multivariateforecasting regression, instead of univariate regressions for each variable separately.

12Vassalou (2003) provides evidence that both SMB and HML convey information about future GDP growthwhile Da and Schaumburg (2011) find that these factors are related to market volatility.

13We use the non-traded liquidity factor (Eq. (8) in Pastor and Stambaugh, 2003).

11

book ratios of small-growth, small-middle BM, small-value, big-growth, big-middle BM, and

big-value portfolios, respectively. Similarly, HML∗ corresponds to the difference between the

monthly market-to-book ratios of value and growth stocks:

HML∗ =MBSH +MBBH

2− MBSL +MBBL

2. (23)

Since SMB ' ∆SMB∗ and HML ' ∆HML∗, this procedure enables us to interpret the orig-

inal factors as innovations in the state variables, which we use in the cross-sectional regressions

conducted later.14

Thus, we conduct multiple regressions corresponding to three different multifactor models:

rt,t+q = aq + bqSMB∗t + cqHML∗t + ut,t+q, (24)

rt,t+q = aq + bqSMB∗t + cqHML∗t + dqCUMDt + ut,t+q, (25)

rt,t+q = aq + bqSMB∗t + cqHML∗t + dqCLt + ut,t+q. (26)

Also we estimate a predictive regression associated with the augmented model estimated in

Fama and French (1993):

rt,t+q = aq + bqSMB∗t + cqHML∗t + dqTERMt + eqDEFt + ut,t+q. (27)

Table 2 presents summary statistics for the state variables described above. Most state

variables are highly persistent, with autoregressive coefficients above 0.90. The least persistent

variable is CP with an autoregressive coefficient of 0.85.


Table 3 presents the results for the single long-horizon regressions associated with TERM ,

DEF , DY , RF , PE, V S, and CP . We can see that TERM , DEF , DY , RF , and CP forecast

positive market returns, although only in the cases ofDY , DEF , RF , and CP do the asymptotic

t-stats indicate statistical significance (and for DEF and RF only at longer horizons). On the

other hand, PE consistently forecasts negative market returns at all horizons, and this effect is

statistically significant at horizons of and beyond three months. The value spread also forecasts

14The relation is only approximate, since we are ignoring the dividend component of returns. The constructionof SMB∗ and HML∗ is similar in spirit to the value spread computed by Campbell and Vuolteenaho (2004).

12

negative market returns at all horizons, and the slopes are significant at the 5% level for horizons

between three and 48 months.15

Basically, all the variables forecast positive market returns with the exception of PE and

V S and the forecasting power of most variables increases with the horizon as indicated by the

approximately monotonic pattern in the R2 estimates.


The results for the multiple long-horizon regressions (17)-(20) are displayed in Table 4. To

save space, we report results only for horizons of 1, 12, and 60 months. The slope estimates

indicate that both TERM and DEF forecast positive market returns, but only the slope as-

sociated with DEF is statistically significant at horizons of 12 and 60 months. In the case of

regression (18), at q = 60, for which there is greater evidence of predictability as indicated by

the adjusted R2 estimates, TERM , DY , and RF are statistically significant and forecast pos-

itive market returns but the slope associated with DEF is negatively estimated, although not

significant. In the case of regression (19), TERM forecasts positive market returns, while PE is

negatively correlated with future expected returns at all three horizons. V S forecasts negative

market returns at q = 1, 12, but the predictive slope is positive at the 60-month horizon. The

three forecasting coefficients are statistically significant at q = 60. Regarding the state variables

in regression (20), CP forecasts positive market returns, conditional on TERM , and the slopes

are significant at the 5% or 1% levels for q = 1 and q = 60. TERM forecasts negative market

returns for q = 1 and positive returns thereafter, but none of the slopes is significant at the 10%

level.


The results for the multiple long-horizon regressions (24)-(27) are displayed in Table 5. Both

SMB∗ and HML∗ forecast positive market returns at all horizons; the slope associated with

HML∗ is statistically significant at all horizons, while SMB∗ is only marginally significant at

q = 60. CUMD forecasts positive market returns at horizons of 12 and 60 months, conditional

on both SMB∗ and HML∗, but the slopes are statistically significant only at very long horizons

(q = 60). The liquidity factor is positively correlated with future expected returns at all horizons,

but the coefficients are statistically significant only at q = 12. In the fourth multiple regression15The negative slopes associated with V S are in line with the results obtained in Campbell and Vuolteenaho

(2004).

13

(27), we can see that the slopes associated with TERM and DEF , conditional on SMB∗ and

HML∗, are positive, although only TERM is marginally significant in predicting the market

return for q = 60 (based on the Newey-West standard errors).16

Overall, these results show that most candidates for ICAPM state variables can forecast

market returns, although the evidence of predictability is much stronger at longer horizons.


4 Are factor risk prices consistent with the ICAPM?

To assess the ICAPM restrictions regarding factor risk prices, we test different multifactor

models in the cross-section of stock returns. In each model, the first factor is the market equity

premium, RM , computed as the monthly return of the value-weighted index in excess of the

one-month Treasury bill rate, available from Kenneth French’s website.

The first group of multifactor models corresponds to models explicitly justified as ICAPM

applications in which the risk factors represent innovations to state variables used in the pre-

dictability of returns literature to forecast aggregate equity returns. The first model we estimate

is the ICAPM version of Hahn and Lee (2006), in which the additional risk factors relative to

the market factor are the innovations in TERM and DEF :

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1)

+γTERM Cov(Ri,t+1 −Rf,t+1,∆TERMt+1) + γDEF Cov(Ri,t+1 −Rf,t+1,∆DEFt+1). (28)

The second model is the ICAPM of Petkova (2006) that consists of the innovations to RF and

DY , in addition to TERM and DEF :

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1) + γTERM Cov(Ri,t+1 −Rf,t+1,∆TERMt+1)

+γDEF Cov(Ri,t+1 −Rf,t+1,∆DEFt+1) + γDY Cov(Ri,t+1 −Rf,t+1,∆DYt+1)

+γRF Cov(Ri,t+1 −Rf,t+1,∆RFt+1). (29)

The third model is an unrestricted version of the Campbell and Vuolteenaho (2004) ICAPM,

16We also conduct multiple predictive regressions for the equity premium. In most cases, the signs of theslopes are the same as in the regressions for the market return. In the few cases in which the sign flips the pointestimates of the slopes are not statistically significant.

14

in which investment opportunities are described by PE, TERM , and V S:17

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1) + γTERM Cov(Ri,t+1 −Rf,t+1,∆TERMt+1)

+γPE Cov(Ri,t+1 −Rf,t+1,∆PEt+1) + γV S Cov(Ri,t+1 −Rf,t+1,∆V St+1). (30)

The fourth model is the three-factor model from Koijen, Lustig, and Van Nieuwerburgh (2010),

in which the state variables are TERM and CP :18

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1)

+γTERM Cov(Ri,t+1 −Rf,t+1,∆TERMt+1) + γCP Cov(Ri,t+1 −Rf,t+1,∆CPt+1). (31)

Next, we use multifactor models with less theoretical justification, but that some authors con-

sider as possible applications of the ICAPM. The fifth model is the Fama and French (1993,

1996) three-factor model (FF3), in which the factors are SMB and HML in addition to the

market risk premium:

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1) + γSMB Cov(Ri,t+1 −Rf,t+1, SMBt+1)

+γHML Cov(Ri,t+1 −Rf,t+1, HMLt+1). (32)

SMB is used to explain the size premium, the positive spread in average returns between

small and big stocks. HML seeks to explain the value premium, that value stocks (stocks

with high book-to-market) have larger average returns than growth stocks (stocks with low

book-to-market).

The sixth model is the Fama-French three-factor model augmented by TERM and DEF

17Campbell and Vuolteenaho estimate a model with two factors, cash flow news (NCF ) and discount ratenews (NDR). Both NCF and NDR are linear functions of the innovations in the state variables, so that the twospecifications are equivalent. For further details see Maio (2012a).

18The Koijen, Lustig, and Van Nieuwerburgh (2010) model is not motivated by the authors as an ICAPMapplication. However, since the respective factors (other than the market factor) are related to state variableswidely used in the predictability literature (TERM and CP ), it makes sense to test whether this model satisfiesthe ICAPM restrictions.

15

(FF5):


+γHML Cov(Ri,t+1 −Rf,t+1, HMLt+1) + γTERM Cov(Ri,t+1 −Rf,t+1,∆TERMt+1)

+γDEF Cov(Ri,t+1 −Rf,t+1,∆DEFt+1). (33)

Fama and French (1993) use this model to explain both equity and bond risk premiums.

The seventh model is Carhart’s (1997) four-factor model (C):


+γHML Cov(Ri,t+1 −Rf,t+1, HMLt+1) + γUMD Cov(Ri,t+1 −Rf,t+1, UMDt+1). (34)

Added to the Fama and French (1993) model is the momentum (UMD) factor. The role of

UMD is to explain the momentum anomaly; that is, past short-term winners tend to have

higher average returns than past losers (Jegadeesh and Titman, 1993).

Finally, we use the four-factor model employed by Pastor and Stambaugh (2003), which

incorporates a liquidity related risk factor (L):


+γHML Cov(Ri,t+1 −Rf,t+1, HMLt+1) + γL Cov(Ri,t+1 −Rf,t+1, Lt+1), (35)

which is denoted as PS.

We use two sets of portfolio returns for testing assets. The first group is the 25 portfolios

sorted on size and book-to-market (SBM25). The second group is 25 portfolios sorted on size

and momentum (SM25). The portfolio return data are obtained from Kenneth French’s website.

We add the market return to each group of portfolios. Adding the market return enables

a more powerful empirical test, and allows us to combine the literature on cross-sectional asset

pricing with the literature on the market risk-return trade-off. Thus, the estimates of risk

aversion and the intertemporal risk prices incorporate information from both the cross-section

of equity premiums and the aggregate equity premium.19 Moreover, adding factors to the menu

19Note that it is common practice in the literature on the aggregate risk-return trade-off to estimate only therisk aversion parameter by assuming (in opposition with the underlying theory of the ICAPM) that the risk pricesassociated with time-varying investment opportunities are negligible, i.e., that hedging motives are marginal. Inour case, the “hedging” factor risk prices are the core of the analysis.

16

of testing assets when the factors are returns themselves represents a more appropriate test of

asset pricing models, as suggested by Lewellen, Nagel, and Shanken (2010).

We estimate each multifactor model in expected return-covariance form by a one-stage gener-

alized method of moments procedure (GMM, Hansen, 1982). This method uses equally weighted

moments, which is conceptually equivalent to running an ordinary least squares (OLS) cross-

sectional regression of average excess returns on factor covariances (right-hand side variables).

The advantage of the GMM procedure is that we don’t need to have previous estimates of the

individual covariances, as these are implied in the GMM moment conditions. This estimation al-

lows us to assess whether each model can explain the returns of a set of economically interesting

portfolios (e.g., SBM25).20

The GMM system includes N + K + 1 moment conditions. The first N sample moments

correspond to the pricing errors for each of the N testing returns:

gT (b) ≡

1T

T−1∑t=0

(Ri,t+1 −Rf,t+1)− γ(Ri,t+1 −Rf,t+1) (RMt+1 − µm)

−γ1(Ri,t+1 −Rf,t+1) (f1,t+1 − µ1)− γ2(Ri,t+1 −Rf,t+1) (f2,t+1 − µ2)

−...− γK(Ri,t+1 −Rf,t+1) (fK,t+1 − µK)

RMt+1 − µm

f1,t+1 − µ1

...

fK,t+1 − µK

= 0,

i = 1, ..., N. (36)

K denotes the number of factors in addition to the market factor (that is, each model has K+1

factors).

In this system, the market return (RMt+1) is the first factor, with unconditional mean

given by µm. The remaining factors are given by (f1,t+1, ..., fK,t+1), and the respective means

are denoted by (µ1, ..., µK). For example, in the case of the Fama-French model, we have

K = 2 with f1,t+1 ≡ SMBt+1, f2,t+1 ≡ HMLt+1. (γ1, ..., γK) denote the (covariance) risk

prices associated with the “hedging” factors. The last K + 1 moment conditions in system (36)20The procedure is also more convenient than the two-pass time-series/cross-sectional regressions approach

(Cochrane, 2005; Brennan, Wang, and Xia, 2004), since we want to estimate the model in expected return-covariance form rather than in expected return-beta form to obtain the covariance risk prices for each factor, andspecifically the market (covariance) risk price, which represents an estimate of the relative risk aversion.

17

enable us to estimate the factor means.

Thus, the estimated covariance risk prices from the first N moment conditions do account

for the estimation error in the factor means, as in Cochrane (2005) (Chapter 13), and Yogo

(2006). In this system, there are N −K − 1 overidentifying conditions (N + K + 1 moments

and 2(K + 1) parameters to estimate). The standard errors for the parameter estimates, and

the remaining GMM formulas are presented in Appendix 8.

To assess the robustness of the asymptotic standard errors, we conduct a bootstrap simula-

tion to produce empirical p-values for the tests of individual significance of the factor risk prices.

The bootstrap simulation consists of 10,000 replications in which the portfolio return data and

the factors are simulated independently, that is, the data are simulated under the hypothesis

that the model is not true. Full details of this bootstrap simulation are available in Appendix

8.

Although our focus is on the analysis of the factor risk prices, for completeness we also

present some measures for the overall explanatory power of each model. The idea is to assess

whether each model satisfies the economic restrictions associated with the ICAPM, in addition

to explaining the dispersion in equity premiums over the cross-section.

