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A survey on the estimation of CCAPMs via moment restrictions : the case of Japan野田, 顕彦(Noda, Akihiko)
Keio Economic Society, Keio University
2013
Keio economic studies Vol.49, (2013. ) ,p.69- 91
The purpose of this paper is to provide brief review of estimation methods for the
standard consumption-based asset pricing model (CCAPM) and some of its
serious empirical problems, namely, the two puzzles in the Japanese financial
market. First, we introduce Hansen's (1982) generalized method of moments(GMM) estimator for estimating the parameter of the standard CCAPM. Second,
we show the superiority of alternative GMM estimator, generalized empirical
likelihood (GEL), by measuring the difference of the higher order bias on thestandard CCAPM and indicate the GEL estimator suggest a possibility for solving
the puzzles. Last, we suggest a few methods to examine the standard CCAPM for
future research.
Journal Article
https://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=AA00260
492-20130000-0069

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KEIO ECONOMICSTUDIES49, 69‑91 (2013)

A

Faculty of Economics,
Keio Economic

The purpose

problems,

we introduce
estimating

namely,Hansen's

University,Wakayama,

2011; final version accepted

asset pricing modelJapan

Septemberand some

generalized

methodof momentsof the standard

CCAPM.

Second,estimator, generalized

2012

of estimation methods(CCAPM)the two puzzles in the Japanese(1982)

the parameterof alternative GMM

Wakayama&of this paper is to provide brief review

the standard consumption‑based
empirical

Noda

Observatory, Keio University,Tokyo, JapanFirst version received August

Abstract:

Akihikofinancial market.

weFirst,(GMM)

estimatorshow

the superiority

empirical likelihood (GEL),

the difference of the higher order bias on the standard CCAPM

forof its serious

forby measuring

and indicate the GEL

estimator suggest a possibility for solving the puzzles. Last, we suggest a few methods
to examine

the standard CCAPM

Key words:

GMM,

GEL, CCAPM,JEL Classification
Number:In economics,
sumptioncanbe

it is expected

allocation
explained

by

in

that

an investorwillchangehis/herrealassetsreturns.

inthe subjective

discount

factorthe relationship(CCAPM)developed

estimate

the parameters

are,

INTRODUCTION

to changes

intertemporal

(GMM)

1.

response

sion (and/or

There

EquityPremium Puzzle,Risk‑freeRate Puzzle.

G12, E21, C13.

ies examine

ments

for future research.

elasticity
using

by Rubinsteinandof substitution).

the standard(1976),

andthe degree

EspeciallyLucas(1978)

CCAPM

using

some

empirical

problems

This

con‑

relationshipof relative risk
in finance,

consumption‑basedof the standard

intertemporalmany

asset pricing
andBreedenthe generalized

aver‑
stud‑
model

(1979),

methodandof mo‑

estimator.
however,serious

that

arise

in

the

standard

Acknowledgments.
The author would like to thank Mikio Ito, Colin McKenzie,
Taisuke Otsu, Naoya
Sueishi, Shunsuke Sugiyama, Tatsuma Wada, Makoto
Yano, and an anonymousreferee for their helpfulcomments

and suggestions. The author would alsolike to thank the financial assistanceprovided by the Japan

Society for the Promotion
Academic

Developmentof Science Grant in Aid for ScientificResearch No.24530364
Funds. All data and programs

E‑mail: noda@eco.wakayama‑u.ac.jp;
Copyrightc2013,by the Keio Economic

noda@sanken.keio.ac.jp
Society69

and the Keio Gijuku

used for thispaper are available on request.

70

KEIO

CCAPM

framework.

