Arbitrage and the Empirical Evaluationof Asset-Pricing Models

Zhenyu WangColumbia University

Xiaoyan ZhangCornell University ∗

September 23, 2003

Abstract

A good asset-pricing model should be arbitrage-free. Consumption-based nonlinearmodels are arbitrage-free, but linear factor models and their time-varying extensionsdo not necessarily preclude arbitrage. In this paper, we introduce a simulation-basedBayesian analysis of Hansen and Jagannathan’s two pricing-error measures, of whichthe second requires the correct models to be arbitrage-free while the first does not. Thearbitrage-free requirement is important to the empirical evaluation of time-varying andmulti-factor models, especially when they are used to price conditional portfolios. Usingthe first measure, we are confident that time-varying extensions improve upon a staticmodel, that the Fama-French three-factor model is better than single-factor models,and that the time-varying Fama-French model has substantially smaller pricing errorsthan the consumption-based nonlinear models. However, we do not have the sameconfidence in these statements if the second measure is used for comparing modelson conditional portfolios, because time-varying and multi-factor models often providearbitrage opportunities.

∗The authors thank Andrew Ang, Mike Chernov, Gur Huberman, Paul Glasserman, Bob Hodrick, MichaelJohannes, and Tano Santos for helpful discussions. Zhang acknowledges support by the Cornell TheoryCenter.

Arbitrage and the Empirical Evaluationof Asset-Pricing Models

Abstract

A good asset-pricing model should be arbitrage-free. Consumption-based nonlinearmodels are arbitrage-free, but linear factor models and their time-varying extensionsdo not necessarily preclude arbitrage. In this paper, we introduce a simulation-basedBayesian analysis of Hansen and Jagannathan’s two pricing-error measures, of whichthe second requires the correct models to be arbitrage-free while the first does not. Thearbitrage-free requirement is important to the empirical evaluation of time-varying andmulti-factor models, especially when they are used to price conditional portfolios. Usingthe first measure, we are confident that time-varying extensions improve upon a staticmodel, that the Fama-French three-factor model is better than single-factor models,and that the time-varying Fama-French model has substantially smaller pricing errorsthan the consumption-based nonlinear models. However, we do not have the sameconfidence in these statements if the second measure is used for comparing modelson conditional portfolios, because time-varying and multi-factor models often providearbitrage opportunities.

Contents

1 Introduction 1

2 The Econometric Framework 52.1 Measuring Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Computing the Measures of Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . 82.3 A Simulation-Based Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Empirical Investigations 143.1 The Test Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 The Asset-Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Effects of the Arbitrage-Free Requirement . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Conclusion 26

5 References 27

6 Tables 30

7 Figures 37

1 Introduction

The stochastic discount factor (SDF) of an asset-pricing model should always be positive

if prices assigned by the model never provide arbitrage opportunities (Hansen and Richard

(1987) and Harrison and Kreps (1979)). In this case, the asset-pricing model is said to be

arbitrage-free. The SDF of the nonlinear consumption CAPM is always positive. Most non-

linear consumption-based models, as extensions of the consumption CAPM, are arbitrage-

free. In contrast, the SDF of the CAPM, in which the return on the market portfolio is the

single factor, can be negative and may thereby provide arbitrage opportunities. Multi-factor

and time-varying linear asset-pricing models are not arbitrage-free either, because their SDFs

may often take large negative values. Although a number of authors report multi-factor and

time-varying linear models are successful in explaining security returns, the empirical evalu-

ation of those models in the literature does not require a correct model to be arbitrage-free.

Our objective in this paper is to investigate the effects of the arbitrage-free requirement on

the evaluation and comparison of asset-pricing models.

Why is the arbitrage-free requirement important to the empirical evaluation of asset-

pricing models? Given a collection of test assets with observed data, our effort in fitting the

data with models might lead us to the SDFs that take negative values. A negative SDF allows

for arbitrage, especially when it is used for pricing derivative securities. A model that allows

for arbitrage is a wrong model and an arbitrage is an important pricing error. Hansen and

Jagannathan (1997) introduce two measures of pricing errors. The first measure, denoted

by δ, is the distance from a pre-specified SDF to the set, denoted by M, of all SDFs that

price the test assets correctly. The second measure, denoted by δ+, is the distance from the

pre-specified SDF to the set, denoted by M+, of all positive SDFs that price the test assets

correctly. These two measures are referred to as HJ distances. The second distance requires

a correct model to be arbitrage-free. Obviously, the second HJ distance δ+ is generally larger

than the first HJ distance δ, because M+ is a subset of M. We may underestimate pricing

1

errors if we do not require the correct models to be arbitrage-free.

It is possible that a pre-specified model has a small δ but a large δ+. Figure 1 illustrates

such a case. The pre-specified SDF yIt in the figure is much closer to M than M+, and the

first HJ distance δI is thus much smaller than the second HJ distance δI+. It is also possible

that the two HJ distances are not very different. In Figure 1, another pre-specified SDF yJt

is such an example. Suppose yJt is the SDF of a static single-factor model. Having observed

that this model has large pricing errors, we may introduce a time-varying model that has yIt

as the SDF. When we ignore arbitrage, the pricing error of yIt seems small as measured by its

first HJ distance δI. If yIt is often negative, the pricing error measured by δI

+ could be much

larger. If we require the correct models to be arbitrage-free and use the second HJ distance

to compare models, we might find that the time-varying model yIt is not really much better

than the static single-factor model yJt . Most of the improvement of the time-varying model

is getting closer to M while it does not necessarily get closer to M+. This is the main point

of our paper. The task of the paper is to compare models using the second HJ distance δ+,

as well as the first HJ distance δ.

In the empirical investigation, we find that the arbitrage-free requirement is important

to the empirical evaluation of time-varying linear factor models. Using the first HJ distance,

which does not require the correct models to be arbitrage-free, we are confident that time-

varying extension improves upon static models and that the Fama-French three-factor model

is better than single-factor models. Using the same measure, we are also confident that the

time-varying Fama-French model has substantially smaller pricing errors than consumption-

based nonlinear models. We no longer have the same confidence, however, if we use the

second HJ distance to measure pricing errors, especially on conditional portfolios. The

reason is that the SDFs in time-varying and multi-factor models take large negative values

frequently and thus provide significant arbitrage opportunities. When we use the second HJ

distance to measure pricing errors these time-varying and multi-factor models are far away

from the set of the correct models that are required to be arbitrage-free. If we ignore the

2

arbitrage-free requirement and use the first HJ distance, these time-varying and multi-factor

models seem to fit the data well.

Our work builds on the HJ distances, because they have three advantages for analyzing

specification errors in stochastic discount factor models. First, instead of assuming a pre-

specified model is correct, HJ distances examine how wrong the model is. Since all the

models are approximations, no models proposed in the literature are correct. It is natural

to ask whether a model is better than an alternative. For this objective, pricing errors serve

as a natural and practical criterion for comparing models, and HJ distances measure pricing

errors. Second, HJ distances incorporate conditioning information conveniently. Evaluating

a model using conditional information allows us to examine conditional restrictions of an

SDF. Finally and most importantly, the second HJ distance requires the correct models to

be arbitrage-free. The two distances therefore allow us to examine the importance of the

arbitrage-free requirement.

We introduce a simple and straightforward methodology, which uses the simulation-based

Bayesian analysis developed in the statistics literature. This methodology allows us to start

with an incorrectly specified model and to consider the arbitrage-free requirement. We ob-

tain the posterior distributions of HJ distances, which are convenient for formal comparison

of models. We also obtain the posterior distributions of many nonlinear measures of interest,

for which classic sampling distributions are not available or are hard to derive. The Bayesian

analysis has another advantage over the classic sampling theories, which are mostly asymp-

totic, requiring a long history of data. Bias due to the mismatch between an asymptotic

theory and a finite set of sample has been a great concern in the literature. The Bayesian

analysis we employ in this paper does not require the sample history to be unrealistically

long. Through posterior distributions, uncertainty associated with finite samples is properly

incorporated into the statistical inference of model evaluation and comparison.

The first HJ distance has been applied to a variety of empirical finance problems in the

3

cross-section of securities.1 Most existing empirical analyses using the first HJ distance are

based on the sampling distribution theory developed by Jagannathan and Wang (1996)2.

Although their sampling theory is very useful, all the studies using the theory ignore the

arbitrage-free requirement. The second HJ distance is largely ignored in the literature. An

important reason is that statistical inference of the second HJ distance is difficult, although

Hansen, Heaton, and Luttmer (1995) present a sampling theory that formulates consistency

and the limiting distribution of sample analogs of the HJ distances.

All studies using the sampling theories developed by Jagannathan and Wang (1996) and

Hansen, Heaton, and Luttmer (1995) miss an important advantage of HJ distances, namely,

the HJ distances are designed for comparing models under the assumption that all models

are wrong. The asymptotic sampling theories are useful for testing whether an HJ distance

is zero but not for comparing models, because in the sampling theories a number (e.g., zero)

for the HJ distance has to be assumed in the null hypothesis. All the studies in the literature

hypothesize that a model is correct and then test whether an HJ distance is significantly

different from zero. There has not been any formal statistical inference of model comparison

using HJ distances.

The remainder of the paper is organized as follows. Section 2 presents the econometric

framework. In this section, we review the measures of pricing errors and discuss the com-

putation of HJ distances. Then we introduce the simulation-based Bayesian analysis of HJ

distances. Section 3 presents our empirical investigation. In this section, we describe the

test assets and the asset-pricing models under our examination. Then we examine the effects

of the arbitrage-free requirement and compare models. Section 4 concludes.