The first measure is the mean absolute pricing error:

MAE =1N

N∑i=1

|αi|, (37)

where αi, i = 1, ..., N represents the first N moments, i.e., the pricing errors associated with the

N testing assets.

The second goodness-of-fit measure is the cross-sectional OLS coefficient of determination:

R2OLS = 1− VarN (αi)

VarN (Ri), (38)

where Ri = 1T

∑T−1t=0 (Ri,t+1 −Rf,t+1) denotes the average excess return for asset i, and VarN (·)

stands for the cross-sectional variance. R2OLS measures the fraction of the cross-sectional vari-

ance in average excess returns explained by the model.21


We start the empirical analysis by estimating two-factor models that include a “hedging”21We do not present the values for the asymptotic χ2 test of overidentifying restrictions, since both the mean

absolute error (MAE) and R2OLS represent more robust measures of the models’ global fit.

18

risk factor in addition to the market return:

E(Ri,t+1 −Rf,t+1) = γ Cov(Ri,t+1 −Rf,t+1, RMt+1) + γz Cov(Ri,t+1 −Rf,t+1,∆zt+1), (39)

where z = TERM , DEF , DY , RF , PE, V S, and CP . The objective is to analyze whether

some of the most relevant predictors of market returns proposed in the predictability literature

can be justified in two-factor ICAPM specifications.

The factor risk price estimates are displayed in Table 6. In the tests with SBM25 (Panel

A), the point estimates for the relative risk aversion (RRA) parameter are negative in most

specifications; the exceptions are the models with ∆DEF , ∆V S, and ∆CP . Moreover, the risk

price estimates associated with the “intertemporal” factor are negative in the specifications with

∆DEF , ∆DY , and ∆RF , which are inconsistent with the positive forecasting slopes over the

market return associated with the level of these variables, as shown in the last section. In the

model with ∆PE, the positive risk price estimate is also inconsistent with the corresponding

negative slope estimated in the single predictive regression (16). Only in the specifications

with ∆TERM , ∆V S, and ∆CP is there consistency in sign between the factor risk price

estimates from the cross-sectional regressions and the slopes from the forecasting regressions

over the market return. Overall, only two two-factor models (including ∆V S or ∆CP ) satisfy

the restrictions on the market (covariance) risk price (RRA parameter) and on the “hedging

factor” risk price.

In the tests with SM25 (Panel B), only in the specifications with ∆RF and ∆V S is there an

implausible negative estimate for γ. The factor risk prices associated with ∆TERM , ∆DEF ,

∆RF , and ∆CP are negative in all cases, which is inconsistent with the corresponding positive

slopes estimated in the single predictive regressions. Moreover, the risk price estimate for ∆V S

is estimated positively, which is inconsistent with the negative slopes from the single forecasting

regressions.

On the other hand, the signs of the risk price estimates associated with ∆DY and ∆PE

are consistent with the forecasting ability of these variables over market returns, although in

both cases the risk price estimates are largely insignificant. Furthermore, in both cases there is

no explanatory power over the cross-section of returns, as illustrated by the negative estimates

for the cross-sectional R2, indicating that these two models perform more poorly than a model

that predicts constant expected risk premiums in the cross-section of equities.

19

The results in Table 6 show overall that most of the state variables in the literature are not

valid risk factors under the ICAPM. Specifically, only two two-factor models satisfy the economic

restrictions underlying the ICAPM, jointly with a reasonable explanatory power over the cross-

section of excess stock returns (∆V S and ∆CP in the test with SBM25, with R2 = 38% and

42%, respectively).

We next estimate factor models presented in the literature as empirical applications of the

ICAPM, which include combinations of the state variables analyzed above, specifically the three-

factor Hahn and Lee (2006) model (henceforth, HL); the five-factor Petkova (2006) model (P);

a four-factor model that corresponds to an unrestricted version of Campbell and Vuolteenaho

(2004) (CV); and the three-factor model from Koijen, Lustig, and Van Nieuwerburgh (2010)

(KLVN). The results are displayed in Table 7.

In the tests with SBM25 (Panel A), we have the usual result that the CAPM cannot price the

SBM25 portfolios, as shown by the negative cross-sectional R2 of -42%, while the four multifactor

models show a considerable explanatory power over the dispersion in risk premiums within these

portfolios, with R2 estimates above 70%. The point estimates for the RRA parameter, however,

are negative and statistically insignificant in all four factor models, compared to a positive

estimate of 2.79 in the CAPM, which is statistically significant.

In the case of HL, the estimates for γTERM and γDEF are positive and negative, respectively,

although only the risk price associated with ∆TERM is statistically significant. Thus, the

sign of γDEF is inconsistent with evidence above that DEF (conditional on TERM) forecasts

positive market returns.

In the case of P, both γDY and γRF are negative, while both γTERM and γDEF are positive.

All four risk price estimates are not statistically significant, however, which indicates evidence

of multicollinearity. Thus, the signs of γDEF , γDY , and γRF are inconsistent with the evidence

from the multiple regressions (18) at a horizon of 60 months (for which there is stronger evidence

of predictability) that both DY and RF (conditional on the other variables) forecast positive

market returns, while DEF forecasts negative market returns (although with no significance).

In the case of CV, the estimate for γPE is positive, and thus inconsistent with the correspond-

ing negative slopes found in the predictive regressions (19), while γV S is estimated negatively,

which also goes against the sign of the predictive slope at q = 60. Both γPE and γV S are not

statistically significant at the 10% level, suggesting the presence of multicollinearity.

Regarding the KLVN model, the point estimates for γTERM and γCP are both positive,

20

and thus consistent with the corresponding slopes in the multiple regression (20) at q = 12, 60.

However, the point estimate for γCP is largely insignificant based on both the asymptotic t-stat

and empirical p-value.


When the testing portfolios are SM25 (Panel B), the explanatory power of the CAPM for

the cross-section of excess returns is also negative (R2 = −10%), and the corresponding RRA

estimate is 2.34, which is significant at the 5% level. The four multifactor models are significantly

better than the CAPM in pricing the SM25 portfolios, with cross-sectional R2 estimates varying

between 50% (HL) and 67% (P). Contrary to the tests with SBM25, the point estimates for

γ are positive for the HL, P, and KLVN models, but only in the cases of HL and KLVN are

there estimates above one (5.56 and 10.77, respectively), which are significant (only marginally

in the case of HL, based on the asymptotic t-stats). For risk price estimates associated with the

intertemporal factor, there are inconsistencies with the slopes from the corresponding predictive

regressions in all four models.

In the case of HL, the point estimates for both γTERM and γDEF are negative, which goes

against the positive slopes estimated from the multiple long-horizon regressions. In the case of

P, the estimates for γTERM , γDY , and γRF are negative; again, these estimates are inconsistent

with the positive correlation between each of the corresponding state variables and future market

return found in the multiple predictive regressions. Thus, only for one factor (DEF ) is the sign

of the risk price consistent with the sign of the slope in the forecasting regressions, but the

point estimate is insignificant (t-stat = -0.11). In the case of CV, the risk price associated with

∆TERM is estimated negatively and is statistically significant, which goes against the positive

slope from the long-horizon regressions, while both γPE and γV S have the wrong sign as in the

test with SBM25. Finally, for KLVN, the negative risk price associated with ∆TERM , which

is significant at the 5% level, is at odds with the positive slopes (for longer horizons) of TERM

in the multiple predictive regressions.

If we compare the hedging factor risk prices with the slopes from the single forecasting

regressions in Table 3, instead of the multiple regressions, then the same qualitative results

hold. That is, in each the four models tested in both sets of portfolios there is inconsistency in

sign with the forecasting slopes for at least one state variable in the model. The only exception

is the KLVN model tested on SBM25, which satisfies the sign consistency (as in the comparison

21

with the multiple regressions). However, this model generates a negative estimate for the risk

aversion coefficient as discussed above.

Overall, these results show that these four multifactor models are inconsistent with the

ICAPM in pricing either the SBM25 or SM25 portfolios, despite significant explanatory power

over the cross-section. This inconsistency shows up in both the risk price estimates associated

with the hedging risk factors and the risk-aversion estimates. Hence, these models can serve as

good empirical models that explain the size, value and momentum anomalies, but there is no

underlying justification in line with the ICAPM theory.

Next, we estimate the four empirical-based multifactor models. The results are displayed in

Table 8, which is similar to Table 7.

In the tests with SBM25, all four models have a significant explanatory power over the cross-

section of equity premiums, with cross-sectional R2 estimates around 70%. On the other hand,

in the tests with SM25, only C and FF5 have relevant explanatory power over the cross-section,

with R2 estimates above 70%, thus confirming that the FF3 model cannot price the momentum

portfolios (Fama and French, 1996; Cochrane, 2007, among others). The RRA estimates are

positive and statistically significant in the cases of FF3 and C, but insignificantly negative in

the cases of PS and FF5, when the test portfolios are SBM25. In the tests with SM25, γ is

implausibly below one in the FF3 model, and negative in the liquidity model, while in the cases

of C and FF5, the RRA estimates are positive and statistically significant (in the case of FF5,

only marginally and based on the asymptotic t-stat).

For hedging risk factors, in the case of FF3 tested with SBM25, the risk price estimates

associated with SMB and HML are both positive and statistically significant (in the case

of SMB, only marginally and based on the Newey-West t-stat), which is consistent with the

positive slopes estimated in the multiple predictive regressions (24). In the tests with SM25,

the point estimate for γHML is negative, although not statistically significant. In the case of

the C model, the three factor risk prices are estimated positively in both tests, with HML and

UMD significantly priced, while γSMB is not statistically significant. Thus, all three risk price

estimates are consistent with the corresponding positive slopes from the long-horizon regressions

in (25). In the case of the liquidity model, the estimates associated with γL are positive in both

tests, which are compatible with the positive slopes estimated in the long-horizon regression

(26) at q = 60. Thus the restriction on the hedging risk prices is satisfied in the test of this

model with SBM25, but the associated RRA estimate is negative as referred above.

22


Finally, in the case of FF5, the point estimates for γTERM and γDEF are positive and

negative, respectively, in the tests with SBM25, and these estimates flip signs in the tests with

SM25. Only γTERM is statistically significant, however. This implies that the signs of γDEF (in

the tests with SBM25) and γTERM (in the tests with SM25) are incompatible with the positive

slopes associated with TERM and DEF in the predictive regressions (27). Therefore, only two

multifactor models admit an ICAPM interpretation – FF3 tested with the SBM25 portfolios,

and C tested with both sets of equity portfolios. When tested on the SM25 portfolios, the FF3

model is no longer consistent with the ICAPM, besides having no explanatory power for the

cross-section of equity premiums (R2OLS = 1%).

Overall, considering the 16 cross-sectional tests (eight models times two sets of test assets)

only in three cases can the multifactor model be justified as an empirical application of the

ICAPM. As for pricing both sets of equity portfolios, only the Carhart (1997) four-factor model

is able to satisfy the ICAPM restrictions. The Fama-French model on the other hand, satisfies

the ICAPM criteria only when pricing the size-value portfolios. Moreover, none of the four

multifactor models a priori more likely to be consistent with the ICAPM (HL, P, CV, and

KLVN), satisfy the ICAPM restrictions.

5 Additional results

To assess the robustness of the results in Sections 3 and 4, we add an intercept to the estimation

of the multifactor models; estimate the models by second-stage GMM; add bond risk premia

to the test assets; estimate the models in expected return-beta form; conduct a bootstrap

simulation for the predictive slopes; use different state variables associated with SMB and

HML; and employ a different measure of expected market return. We also conduct additional

robustness checks that are available in an addendum to this paper: we estimate the multifactor

models by first orthogonalizing the “hedging” factors relative to the market factor; exclude the

excess market return from the menu of test assets; estimate the models with alternative equity

portfolios; and employ a different measure of innovation in the state variables. Overall, the

qualitative results are similar to the benchmark results.

23

5.1. Including an intercept in the cross-sectional tests

As a robustness check we estimate the multifactor models by including an intercept in the pricing

equation. If a given model is specified correctly, the intercept should be indistinguishable from

zero. Moreover, the factor risk price estimates and cross-sectional R2 should not be too different

from the corresponding estimates in the benchmark or restricted specification without intercept.

Results available on the addendum to this paper show that in tests with either set of portfo-

lios, the CAPM intercept is strongly statistically significant at around 1% per month (approx-

imately 12% per year), which is economically significant. These estimates represent evidence

that the CAPM is misspecified when tested over SBM25 or SM25.

For the ICAPM candidates, in most cases the intercept is not statistically significant. The

exceptions are HL and CV in the tests with SM25 with excess zero beta rates of around 1% and

2% per month, respectively, which are significant at the 5% and 1% levels.

On the other hand, the relative risk aversion (RRA) estimates are negative in the cases of

the CAPM and most ICAPM specifications, with the exceptions of P (although even there the

magnitude is lower than one) and KLVN (around eight), both cases in the test with SM25.

The negative estimates associated with the market risk price are likely a consequence of mul-

ticollinearity related to the fact that the market covariances (betas) do not vary significantly

across the test portfolios (Fama and French, 1993; and Jagannathan and Wang, 2007). Regard-

ing the factor risk prices associated with the innovations in the state variables, we have the

same signs as in the benchmark test without intercept. The sole exception is for γV S , which is

now estimated positively in the tests with SM25.

Thus, we conclude that the four models are rejected due to inconsistency between the factor

risk prices and the predictive slopes, added to the negative estimates for the coefficient of risk

aversion.

For the four empirical-based multifactor models the point estimates of the intercept are

economically significant for most models, varying between 1% and 3% per month. The sole

exception is C in the test with SM25, with a point estimate of zero. Moreover, these point

estimates are statistically significant in the cases of FF3, PS, and FF5 for both tests. Hence,

this represents evidence that these three models are potentially misspecified.

On the other hand, the RRA estimates are negative in the cases of FF3, PS, and FF5 for

both sets of portfolios. Regarding the risk price estimates for the “hedging” factor, all four

models pass the consistency restriction with the slopes of the times series regression in tests

24

with SBM25, although the RRA estimates are clearly negative in the cases of FF3, PS, and

FF5, as referred above. In tests with SM25, only C satisfies the sign restrictions on the hedging

factor risk prices.

Overall, then, only the C model produces both reasonable RRA estimates and intertemporal

risk prices that are consistent with slopes from the time-series regressions.

5.2. Estimating by efficient GMM

We reestimate the multifactor models analyzed in the previous section by second-stage or ef-

ficient GMM, which allows us to obtain the most efficient estimates of the parameters, i.e.,

estimates with the lowest standard errors. Thus, in the second step the moments are weighted

according to the inverse of the spectral density matrix, which is equivalent to a generalized least

squares (GLS) cross-sectional regression of average excess returns on factor covariances.

We compute the WLS (weighted least squares) coefficient of determination, which is equal

to

R2WLS = 1− α

′S∗−1

N α

R′S∗−1

N R, (40)

where α is the vector of demeaned pricing errors; S∗N contains the diagonal elements of the

block of the spectral density matrix associated with the N pricing errors; and R is the vector

of demeaned (average) excess returns. The WLS R2 assigns less weight to the noisier pricing

errors, i.e., the pricing errors with greater variance (see Ferson and Harvey, 1999; and Shanken

and Zhou, 2007).22

Lewellen, Nagel, and Shanken (2010) justify the use of efficient GMM (or, equivalently,

GLS/WLS cross-sectional regressions of average returns on factor betas/covariances) over first-

stage GMM (or, equivalently, OLS cross-sectional regressions) with the argument that it is

harder to obtain a high cross-sectional GLS/WLS R2 than a high OLS R2, especially when

pricing the SBM25 portfolios. Furthermore, the GLS/WLS R2 represents a measure of how

close the factor-mimicking portfolios are to the mean-variance frontier constructed from the

test assets. Yet Cochrane (1996, 2005) argues that in the efficient GMM estimation one is

pricing repackaged portfolios, which often represent extreme linear combinations of the original

portfolios. Thus, not only does one lose the economic content of the original portfolios (e.g.,

22The WLS R2 should be more robust than the GLS R2 (that is based on the full SN matrix) since the inverseof the spectral density matrix is potentially misspecified when the number of moment conditions is relativelylarge and the size of the time-series is not very large (see Shanken and Zhou, 2007).

25

explaining the size, value, and momentum anomalies), but these “efficient” portfolios may also

be uninteresting for the average investor because of high transactions costs and potential risk.23

Results available on the paper’s addendum indicate that the CAPM cannot price the repack-

aged portfolios as illustrated by the negative estimates of the WLS cross-sectional coefficient

of determination. At the same time, the ICAPM specifications significantly improve the fit

associated with the CAPM when the test portfolios are SBM25, with R2WLS estimates varying

between 54% (P) and 71% (CV). On the other hand, in the tests with SM25, the fit is quite

low, with the exception of P (R2WLS = 0.38). Most models are, however, inconsistent with the

ICAPM because at least one hedging risk price has the wrong sign. The sole exception is the

KLVN model in the test with SBM25, which meets the criteria on both the market and state

variable risk prices. The second-stage point estimates for γ and γCP in the KLVN model are in

any case largely insignificant.

As in the case of the ICAPM applications, all the four empirical multifactor models present

large explanatory ratios (above 65%) in the test with SBM25. Yet, in the test with SM25,

only the C model delivers a positive R2WLS estimate (66%), with the remaining three models

presenting negative explanatory ratios. In the test with SBM25, the sign consistency of the

hedging risk prices with the predictive slopes and the plausibility of the risk aversion estimates

are met by the four models. The point estimates for the risk prices in the PS and FF5 models

are, however, not significant in most cases, suggesting the presence of multicollinearity.

As in the first-stage estimation, C is consistent with the ICAPM criteria when tested on

the SM25 portfolios. In contrast with the first-stage results, the FF3 model satisfies the sign

restrictions on both the market and hedging risk prices. However, the explanatory power of

FF3 over the SM25 portfolios is around zero (R2WLS = −1%). In the case of PS the RRA

estimate is negative while the explanatory ratio is also negative (-37%). Only the C model

satisfies the ICAPM restrictions in both tests while still having positive explanatory power over

the “transformed” portfolios, while the other three models meet these criteria in the test with

the SBM25 portfolios.

23An alternative estimation procedure is the second-moment matrix of returns from the test assets (Hansenand Jagannathan, 1997), also used by Hodrick and Zhang (2001), Jacobs and Wang (2004), and Kan and Robotti(2008), among others. As Cochrane (2005) points out, often the second-moment matrix of returns is closer tobeing singular than the spectral density matrix, implying that the resulting portfolios are even more “extreme.”

26

5.3. Pricing bond risk premia

We add excess bond returns to the equity portfolios to assess whether the factor risk prices

change in a significant way by forcing each model to price jointly the cross-section of stock

and bond returns. This becomes a more demanding test for each model since we are pricing

simultaneously equity and bond risk premia. We add to each equity portfolio group the excess

returns on seven Treasury bonds with maturities of 1, 2, 5, 7, 10, 20, and 30 years, available

from CRSP. Thus, we have a total of 33 test assets in each asset pricing test (SBM25 or SM25).

Regarding the ICAPM specifications, results available in the addendum show that in the

test with SBM25 the only changes relative to the benchmark test are that γDY becomes positive

and the estimate for γ in the KLVN model is now positive (but smaller than one). Yet, both

estimates are largely insignificant. When the equity portfolios are SM25, there are a number of

changes to the equity-only test. Specifically, the point estimates for γDEF , γDY , and γV S are

now positive, while γCP is estimated negatively. Moreover, the RRA parameter in the CV model

is now positive (3.47). All the point estimates for these risk prices, are however, not significant

at the 10% level. By comparing the risk price estimates with the corresponding slopes in the

predictive regressions, in only one case (KLVN tested on SBM25) there is consistency in signs.

However, the corresponding RRA estimate is implausibly below one (0.51). In sum, all four

models continue to be inconsistent with the ICAPM, when we add bond risk premia to the asset

pricing tests.

In the case of the empirical factor models tested on SBM25, the point estimates for γ in the

PS and FF5 models are now positive (4.13 and 2.08, respectively), and in the first model there

is significance at the 10% level. In these two models, the risk price estimates for the liquidity

and ∆DEF factors flip signs relative to the benchmark test. Thus, the PS model does not meet

anymore the sign restriction in the hedging factor risk prices, while FF5 meets this criteria,

however, both γL and γDEF are highly insignificant. When the equity portfolios are SM25, the

only change relative to the benchmark test is that the risk price for SMB becomes positive in

the PS model. Yet, such model is still inconsistent with the ICAPM due to the negative RRA

estimate. Thus, when we add bond risk premia to the test with the size-momentum portfolios,

it is still only the C model that meets the ICAPM criteria.

27

5.4. Estimating the models in expected return-beta representation

We estimate the multifactor models in expected return-beta form by using the time-series/cross-

sectional regression approach taken by Brennan, Wang, and Xia (2004), and Cochrane (2005),

among others. In the first step, we conduct time-series regressions to estimate the factor loadings

for each asset. For example, in the case of FF3, we have

Ri,t+1 −Rf,t+1 = δi + βi,MRMt+1 + βi,SMBSMBt+1 + βi,HMLHMLt+1 + εi,t, (41)

where the coefficients βi stand for the factor loadings. In the second step, we conduct an OLS

cross-sectional regression of average excess returns on the factor loadings to obtain estimates

for the factor (beta) risk prices (λ):

Ri −Rf = λMβi,M + λSMBβi,SMB + λHMLβi,HML + αi. (42)

We use the same approach for the other multifactor models. To compute the t-statistics associ-

ated with the risk prices, we use the Shanken (1992) approach, which accounts for the estimation

error in the betas. The implied relative risk aversion (RRA) estimates can be obtained from

the market beta risk prices:24

γ =λM

Var(RMt+1). (43)


The results for the ICAPM applications are displayed in Table 9. The estimates for λM

are positive in all four multifactor models and in both tests. Thus, the implied risk aversion

estimates are also positive in all cases, in contrast to the benchmark test with GMM. Specifically,

the implied estimates for γ (not tabulated) vary between 2.07 and 2.23 in the tests with SBM25,

and between 2.63 and 2.88 in the tests with SM25. For the hedging beta risk prices, the signs

are nearly the same as for the corresponding covariance risk prices in the benchmark test with

first-stage GMM. The sole exception is HL tested with SBM25, in which λDEF is now positive,

in contrast to the negative estimate for γDEF in the benchmark test, which is in line with the

positive slope in the multiple forecasting regressions. Thus, in the test with SBM25, HL satisfies

the restrictions on the signs of the “intertemporal” risk prices, λTERM and λDEF , but λDEF is24This relation is only true when the cross-sectional regression is estimated with single-regression betas instead

of multiple-regression betas. However, since in most of the multifactor models tested in this paper the factorsare only weakly correlated, this equation represents a good approximation.

28

insignificant (t-ratio = 0.20). The KLVN model also satisfies the ICAPM criteria on the risk

prices (as in the first-stage GMM estimation with orthogonalized hedging factors against the

market factor and second-stage GMM test) but the point estimate of λCP is largely insignificant.

The results for the empirical models are displayed in Table 10. As in the case of Models HL,

P, CV, and KLVN, the market risk prices are positive for all four models. The (untabulated)

implied RRA estimates vary between 1.72 and 2.18 in the test with SBM25; in the test with

SM25, they range between 1.77 and 2.28. The signs of the hedging risk prices are in most cases

the same as the covariance risk prices in the benchmark GMM tests. The exceptions are λDEF ,

which changes its sign relative to γDEF in the estimation with either SBM25 or SM25, and

λSMB, which is estimated positively in PS, in the test with SM25.

Consequently, in addition to FF3 (in the test with SBM25) and C (for both sets of portfolios),

which satisfy the risk price restrictions in the GMM benchmark test, the risk price estimates

for PS and FF5 in the test with SBM25 are also in line with the ICAPM criteria. The point

estimate for λDEF is strongly insignificant, however (t-ratio = 0.66). Overall, the qualitative

results are very similar to the benchmark test with first-stage GMM.


We also estimate a GLS cross-sectional regression in which the observations are weighted

according to the inverse of the covariance matrix of the residuals from the time-series regressions

(see Cochrane, 2005, Chapter 12). When one of the factors is also a testing asset (as is the case

of the excess market return in all of the models tested) this method allows us to incorporate

fully the restrictions associated with the pricing equation for the market return when pricing

the other assets. Put differently, the GLS regression assigns a zero pricing error for the excess

market return and the market risk price is numerically equal to the average excess market return

(0.38% per month in this sample).

Results available in the addendum show that in most cases the signs of the hedging factor

risk prices are the same as in the corresponding OLS cross-sectional regressions. Among the

exceptions are the point estimates for λDEF in the tests of HL and P with SM25, which are now

positive, although not statistically significant. The risk prices for HML are also now positive

in the case of both FF3 and PS tested on SM25 (contrary to the OLS regressions), and these

point estimates are significant at the 5% and 10% levels, respectively. Thus, contrary to the

OLS cross-sectional regression, the FF3 and PS models are consistent with the ICAPM when

29

tested on the SM25 portfolios using GLS. However, the point estimate for λL is not significant

at the 10% level.

5.5. Bootstrap simulation in the predictive regressions

We conduct a bootstrap simulation that produces an empirical distribution that better approx-

imates the finite sample distribution of the coefficient estimates in the predictive regressions in

Section 3. In this simulation, the market return and forecasting variables are simulated (10,000

times) under the null of no predictability of the market return and assuming that the predictor,

zt, follows an AR(1) process,

rt,t+q = aq + ut,t+q, (44)

zt+1 = ψ + φzt + εt+1. (45)

This bootstrap procedure allows for the high persistence of the forecasting variable and the

cross-correlation between the two residuals above correcting for the Stambaugh (1999) bias.

Details of the bootstrap algorithm are provided in Appendix 8.

Results in the addendum show that all the slopes in the predictive regressions (17)-(20) are

significant at the 5% level for the 60-month horizon. In contrast, at the one-month horizon most

of the coefficients are non significant at the 10% level according to the simulated p-values. The

sole exception is the coefficient associated with CP .

For the predictive regressions (24)-(27) the results are qualitatively similar. At the 60-

month horizon, all the slopes are significant at the 1% level, with the exception of DEF (p-

value = 0.19) and CL (p-value = 0.22) while at the one-month horizon all the estimates are

largely insignificant, with p-values well above 10%. These results complement the asymptotic

t-statistics in that the point estimates for the predictive slopes are more reliable at long horizons

(60 months).