1For instance, Jagannathan and Wang (1996) apply the first HJ distance to study the conditional CAPMwith human capital, Buraschi and Jackwerth (2001) apply the first HJ distance to examine option prices,Lettau and Ludvigson (2002) apply the first HJ distance to study the consumption CAPM, conditioningon the consumption-wealth ratio, and Hodrick and Zhang (2001) apply the first HJ distance to show thatall the recently proposed linear asset-pricing models have significant pricing errors. The complete list ofapplications of the first HJ distance is too long to present here.

2Although they presented the sampling theory in the context of linear asset-pricing models, their theoryapplies to nonlinear models with the help of a Taylor’s series expansion, which is often referred to as thedelta method in the econometrics literature.

4

2 The Econometric Framework

2.1 Measuring Pricing Errors

In our empirical investigation, there are n assets, which are referred to as test assets. We

observe returns on the assets and use an n× 1 vector, rt, to denote the asset returns during

period t. We also observe k factors and l state variables. At the end of period t, the

vector of the observed factors is ft, and the vector of the realized state variables is xt. Let

zt = (r′t, f′t, x

′t)

′. We assume zt follows a stationary stochastic process with finite second

moment. An SDF is denoted by mt. We assume mt ∈ L2, where L2 is the space of random

variables with finite second moments. If the SDF is correct, it satisfies:

Et−1[mtrt] = 1n , (1)

where Et−1[·] is the expectation conditioning on the information in period t − 1, and 1n is

an n × 1 vector of 1s.

To incorporate conditional asset pricing restrictions, a common practice is to scale the

returns by some lagged variables before taking the unconditional expectation. In general, we

introduce a matrix, H(zt−1), whose elements are functions of zt−1. Multiplying both sides of

equation (1) by H(zt−1) and taking unconditional expectations, the resulting asset pricing

restriction is

E[mtH(zt−1)rt] = E[H(zt−1)1n] . (2)

The vector, H(zt−1)rt, is referred to as the scaled returns, which can be viewed as payoffs

of conditional portfolios of the test assets. If we normalize each row of H(zt−1) to a vector

of weights that sum up to 1, H(zt−1)rt is the vector of returns on conditional portfolios.

Evaluating a conditional SDF model using conditioning variables is equivalent to evaluating

an unconditional SDF model on conditional portfolios.

If we let H(zt−1) = zt−1 ⊗ In+k+l, then H(zt−1)rt = vec(zt−1 ⊗ rt), which is a common

5

form of scaled returns.3 Suppose the first element of rt is the return on Treasury Bills and

the other elements are stock returns. If the state variable xt is a scaler and we use lagged

observation xt−1 to scale only the stock returns, the matrix H(zt−1) should be set to

Hx =

1 0 0 · · · 00 xt−1 0 · · · 00 0 xt−1 · · · 0...

...... · · · ...

0 0 0 · · · xt−1

. (3)

The matrix H(zt−1) also handles transformations of returns. If we choose H(zt−1) to be

H0 =

1 0 0 · · · 0−1 1 0 · · · 0−1 0 1 · · · 0...

...... · · · ...

−1 0 0 · · · 1

, (4)

then all but the first elements of H(zt−1)rt are excess stock returns over the Treasury Bills.

In this case, we have H(zt−1)1n = H01n = (1, 0, · · · , 0)′. If we are interested in only the

unconditional asset-pricing restriction on the original test assets, we choose H(zt−1) = In.

Let M be the set of SDFs that correctly price the portfolios on average, that is,

M =mt : mt ∈ L2, E[mtH(zt−1)rt] = E[H(zt−1)1n]

. (5)

Although the SDFs in M assign the correct price to the payoffs of the conditional portfolios

of the test assets, they may provide arbitrage opportunities, because M does not require

its members to be nonnegative.4 In an equilibrium model, the SDF should not be negative

because the SDF is the marginal rate of substitution of consumption between today and

3The idea of scaling asset returns by conditioning variables is originated from Hansen and Singleton(1982) and explored by numerous researchers. For a discussion on the efficient use of conditioning variables,see Ferson and Siegel (2001).

4To see how a negative mt provides arbitrage, let vt be the payoff of a security such that vt = 1 if mt < 0and vt = 0 otherwise. This security is a contingent claim and can be considered as a hypothetical derivativesecurity. Using mt to value vt, the price of the security is Et−1[mtvt]. If the conditional probability ofmt < 0 is nonzero, we have Et−1[mtvt] < 0. In this case, the security is an arbitrage because its price isnegative while its payoff is never negative. A necessary and sufficient condition for preventing such arbitrageis mt ≥ 0 with probability 1.

6

tomorrow. Let M+ be the set of nonnegative SDFs that, on average, correctly price the

conditional portfolios:

M+ =mt : mt ∈ L2, mt ≥ 0, E[mtH(zt−1)rt] = E[H(zt−1)1n]

. (6)

We assume that M+ is nonempty. This assumption holds if the observed prices of the

conditional portfolios do not provide arbitrage.

Let yt be the SDF of a pre-specified asset-pricing model in our empirical investigation. In

general, the prices assigned by the pre-specified asset pricing model yt are not consistent with

the observed prices, i.e., yt ∈ M. Hansen and Jagannathan (1997) introduce two measures

of the pricing errors:

δ = minmt∈M

√E[(yt − mt)2] (7)

δ+ = minmt∈M+

√E[(yt − mt)2] . (8)

The measure δ+ requires a correct model to be arbitrage-free while the measure δ does not.

The measures are the least-square distances from the pre-specified SDF to the set of the

SDFs that we consider to be correct. If we consider M to be the set of correct SDFs, we

measure the pricing errors of yt by δ. If we consider M+ to be the set of correct models,

we measure the pricing errors of yt by δ+. The finance literature refers to δ+ and δ as HJ

distances.5 Hansen and Jagannathan show that δ and δ+ are the maximum pricing errors

over normalized payoffs. Obviously, δ ≤ δ+ because M+ ⊂ M. Since δ and δ+ are different

in general, the arbitrage-free requirement affects our measure of pricing errors. To examine

the importance of the arbitrage-free requirement, we can look at the difference δ+ − δ and

the ratio (δ+ − δ)/δ.

In most empirical analyses of asset-pricing models, the pre-specified model often has

unknown parameters. A general form of the pre-specified SDF is yt = g(θ, ft, zt−1). Note

that we allow the SDF in the specified model to depend on lagged variables. The functional

5If yt is a constant discount factor, δ and δ+ become the two volatility bounds derived by Hansen andJagannathan (1991).

7

form g(·, ·, ·) is pre-specified and the vector of parameters, θ, is unknown. The HJ distances

of yt are therefore functions of θ and should be denoted by δ(θ) and δ+(θ). With free

parameters θ, the function g specifies a class of SDFs. The distance from the class of SDFs

to M+ (or M) is defined as the minimum of δ+(θ) (or δ(θ)) over all possible θ, as suggested

by Hansen and Jagannathan (1997). That is, we define δ = minθ δ(θ) and δ+ = minθ δ+(θ).

By θ and θ+, we denote the solutions to the two minimization problems, respectively.

When an SDF yt is specified, we can theoretically assess the arbitrage opportunity by

calculating the probability that yt is negative. Let us denote such a probability by π; i.e.,

π = Probyt < 0. We refer to π as the negativity rate of yt. When a set of SDFs is specified

in the form yt = g(θ, ft, zt−1) with free parameters, the negativity rate of yt depends on θ.

Let us denote it by π(θ). A particularly interesting θ is the one that minimizes δ(θ). Defining

π ≡ π(θ), we refer to π as the negativity rate of yt = g(θ, ft, zt) with free parameters θ. The

negativity rate π indicates the probability for yt to be negative after choosing the parameters

θ to minimize yt’s distance to M.

2.2 Computing the Measures of Pricing Errors

We assume that zt = (rt, ft, xt) follows a VAR process.6 That is,

zt = C + Azt−1 + εt , εt ∼ N(0m, Ω) , (9)

where m = n+k+ l is the dimension of vector zt and Ω is an m×m positive definite matrix.

The noise term εt is independent across time. Using the partitioned vectors and matrices,

the process zt can be expressed as rt

ft

xt

=

C1

C2

C3

+

A11 A12 A13

A21 A22 A23

A31 A32 A33

rt−1

ft−1

xt−1

+

ε1t

ε2t

ε3t

. (10)

6We can also assume a more complicated process like the multivariate GARCH for zt to allow conditionalheteroscedasticity. The VAR process is chosen here because it is relatively simple to explain but still generalenough to allow returns to be predictable, as evidenced by many studies in the literature. We also conductedour analysis using the multivariate GARCH process. Switching from the VAR process to the multivariateGARCH process does not change our empirical results materially.

8

The unconditional distribution of zt is a normal distribution, with mean µ and variance Σ,

which are given as

µ = (Im − A)−1C (11)

vec(Σ) = (Im2 − A ⊗ A)−1vec(Ω) . (12)

Given the unconditional distribution of zt = (r′t, f′t , x

′t)

′, we can derive the unconditional

distribution of vt = (r′t, f′t , z

′t−1)

′, which is necessary for the calculation of HJ distances. The

vector vt is linearly related to zt−1 and εt as follows:

vt = C + Azt−1 + Dεt , (13)

where

C =

C1

C2

C1

C2

C3

, A =

A11 A12 A13

A21 A22 A23

In 0n×k 0n×l

0k×n Ik 0k×l

0l×l 0l×l Il

, D =

In×n 0n×k 0n×l

0k×n Ik×k 0k×l

0n×n 0n×k 0n×l

0k×n 0k×k 0k×l

0l×n 0l×k 0l×l

. (14)

Therefore, the unconditional distribution of vt is normal and the mean and variance are,

respectively,

µ = C + Aµ (15)

Σ = AΣA′ + DΩD′ . (16)

The unknown parameters in the data-generating process (9) are the initial value z0, the

coefficient B = (C, A) in the autoregressive regression, and the variance Ω of the noise term.

Let Ψ = (z′0, vec(B)′, vec(Ω)′)′. It is the vector of parameters in the VAR process of zt. We

have T observations on zt, and the set of observed data is Z = (z1, · · · , zT )′. We treat z0 as

part of the unknown parameters because z0 is not in our observed data Z.

For computing δ, Hansen and Jagannathan (1997) show that the square of the HJ distance

ignoring arbitrage can be written as the weighted average of squared pricing errors. For the

9

pre-specified SDF yt = g(θ, ft, zt−1), we can apply Hansen and Jagannathan’s formula to

obtain

δ2(θ) = E[ytH(zt−1)rt − H(zt−1)1n]′(E[H(zt−1)rtr

′tH(zt−1)

′])−1

E[ytH(zt−1)rt − H(zt−1)1n] . (17)

If g is a linear function, the above formula allows us to calculate δ(θ) analytically for a given

Ψ because we can calculate the expectations in (17) analytically. However, if g is a nonlinear

function, we must calculate the expectations numerically as described later. In order to

calculate the second HJ distance, we can use the following formula, which is also obtained

by applying an equation for δ+ derived by Hansen and Jagannathan:

δ2+(θ) = max

λ∈RnE[y2

t −([yt − λ′H(zt−1)rt]

+)2 − 2λ′H(zt−1)1n

], (18)

where n is the number of rows in H(zt−1), and Rn is the space of n × 1 real vectors. The

function [·]+ is defined as [x]+ = x if x ≥ 0 and [x]+ = 0 if x < 0. In equation (18), we

cannot analytically calculate the expectation for a given Ψ. In addition, we must do the

maximization numerically to obtain δ+(θ).

We can however calculate the expectations approximately from a given Ψ. This allows

us to obtain approximations of HJ distances. Since the unconditional distribution of vt is

normal with mean µ and variance Σ, we can generate independent draws of vt from the

unconditional distribution. Then it is easy to compute HJ distances. Let the independent

draws be

v(j) =

r(j)

f (j)

z(j)

, j = 1, · · · , J . (19)

For a set of given parameters θ and a given function g in the pre-specified model yt =

g(θ, ft, zt−1), we can generate independent draws of y(j) by letting y(j) = g(θ, f (j), z(j)).

We can approximate δ(θ) using the formula

δ2(θ) = EJ [y(j)H(z(j))r(j) − H(z(j))1n]′(EJ [H(z(j))r(j)r(j)′H(z(j))′])−1

EJ [y(j)H(z(j))r(j) − H(z(j))1n] , (20)

10

where EJ [·] is defined as J−1∑Jj=1[·] and J is a large integer. Similarly, we can approximate

δ+(θ) using the formula

δ2+(θ) = max

λ∈RnEJ

[(y(j))2 −

([y(j) − λ′H(z(j))r(j)]+

)2 − 2λ′H(z(j))1n

]. (21)

The convergence of the approximation can be established by the Law of Large Numbers, and

the precision of the approximation can be assessed by applying the Central Limit Theorem.

Note that the approximation can be arbitrarily precise by making J large. The two HJ

distances δ and δ+ can then be obtained by minimizing δ(θ) and δ+(θ) over all the choices of

parameters θ. Using the simulation approach, we can also approximate the negativity rate

using the formula

π = limJ→+∞

EJ

[I[g(θ, f

(j)t , z

(j)t−1)]

], (22)

where I[x] equals 1 if x < 0 and 0 otherwise, and θ minimizes δ(θ).

2.3 A Simulation-Based Bayesian Inference

This section develops the simulation-based Bayesian inference of HJ distances. The basic

idea of our approach is as follows. We assume zt follows a general stochastic process, which

depends on some unknown parameters Ψ. Since this stochastic process should fit the data,

we specify a non-informative prior distribution for Ψ. The likelihood of the data is the

probability, denoted by p(Z|Ψ) of Z conditioning on Ψ. We want to obtain the posterior

distribution of the parameters given Z, denoted by p(Ψ|Z). For a given set of SDFs in

the form of yt = g(θ, ft, zt−1), the distribution of zt conditioning on parameters Ψ should

determine the negativity rates and HJ distances. That is, Ψ determines π, δ, and δ+ for

the given form of SDFs. Therefore, the posterior distribution of Ψ determines the posterior

distributions of π, δ, and δ+. The latter posterior distributions, denoted by p(π|Z), p(δ|Z),

and p(δ+|Z), respectively, are what we need for our analysis. The rest of this subsection

details the idea just described.

11

Imposing no prior opinion on parameters in the data-generating process, we assume the

following standard non-informative prior distribution for Ψ = (z′0, vec(B)′, vec(Ω)′)′ in the

data-generating process (9). The prior distributions of the three parts of Ψ are independent;

i.e.,

p(Ψ) = p(z0) p(B) p(Ω) , (23)

where

p(z0) ∝ constant , p(B) ∝ constant , p(Ω) ∝ |Ω|−(m+1)/2 . (24)

The conditional structure of the posterior distribution is

z0 |B, Ω, Z ∼ N(A−1(z1 − C), A−1ΩA′−1

)(25)

Ω | z0, Z ∼ IW(T Ω(z0), T − 1, m

)(26)

vec(B) |Ω, z0, Z ∼ Truncated N(vec(B(z0)), Ω ⊗ (X(z0)

′X(z0))−1)

, (27)

where IW is the inverted Wishert distribution and the functions B(z0), Ω(z0), and X(z0)

are defined as

X(z0) =((1, z′0), (1, z

′1), · · · , (1, z′T−1)

)′(28)

B(z0) = [X(z0)′X(z0)]

−1X(z0)′Z (29)

Ω(z0) =1

T[Z − B(z0)X(z0)]

′[Z − B(z0)X(z0)] . (30)

The normal distribution of vec(B) is truncated because the norm of eigenvalues of A must

be less than 1 for the VAR to be stationary.

It is analytically difficult to derive the posterior distribution of Ψ = (z′0, vec(B)′, vec(Ω)′)′,

and it is impossible to derive the posterior distribution of the HJ distances δ and δ+. The

Markov Chain Monte Carlo (MCMC) simulation method provides a way to estimate the

posterior distributions numerically. To estimate the posterior distributions of negativity

rates and HJ distances, the MCMC procedure is as follows.

12

1. Start from an arbitrary z(0)0 .

2. For i = 1, · · · , N0 + N , do the following:

(a) Obtain the ith sample of VAR parameters:

• Draw Ω(i) from IW(T Ω(z(i−1)0 ), T − 1, m);

• Draw vec(B(i)) from

truncated N(vec(B(z

(i−1)0 )), Ω(i) ⊗ [X(z

(i−1)0 )′X(z

(i−1)0 )]−1

).

• Draw z(i)0 from N

([A(i)]−1(z1 − C(i)), [A(i)]−1Ω(i)[A(i)′]−1

).

(b) Obtain the ith sample of the unconditional mean and variance of zt:

µ(i) =(Im − A(i)

)−1C(i)

vec(Σ(i)) =(Im2 − A(i) ⊗ A(i)

)−1vec(Ω(i)) .

(c) Obtain the ith sample of the unconditional mean and variance of vt:

µ(i) = C(i) + A(i)µ(i)

Σ(i) = A(i)Σ(i)A(i)′ + DΩ(i)D′ ,

where C(i), A(i), and D are constructed in the same way as in equation (14).

(d) Calculate the ith samples, δ(i), δ(i)+ , and π(i), with the help of equations (20)–(22).

3. Discard the first N0 samples.

4. Approximate the posterior distributions of HJ distances, the negativity rates, and the

model parameters can be approximated by the distribution of the samples δ(i)Ni=1,

δ(i)+ N

i=1, and π(i)Ni=1. For example, the posterior cumulative probability distribution

of δ(i)+ can be estimated by

Prob(δ+ ≤ X) ≈ 1

N

N∑i=1

I[δ(i)+ − X] , (31)

13

where I[·] is defined in equation (22). The posterior mean of δ+ can be approximated

by

E[δ+ |Z] ≈ 1

N

N∑i=1

δ(i)+ . (32)

The standard deviation and median of the posterior distribution of δ+ can be estimated

by their sample analog. The posterior distributions of π, δ, δ+ − δ, and (δ+ − δ)/δ can

be approximated similarly.

The approximation of the posterior distributions is more precise if the number of simulations,

N , is larger. We choose N = 5, 000 for this paper. We discard the first N0 simulations as

the usual MCMC practice to help the distribution of the draws converge to the posterior

distribution. We choose N0 to be 1,000.

Although HJ distances were introduced by Hansen and Jagannathan (1997) for comparing

pricing errors across models, no application in the literature has formal statistical inference

of model comparison in terms of the distances. In fact, it is straightforward to conduct

Bayesian inference of model comparison using HJ distances. For example, let δI+ and δJ

+ be

the second HJ distance for the SDFs yIt and yJ

t , respectively. The question is whether yIt has

substantially smaller pricing errors than yJt . If we think a reduction of pricing errors by 100q

percent is substantial, we are interested in the posterior probability of δI+ < (1− q)δJ

+, which

can be easily calculated using the sample draws from the posterior distributions of δI+ and

δJ+. If the question is whether yI

t is simply better than yJ+, we should set q to zero.

3 Empirical Investigations

3.1 The Test Assets

The asset returns rt considered in our empirical investigation are the monthly return on

Treasury Bills and the monthly returns on nine stock portfolios sorted by firm size and

14

book-to-market ratio. The sample consists of 456 monthly observations from January 1964

to December 2001. The nine stock portfolios are constructed in the same way as in Fama

and French (1993). We construct the excess returns on the above nine stock portfolios by

subtracting the Treasury Bill rate. This is equivalent to choosing H(zt−1) = H0 as in equation

(4). Table 1 presents the summary statistics for the excess returns on nine stock portfolios.