5.6. Using different proxies for the state variables associated with SMB and HML

We conduct the multiple long-horizon regressions by using different proxies for the state variables

associated with SMB and HML. As in the proxy for CL, we construct CSMB as

CSMBt =t∑

s=t−59

SMBs, (46)

30

and CHML is defined in an analogous way. Results available on the paper’s addendum show

that both CSMB and CHML forecast positive market returns at all horizons, but both co-

efficients are not significant at the 10% level in most cases. The sole exception is the slope

for CHML, which is significant in the regression for q = 12. Conditional on both CSMB

and CHML, CUMD is negatively correlated with expected market returns at q = 1 and is

positively correlated at q = 12, 60, but all the coefficients are highly insignificant. Regard-

ing the liquidity factor, it forecasts positive market returns at q = 1, 12 and negative returns

for q = 60, conditional on both CSMB and CHML. However, all the predictive slopes are

largely insignificant. Finally, both TERM and DEF forecast positive market returns at all

horizons, conditional on CSMB and CHML, although only DEF is a significant predictor (at

q = 12, 60).

When we compare the predictive slopes from the regressions at q = 60 (for which there is

greater evidence of predictability) with the factor risk price estimates in the benchmark GMM

test, only FF3 (in the test with SBM25) and C (both tests) satisfy the sign restrictions. However,

there is no evidence that the state variables associated with SMB and UMD forecast market

returns at any horizon. Hence, the results in the benchmark case that the FF3 and C models

satisfy the ICAPM criteria rely crucially on the way we construct the proxies for SMB and

HML in the predictive regressions.

5.7. Alternative proxy for the expected market return

There is a long literature discussing the biases and low statistical power of predictive regressions

of stock market returns as those analyzed in Section 3.25 On the other hand, Cochrane (2008)

argues that predictive regressions for stock returns have greater statistical power at very long

horizons and that the current log market dividend yield (DY ) is a very good proxy for expected

long-run market returns. By using a variance decomposition for DY , Cochrane shows that

nearly all the variation in DY is due to the predictability of long-run returns by the dividend

yield. In other words, the long-run return coefficient of expected returns on DY is close to one.

Thus we can use DY as a proxy for long-run expected market returns in our analysis.

We conduct the following contemporaneous multiple regressions of DY on other state vari-

25An incomplete list includes Nelson and Kim (1993), Stambaugh (1999), Kilian (1999), Valkanov (2003),Lewellen (2004), Torous, Valkanov, and Yan (2004), Campbell and Yogo (2006), Ang and Bekaert (2007), andBoudoukh, Richardson, and Whitelaw (2008).

31

ables:

DYt = η0 + η1TERMt + η2DEFt + vt, (47)

DYt = η0 + η1TERMt + η2PEt + η3V St + vt, (48)

DYt = η0 + η1TERMt + η2CPt + vt. (49)

Results available in the addendum show that in all three regressions the coefficients asso-

ciated with TERM are negative. Conditional on TERM , DEF is positively correlated with

DY , while conditional on TERM , the slopes for PE and V S are negative and positive, re-

spectively. Finally, conditional on TERM , CP forecasts positive market returns. Based on

heteroskedasticity-robust standard errors (White, 1980) all the coefficient estimates are signifi-

cant at the 1% level, with the sole exception of V S which is not significant at the 10% level. The

implication of these results is that in most cases the consistency in sign with the cross-sectional

risk prices is not satisfied. The exception is for KLVN tested on SM25, which now meets all the

ICAPM criteria.

We also conduct regressions of DY on the state variables associated with the empirical

factors:

DYt = η0 + η1SMB∗t + η2HML∗t + vt, (50)

DYt = η0 + η1SMB∗t + η2HML∗t + η3CUMDt + vt, (51)

DYt = η0 + η1SMB∗t + η2HML∗t + η3CLt + vt, (52)

DYt = η0 + η1SMB∗t + η2HML∗t + η3TERMt + η4DEFt + vt. (53)

The results reported in the paper’s addendum show that the signs of the coefficients are

basically the same as the corresponding slopes in the forecasting regressions on stock returns

and most point estimates are strongly significant (1% or 5% levels). The sole exceptions are the

coefficients associated with TERM and CL, which now become negative, although the slope for

CL is highly insignificant. This implies that the FF5 model when tested on the SM25 portfolios

(but not on SBM25) turns out to be consistent with the ICAPM. In contrast, the PS model no

longer satisfies the hedging risk price criteria in the test with SBM25.

32

6 Higher order moments of the investment opportunity set

6.1. The ICAPM with volatility forecasts

We have observed that a given risk factor is compatible with the ICAPM framework if its

associated state variable forecasts changes in either the first or the second moments of market

returns. We now analyze whether the candidate risk factors we have discussed forecast the

volatility of market returns and whether the predictive slopes are consistent with the factor risk

prices from all the various cross-sectional tests.

To see if each of the candidate state variables individually forecasts the future volatility of

aggregate stock returns, we conduct single long-horizon regressions on the realized stock market

variance (SV AR):

SV ARt,t+q = aq + bqzt + vt,t+q, (54)

where SV ARt,t+q = SV ARt+1 + ...+SV ARt+q is the cumulative sum of SV AR over q periods,

and vt,t+q represents a forecasting error with zero conditional mean.26 From (54) it follows that

Et(SV ARt,t+q) = aq + bqzt. If bq > 0 (bq < 0), the state variable is associated with positive

(negative) changes in the future volatility of aggregate returns.


The results reported in Table 11 show that PE and V S forecast positive variability in future

market returns at all horizons, but the slopes are statistically significant only at short horizons

(from one to 12 months). The slopes associated with DY are significantly negative at all

horizons. In the case of TERM and DEF , the slopes are positive at short horizons and become

negative for horizons greater than three months (TERM) or 24 months (DEF ). However, these

estimates are not statistically significant, with the exception of DEF at very short horizons.

In the cases of RF and CP we have the opposite pattern, with negative coefficients at short

horizons and positive slopes for longer horizons, but there is statistical significance only at very

short horizons.

Thus, if we restrict ourselves to the predictive slopes at shorter horizons (for which there

is greater statistical significance) the forecasting coefficients in the regressions for SV AR have

opposite signs relative to the single predictive regressions for expected market returns when

26SV AR is computed as the sum of squared daily returns on the S&P 500 (Guo, 2006b; and Goyal and Welch,2008). The data on SV AR are obtained from Amit Goyal’s webpage.

33

the state variables are DY , RF , PE, V S, and CP . Since a positive slope in the predictive

regressions of the market volatility corresponds to a negative risk price in the cross-sectional

regression as shown in Section 2, the slopes associated with TERM , DEF , DY , PE, V S, and

CP are consistent with the corresponding factor risk prices.27 However, the risk price estimates

associated with ∆DY and ∆PE (test with SM25) and ∆DEF (test with SBM25) are highly

insignificant. When we account for the restriction of a risk aversion coefficient (RRA) above

one the two-factor models based on DEF , V S, and CP satisfy the ICAPM criteria in the tests

with SBM25, while the specifications based on TERM , DY , and PE meet these criteria in the

tests with SM25.

When we compare the slopes from the single forecasting regressions for the market variance

with the hedging risk prices associated with the HL, P, CV, and KLVN models in Section 4, we

find that the HL and KLVN models now meet the sign restrictions in the hedging risk prices

when tested on the SM25 (but not on SBM25) portfolios. However, as referred above, TERM

is not a significant predictor of the volatility of aggregate returns.

We also conduct multiple long-horizon regressions for the market variance with the variables

associated with Models HL, P, CV, and KLVN:

SV ARt,t+q = aq + bqTERMt + cqDEFt + ut,t+q, (55)

SV ARt,t+q = aq + bqTERMt + cqDEFt + dqDYt + eqRFt + ut,t+q, (56)

SV ARt,t+q = aq + bqTERMt + cqPEt + dqV St + ut,t+q, (57)

SV ARt,t+q = aq + bqTERMt + cqCPt + ut,t+q. (58)

The slope estimates are displayed in Table 12, which is similar to Table 4. We can see

that most of the slopes in the four long-horizon regressions flip signs relative to the predictive

regressions for expected returns discussed in Section 3. If we focus on the results at q = 60

(for which the evidence of predictability is stronger, as indicated by the adjusted R2 estimates),

all four models do not satisfy the consistency criteria on the hedging risk prices, since there

is at least one state variable in each model with the wrong sign. Focusing on the one-month

forecasting regressions for market volatility the predictive slopes in the HL and KLVN models

satisfy the sign consistency with the corresponding risk prices when the testing portfolios are

27However, in the case of V S and CP there is consistency only in the tests with SBM25, while in the case ofTERM , DY , and PE the consistency only holds for the tests with the SM25 portfolios.

34

SM25. However, the RRA estimate is relatively large (around 11) in the case of KLVN and there

is no evidence that TERM (conditional on DEF ) forecasts SV AR at the one-month horizon.

In the case of the predictive regressions at q = 12, there is consistency in the hedging risk prices

for the HL and KLVN models tested on SBM25, but these specifications do not satisfy the

restrictions on the RRA parameter as discussed in Section 4.


Next, we conduct multiple long-horizon regressions for the variance of returns corresponding

to the four empirical-based multifactor models, FF3, C, PS, and FF5,

SV ARt,t+q = aq + bqSMB∗t + cqHML∗t + ut,t+q, (59)

SV ARt,t+q = aq + bqSMB∗t + cqHML∗t + dqCUMDt + ut,t+q, (60)

SV ARt,t+q = aq + bqSMB∗t + cqHML∗t + dqCLt + ut,t+q, (61)

SV ARt,t+q = aq + bqSMB∗t + cqHML∗t + dqTERMt + eqDEFt + ut,t+q. (62)

The results are presented in Table 13, which is similar to Table 5. We can see that both

SMB∗ and HML∗ are negatively correlated with the future market variance, although SMB∗

is statistically significant (10% level) only at q = 12. Conditional on both SMB∗ and HML∗,

CUMD forecasts positive market volatility, while CL is negatively correlated with future SV AR

at q = 1 and positively correlated at longer horizons. However, the slope estimates associated

with both CUMD and CL are not statistically significant at any horizon. Conditional on both

SMB∗ and HML∗, TERM is negatively correlated with SV AR at q = 12 and q = 60, but this

effect is statistically significant only for q = 12. On the other hand, the slopes associated with

DEF are positive at the three horizons, but there is statistical significance only for q = 1 and

q = 12.

When we compare these predictive slopes with the corresponding risk price estimates in

Table 8, it follows that the FF3 and FF5 models satisfy the sign restrictions on the hedging risk

prices from the test with SBM25. However, the RRA estimate in the FF5 model is negative.

Thus, only the FF3 model satisfies the ICAPM criteria in the test with SBM25. When the

comparison is made against the risk price estimates from the test with SM25, none of the four

multifactor models meets the criteria.


35

Thus, when future investment opportunities are driven by market volatility, only the FF3

model meets the ICAPM criteria among the four empirical factor models (and only in the test

with the SBM25 portfolios). This means that none of the other three models can be justified

on the argument that the respective state variables forecast changes in future stock market

volatility. Specifically, the Carhart (1997) model can be justified as an ICAPM application only

when we consider future expected market return as the single dimension of future investment

opportunities.28

We also conduct the predictive regressions for SV AR by using the alternative state variable

proxies CSMB and CHML in place of SMB∗ and HML∗, respectively. In results available

in the addendum to this paper, in most cases the predictive slopes have the same sign as in

the benchmark regressions (59)-(62). The only exception is the regression for the FF5 model at

q = 60, in which the coefficients associated with CSMB, TERM , and DEF flip sign relative

to the benchmark regression. However, these estimates are highly insignificant.

6.2. Simultaneous variation in the first two moments of aggregate returns

In the above analysis we test separately the consistency between the cross-sectional risk prices

and the predictive slopes for the first two moments of aggregate returns. In principle, a state

variable can forecast both the mean and variance of market returns in such a way that the two

“cancel out” maintaining a constant conditional Sharpe ratio with no predictive power for the

investment opportunity set. According to the ICAPM this variable should not be priced in the

cross-section although the comparison between the factor risk price and each of the predictive

slopes would point to an inconsistency with the ICAPM.

To address this issue we measure the impact of each state variable on a conditional Sharpe

ratio, which proxies for the net change in the investment opportunity set.29 A rise in this

ratio signals an improvement in future investment opportunities (better mean and/or lower

variance). Following the discussion in Section 2, if a given state variable is positively (negatively)

correlated with the conditional Sharpe ratio, the corresponding factor risk price should be

positive (negative). In other words, the signs of the slopes are interpreted in the same way as

the coefficients in the regressions for the market return.28We conduct predictive regressions by using alternative proxies for the stock market return variance. Results

available on the addendum to this paper show that the finding that the FF3 model can be consistent with thecorresponding state variables forecasting SV AR does not generalize to the alternative volatility measures.

29Under some assumptions (mean-variance objective function and that the state variables follow Markov pro-cesses), Brennan and Xia (2006), and Nielsen and Vassalou (2006) show that the slope and the intercept of thecapital market line are two sufficient statistics for all ICAPM state variables.