Our nine portfolios exhibit a large cross-sectional variation in average excess returns, which

are very similar to the 25 portfolios constructed by Fama and French. Portfolios with higher

book-to-market ratio have higher average excess returns. We choose nine portfolios rather

than 25 portfolios so we can quickly explore a variety of model specifications with a large

number of simulations in short time. We do try some analysis with 25 portfolios and results

are similar.

To incorporate conditioning information, we consider the yield spread between 30-year

Treasury bonds and 1-month Treasury Bills. This variable is the term premium and has

been used in the literature as a proxy for the changes of risk in the markets. It is shown to

correlate with the business cycle. To ensure stationarity, we filter the variable as in Hodrick

and Prescott (1997). To avoid forward-looking bias, we use only the information up to time

t when we generate the filtered value of the variable at time t. Figure 2 plots the time series

of the filtered term premium. We use the filtered variable as a state variable and refer to it

as TERM.

Other variables have been used as state variables in the literature. A popular state

variable is the default premium, which is the yield spread between Aaa-rated corporate

bonds and Baa-rated corporate bonds. We have repeated all of our analysis using this state

variable and the results are similar to the results obtained by using TERM. Therefore, we do

not report the results for the default premium. Another conditioning variable CAY, which

is the consumption-to-wealth ratio, has been suggested by Lettau and Ludvigson (2002).

We do not use CAY because it is observed quarterly while all our analyses use monthly

observations.

15

Following Hansen and Singleton (1982), we use past realizations of the state variables

to scale the asset returns in order to bring conditioning information into our examination

of pricing equations. For example, we use TERM to scale the excess returns of the stock

portfolios. This is equivalent to choosing H(zt−1) to be HxH0, with xt−1 = TERM, where

Hx and H0 are defined in equations (3) and (4). We can also scale the portfolio returns by

DEF. In this paper, we choose TERM because it exhibits larger variation over time than DEF as

shown in Figure 2. In the rest of the paper, we refer to H(zt−1)rt = H0rt as the non-scaled

asset returns and H(zt−1)rt = HxH0rt as the scaled asset returns.

3.2 The Asset-Pricing Models

To demonstrate the importance of the arbitrage-free requirement, we choose to investigate a

few basic linear and nonlinear models. The linear models we choose are the CAPM, the linear

consumption CAPM, and the Fama-French model, as well as the time-varying extensions of

these models. The nonlinear models are the consumption CAPM, the Abel model, and the

Epstein-Zin model. In the following, we provide the specifications of these models.

The classic asset-pricing model in finance is the CAPM developed by Sharpe (1964). The

SDF of this model is

yCAPMt = b0 + b1rMKT,t , (33)

where rMKT,t is the excess return on the market portfolio, and b0 and b1 are constant para-

meters in the model. The CAPM is often referred to as the unconditional or static CAPM

because it is derived in a single-period setting. To extend it to a multi-period setting, two

versions of the conditional CAPM have been introduced in the literature. The first version

adds the past realization of a state variable as a potential factor, as done in Jagannathan

and Wang (1996), except that they use the default premium as the state variable. The SDF

in this version of the conditional CAPM is

yCAPM+IVt = b0 + b1rMKT,t + c0xt−1 , (34)

16

where xt−1, referred to as the instrument variable (IV), is the past realization of the state

variable. In this paper, we use TERM as the instrument variable x. For convenience, we

denote this model as CAPM+IV. The other version of the conditional CAPM assumes that

b0 and b1 are linear functions of the instrument variable, as suggested by Cochrane (1996).

The SDF of this version of the time-varying CAPM is

yCAPM∗IVt = b0 + b1rMKT,t + c0xt−1 + c1xt−1rMKT,t . (35)

For convenience, we denote this model as CAPM*IV. The CAPM and its time-varying

extensions are not arbitrage-free because their SDFs can be negative.

After noting the large pricing errors of the CAPM on portfolios sorted by firm size and

book-to-market ratio, Fama and French (1993) propose a three-factor model that specifies

the SDF as

yFF3t = b0 + b1rMKT,t + b2rSMB,t + b3rHML,t , (36)

where rSMB,t and rHML,t are the factors constructed by Fama and French to mimic the risks

related to firm size and book-to-market ratio. We refer to this model as FF3. The Fama-

French model has some success, but the pricing errors are still substantial (Fama and French

(1997) and Daniel and Titman (1997)), especially on portfolios incorporating conditioning

information about the business cycle (Hodrick and Zhang (2001)). To improve the FF3, we

consider the following time-varying extension:

yFF3+IVt = b0 + b1rMKT,t + b2rSMB,t + b3rHML,t + c0xt−1 , (37)

which is referred to as FF3+IV in the rest of this paper. Another version of the time-varying

extension of the FF3 is

yFF3∗IVt = b0 + b1rMKT,t + b2rSMB,t + b3rHML,t (38)

+ c0xt−1 + c1xt−1rMKT,t + c2xt−1rSMB,t + b3xt−1rHML,t , (39)

which is referred to as FF3*IV. This type of extension of the Fama-French model is explored

by Kirby (1997). The FF3 and its time-varying extensions are not arbitrage-free.

17

The classic arbitrage-free model is the consumption CAPM, which specifies the SDF as

yCCt = ρ

(Ct

Ct−1

)−γ

, 0 ≤ ρ < 1, γ > 0, (40)

where ρ is the parameter for time preference and γ is the parameter for risk aversion. This

model is referred to as CC and its SDF is positive by construction. Since Mehra and Prescott

(1985), it has been well known that this model has difficulties in explaining equity returns.

Numerous studies look for ways of improving upon the CC. We consider two extensions to

the CC. The first is the model proposed by Abel (1990). The SDF of the model is

yAbelt = ρ

(Ct−1

Ct−2

)η(γ−1) (Ct

Ct−1

)−γ

, 0 ≤ ρ < 1, γ > 0, η ≥ 0, (41)

where η is the parameter for habit persistence. If η > 0 and γ = 1, yAbelt is different from

yCCt because yAbel

t depends on the past growth rate of consumption. The second extension

considered in this paper is the model proposed by Epstein and Zin (1989), which generalizes

the time-additive expected utility function in the CC. In this extension, the SDF is

yEZt = ργ/α

(Ct

Ct−1

)γ α−1α

(1 + rMKT,t)γ−α

α , 0 ≤ ρ < 1, γ > 0, α ≥ 0, (42)

where rMKT,t is the real return on the market portfolio. The parameter α is the intertemporal

rate of substitution. The deviation of parameter α from γ controls the model’s deviation

from the CC. If α = γ, the model is equivalent to the CC. In the rest of this paper, we

refer to the first extension as the Abel model and to the second as the Epstein-Zin model

or simply the EZ model. Like the CC, both models are nonlinear and exclude arbitrage

by construction. We use real per-capita purchases of non-durable goods and services as

the proxy for Ct. We obtain the consumption data from Citibase and divide them by the

monthly estimates of population from the Bureau of the Census to derive per-capita figures.

The finance literature sometimes approximates the CC by using a linear factor model with

the growth rate of consumption as the factor. This model is studied in Breeden, Gibbons and

Litzenberger (1989) and Chen, Roll and Ross (1986). We refer to the linear approximation

18

of the CC as LCC. The SDF of the model is

yLCCt = b0 + b1 ln(Ct/Ct−1) . (43)

Recent literature has also extended the LCC into time-varying models by adding a condi-

tioning variable and its interaction with the growth rate of consumption (see Hodrick and

Zhang (2001) and Lettau and Ludvigson (2002)). The SDFs of these types of models are

yLCC+IVt = b0 + b1 ln(Ct/Ct−1) + c0xt−1 (44)

yLCC∗IVt = b0 + b1 ln(Ct/Ct−1) + c0xt−1 + c1xt−1 ln(Ct/Ct−1) . (45)

As above, we choose xt−1 to be the variable TERM, and we refer to the above two models as

LCC+IV and LCC*IV, respectively. Unlike the nonlinear CC, the linear approximation and

time-varying extensions such as LCC, LCC+IV, and LCC*IV are not arbitrage-free.

3.3 Effects of the Arbitrage-Free Requirement

To examine the effects of the arbitrage-free requirement on the empirical evaluation of the

CAPM, CAPM+IV, and CAPM*IV, the posterior distributions of HJ distances for these

models are presented in Figure 3. For the non-scaled returns, the posterior distributions of

δ and δ+ are almost the same for the CAPM. This is also true for the CAPM+IV. For the

CAPM*IV, there is a slight difference between the posterior distributions of δ and δ+. For

the scaled returns, the difference between the posterior distributions of δ and δ+ is getting

larger when we move from the CAPM to the CAPM+IV, and then to the CAPM*IV. Table

2 reports the mean, median, and standard deviation of the posterior distributions of HJ

distances and their differences. For the non-scaled returns, the posterior means of (δ+−δ)/δ

are only 1% for the CAPM, 2% for the CAPM+IV, and 6% for the CAPM*IV. For the scaled

returns, the posterior means of (δ+ − δ)/δ are 7% for the CAPM, 10% for the CAPM+IV,

and 20% for the CAPM*IV. Therefore, the arbitrage-free requirement does not substantially

affect the evaluation of the CAPM, but it does moderately affect the evaluation of its time-

varying extensions. Since the conditioning variable TERM is volatile over time as shown in

19

Figure 2, it is possible that the SDFs of the time-varying models are negative more frequently

than the SDF of the static CAPM. Table 2 reports the summary statistics for the posterior

distributions of the negativity rate π. The posterior mean of π is higher for the CAPM+IV

and the CAPM*IV than the posterior mean of π for the CAPM.

A concern is whether the concentrated posterior distributions in Figure 3 are results of

data or artifacts of the non-informative prior distribution of Ψ. The posterior distributions

we discussed above contain a large amount of information in the data. To demonstrate

this, we estimate posterior distributions conditioning on observing only the monthly returns

during the last three years. These posterior distributions are presented in Figure 4. With

only 36 observations, the posterior distributions are much more dispersed and thus much less

informative. Unfortunately, we cannot estimate the distributions of HJ distances implied by

the prior distribution of Ψ because the prior is non-informative and improper. However, the

fact that the posterior distributions with 36 observations are far more dispersed indicates

that the distributions of HJ distances implied by the prior are not concentrated on some

finite values. Therefore, the concentrated posterior distributions conditioning on the 456

observations, as in Figure 3, exhibit the information contained in the data. For the CAPM,

the posterior distributions of the two HJ distances in Figure 4 are almost the same for

both the non-scaled and scaled returns. For the CAPM+IV and CAPM*IV, the posterior

distributions of the two HJ distances are also almost the same for the non-scaled returns

but very different for the scaled returns. These are further evidence that the locations of

the posterior distributions are determined by the data and the model but not by the prior

distribution, because all the posterior distributions presented in Figure 4 are derived from

the same prior distribution.

For the LCC, LCC+IV, and LCC*IV, the effects of the arbitrage-free requirement are

larger than the effects for the CAPM, CAPM+IV, and CAPM*IV, respectively. This is

true for both the non-scaled and scaled returns. The posterior distributions of δ and δ+

for the LCC, LCC+IV, and LCC*IV are plotted in Figure 5. The difference between the

20

poster distributions of δ and δ+ is clearly visible. Table 3 reports the posterior means of the

negativity rates for LCC, LCC+IV and LCC*IV, which are above zero in most cases. Table

3 also reports the summary statistics for the posterior distributions of HJ distances. For the

non-scaled returns, the posterior mean of (δ+−δ)/δ is 5% for the LCC, 8% for the LCC+IV,

and 15% for the LCC*IV. For the scaled returns, the posterior mean of (δ+− δ)/δ is 11% for

the LCC, 17% for the LCC+IV, and 24% for the LCC*IV. These numbers are larger than

the numbers for the CAPM, CAPM+IV, and CAPM*IV, respectively.

The arbitrage-free requirement has huge effects on the evaluation of the time-varying ex-

tensions of the FF3. For the FF3 and its time-varying extensions, the posterior distributions

of HJ distances are plotted in Figure 6. The summary statistics for the posterior distribu-

tions are reported in Table 4. For the non-scaled returns, the arbitrage-free requirement has

almost no effects on the evaluation of FF3 and FF3+IV but a large effect on FF3*IV. The

posterior mean and median of the negativity rate for the FF3*IV are both about 29%, and

the posterior mean and median of (δ+ − δ)/δ are 53% and 45%, respectively. For the scaled

returns, the arbitrage-free requirement has significant effects on each of the three models.

In particular, the posterior distributions of δ and δ+ for FF3*IV are drastically different, as

shown in Panel F of Figure 6. The posterior mean and median of (δ+−δ)/δ are, respectively,

127% and 129%!

Although the factors scaled by the instrument variable make the SDF vary over time and

fit the data well, they also make the model provide big or frequent arbitrage opportunities,

because the fluctuation of the instrument variable often drags the SDF to the negative region.

To fit the data, economists usually apply the generalized method of moments (GMM) to

the restriction7 E[g(θ, ft, zt−1)H(zt−1)rt] = H(zt−1)rt. If θ is the GMM estimate of θ, the

GMM estimate of the SDF at t is yt = g(θ, ft, zt−1). For the FF3 and its time-varying

extensions, we plot the GMM estimates of the SDFs in Figure 7. It shows that the SDF

7In the literature, economists often write linear factor models in terms of betas and estimate them usingthe regression method. According to Jagannathan and Wang (2002), applying the regression method to amodel in terms of betas is equivalent to applying the GMM to the restriction in terms of the model’s SDF.

21

of the FF3 estimated for the non-scaled returns is mostly positive while the SDFs of the

FF3*IV, estimated for both the non-scaled and scaled returns, take large negative values

frequently. This is consistent with the large negativity rate of the FF3*IV reported in Table

4.

The arbitrage-free requirement has little effect on the evaluation of the consumption-

based nonlinear models, namely, the CC, the Abel model, and the EZ model. In Figure 8,

for each model the posterior distributions of δ and δ+ are almost the same. In Table 5, the

posterior mean and median of (δ+ − δ)/δ are small for all the three models and for both

the non-scaled and scaled returns. Since these models are arbitrage-free by definition, the

negativity rate is always zero and thus not reported in Table 5.

3.4 Model Comparison

The main purpose of HJ distances is model comparison. Table 6 reports the posterior mean

of the HJ-distance ratios and the posterior probability that the ratios are less than 1− q for

q = 0, 10%, and 20%. All the numbers referenced in this subsection are from Table 6.

Let us first compare the CAPM with its time-varying extensions. The posterior mean

of δCAPM+IV/δCAPM is 0.95 for the non-scaled returns and 0.93 for the scaled returns. The

posterior probability that δCAPM+IV is smaller than δCAPM is 1.00 for both the non-scaled and

scaled returns. Therefore, measured by the first HJ distance, which ignores the arbitrage-

free requirement, we are sure that CAPM+IV improves upon the CAPM. The improvement

seems still clear if measured by the second HJ distance that requires the correct models to be

arbitrage-free. The posterior mean of δCAPM+IV+ /δCAPM

+ is 0.96 for the non-scaled returns and

0.95 for the scaled returns. The posterior probability that δCAPM+IV+ is smaller than δCAPM

+

is 0.97 for the non-scaled returns and 0.93 for the scaled returns. Although it is clear that

the CAPM+IV improves upon the CAPM, the improvement is unlikely to be substantial if

we regard 10% or 20% pricing error reduction as substantial reduction. The probability that

22

δCAPM+IV is smaller by at least 10% than δCAPM is 0.17 for the non-scaled returns and 0.26

for the scaled returns. The probability that δCAPM+IV+ is smaller by at least 10% than δCAPM

+

is 0.13 for the non-scaled returns and 0.15 for the scaled returns. The probability of a 20%

improvement is even lower, as shown by the probabilities for q = 20% reported in Table 6.

The results for the comparison of the CAPM and the CAPM*IV are similar.

Let us next compare the LCC and its time-varying extensions. The posterior mean of

δLCC+IV/δLCC is 0.95 for the non-scaled returns and 0.90 for the scaled returns. The ratio of

δLCC∗IV to δLCC has an even lower posterior mean. The posterior probability that δLCC+IV is

smaller than δLCC is 1.00 for both the non-scaled and scaled returns. The same is true for the

probability that δLCC∗IV is smaller than δLCC. These indicate that time-varying extensions

improve upon the LCC if the first HJ distance is used for the comparison. However, if the

second HJ distance δ+ is used, the results of the comparison are different. The posterior

means of δLCC+IV+ /δLCC

+ and δLCC∗IV+ /δLCC

+ are much closer to 1. The posterior probability that

δLCC+IV+ is smaller than δLCC

+ is 0.76 for the non-scaled returns and 0.75 for the scaled returns.

The probability that δLCC∗IV+ is smaller than δLCC

+ is 0.75 for the non-scaled returns and 0.79

for the scaled returns. Therefore, if we require the correct models to be arbitrage-free, we are

not confident that the time-varying extension improves upon the linear approximation of the

consumption CAPM. Moreover, the improvement of the time-varying extension to the LCC

is unlikely to be substantial. No matter which HJ distance is used, the posterior probability

that the LCC+IV has substantially (namely, at least 10% or 20%) smaller pricing errors

than the LCC is well bellow 0.50 for both the non-scaled and scaled returns. The same is

true for LCC*IV.

The FF3 is originally suggested by Fama and French (1993) to explain the pricing errors

of the CAPM. It is therefore interesting to compare the FF3 with the CAPM. For the non-

scaled returns, the FF3 clearly has smaller pricing errors than the CAPM no matter which

HJ distance is used to measure pricing errors. The posterior mean of δFF3/δCAPM is 0.77,

and the posterior mean of δFF3+ /δCAPM

+ is 0.78. The posterior probability that the FF3 is

23

better than the CAPM is 1.00 when either δ or δ+ is used for the comparison. It is less

certain that the improvement of the FF3 upon the CAPM is substantial. For q = 10%,

the posterior probabilities of δFF3 < (1 − q)δCAPM and δFF3+ < (1 − q)δCAPM

+ are lower than

0.90. For q = 20%, the above two probabilities are lower than 0.60. It is also interesting to

compare the FF3 with the LCC. The results of the comparison between FF3 and LCC are

very similar to the results of the comparison between FF3 and CAPM.

Note that for the non-scaled returns, the above comparison of FF3 with CAPM and LCC

are not affected by the arbitrage-free requirement. This is not true for the scaled returns.