36

We follow Whitelaw (1994), and Tang and Whitelaw (2011) in estimating the conditional

Sharpe ratio for each model and at each forecasting horizon. To exemplify our approach, consider

the HL model. First, we compute the conditional expected return as the fitted value, rt,t+q from

the predictive regression on the market return:

rt,t+q = a1q + b1qTERMt + c1qDEFt + ut,t+q. (63)

Then, we compute the fitted conditional variance, SV ARt,t+q from the regression:

SV ARt,t+q = a2q + b2qTERMt + c2qDEFt + vt,t+q. (64)

The pseudo conditional Sharpe ratio is then equal to:30

SRt,t+q =rt,t+q√

SV ARt,t+q

=a1q + b1qTERMt + c1qDEFt(a2q + b2qTERMt + c2qDEFt

) 12

. (65)

Finally, we compute the time series average of the partial derivatives of SRt,t+q with respect to

each state variable in the factor model. In the case of TERM this partial derivative is equal to

∂SRt,t+q

∂TERMt= b1qSV AR

− 12

t,t+q −12SV AR

− 32

t,t+q b2q rt,t+q. (66)

Results available on the paper’s addendum show that for the eight factor models most

implied slopes on the pseudo Sharpe ratio have the same signs as the corresponding slopes from

the predictive regressions for the market return in Section 3 at q = 60. The sole exception is the

coefficient associated with DEF in the FF5 model which is now negative. Thus, these results

show that the effect of the state variables on the mean aggregate return dominate the impact on

the market volatility. This implies that the qualitative conclusions regarding the factor models

satisfying the ICAPM criteria are mostly the same as in Section 4. The exception is the case of

the FF5 model estimated with SBM25, which satisfies the sign restrictions on the hedging risk

prices. Yet, the corresponding negative estimate for the risk aversion parameter invalidates this

model as a plausible ICAPM application.

30To obviate the problem of fitted negative variance due to the linear specification of the predictive regression,

we censorize the fitted variance at the minimum sample value of SV AR, that is, SV ARt,t+q = max[a2q +

b2qTERMt + c2qDEFt,min(SV ARt)].

37

7 Power and size

In this section, we conduct different Monte Carlo simulations to assess the power and size of the

ICAPM tests in the previous sections. In the first simulation we want to check if the ICAPM

criteria are satisfied for artificial return and factor data constructed under the assumption that

the ICAPM does not hold.

For that, we simulate 10,000 pseudo samples of an artificial state variable according to an

AR(1) process:

zbt+1 = ψ + φzb

t + εbz,t+1, b = 1, ..., 10000, (67)

where εbz,t+1 denotes a simulated zero-mean normally-distributed error. In the simulation of

εbz,t+1 we use the sample variance of the residuals from the AR(1) process of TERM . We

calibrate the auto-regressive coefficient at two possible values, 0.95 and 0.99, which are consistent

with the estimated coefficients obtained for our state variables in Table 2. We fix the drift ψ at

the sample mean of TERM . For each pseudo sample, the initial realization of zbt is constructed

as

zb1 ∼ N

(ψ

1− φ,

11− φ2

), (68)

and the innovation in the state variable is defined as ∆zbt+1 = zb

t+1 − zbt .

The excess returns on the test assets are simulated according to the following equation:

(Ri,t+1 −Rf,t+1)b = µi + εbi,t+1, b = 1, ..., 10000, i = 1, ..., N, (69)

where µi represents the expected excess return for asset i which is assumed to be constant

over time and εbi,t+1 denotes a simulated zero-mean normal error for asset i. These errors are

simulated from the sample covariance matrix of the excess returns of the SBM25 portfolios and

the market portfolio (the Nth asset in our cross-sectional tests).

Under this data-generating process for the non-ICAPM world we assume that the CAPM

holds, that is, the individual expected excess returns are given by

µi = γ Cov(Ri,t+1 −Rf,t+1, RMt+1), (70)

where Cov(Ri,t+1−Rf,t+1, RMt+1) denotes the covariances estimated with the original sample,

and γ is fixed at 2.80, which represents the RRA estimate obtained in the test of the CAPM

38

with SBM25 for the original sample.

The log market return used in the predictive regressions, rbt+1 = ln(1 + Rb

m,t+1) is obtained

from the simulated simple market return:

Rbm,t+1 = (Rm,t+1 −Rf,t+1)b +Rf,t+1, (71)

where Rf,t+1 denotes the sample mean of the risk-free rate.

Armed with the simulated data, we run single predictive regressions (16) at the one-month

and 60-month horizons and obtain an empirical distribution of the predictive slopes (b1, b60)

and associated asymptotic t-statistics. Similarly, we estimate the two-factor ICAPM pricing

equation (39) with the pseudo data and obtain an empirical distribution of the hedging risk

price (γz) and associated t-statistic.

To check whether the restriction on the hedging risk price is satisfied we compute the fraction

of pseudo samples in which the signs of the predictive slope and risk price are coincident requiring

also that both coefficients be significant at the 5% level on those samples (we call this fraction

the ICAPM acceptance rate). The results in Table 14 (last two rows) show that for both values

of φ, these acceptance rates are virtually zero when the comparison is made against both the

one-month and 60-month predictive slopes. These low fractions are the result of low individual

statistical significance for both the predictive coefficient (below 7% in the one-month predictive

regressions) and the hedging risk price estimates (1%). These results show that in an world in

which the ICAPM does not hold, the sign restrictions on the intertemporal risk price are almost

never satisfied.


Next, we conduct an alternative Monte Carlo simulation in which we assume that the ICAPM

does hold. The question we want to address is how likely it is that the restrictions on the ICAPM

criteria are found significant in this ICAPM world.

The first difference relative to the previous data generating process is that the errors for the

state variable, εbz,t+1, and the excess returns, εbi,t+1 are simulated from the sample augmented

covariance matrix of the state variable innovations and excess returns, that is, we allow for

the contemporaneous correlation between the state variable and the test assets, including the

market equity premium.

39

The second major difference is that we allow expected returns to be time-varying by assuming

that the conditional covariances with the market factor are linear in the lagged state variable:

Covt(Ri,t+1 −Rf,t+1, RMt+1) = ai + bizt, i = 1, ..., N, (72)

where the coefficients ai and bi are estimated from the following time-series regressions for the

original sample:

(Ri,t+1 −Rf,t+1)RMt+1 = ai + biTERMt + νi,t+1, i = 1, ..., N. (73)

Thus, the pseudo return generating process is given by

(Ri,t+1 −Rf,t+1)b = µbit + εbi,t+1, b = 1, ..., 10000, i = 1, ..., N, (74)

µbit = γai + γbiz

bt + γz Cov(Ri,t+1 −Rf,t+1,∆zt+1), (75)

where the covariances with the innovation in the state variable, Cov(Ri,t+1 − Rf,t+1,∆zt+1)

are estimated from the original sample for TERM . In the above equation, we calibrate three

alternative values for γ (3, 4, and 5), where three is close to the estimate obtained for the

CAPM. We also calibrate γz at three alternative values, 300, 600, and 1,200, where 600 is close

to the estimate obtained for the two-factor ICAPM with TERM in the original sample. This

gives a total of 18 alternative Monte Carlo simulations.

The results in Table 14 show that the acceptance rates of the ICAPM are relatively high,

varying between 86% and 96% in the case of the one-month predictive regression, while in the

case of the 60-month regression, these rates vary between 85% and 96%. These high acceptance

rates are the result of large individual statistical significance for the predictive slope (above 89%

of the simulations) and the factor risk price (over 96%), in conjunction with the fact that the

slope and risk price estimates in the pseudo samples have the same sign most of the time (over

91% in untabulated results).

Overall, this simulation shows that in an ICAPM world there is a very high probability that

the restrictions on the intertemporal risk prices will be true in a sample similar to ours.

We conduct a third Monte Carlo simulation to control for the bias associated with a missing

risk factor in the ICAPM pricing equation.31 Specifically, we assume that the true ICAPM

31We thank the referee for this suggestion.

40

specification contains two hedging risk factors: one that is observable and hence used in the

empirical test and another state variable that is not observable to the econometrician. Re-

sults presented in the addendum show that the acceptance rates, although lower than in the

benchmark simulation discussed above, are still above 50%, thus showing that the ICAPM is

reasonably robust to the bias caused by a missing state variable in the pricing equation.

8 Conclusion

Is the ICAPM a “fishing license” for empirical multifactor models as Fama (1991) claims? We

have studied the restrictions associated with the ICAPM for a time-series of the market return

and a cross-section of portfolios. By using a simple version of the Merton (1973) ICAPM, we

identify three main conditions for a multifactor model to meet to be justifiable by the ICAPM.

First, the candidates for ICAPM state variables must forecast the first or second moments

of stock market returns. Second, and most importantly, the state variables should forecast

changes in investment opportunities with the same sign as its innovation prices the cross-section.

Specifically, if a given state variable forecasts positive expected returns it should earn a positive

risk price in the cross-sectional test of the respective multifactor model. The third restriction

associated with the ICAPM is that the market (covariance) risk price estimated from the cross-

sectional tests must be economically plausible as an estimate of the coefficient of relative risk

aversion (RRA) of a representative investor.

We apply our ICAPM criteria to 8 multifactor models, tested over 25 portfolios sorted on

size and book-to-market (SBM25), and 25 portfolios sorted on size and momentum (SM25). Our

results show that only in three out of 16 tests are factor risk prices consistent with the ICAPM

theory: the Fama and French (1993) three-factor model tested over SBM25, and the Carhart

(1997) model tested over SBM25 and SM25. Thus these models can be justified as empirical

applications of the ICAPM. When we consider changes in the investment opportunity set driven

by the second moment of aggregate returns, only the Fama and French (1993) model tested with

the SBM25 portfolios satisfies the ICAPM criteria. In most models, there are inconsistencies in

both the risk price estimates associated with the “hedging” risk factors and the risk-aversion

estimates. These findings are robust to many different specifications of our tests.

Overall, the Fama and French (1993) three-factor model performs the best in consistently

meeting the ICAPM restrictions when investment opportunities are driven by the first two

moments of aggregate returns when tested with the SBM25 portfolios. Apart from this model

41

and the Carhart (1997) model, the other models cannot be justified with the ICAPM theory.

The ICAPM is not really a “fishing license” after all.

42

Appendix A. GMM formulas

Following Cochrane (2005), the weighting matrix associated with the GMM system (36) is givenby

W =[W∗ 00 IK+1

], (1)

where W∗ (N ×N) is the weighting matrix associated with the first N moments; 0 denotes aconformable matrix of zeros; and IK+1 denotes a K + 1-dimensional identity matrix. In thisspecification, W∗ is the weighting matrix for the first N moment conditions (correspondingto the N pricing errors), while IK+1 is the weighting matrix associated with the last K + 1orthogonality conditions that identify the factor means.

In the first-step GMM (OLS cross-sectional regression), W∗ corresponds to the identitymatrix, W∗ = IN , and in the second-step GMM (GLS cross-sectional regression), W∗ is theinverse of the first (N ×N) block of the spectral density matrix, W∗ = S−1

N .The risk price estimates b have variance formulas given by

Var(b) =1T

(d′Wd)−1d′WSWd(d′Wd)−1, (2)

where d ≡ ∂gT (b)∂b′ represents the matrix of moments’ sensitivities to the parameters; S is an

estimator for the spectral density matrix S derived under the heteroskedasticity-robust or White(1980) standard errors, that is, no lags of the moment functions are considered in the compu-tation of S.32

Appendix B. Bootstrap simulation for cross-sectional test

The bootstrap algorithm used in the estimation of the pricing equations in the cross-section ofstock returns consists of the following steps:

1. Each model is estimated by first-stage GMM, and we save the t-statistics associated withthe market and hedging risk prices, [t(γ), t(γ1), ..., t(γK)].

2. In each replication b = 1, ..., 10000, we construct a pseudo-sample of excess returns foreach asset (of size T ) by drawing with replacement:

{(Ri,t+1 −Rf,t+1)b, t = sb1, s

b2, ..., s

bT }, i = 1, ..., N, (3)

where the time indices sb1, s

b2, ..., s

bT are created randomly from the original time sequence

1, ..., T . Notice that all excess returns have the same time sequence to preserve the con-temporaneous cross-correlation between asset returns.

3. For each replication b = 1, ..., 10000, we also construct an independent pseudo-sample ofthe factors:

{RM bt+1, f

b1,t+1, ..., f

bK,t+1, t = rb

1, rb2, ..., r

bT }, (4)

where the time sequence (rb1, r

b2, ..., r

bT ) is independent from sb

1, sb2, ..., s

bT . Notice that the

time sequence is the same for all factors to preserve their cross-correlations.

4. In each replication, we estimate the factor model by first-stage GMM, but using the

32As stressed by Cochrane (2005), if the asset pricing model is true, then the moments that define the pricingerrors will be orthogonal to all past information, including the past pricing errors. This is equivalent to using theNewey and West (1987) algorithm with zero lags.

43

artificial data rather than the original data. The moment conditions are given by:

gT (b) ≡ (5)

1T

T−1∑t=0

(Ri,t+1 −Rf,t+1)b − γb(Ri,t+1 −Rf,t+1)b(RM b

t+1 − µbm

)−γb

1(Ri,t+1 −Rf,t+1)b(f b1,t+1 − µb

1

)− γb

2(Ri,t+1 −Rf,t+1)b(f b2,t+1 − µb

2

)−...− γb

K(Ri,t+1 −Rf,t+1)b(f b

K,t+1 − µbK

)RM b

t+1 − µbm

f b1,t+1 − µb

1...

f bK,t+1 − µb

K

= 0,

i = 1, ..., N.

We save the t-statistics associated with the individual factor risk prices, [t(γb), t(γb1), ..., t(γb

K)]leading to an empirical distribution of the t-statistics.

5. The empirical p-value associated with the market risk price (for a two-sided test) is com-puted as:

p(γ) ={ [

#{t(γb) ≥ t(γ)

}+ #

{t(γb) ≤ −t(γ)

}]/10000, if γ ≥ 0[

#{t(γb) ≤ t(γ)

}+ #

{t(γb) ≥ −t(γ)

}]/10000, if γ < 0

, (6)

and similarly for the other factor risk prices. In the above expression, #{t(γb) ≥ t(γ)

}denotes the number of replications in which the pseudo t-stats are greater or equal thanthe t-ratio from the original sample.

Appendix C. Bootstrap algorithm for predictive regressions

The bootstrap algorithm associated with the long-horizon regressions consists of the followingsteps:

1. The long-horizon regression is estimated by OLS, and we save the slope estimate, bq:

rt,t+q = aq + bqzt + ut,t+q. (7)

We also estimate by OLS the following system that imposes the joint null of no predictabil-ity of returns and a persistent predictor that follows an AR(1) process:

rt,t+q = aq + ut,t+q, (8)zt+1 = ψ + φzt + εt+1. (9)

The time-series of OLS residuals, ut,t+q and εt+1 and the OLS estimates, aq, ψ, φ are saved.

2. In each replication m = 1, ..., 10000, we construct pseudo-samples for the innovations inthe market return and the predictor by drawing with replacement from the two residuals:

{umt,t+q}, t = sm

1 , sm2 , ..., s

mT , (10)

{εmt+1}, t = sm1 , s

m2 , ..., s

mT , (11)

where the time indices sm1 , s

m2 , ..., s

mT are created randomly from the original time sequence

1, ..., T . Notice that the innovations in both the return and predictor have the same timesequence to account for their contemporaneous cross-correlation.

44

3. For each replication m = 1, ..., 10000, we construct a pseudo-sample of the market returnand predictor, by imposing the null:

rmt,t+q = aq + um

t,t+q, (12)

zmt+1 = ψ + φzm

t + εmt+1. (13)

4. In each replication, we estimate the long-horizon regression, but using the artificial datarather than the original data:

rmt,t+q = am

q + bmq zmt + vm

t,t+q. (14)

The initial value for zt, z0, is picked at random from one of the observations of zt. In

result, we have an empirical distribution of the regression slope estimates,{bmq

}10,000

m=1(as

opposed to the asymptotic theoretical distribution).

5. The p-value associated with the slope estimate is calculated as

p(bq) =

[#{bmq ≥ bq

}+ #

{bmq ≤ −bq

}]/10000, if bq ≥ 0[

#{bmq ≤ bq

}+ #

{bmq ≥ −bq

}]/10000, if bq < 0

, (15)

where #{bmq ≥ bq

}denotes the number of bootstrapped slope estimates that are higher

than the original slope estimate.

45

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49

List of Tables

1 Consistency of multifactor models with the ICAPMThis table reports the consistency of the factor risk prices from multifactor models withthe ICAPM criteria. The criteria is associated with the magnitude of the market riskprice (or risk aversion coefficient) (γ), and the consistency in sign of the risk prices of thehedging factors with the corresponding predictive slopes over the excess market return(γz,E(r)) and the market variance (γz, σ

2(r)). The multifactor models are Hahn and Lee(2006) (HL), Petkova (2006) (P), Campbell and Vuolteenaho (2004) (CV), Koijen, Lustig,and Van Nieuwerburgh (2010) (KLVN), Fama and French (1993) (FF3), Carhart (1997)(C), Pastor and Stambaugh (2003) (PS), and the Fama and French (1993) five-factormodel (FF5). The testing assets in the cross-sectional tests are the 25 size/book-to-market portfolios (SBM25, Panel A), and 25 size/momentum portfolios (SM25, PanelB). A “X” indicates that the ICAPM criteria is satisfied. . . . . . . . . . . . . . . . . 51

2 Descriptive statistics for state variablesThis table reports descriptive statistics for the state variables used in the predictiveregressions. The forecasting variables are the term-structure spread (TERM); defaultspread (DEF ); market dividend yield (DY ); one-month Treasury bill rate (RF ); marketprice-earnings ratio (PE); value spread (V S); Cochrane-Piazzesi factor (CP ); size pre-mium (SMB∗); value premium (HML∗); momentum premium (CUMD); and liquidityfactor (CL). The sample is 1963:07–2008:12. φ designates the first order autocorrelationcoefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Single predictive regressions for ICAPM state variables . . . . . . . . . . . . . . . 534 Multiple predictive regressions for ICAPM state variables . . . . . . . . . . . . . 545 Multiple predictive regressions for state variables constructed from empirical factors 556 Factor risk premiums for ICAPM state variables . . . . . . . . . . . . . . . . . . 567 Factor risk premiums for ICAPM specifications . . . . . . . . . . . . . . . . . . . 578 Factor risk premiums for empirical risk factors . . . . . . . . . . . . . . . . . . . 589 Beta factor risk premiums for ICAPM specifications . . . . . . . . . . . . . . . . 5910 Beta factor risk premiums for empirical risk factors . . . . . . . . . . . . . . . . . 6011 Single predictive regressions for ICAPM state variables (SV AR) . . . . . . . . . 6112 Multiple predictive regressions for state variables (SV AR) . . . . . . . . . . . . . 6213 Multiple predictive regressions for state variables constructed from empirical fac-

tors (SV AR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6314 Monte Carlo simulation

This table reports results for a Monte Carlo simulation with 10,000 replications of anartificial two-factor ICAPM. φ, γ and γz denote the calibrated values for the autore-gressive coefficient of the state variable, risk aversion coefficient and hedging risk price,respectively. t(b1) (t(b60)) denotes the fraction of replications in which the predictiveslope in the one-month (60-months) regression is statistically significant at the 5% level,while t(γz) denotes the percentage of replications in which the risk price for the hedgingfactor is statistically significant. Sign(b1) (Sign(b60)) denotes the fraction of replicationsfor which the predictive slope in the one-month (60-months) regression shares the samesign as the hedging risk price, and both coefficients are significant at the 5% level. Thelast two rows are associated with a simulation in which the ICAPM is not true. . . . . . 64

50

Table 1: Consistency of multifactor models with the ICAPMThis table reports the consistency of the factor risk prices from multifactor models with the ICAPMcriteria. The criteria is associated with the magnitude of the market risk price (or risk aversion coefficient)(γ), and the consistency in sign of the risk prices of the hedging factors with the corresponding predictiveslopes over the excess market return (γz,E(r)) and the market variance (γz, σ

2(r)). The multifactormodels are Hahn and Lee (2006) (HL), Petkova (2006) (P), Campbell and Vuolteenaho (2004) (CV),Koijen, Lustig, and Van Nieuwerburgh (2010) (KLVN), Fama and French (1993) (FF3), Carhart (1997)(C), Pastor and Stambaugh (2003) (PS), and the Fama and French (1993) five-factor model (FF5). Thetesting assets in the cross-sectional tests are the 25 size/book-to-market portfolios (SBM25, Panel A),and 25 size/momentum portfolios (SM25, Panel B). A “X” indicates that the ICAPM criteria is satisfied.

γ γz,E(r) γz, σ2(r)

Panel A (SBM25)HL × × ×P × × ×

CV × × ×KLVN × X ×FF3 X X XC X X ×PS × X ×

FF5 × × XPanel B (SM25)

HL X × ×P × × ×

CV × × ×KLVN × × ×FF3 × × ×C X X ×PS × × ×

FF5 × × ×

51

Table 2: Descriptive statistics for state variablesThis table reports descriptive statistics for the state variables used in the predictive re-gressions. The forecasting variables are the term-structure spread (TERM); default spread(DEF ); market dividend yield (DY ); one-month Treasury bill rate (RF ); market price-earnings ratio (PE); value spread (V S); Cochrane-Piazzesi factor (CP ); size premium(SMB∗); value premium (HML∗); momentum premium (CUMD); and liquidity factor (CL).The sample is 1963:07–2008:12. φ designates the first order autocorrelation coefficient.

Mean Stdev. Min. Max. φTERM 0.009 0.011 −0.031 0.033 0.967DEF 0.010 0.005 0.003 0.034 0.992DY −3.580 0.413 −4.495 −2.801 0.996RF 0.005 0.002 0.000 0.014 0.955PE 2.871 0.448 1.893 3.789 0.997V S 1.559 0.159 1.201 2.222 0.938CP 0.010 0.016 −0.052 0.070 0.845

SMB∗ −0.122 0.250 −1.033 0.780 0.924HML∗ −2.837 0.981 −6.445 −0.895 0.966CUMD 0.508 0.199 −0.132 1.081 0.961CL 0.033 0.582 −1.496 0.904 0.991

52

Table 3: Single predictive regressions for ICAPM state variablesThis table reports the results for single long-horizon regressions for the monthly continuously compoundedreturn on the value-weighted market index, at horizons of 1, 3, 12, 24, 36, 48, and 60 months ahead.The forecasting variables are the current values of the term-structure spread (TERM); default spread(DEF ); market dividend yield (DY ); one-month Treasury bill rate (RF ); market price-earnings ratio(PE); value spread (V S); and the Cochrane-Piazzesi factor (CP ). The original sample is 1963:07-2008:12,and q observations are lost in each of the respective q-horizon regressions, for q = 1, 3, 12, 24, 36, 48, 60.For each regression, in line 1 are reported the slope estimates, and in lines 2 and 3 are reported Newey-West (in parenthesis) and Hansen-Hodrick t-ratios (in brackets) computed with q lags. Italic, underlined,and bold t-statistics denote statistical significance at the 10%, 5%, and 1% levels, respectively. R2(%)denotes the adjusted coefficient of determination (in %).

q = 1 q = 3 q = 12 q = 24 q = 36 q = 48 q = 60TERM 0.16 0.37 1.78 2.28 3.65 5.43 7.20

(0.88) (0.79) (1.07) (1.15) (1.59) (1.44) (1.28)[0.89] [0.71] [0.98] [1.16] [1.44] [1.33] [1.16]

R2(%) 0.16 0.26 1.56 1.45 2.71 4.83 6.29

DEF 0.67 2.04 7.30 7.89 11.58 17.96 27.82(1.30) (1.41) (1 .85 ) (1.49) (1.50) (1 .93 ) (2.91)[1.45] [1.25] [1 .74 ] [1.30] [1.27] [1 .82 ] [3.23]

R2(%) 0.43 1.12 3.74 2.49 4.00 7.71 14.23

DY 0.01 0.03 0.11 0.19 0.25 0.33 0.43(1 .87 ) (2.27) (2.32) (2.23) (2.50) (3.39) (5.66)[1 .88 ] [2.01] [1.96] [2.00] [2.75] [5.57] [5.61]

R2(%) 0.77 2.23 8.18 13.02 17.25 21.83 27.55

RF 0.65 1.88 6.78 18.87 25.55 30.54 41.95(0.68) (0.74) (0.79) (2.28) (2.10) (2.04) (2.13)[0.72] [0.66] [0.78] [2.27] [2.00] [1 .92 ] [2.13]

R2(%) 0.11 0.27 0.89 3.91 5.33 5.99 8.16

PE −0.01 −0.02 −0.09 −0.17 −0.24 −0.32 −0.41(−1.63) (−2.01) (−2.21) (−2.35) (−2.76) (−3.83) (−6.25)[−1.63] [−1 .78 ] [−1 .89 ] [−2.09] [−2.78] [−4.75] [−9.67]

R2(%) 0.57 1.73 7.12 13.16 19.49 25.77 32.42

V S −0.02 −0.06 −0.28 −0.39 −0.51 −0.44 −0.45(−1.41) (−2.37) (−2.94) (−2.13) (−2.07) (−1 .85 ) (−1 .81 )[−1.41] [−2.12] [−2.59] [−1 .94 ] [−2.14] [−2.12] [−1 .92 ]

R2(%) 0.37 1.59 8.09 8.69 10.80 6.20 4.98

CP 0.35 0.84 1.84 2.34 3.59 5.47 6.39(2.63) (2.51) (2.00) (1 .82 ) (2.13) (2.58) (2.08)[2.62] [2.25] [1 .83 ] [1 .71 ] [2.07] [2.44] [1 .87 ]

R2(%) 1.47 2.61 3.24 2.88 4.98 9.29 9.78

53

Table 4: Multiple predictive regressions for ICAPM state variablesThis table reports the results for multiple long-horizon regressions for the monthly continuously com-pounded return on the value-weighted market index, at horizons of 1, 12, and 60 months ahead. Theforecasting variables are the current values of the term-structure spread (TERM); default spread (DEF );market dividend yield (DY ); one-month Treasury bill rate (RF ); market price-earnings ratio (PE); valuespread (V S); and the Cochrane-Piazzesi factor (CP ). The original sample is 1963:07-2008:12, and q ob-servations are lost in each of the respective q-horizon regressions, for q = 1, 12, 60. For each regression,in line 1 are reported the slope estimates, and in lines 2 and 3 are reported Newey-West (in parenthesis)and Hansen-Hodrick t-ratios (in brackets) computed with q lags. Italic, underlined, and bold t-statisticsdenote statistical significance at the 10%, 5%, and 1% levels, respectively. R2(%) denotes the adjustedcoefficient of determination (in %).