For example, for the scaled returns, the posterior probabilities of δFF3 < δCAPM is 1.00,

but the posterior probability of δFF3+ < δCAPM

+ is only 0.71. The arbitrage-free requirement

significantly reduces the posterior probability that the pricing error of the FF3 is smaller by

at least 10% than the pricing error of CAPM. The same is true if we look at the 20% error

reduction or if we compare the FF3 with the LCC. Therefore, when the models are evaluated

by δ+ for the scaled returns, we are not confident that the FF3 is better or substantially

better than the CAPM or the LCC. A possible reason for this is as follows. Since the non-

scaled returns are sorted by firm size and book-to-market ratio, the SDF with the SMB and

HML factors do not need to take many large negative values to price the returns. However,

the scaled returns are different from the returns on the portfolios sorted by firm size and

book-to-market ratio, the SDF with the SMB and HML factors need to be negative in order

to price the returns well. This can be seen by comparing the GMM estimates of the SDFs

plotted in Panels A and D of Figure 7. When the arbitrage-free requirement is in place, the

SDF of the FF3 is thus far away from being a correct model for the scaled returns.

The FF3*IV is the model that clearly out-performs the CAPM and the LCC for both the

non-scaled and scaled returns and in terms of both HJ distances. The posterior probabilities

of δFF3∗IV < δCAPM, δFF3∗IV < δLCC, δFF3∗IV+ < δCAPM

+ , and δFF3∗IV+ < δLCC

+ are all equal to

1.00 for the non-scaled returns and above 0.96 for the scaled returns. It is also very likely

that the pricing errors of the FF3 is substantially smaller than the CAPM and the LCC,

24

except when the pricing errors are measured by δ+ for the scaled returns. If we set q = 10%,

the posterior probabilities of δFF3∗IV < (1−q)δCAPM and δFF3∗IV < (1−q)δLCC are above 0.99

for the non-scaled and scaled returns. The posterior probabilities of δFF3∗IV+ < (1− q)δCAPM

+

and δFF3∗IV+ < (1 − q)δLCC

+ are above 0.98 for the non-scaled returns but below 0.90 for the

scaled returns. If we set q = 20%, these probability are above 0.92 except the probabilities

of δFF3∗IV+ < (1− q)δCAPM

+ and δFF3∗IV+ < (1− q)δLCC

+ for the scaled returns, which are bellow

0.54. If fact, the FF3*IV is the one that has the smallest pricing errors among the time-

varying and multi-factor models considered in this paper, although we do not present formal

statistical inference of the comparisons.

We present the formal comparison of the FF3*IV with the consumption-based nonlin-

ear models, which are arbitrage-free. The posterior distributions exhibit strong confidence

that the FF3*IV is better than the three nonlinear models. The posterior probabilities of

δFF3∗IV < δCC, δFF3∗IV < δAbel, and δFF3∗IV < δEZ are all above 0.99 for both the non-scaled

and scaled returns. Similar probabilities for δ+ are all above 0.97 for the non-scaled returns

and around 0.90 for the scaled returns. The inference of the magnitude of the difference

between the pricing errors of the FF3 and a consumption-based non-linear model, however,

depends on the arbitrage-free requirement. If the arbitrage-free requirement is ignored, we

are confident that FF3*IV is substantially better than the consumption-based nonlinear

models, but this confidence may disappear if we require the correct models to be arbitrage-

free. The posterior probability that by at least 20% δFF3∗IV is smaller than δCC is above 0.95

for both the non-scaled and scaled returns. This is true if we replace the CC by the Able or

EZ model, but no longer true if we replace δ by δ+. For q = 20%, the posterior probabilities

of δFF3+ < (1 − q)δCC

+ , δFF3+ < (1 − q)δAbel

+ , and δFF3+ < (1 − q)δEZ

+ are all lower than 0.90 for

the non-scaled returns and 0.50 for the scaled returns. If we set q = 10%, these probabilities

are bellow 0.80 for the scaled returns.

25

4 Conclusion

A good asset-pricing model should satisfy three criteria. First, it should have small pric-

ing errors. Second, it should price conditionally managed portfolios. Third, it should be

arbitrage-free. In this paper, we emphasize the third criterion, which is mostly ignored in

the literature. The third criterion will reduce pricing errors of an estimated model while be-

ing used for pricing securities that are not included in the test assets and is indispensable if

the estimated model is to be used for pricing derivative securities. We introduce a straight-

forward methodology for analyzing two HJ distances, which allow us to either ignore or

incorporate the arbitrage-free requirement when measuring pricing errors. Our methodology

provides formal statistical inference of the model comparison based on HJ distances.

To demonstrate the importance of the arbitrage-free requirement to the empirical evalua-

tion of asset-pricing models, we focus on the comparison of static models to their time-varying

extensions, the comparison of single-factor models to multi-factor extensions, and the com-

parison of the linear factor models to consumption-based non-linear models. Although the

time-varying and multi-factor models are often successful in explaining asset returns, they

are not arbitrage-free. In contrast, the static single-factor linear models are not successful but

their SDFs are mostly positive. The consumption-based nonlinear models are not successful

but arbitrage-free. Using the first HJ distance, which ignores the arbitrage-free requirement,

we are confident that time-varying extension improves upon static models, that the Fama-

French three-factor model is better than single-factor models, and that the time-varying

Fama-French model has substantially smaller pricing errors than the consumption-based

nonlinear models. However, the confidence disappear, especially for conditional portfolios,

if we use the second HJ distance, which requires the correct models to be arbitrage-free.

Therefore, arbitrage is particularly important to the empirical evaluation of time-varying

linear factor models.

Many other important asset-pricing models should be evaluated under the arbitrage-

26

free requirement but are not considered in this paper. The model proposed by Chen,

Roll and Ross (1986) includes many economic factors. The model proposed by Bansal

and Viswanathan (1993) specifies the SDF as a polynomial function of economic variables.

The model with the momentum factor is suggested by Carhart (1997) based on the findings

of Jegadeesh and Titman (1993). The model proposed by Jagannathan and Wang is sim-

ilar to the CAPM+IV but has the labor-income growth rate as an additional factor. The

model proposed by Lettau and Ludvigson (2002) resembles the LCC*IV but uses a different

conditioning variable, which measures the consumption-wealth ratio quarterly. The model

proposed by Campbell and Cochrane (1999) is an extension to the Abel and EZ models.

Motivated by the work of Huberman and Halka (2001), the model proposed by Pastor and

Stambaugh (2003) introduces the factor of the systematic liquidity. Observing that volatil-

ity is priced in asset returns, Ang, Hodrick, Xing and Zhang (2003) construct a model that

contains a volatility factor. None of these models, except the one proposed by Campbell and

Cochrane, is arbitrage-free. Using HJ distances, especially the second distance, to compare

these models will be interesting. Such comparison deserves a separate study, because many

special issues related to these models must be addressed and the special data must be used

for these models.

5 References

Ang, Andrew, Robert Hodrick, Yuhang Xing, and Xiaoyan Zhang, 2003, The Cross-Sectionof Volatility and Expected Returns, Working paper, Columbia University.

Abel, Andrew B., 1990, Asset prices under habit formation and catching up with the Joneses,American Economic Review 80, 38–42.

Bansal, Ravi, and S. Viswanathan, 1993, No arbitrage and arbitrage pricing: A new ap-proach, Journal of Finance 48, 1231–62.

Breeden, Douglas T., 1979, An intertemporal asset pricing model with stochastic consump-tion and investment opportunities, Journal of Financial Economics 7, 265–296.

Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger, 1989, Empiricaltests of the consumption-oriented CAPM, Journal of Finance 44, 231–262

27

Buraschi, Andrea, and Jens Carsten Jackwerth, 2001, The price of a smile: Hedging andspanning in option markets, Review of Financial Studies 14, 495–527.

Campbell, John Y., and John H. Cochrane, 1999, By force of habit: A consumption-basedexplanation of aggregate stock market behavior, Journal of Political Economy 107, 205–251.

Carhart, Mark M., 1997, On persistence in mutual fund performance, Journal of Finance 52,57-82.

Chen, Nai-Fu, Richard Roll, and Stephen A. Ross, 1986, Economic forces and the stockmarket, Journal of Business 59, 383-403.

Cochrane, John H., 1996, A cross-sectional test of an investment-based asset pricing model,Journal of Political Economy 104, 572–621.

Daniel, Kent, and Sheridan Titman, 1997, Evidence on the characteristics of cross-sectionalvariation in stock returns, Journal of Finance 52, 1–33.

Epstein, Larry G., and Stanley E. Zin, 1989, Substitution, risk aversion, and the temporalbehavior of consumption and asset returns: A theoretical framework, Econometrica 57, 937–969.

Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns onstocks and bonds, Journal of Financial Economics 33, 3–56.

Fama, Eugene, and Kenneth R. French, 1997, Industry costs of equity, Journal of FinancialEconomics 43, 153–193.

Ferson, Wayne E., and Andrew F. Siegel, 2001, The efficient use of conditioning informationin portfolios, Journal of Finance 56, 967–982

Hansen, Lars Peter, John Heaton, and Erzo G. J. Luttmer, 1995, Econometric evaluation ofasset pricing models, Review of Financial Studies 8, 237–274.

Hansen, Lars Peter, and Ravi Jagannathan, 1991, Implications of security market data formodels of dynamic economies, Journal of Political Economy 99, 225–262.

Hansen, Lars Peter, and Ravi Jagannathan, 1997, Assessing specification errors in stochasticdiscount factor models, Journal of Finance 52, 557–590.

Hansen, Lars Peter, and Scott F. Richard, 1987, The role of conditioning information indeducing testable restrictions implied by dynamic asset pricing models, Econometrica 55,587–613.

Hansen, Lars Peter, and Kenneth Singleton, 1982, Generalized instrumental variable esti-mation of nonlinear rational expectation models, Econometrica 50, 1269–1286.

Harrison, J. Michael, and David M. Kreps, 1979, Martingales and arbitrage in multiperiod

28

securities markets, Journal of Economic Theory 20, 381–408.

Hodrick, Robert J., and Edward C. Prescott, 1997, Postwar U.S. business cycles: An empir-ical investigation, Journal of Money, Credit and Banking 29, 1–16.