Row TERM DEF DY RF PE V S CP R2(%)Panel A (q = 1)

1 0.12 0.62 0.34(0.69) (1.23)[0.70] [1.35]

2 0.21 0.18 0.01 −0.01 0.55(0.81) (0.23) (1.43) (−0.01)[0.86] [0.27] [1.43] [−0.01]

3 0.21 −0.01 −0.01 0.56(1.18) (−1.20) (−0.84)[1.19] [−1.22] [−0.86]

4 −0.10 0.38 1.34(−0.48) (2.49)[−0.50] [2.54]

Panel B (q = 12)1 1.48 6.84 4.62

(0.88) (1 .76 )[0.81] [1 .70 ]

2 2.09 1.75 0.12 −3.26 10.85(1.13) (0.39) (2.01) (−0.29)[0.98] [0.35] [1.64] [−0.25]

3 2.64 −0.06 −0.24 13.34(1 .82 ) (−1.37) (−2.27)[1 .70 ] [−1.22] [−2.19]

4 0.61 1.62 3.19(0.29) (1.29)[0.26] [1.15]

Panel C (q = 60)1 5.86 25.81 18.15

(1.10) (2.56)[0.96] [2.78]

2 17.11 −10.04 0.37 62.35 45.40(2.75) (−0.74) (4.97) (2.06)[2.76] [−0.74] [4.97] [2.06]

3 8.77 −0.48 0.23 43.08(2.49) (−7.74) (2.04)[2.49] [−7.71] [2.05]

4 3.39 5.11 10.60(0.68) (2.45)[0.63] [3.14]

54

Table 5: Multiple predictive regressions for state variables constructed from empirical factorsThis table reports the results for multiple long-horizon regressions for the monthly continuously com-pounded return on the value-weighted market index, at horizons of 1, 12, and 60 months ahead. Theforecasting variables are the current values of the term-structure spread (TERM); default spread (DEF );size premium (SMB∗); value premium (HML∗); momentum factor (CUMD); and liquidity factor (CL).The original sample is 1963:07-2008:12, and q observations are lost in each of the respective q-horizonregressions, for q = 1, 12, 60. For each regression, in line 1 are reported the slope estimates, and in lines2 and 3 are reported Newey-West (in parenthesis) and Hansen-Hodrick t-ratios (in brackets) computedwith q lags. Italic, underlined, and bold t-statistics denote statistical significance at the 10%, 5%, and1% levels, respectively. R2(%) denotes the adjusted coefficient of determination (in %).

Row SMB∗ HML∗ CUMD CL TERM DEF R2(%)Panel A (q = 1)

1 0.00 0.00 0.65(0.00) (1.98)[0.00] [1 .92 ]

2 0.00 0.00 −0.00 0.50(0.02) (1 .92 ) (−0.42)[0.02] [1 .87 ] [−0.44]

3 −0.01 0.01 0.01 0.98(−0.58) (2.50) (1.52)[−0.61] [2.53] [1.58]

4 −0.00 0.00 0.16 0.27 0.53(−0.02) (1.63) (0.90) (0.47)[−0.02] [1.62] [0.91] [0.53]

Panel B (q = 12)1 0.01 0.05 9.91

(0.12) (3.04)[0.11] [2.56]

2 0.01 0.05 0.02 9.81(0.12) (3.01) (0.22)[0.10] [2.56] [0.20]

3 −0.05 0.08 0.07 14.64(−0.64) (4.02) (2.11)[−0.56] [3.70] [1 .88 ]

4 0.01 0.05 1.90 1.69 11.68(0.14) (2.70) (1.24) (0.41)[0.12] [2.28] [1.16] [0.40]

Panel C (q = 60)1 0.32 0.15 36.99

(1 .71 ) (3.27)[1 .77 ] [3.82]

2 0.27 0.17 0.45 42.49(1 .69 ) (3.67) (2.97)[1 .69 ] [3.67] [2.99]

3 0.27 0.17 0.05 37.38(1 .85 ) (5.12) (0.90)[1 .85 ] [5.08] [0.90]

4 0.31 0.14 7.99 6.11 45.79(1 .76 ) (4.89) (1 .72 ) (0.60)[1 .79 ] [6.49] [1.43] [0.57]

55

Table 6: Factor risk premiums for ICAPM state variablesThis table reports the estimation of the factor risk premiums from first-stage GMM with equallyweighted errors. The testing assets are the 25 size/book-to-market portfolios (SBM25, Panel A) and25 size/momentum portfolios (SM25, Panel B). γ represents the risk price for the market factor.γTERM , γDEF , γDY , γRF , γPE , γV S , γCP represent the risk prices associated with the Term-structurespread, default spread, market dividend yield, one-month Treasury bill rate, market price-earnings ratio,value spread, and the Cochrane-Piazzesi factor, respectively. The first line associated with each rowpresents the covariance risk price estimates, and the second line reports the asymptotic GMM robustt-statistics (in parenthesis). The column MAE(%) presents the average absolute pricing error (in %).The column R2

OLS denotes the OLS cross-sectional R2. The sample is 1963:07-2008:12. Italic, underlined,and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively.

Row γ γTERM γDEF γDY γRF γPE γV S γCP MAE(%) R2OLS

Panel A (SBM25)1 −1.86 614.12 0.11 0.72

(−0.61) (1.98)2 2.28 −272.13 0.22 −0.38

(1 .76 ) (−0.93)3 −7.62 −17.13 0.17 −0.01

(−2.01) (−2.72)4 −3.71 −2675.52 0.12 0.43

(−0.96) (−1.52)5 −7.66 17.00 0.18 −0.04

(−1 .87 ) (2.50)6 4.73 −6.21 0.17 0.38

(3.96) (−3.35)7 4.54 227.39 0.14 0.42

(1.62) (2.33)Panel B (SM25)

1 6.78 −547.92 0.27 0.43(2.52) (−2.22)

2 0.99 −763.32 0.31 0.08(0.43) (−2.32)

3 4.48 3.53 0.34 −0.09(1 .74 ) (0.79)

4 −2.29 −1947.25 0.31 0.05(−0.99) (−2.32)

5 4.89 −4.15 0.34 −0.08(1 .86 ) (−0.92)

6 −0.16 8.14 0.32 0.04(−0.12) (2.87)

7 2.10 −26.81 0.34 −0.09(1.59) (−0.52)

56

Table 7: Factor risk premiums for ICAPM specificationsThis table reports the estimation of the factor risk premiums from first-stage GMM with equallyweighted errors. The testing assets are the 25 size/book-to-market portfolios (SBM25, Panel A) and25 size/momentum portfolios (SM25, Panel B). γ represents the risk price for the market factor.γTERM , γDEF , γDY , γRF , γPE , γV S , γCP represent the risk prices associated with the Term-structurespread, default spread, market dividend yield, one-month Treasury bill rate, market price-earnings ra-tio, value spread, and the Cochrane-Piazzesi factor, respectively. The first line associated with eachrow presents the covariance risk price estimates, the second line reports the asymptotic GMM robustt-statistics (in parenthesis), and the third line shows empirical p-values from a bootstrap simulation (inbrackets). The column MAE(%) presents the average absolute pricing error (in %). The column R2

OLS

denotes the OLS cross-sectional R2. The sample is 1963:07-2008:12. Italic, underlined, and bold numbersdenote statistical significance at the 10%, 5%, and 1% levels, respectively.

Row γ γTERM γDEF γDY γRF γPE γV S γCP MAE(%) R2OLS

Panel A (SBM25)1 2.79 0.23 −0.42

(2.52)[0.03]

2 −2.15 608.84 −173.10 0.10 0.74(−0.85) (2.45) (−0.30)[0.58] [0.04] [0.85]

3 −4.66 436.86 234.60 −3.35 −1049.02 0.09 0.77(−0.56) (1.52) (0.57) (−0.19) (−0.90)[0.66] [0.20] [0.66] [0.89] [0.46]

4 −4.83 380.65 8.88 −2.29 0.08 0.78(−0.85) (2.66) (0.98) (−0.72)[0.52] [0.02] [0.46] [0.60]

5 −0.28 485.40 78.68 0.08 0.77(−0.08) (1 .69 ) (0.76)[0.96] [0.21] [0.61]

Panel B (SM25)1 2.34 0.34 −0.10

(2.16)[0.05]

2 5.56 −503.90 −490.50 0.25 0.50(1 .72 ) (−1 .96 ) (−0.98)[0.17] [0.09] [0.46]

3 0.51 −614.81 −56.50 −2.29 −2242.96 0.20 0.67(0.08) (−2.16) (−0.11) (−0.17) (−1.61)[0.95] [0.04] [0.93] [0.89] [0.15]

4 −0.90 −882.29 19.58 −5.37 0.21 0.60(−0.09) (−2.12) (1.22) (−0.77)[0.94] [0.05] [0.34] [0.53]

5 10.77 −778.67 241.79 0.21 0.62(2.13) (−2.35) (1.53)[0.08] [0.04] [0.23]

57

Table 8: Factor risk premiums for empirical risk factorsThis table reports the estimation of the factor risk premiums from first-stage GMM with equallyweighted errors. The testing assets are the 25 size/book-to-market portfolios (SBM25, Panel A)and 25 size/momentum portfolios (SM25, Panel B). γ represents the risk price for the market fac-tor. γTERM , γDEF , γSMB , γHML, γUMD, γL represent the risk prices associated with the Term-structurespread, default spread, size factor, value factor, momentum factor, and liquidity factor, respectively. Thefirst line associated with each row presents the covariance risk price estimates, the second line reports theasymptotic GMM robust t-statistics (in parenthesis), and the third line shows empirical p-values froma bootstrap simulation (in brackets). The column MAE(%) presents the average absolute pricing error(in %). The column R2

OLS denotes the OLS cross-sectional R2. The sample is 1963:07-2008:12. Italic,underlined, and bold numbers denote statistical significance at the 10%, 5%, and 1% levels, respectively.

Row γ γSMB γHML γUMD γL γTERM γDEF MAE(%) R2OLS

Panel A (SBM25)1 3.31 2.84 8.67 0.10 0.69

(2.69) (1 .87 ) (5.49)[0.02] [0.13] [0.00]

2 7.14 2.59 15.37 22.64 0.09 0.78(3.38) (0.94) (4.45) (3.23)[0.00] [0.47] [0.00] [0.00]

3 −0.72 1.88 7.56 8.78 0.10 0.73(−0.33) (0.90) (3.81) (1 .86 )[0.81] [0.49] [0.00] [0.10]

4 −0.32 1.57 3.27 407.79 −1.51 0.09 0.76(−0.12) (0.61) (0.91) (2.54) (−0.01)[0.92] [0.63] [0.47] [0.02] [1.00]

Panel B (SM25)1 0.25 2.76 −4.62 0.32 0.01

(0.19) (1 .78 ) (−1.58)[0.90] [0.15] [0.22]

2 5.00 2.08 10.57 7.24 0.13 0.84(3.10) (1.14) (3.36) (4.24)[0.00] [0.36] [0.00] [0.00]

3 −21.17 −2.63 −18.04 43.01 0.31 0.19(−2.10) (−0.49) (−1.24) (2.56)[0.06] [0.71] [0.32] [0.01]

4 10.58 6.16 16.41 −865.52 183.26 0.16 0.73(1 .81 ) (1.16) (1 .76 ) (−2.17) (0.25)[0.11] [0.32] [0.12] [0.04] [0.83]

58

Table 9: Beta factor risk premiums for ICAPM specificationsThis table reports the estimation of the beta factor risk premiums from OLS cross-sectional re-gressions. The testing assets are the 25 size/book-to-market portfolios (SBM25, Panel A) and 25size/momentum portfolios (SM25, Panel B). λM represents the beta risk price for the market fac-tor. λTERM , λDEF , λDY , λRF , λPE , λV S , λCP represent the beta risk prices associated with the Term-structure spread, default spread, market dividend yield, one-month Treasury bill rate, market price-earnings ratio, value spread, and the Cochrane-Piazzesi factor, respectively. The first line associatedwith each row presents the beta risk price estimates (multiplied by 100), and the second line reports theShanken t-statistics (in parenthesis). The column R2

OLS denotes the OLS cross-sectional R2. The sampleis 1963:07-2008:12. Italic, underlined, and bold numbers denote statistical significance at the 10%, 5%,and 1% levels, respectively.

Row λM λTERM λDEF λDY λRF λPE λV S λCP R2OLS

Panel A (SBM25)1 0.55 −0.42

(2.72)2 0.44 0.50 0.02 0.74

(2.01) (2.43) (0.20)3 0.41 0.47 0.06 −0.35 −0.08 0.77

(2.06) (1 .95 ) (1.04) (−0.30) (−1 .65 )4 0.41 0.32 0.74 −0.96 0.78

(2.07) (2.74) (1.20) (−1.44)5 0.41 0.39 0.50 0.77

(1.97) (1 .67 ) (0.63)Panel B (SM25)

1 0.46 −0.09(2.29)

2 0.57 −0.45 −0.10 0.50(2.73) (−2.51) (−1.59)

3 0.52 −0.34 −0.05 −0.95 −0.06 0.67(2.48) (−1.28) (−0.76) (−1.03) (−1.04)

4 0.52 −0.74 2.04 −0.82 0.60(2.38) (−2.19) (1 .82 ) (−0.47)

5 0.55 −0.70 2.02 0.62(2.20) (−1 .85 ) (1.37)

59

Table 10: Beta factor risk premiums for empirical risk factorsThis table reports the estimation of the beta factor risk premiums from OLS cross-sectional re-gressions. The testing assets are the 25 size/book-to-market portfolios (SBM25, Panel A) and 25size/momentum portfolios (SM25, Panel B). λM represents the beta risk price for the market factor.λTERM , λDEF , λSMB , λHML, λUMD, λL represent the beta risk prices associated with the term-structurespread, default spread, size factor, value factor, momentum factor, and liquidity factor, respectively. Thefirst line associated with each row presents the beta risk price estimates (multiplied by 100), and thesecond line reports the Shanken t-statistics (in parenthesis). The column R2

OLS denotes the OLS cross-sectional R2. The sample is 1963:07-2008:12. Italic, underlined, and bold numbers denote statisticalsignificance at the 10%, 5%, and 1% levels, respectively.