Hodrick, Robert J., and Xiaoyan Zhang, 2001, Evaluating the specification errors of asset-pricing models, Journal of Financial Economics 62, 327–376.

Huberman, Gur, and Dominika Halka, 2001, Sytematic liquidity, Journal of Financial Re-search 24, 161-178.

Jagannathan, Ravi, and Zhenyu Wang, 1996, The conditional CAPM and the cross-sectionof expected returns, Journal of Finance 51, 3–53.

Jagannathan, Ravi, and Zhenyu Wang, 2002, Empirical evaluation of assetpricing models:A comparison of the SDF and beta methods, Journal of Finance 57, 2337–2367.

Jegadeesh, Narasimhan, and Sheridan Titman, 1993, Returns to buying winners and sellinglosers: Implications for stock market efficiency, Journal of Finance 48, 65-91.

Kirby, Chris, 1997, Measuring the predictable variation in stock and bond returns, Reviewof Financial Studies 10, 579–630

Lettau, Martin, and Sydney Ludvigson, 2002, Resurrecting the (C)CAPM: A cross-sectionaltest when risk premia are time-varying, Journal of Political Economy 109, 1238–1287.

Mehra, Rajnish, and Edward C. Prescott, 1985, The equity premium: a puzzle, Journal ofMonetary Economics 15, 145–162.

Pastor, Lubos, and Robert F. Stambaugh, 2003, Liquidity risk and expected stock returns,Journal of Political Economy 111, 642-685.

Sharpe, William F., 1964, Capital asset prices: A theory of market equilibrium under con-ditions of risk, Journal of Finance 19, 425–442.

29

6 Tables

Table 1: Summary Statistics for the Stock Portfolios

The reported results are the means and standard deviations of the monthly excess returns(in percent) on size and book-to-market (B/M) portfolios. The data are from CRSP andCompustat, and the sample period is 1964-2001. Individual firms are sorted into three sizeand B/M groups independently, and the portfolios are constructed as intersections of sizeand B/M groups, as in Fama and French (1993). The ranks are in ascending orders.

Rank by Size Rank by B/M Mean Standard Deviation1 1 0.29 0.331 2 0.62 0.281 3 0.86 0.272 1 0.52 0.292 2 0.61 0.252 3 0.94 0.233 1 0.47 0.233 2 0.44 0.213 3 0.78 0.20

30

Table 2: The CAPM and Its Time-Varying Extensions

This table reports the summary statistics for the estimated posterior distributions of π, δ,δ+, δ+ − δ, and (δ+ − δ)/δ for the CAPM and its time-varying extensions. In Panel A, assetreturns are not scaled by any conditioning variables. In Panels B, asset returns are scaledby the variable TERM.

Panel A: For the Non-Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe CAPM:mean 0.000 0.316 0.320 0.004 1%median 0.000 0.315 0.317 0.000 0%stdev 0.000 0.064 0.065 0.013The CAPM+IV:mean 0.036 0.300 0.306 0.006 2%median 0.002 0.298 0.303 0.001 0%stdev 0.062 0.065 0.065 0.015The CAPM*IV:mean 0.135 0.276 0.294 0.018 6%median 0.131 0.275 0.291 0.007 2%stdev 0.088 0.067 0.066 0.027

Panel B: For the Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe CAPM:mean 0.075 0.223 0.239 0.015 7%median 0.044 0.222 0.236 0.009 4%stdev 0.084 0.051 0.054 0.018The CAPM+IV:mean 0.090 0.207 0.228 0.021 10%median 0.055 0.206 0.225 0.015 7%stdev 0.095 0.051 0.054 0.022The CAPM*IV:mean 0.105 0.182 0.218 0.036 20%median 0.082 0.182 0.216 0.028 15%stdev 0.090 0.049 0.053 0.031

31

Table 3: The LCC and Its Time-Varying Extensions

This table reports the summary statistics for the estimated posterior distributions of π, δ,δ+, δ+ − δ, and (δ+ − δ)/δ for the LCC and its time-varying extensions. In Panel A, assetreturns are not scaled by any conditioning variables. In Panel B, asset returns are scaled bythe variable TERM.

Panel A: For the Non-Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe LCC:mean 0.028 0.319 0.334 0.014 5%median 0.004 0.317 0.332 0.002 1%stdev 0.045 0.066 0.068 0.026The LCC+IV:mean 0.074 0.302 0.327 0.025 8%median 0.047 0.300 0.323 0.009 3%stdev 0.080 0.066 0.069 0.035The LCC*IV:mean 0.164 0.279 0.322 0.043 15%median 0.163 0.277 0.317 0.025 9%stdev 0.089 0.068 0.072 0.046

Panel B: For the Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe LCC:mean 0.088 0.233 0.259 0.026 11%median 0.039 0.232 0.257 0.016 7%stdev 0.103 0.052 0.060 0.029The LCC+IV:mean 0.114 0.209 0.246 0.036 17%median 0.085 0.208 0.242 0.027 13%stdev 0.109 0.051 0.058 0.034The LCC*IV:mean 0.138 0.195 0.242 0.047 24%median 0.117 0.193 0.238 0.037 19%stdev 0.110 0.053 0.059 0.038

32

Table 4: The FF3 and Its Time-Varying Extensions

This table reports the summary statistics for the estimated posterior distributions of π, δ,δ+, δ+ − δ, and (δ+ − δ)/δ for the FF3 and its time-varying extensions. In Panel A, assetreturns are not scaled by any conditioning variables. In Panel B, asset returns are scaled bythe variable TERM.

Panel A: For the Non-Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe FF3:mean 0.001 0.243 0.248 0.005 2%median 0.000 0.242 0.246 0.000 0%stdev 0.002 0.060 0.060 0.017The FF3+IV:mean 0.056 0.215 0.226 0.011 5%median 0.017 0.213 0.223 0.004 2%stdev 0.077 0.060 0.061 0.020The FF3*IV:mean 0.287 0.124 0.189 0.065 53%median 0.286 0.117 0.185 0.052 45%stdev 0.083 0.062 0.065 0.052

Panel B: For the Scaled Asset Returns

π δ δ+ δ+ − δ (δ+ − δ)/δThe FF3:mean 0.187 0.188 0.232 0.044 23%median 0.189 0.186 0.229 0.035 19%stdev 0.103 0.052 0.055 0.034The FF3+IV:mean 0.218 0.168 0.223 0.055 33%median 0.226 0.166 0.222 0.047 28%stdev 0.107 0.052 0.056 0.038The FF3*IV:mean 0.281 0.088 0.200 0.112 127%median 0.291 0.083 0.197 0.107 129%stdev 0.104 0.046 0.052 0.053

33

Table 5: The Consumption-Based Nonlinear Models

This table reports the summary statistics for the estimated posterior distributions of δ, δ+,δ+−δ, and (δ+−δ)/δ for the consumption-based nonlinear models. In Panel A, asset returnsare not scaled by any conditioning variables. In Panel B, asset returns are scaled by thevariable TERM.

Panel A: For the Non-Scaled Asset Returns

δ δ+ δ+ − δ (δ+ − δ)/δThe CC:mean 0.300 0.311 0.011 4%median 0.297 0.308 0.004 1%stdev 0.064 0.067 0.021The Abel Model:mean 0.289 0.299 0.011 4%median 0.287 0.297 0.004 4%stdev 0.066 0.069 0.020The EZ Model:mean 0.303 0.310 0.007 2%median 0.302 0.307 0.001 2%stdev 0.063 0.065 0.015

Panel B: For the Scaled Asset Returns

δ δ+ δ+ − δ (δ+ − δ)/δThe CC:mean 0.237 0.244 0.007 3%median 0.235 0.240 0.003 1%stdev 0.052 0.056 0.012The Abel Model:mean 0.223 0.237 0.015 7%median 0.221 0.235 0.007 6%stdev 0.053 0.058 0.020The EZ Model:mean 0.219 0.233 0.014 6%median 0.217 0.231 0.008 6%stdev 0.053 0.056 0.019

34

Table 6: Model Comparison

This table compares multi-factor and time-varying models with single-factor and nonlinearmodels. In Panel A, each number in the first part is the posterior mean of the ratio of δ formodel I to δ for model J, where model I is listed in the first column and model J is listedin the second row. Each number in the second part is the posterior probability of δI < δJ.Each number in the last two parts is the posterior probability that δ for model I is smallerthan δ for model J by at least 100q percent. Panel B is similar to Panel A except using δ+.