Row λM λSMB λHML λUMD λL λTERM λDEF R2OLS

Panel A (SBM25)1 0.35 0.22 0.49 0.69

(1 .82 ) (1.58) (3.86)2 0.43 0.22 0.52 3.34 0.78

(2.25) (1.58) (4.02) (3.47)3 0.34 0.25 0.45 2.72 0.73

(1 .75 ) (1 .78 ) (3.55) (2.18)4 0.39 0.21 0.44 0.35 0.03 0.76

(2.00) (1.51) (3.39) (2.81) (0.66)Panel B (SM25)

1 0.39 0.40 −0.46 0.01(2.04) (2.69) (−2.12)

2 0.45 0.18 0.47 0.94 0.84(2.34) (1.20) (2.41) (5.38)

3 0.35 0.59 −1.23 12.47 0.19(1 .68 ) (2.50) (−1 .85 ) (2.80)

4 0.35 0.32 0.28 −0.70 −0.04 0.73(1 .65 ) (1.61) (0.66) (2.39) (−0.50)

60

Table 11: Single predictive regressions for ICAPM state variables (SV AR)This table reports the results for single long-horizon regressions for the stock market variance (SV AR),at horizons of 1, 3, 12, 24, 36, 48, and 60 months ahead. The forecasting variables are the current valuesof the term-structure spread (TERM); default spread (DEF ); market dividend yield (DY ); one-monthTreasury bill rate (RF ); market price-earnings ratio (PE); value spread (V S); and the Cochrane-Piazzesifactor (CP ). The original sample is 1963:07-2008:12, and q observations are lost in each of the respectiveq-horizon regressions, for q = 1, 3, 12, 24, 36, 48, 60. For each regression, in line 1 are reported the slopeestimates, and in lines 2 and 3 are reported Newey-West (in parenthesis) and Hansen-Hodrick t-ratios(in brackets) computed with q lags. Italic, underlined, and bold t-statistics denote statistical significanceat the 10%, 5%, and 1% levels, respectively. R2(%) denotes the adjusted coefficient of determination (in%).

q = 1 q = 3 q = 12 q = 24 q = 36 q = 48 q = 60TERM 0.02 0.03 −0.16 −0.50 −0.48 −0.28 0.00

(1.05) (0.62) (−0.89) (−1 .68 ) (−1.06) (−0.48) (0.00)[1.34] [0.55] [−0.73] [−1.51] [−0.92] [−0.41] [0.00]

R2(%) 0.32 0.16 0.93 3.90 2.19 0.55 0.00

DEF 0.18 0.31 0.31 0.07 −0.33 −0.68 −0.73(1.99) (1 .83 ) (0.82) (0.09) (−0.23) (−0.36) (−0.30)[2.62] [1.63] [0.67] [0.07] [−0.19] [−0.30] [−0.26]

R2(%) 3.78 1.90 0.49 0.01 0.15 0.47 0.42

DY −0.00 −0.00 −0.02 −0.03 −0.05 −0.06 −0.07(−4.08) (−3.85) (−3.30) (−2.80) (−2.51) (−2.46) (−2.58)[−4.91] [−3.41] [−2.66] [−2.23] [−2.11] [−2.25] [−2.51]

R2(%) 1.58 3.68 14.79 22.97 27.33 29.31 30.73

RF −0.17 −0.35 0.10 0.90 0.77 0.35 −0.60(−1.38) (−0.99) (0.14) (0.73) (0.46) (0.15) (−0.22)[−1 .83 ] [−0.88] [0.11] [0.64] [0.42] [0.16] [−0.24]

R2(%) 0.84 0.67 0.02 0.50 0.23 0.03 0.07

PE 0.00 0.00 0.01 0.02 0.03 0.04 0.04(2.70) (2.83) (2.30) (1 .89 ) (1.64) (1.50) (1.44)[3.29] [2.51] [1 .83 ] [1.48] [1.32] [1.26] [1.24]

R2(%) 0.54 1.50 6.94 11.68 14.51 15.60 15.90

V S 0.00 0.01 0.03 0.04 0.04 0.03 0.03(4.01) (4.11) (2.19) (1.25) (0.83) (0.50) (0.59)[4.81] [3.63] [1 .77 ] [1.02] [0.73] [0.49] [0.61]

R2(%) 1.12 2.72 6.77 5.02 3.34 1.13 0.96

CP −0.03 −0.07 −0.19 −0.04 0.12 0.28 0.46(−1 .84 ) (−1 .66 ) (−1.61) (−0.15) (0.31) (0.68) (1.16)[−2.24] [−1.47] [−1.36] [−0.13] [0.28] [0.59] [1.00]

R2(%) 1.02 1.39 2.62 0.06 0.25 0.98 2.15

61

Table 12: Multiple predictive regressions for state variables (SV AR)This table reports the results for multiple long-horizon regressions for the stock market variance (SV AR),at horizons of 1, 12, and 60 months ahead. The forecasting variables are the current values of the term-structure spread (TERM); default spread (DEF ); market dividend yield (DY ); one-month Treasury billrate (RF ); market price-earnings ratio (PE); value spread (V S); and the Cochrane-Piazzesi factor (CP ).The original sample is 1963:07-2008:12, and q observations are lost in each of the respective q-horizonregressions, for q = 1, 12, 60. For each regression, in line 1 are reported the slope estimates, and in lines2 and 3 are reported Newey-West (in parenthesis) and Hansen-Hodrick t-ratios (in brackets) computedwith q lags. Italic, underlined, and bold t-statistics denote statistical significance at the 10%, 5%, and1% levels, respectively. R2(%) denotes the adjusted coefficient of determination (in %).

Row TERM DEF DY RF PE V S CP R2(%)Panel A (q = 1)

1 0.01 0.18 3.68(0.69) (2.03)[0.85] [2.68]

2 −0.07 0.39 −0.00 −0.48 11.31(−1 .82 ) (2.48) (−4.07) (−1.37)[−2.34] [3.29] [−5.01] [−1 .80 ]

3 0.01 0.00 0.00 0.97(0.69) (1.09) (2.83)[0.87] [1.31] [3.37]

4 0.05 −0.04 2.30(1.59) (−1 .90 )[2.10] [−2.45]

Panel B (q = 12)1 −0.18 0.36 1.41

(−1.02) (1.02)[−0.84] [0.82]

2 −0.22 1.37 −0.03 1.78 29.73(−1.27) (2.87) (−6.53) (1 .78 )[−1.10] [2.58] [−5.57] [1.58]

3 −0.25 0.01 0.02 11.19(−1.53) (1 .94 ) (1 .78 )[−1.24] [1.64] [1.50]

4 −0.03 −0.18 2.46(−0.12) (−1.23)[−0.10] [−1.07]

Panel C (q = 60)1 0.04 −0.74 0.22

(0.06) (−0.32)[0.06] [−0.28]

2 0.49 0.96 −0.11 11.41 46.30(0.53) (0.51) (−4.20) (2.10)[0.53] [0.51] [−4.19] [2.10]

3 −0.10 0.06 −0.06 18.43(−0.12) (1.54) (−1.42)[−0.12] [1.55] [−1.41]

4 −0.47 0.63 2.78(−0.76) (1.97)[−0.80] [1.97]

62

Table 13:Multiple predictive regressions for state variables constructed from empirical factors (SV AR)This table reports the results for multiple long-horizon regressions for the stock market variance (SV AR),at horizons of 1, 12, and 60 months ahead. The forecasting variables are the current values of the term-structure spread (TERM); default spread (DEF ); size premiums (SMB∗); value premiums (HML∗);momentum factor (CUMD); and liquidity factor (CL). The original sample is 1963:07-2008:12, and qobservations are lost in each of the respective q-horizon regressions, for q = 1, 12, 60. For each regression,in line 1 are reported the slope estimates, and in lines 2 and 3 are reported Newey-West (in parenthesis)and Hansen-Hodrick t-ratios (in brackets) computed with q lags. Italic, underlined, and bold t-statisticsdenote statistical significance at the 10%, 5%, and 1% levels, respectively. R2(%) denotes the adjustedcoefficient of determination (in %).

Row SMB∗ HML∗ CUMD CL TERM DEF R2(%)Panel A (q = 1)

1 −0.00 −0.00 0.71(−0.93) (−1 .72 )[−1.10] [−2.11]

2 −0.00 −0.00 0.00 0.71(−0.96) (−1.60) (0.93)[−1.12] [−1 .94 ] [1.09]

3 −0.00 −0.00 −0.00 0.58(−0.66) (−1.44) (−0.54)[−0.80] [−1 .72 ] [−0.60]

4 −0.00 −0.00 0.00 0.27 7.08(−1 .84 ) (−2.63) (0.06) (2.62)[−2.09] [−3.43] [0.07] [3.46]

Panel B (q = 12)1 −0.01 −0.00 11.36

(−1 .93 ) (−2.33)[−1 .69 ] [−1.98]

2 −0.01 −0.00 0.00 11.27(−1 .92 ) (−2.27) (0.19)[−1 .69 ] [−1 .94 ] [0.17]

3 −0.01 −0.00 0.00 11.35(−2.39) (−1.34) (0.31)[−2.32] [−1.14] [0.25]

4 −0.02 −0.01 −0.27 1.33 19.54(−2.62) (−3.38) (−1 .95 ) (3.79)[−2.31] [−2.96] [−1.63] [3.22]

Panel C (q = 60)1 −0.01 −0.02 23.34

(−0.36) (−2.42)[−0.41] [−2.27]

2 −0.01 −0.02 0.01 23.33(−0.42) (−2.15) (0.36)[−0.42] [−2.16] [0.36]

3 −0.02 −0.02 0.02 25.36(−1.05) (−2.08) (1.02)[−1.13] [−2.20] [1.02]

4 −0.01 −0.03 −0.26 2.14 25.96(−0.69) (−2.71) (−0.37) (1.22)[−0.69] [−2.72] [−0.37] [1.22]

63

Table 14: Monte Carlo simulationThis table reports results for a Monte Carlo simulation with 10,000 replications of an artificialtwo-factor ICAPM. φ, γ and γz denote the calibrated values for the autoregressive coefficient ofthe state variable, risk aversion coefficient and hedging risk price, respectively. t(b1) (t(b60)) de-notes the fraction of replications in which the predictive slope in the one-month (60-months) re-gression is statistically significant at the 5% level, while t(γz) denotes the percentage of replicationsin which the risk price for the hedging factor is statistically significant. Sign(b1) (Sign(b60)) de-notes the fraction of replications for which the predictive slope in the one-month (60-months) re-gression shares the same sign as the hedging risk price, and both coefficients are significant at the5% level. The last two rows are associated with a simulation in which the ICAPM is not true.

t(b1) t(b60) t(γz) Sign(b1) Sign(b60)φ = 0.95, γ = 5, γz = 600 0.94 0.94 0.99 0.91 0.91φ = 0.99, γ = 5, γz = 600 0.99 0.99 1.00 0.96 0.96φ = 0.95, γ = 4, γz = 600 0.93 0.93 0.98 0.90 0.89φ = 0.99, γ = 4, γz = 600 0.98 0.99 1.00 0.96 0.96φ = 0.95, γ = 3, γz = 600 0.89 0.91 0.97 0.86 0.85φ = 0.99, γ = 3, γz = 600 0.97 0.98 1.00 0.95 0.95φ = 0.95, γ = 5, γz = 1200 0.95 0.94 0.99 0.91 0.91φ = 0.99, γ = 5, γz = 1200 0.99 0.99 1.00 0.96 0.96φ = 0.95, γ = 4, γz = 1200 0.93 0.93 0.99 0.89 0.88φ = 0.99, γ = 4, γz = 1200 0.99 0.99 1.00 0.96 0.96φ = 0.95, γ = 3, γz = 1200 0.91 0.91 0.98 0.87 0.86φ = 0.99, γ = 3, γz = 1200 0.98 0.99 1.00 0.95 0.95φ = 0.95, γ = 5, γz = 300 0.94 0.94 0.98 0.92 0.91φ = 0.99, γ = 5, γz = 300 0.99 0.99 1.00 0.96 0.96φ = 0.95, γ = 4, γz = 300 0.93 0.92 0.98 0.90 0.89φ = 0.99, γ = 4, γz = 300 0.99 0.99 1.00 0.96 0.96φ = 0.95, γ = 3, γz = 300 0.90 0.91 0.96 0.86 0.85φ = 0.99, γ = 3, γz = 300 0.98 0.99 0.97 0.92 0.92

φ = 0.95, γ = 2.80 0.07 0.49 0.01 0.00 0.00φ = 0.99, γ = 2.80 0.05 0.31 0.01 0.00 0.00

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multifactor models and their consistency with the icapm · 2018. 8. 6. · icapm and nd that most...

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