Panel A: Comparing Models Using the First HJ DistanceNon-Scaled Returns Scaled Returns

Model J: CAPM LCC CC Abel EZ CAPM LCC CC Abel EZModel I

Posterior Means of δI/δJ

CAPM+IV 0.95 0.95 1.01 1.06 0.99 0.93 0.90 0.88 0.95 0.96CAPM*IV 0.87 0.87 0.93 0.98 0.92 0.82 0.80 0.78 0.84 0.85LCC+IV 0.96 0.95 1.01 1.06 1.00 0.95 0.90 0.89 0.96 0.97LCC*IV 0.89 0.87 0.93 0.98 0.93 0.88 0.84 0.83 0.89 0.91FF3 0.77 0.77 0.82 0.86 0.81 0.84 0.82 0.80 0.86 0.87FF3+IV 0.68 0.68 0.73 0.76 0.72 0.75 0.73 0.72 0.77 0.78FF3*IV 0.39 0.39 0.42 0.44 0.41 0.40 0.39 0.38 0.41 0.41

Posterior Probabilities of δI < δJ

CAPM+IV 1.00 0.71 0.54 0.44 0.61 1.00 0.79 0.87 0.69 0.68CAPM*IV 1.00 0.84 0.71 0.61 0.79 1.00 0.91 0.94 0.83 0.83LCC+IV 0.67 1.00 0.51 0.39 0.50 0.67 1.00 0.87 0.66 0.61LCC*IV 0.82 1.00 0.70 0.59 0.69 0.79 1.00 0.93 0.78 0.75FF3 1.00 0.95 0.88 0.82 0.92 1.00 0.89 0.94 0.81 0.82FF3+IV 1.00 0.97 0.95 0.91 0.97 0.99 0.95 0.97 0.89 0.90FF3*IV 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00 0.99 0.99

Posterior Probabilities of δI < (1 − q)δJ for q = 10%CAPM+IV 0.17 0.31 0.18 0.15 0.15 0.26 0.45 0.49 0.39 0.32CAPM*IV 0.47 0.54 0.40 0.34 0.40 0.65 0.71 0.75 0.64 0.61LCC+IV 0.28 0.18 0.13 0.10 0.21 0.34 0.35 0.45 0.34 0.31LCC*IV 0.50 0.47 0.36 0.30 0.41 0.53 0.57 0.65 0.53 0.48FF3 0.86 0.83 0.72 0.63 0.75 0.59 0.67 0.72 0.60 0.58FF3+IV 0.95 0.91 0.85 0.80 0.88 0.81 0.82 0.86 0.75 0.75FF3*IV 1.00 0.99 0.99 0.98 0.99 1.00 0.99 1.00 0.98 0.99

Posterior Probabilities of δI < (1 − q)δJ for q = 20%CAPM+IV 0.05 0.12 0.05 0.04 0.04 0.10 0.24 0.26 0.19 0.15CAPM*IV 0.22 0.29 0.19 0.16 0.18 0.36 0.48 0.51 0.42 0.38LCC+IV 0.11 0.05 0.04 0.03 0.08 0.16 0.17 0.22 0.17 0.14LCC*IV 0.26 0.21 0.16 0.14 0.21 0.28 0.31 0.37 0.29 0.26FF3 0.55 0.57 0.45 0.39 0.47 0.31 0.43 0.46 0.38 0.33FF3+IV 0.77 0.77 0.68 0.61 0.70 0.55 0.61 0.65 0.55 0.53FF3*IV 0.99 0.98 0.97 0.95 0.98 0.98 0.97 0.98 0.96 0.96

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Panel B: Comparing Models Using the Second HJ DistanceNon-Scaled Returns Scaled Returns

Model J: CAPM LCC CC Abel EZ CAPM LCC CC Abel EZModel I

Posterior Means of δI+/δJ

+

CAPM+IV 0.96 0.92 1.00 1.04 0.99 0.95 0.89 0.94 0.97 0.99CAPM*IV 0.92 0.89 0.96 1.00 0.95 0.91 0.85 0.90 0.93 0.94LCC+IV 1.02 0.98 1.06 1.10 1.06 1.03 0.96 1.01 1.05 1.07LCC*IV 1.01 0.97 1.04 1.09 1.04 1.01 0.94 1.00 1.03 1.05FF3 0.78 0.75 0.81 0.84 0.81 0.97 0.91 0.96 0.99 1.00FF3+IV 0.71 0.68 0.74 0.77 0.73 0.93 0.87 0.92 0.95 0.97FF3*IV 0.59 0.57 0.61 0.64 0.61 0.84 0.78 0.82 0.85 0.86

Posterior Probabilities of δI+ < δJ

+

CAPM+IV 0.97 0.83 0.59 0.48 0.61 0.93 0.90 0.83 0.71 0.68CAPM*IV 0.98 0.91 0.71 0.59 0.77 0.98 0.96 0.90 0.80 0.80LCC+IV 0.47 0.76 0.37 0.26 0.32 0.48 0.75 0.60 0.50 0.42LCC*IV 0.54 0.75 0.45 0.34 0.41 0.56 0.79 0.65 0.55 0.49FF3 1.00 0.99 0.91 0.85 0.95 0.71 0.84 0.75 0.64 0.59FF3+IV 1.00 0.99 0.95 0.92 0.98 0.82 0.90 0.83 0.73 0.72FF3*IV 1.00 1.00 0.98 0.97 0.99 0.96 0.98 0.95 0.89 0.90

Posterior Probabilities of δI+ < (1 − q)δJ

+ for q = 10%CAPM+IV 0.13 0.35 0.20 0.15 0.11 0.15 0.44 0.29 0.25 0.16CAPM*IV 0.33 0.51 0.33 0.26 0.26 0.34 0.61 0.47 0.41 0.32LCC+IV 0.08 0.09 0.05 0.03 0.06 0.10 0.24 0.15 0.13 0.09LCC*IV 0.16 0.19 0.11 0.08 0.11 0.14 0.31 0.20 0.17 0.13FF3 0.85 0.90 0.76 0.68 0.76 0.11 0.39 0.24 0.20 0.11FF3+IV 0.93 0.96 0.86 0.80 0.88 0.27 0.53 0.39 0.34 0.25FF3*IV 0.98 0.99 0.95 0.92 0.97 0.74 0.85 0.78 0.70 0.67

Posterior Probabilities of δI+ < (1 − q)δJ

+ for q = 20%CAPM+IV 0.02 0.10 0.04 0.03 0.02 0.03 0.19 0.09 0.08 0.03CAPM*IV 0.08 0.18 0.10 0.08 0.06 0.07 0.28 0.15 0.13 0.07LCC+IV 0.01 0.01 0.01 0.01 0.01 0.02 0.10 0.04 0.03 0.02LCC*IV 0.03 0.04 0.02 0.02 0.02 0.03 0.13 0.05 0.05 0.03FF3 0.54 0.63 0.49 0.41 0.47 0.02 0.16 0.06 0.06 0.02FF3+IV 0.74 0.80 0.67 0.60 0.67 0.07 0.24 0.13 0.11 0.06FF3*IV 0.92 0.95 0.87 0.82 0.88 0.35 0.54 0.42 0.37 0.30

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7 Figures

Figure 1: A Graphic Illustration of HJ Distances

In the illustration, there are two pre-specified stochastic discount factors (SDF), yIt and yJ

t .The line labeled M represents the set of SDFs that correctly price a given set of assets. Thehalf-line labeled M+ represents the set of non-negative SDFs that correctly price the givenset of assets. For yI

t, the first HJ distance is the shortest distance between yIt and M, and

is denoted δI. The second HJ distance is the shortest distance between yIt and M+, and is

denoted by δI+. The HJ distances for yJ

t are defined similarly and denoted by δJ and δJ+.

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Figure 2: Time Series of Conditioning Variables

The conditioning variable, TERM, is plotted in the graph. The variable TERM is the HP-filteredterm spread, the difference in yields between 30-year government bonds and 1-month T-bills.The data series are observed at the end of each month from January 1964 to December 2001.

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Figure 3: The CAPM and Its Time-Varying Extensions

For the CAPM and its time-varying extensions, this figure presents the estimated posteriordistributions of HJ distances. The gray curves are the estimated posterior probability densityfunctions of δ. The black curves are the estimated posterior probability density functionsof δ+. In the panels on the left column, asset returns are not scaled by any conditioningvariables. In the panels on the right column, asset returns are scaled by the variable TERM.

Panel A: CAPM, Non-Scaled Panel D: CAPM, Scaled

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Figure 4: The Small-Sample Posterior Distributions of HJ Distances

For the CAPM and its time-varying extensions, this figure presents the estimated posteriordistributions of HJ distances, conditioning on a small number observations. The gray curvesare the estimated posterior probability density functions of δ. The black curves are theestimated posterior probability density functions of δ+. In Panels A and B, asset returns arenot scaled by any conditioning variables. In Panels C and D, asset returns are scaled by thevariable TERM.

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Figure 5: The LCC and Its Time-Varying Extensions

For the LCC and its time-varying extensions, this figure presents the estimated posteriordistributions of HJ distances. The gray curves are the estimated posterior probability densityfunctions of δ. The black curves are the estimated posterior probability density functionsof δ+. In the panels on the left column, asset returns are not scaled by any conditioningvariables. In the panels on the right column, asset returns are scaled by the variable TERM.

Panel A: LCC, Non-Scaled Panel D: LCC, Scaled

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Figure 6: The FF3 and Its Time-Varying Extensions

For the Fama-French model and its time-varying extensions, this figure presents the esti-mated posterior distributions of HJ distances. The gray curves are the estimated posteriorprobability density functions of δ. The black curves are the estimated posterior probabilitydensity functions of δ+. In panels on the left column, asset returns are not scaled by anyconditioning variables. In the panels on the right column, asset returns are scaled by thevariable TERM.

Panel A: FF3, Non-Scaled Panel D: FF3, Scaled

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Panel C: FF3*IV, Non-Scaled Panel F: FF3*IV, Scaled

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Figure 7: The GMM Estimates of the FF3 and Its Time-Varying Extensions

For the Fama-French model and its time-varying extensions, the GMM estimates of the SDFsare plotted over the time from January 1964 to December 2001. In panels on the left column,the asset returns used for the GMM estimation are not scaled by any conditioning variables.In the panels on the right column, the asset returns are scaled by the variable TERM.

Panel A: FF3, Non-Scaled Panel D: FF3, Scaled

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Figure 8: The Consumption-Based Nonlinear Models

For the consumption-based nonlinear models, this figure presents the estimated posteriordistributions of HJ distances. The gray curves are the estimated posterior probability densityfunctions of δ. The black curves are the estimated posterior probability density functionsof δ+. In the panels on the left column, asset returns are not scaled by any conditioningvariables. In the panels on the right column, asset returns are scaled by the variable TERM.

Panel A: CC, Non-Scaled Panel D: CC, Scaled

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Panel B: Abel, Non-Scaled Panel E: Abel, Scaled

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