ECONOMIC

Initially,Mehra

the inability of standard
U.S. financial markets

CCAPM

and Prescott (1985)

point out a puzzle, namely,

to rationalize the statisticsthat haveover the past century. Specifically, they show

fail to explain the difference between
in U.S. financial markets.
an equation

STUDIEScharacterizedthat the models

the average returns of risky and risk‑free assets

This puzzle, called the equity premium

describing the intertemporal

rational behavior

puzzle, comesfromof participants in financial

markets. It is easy to verify the puzzle using several statisticscalculated fromfinancialdata and estimates of the subjective discount factor and the degree of risk aversion. In‑
spired by this puzzle, Weil (1989) points out another puzzle that is a variant of the equity
premiumpuzzle. In turn, economists

confront the inability of these models

the average return of safe assets. The puzzles remainindustrialized countries, including
Prescott (2003)

Japan (see also Kocherlakota

economists

attention has been
CCAPM.

have

paid to the methods

suggested.

of GMMOneof estimating

are weakly

in the standard

in small samples,a numbercompared

proce‑

variable regressions breaks down

to the GMM

estimator, we

theof alternative estimators haveempirical likelihood

estimators are attracting the attention of manyof their better performance

GMM

identified. In an effort to improveshouldeconometricians

estimator. This paper
employWe introduce

GEL

estimate the parametersof the standard CCAPM.

cuss alternative methods

to resolve the puzzles on the standard CCAPM

ar‑estimators toearlier studies and dis‑frameworkindetail below.

Theoutline of this paper is as follows. In section 2, we introduce

GMM

estimator, and how

section 3, wethe GMMgive a brief review

framework
estimator

in Japan. In section 4, weand introduce

estimator, to avoid poor small sample

to examine

GMM

AND

discuss small

estimator. In section

using Japanesefinancial data andsuggest a possibility for solving the puzzles. In the last section, we

2.

In

an alternative estimator, theproperties of the GMM

5, we present an application to the standard CCAPM

paper and suggest a few methods(1982)of the puzzles and several existing attempts in thesampleproperties of the GMM

Hansen's

estimator is applied to the standard CCAPM.

context of the standard CCAPM

GEL

and

To date, not so muchthe parameters

report, the analysis of conventionalgues that instead of using the GMM

more

and Mehra

predictions and empiricalof the alternatives is the class of generalized

estimators. GEL

becausethe model's

and linear instrumental

or all of the parameterspoor performance(GEL)(1996)

their theoretical models.

As Stock and Wright (2000)

some

been

betweenmodifieddures of the standard CCAPM
when

to explain

for the U.S. and otherfor details).

In order to resolve the discrepancy
data, someunsolved

the standard CCAPM

summarize

thisfor future research.

CCAPM1In this section,we present the traditionalempirical method, the GMM
estimator ofHansen (1982), to estimate the parameters of the standard CCAPM.
First,we introduce
1 This section and section 4 rely on Otsu (2007), which contains excellentlecture notes on the GMM
GEL

estimators.

andNODA:

the Hansen's(1982)A SURVEY

GMM

eters, and undertake

2.1.

Hansen's(1982)

Here, we introduce
Beforewe review

GMM

71

CCAPMS

provides us with a way to estimate param‑
then indicate how

this estimator is used to

Estimatora brief review

the estimatorsof the estimators of a momentof the parametersWeWe

6q that satisfiesthe following￡[･] denotes expectations

sometimes

assumeof the data, and p denotes
write <?(･)as a m vector ofthat m>p, and the model

=

0,

(1)

taken with respect to the distribution of jq's. First, we
two‑step GMM(2S‑GMM)estimator. Let

9(0)

(2)

where n is the sample size.The GMM
Ogmm

estimator,Ogmm,

E[g(xt, Oo)g(xt, 9o)f], then the asymptotic

are as follows (see Neweyand McFadden

Consistency:･

Asymptotic

normality:

Vn(eGMM‑ 0o)iefficiency:

if Wl, ^i(§GMM

Q

The optimal weighting(1994)‑^ W. Denoting G := ￡'[9g(fi;6>o)]properties of the GMM

estimator

for details):

A/"(0,(G'WG)‑lG'WQWG(G'WG)‑1).

Asymptotic
=

(3)

p
―^ @o‑

>･･
Qgum

･

･

minimizes the quadraticform:= argminng (6)'Wg (6)0G&

where W is an m x m weighting matrix such that W
and Q :=

has a

condition:

E[g(xi,00)]

briefly present Hansen's (1982)

restriction model.

in such a model, let us define someof i.i.d. observations

to be estimated.

functions of the data and the parameters.

where

OF

of the standard CCAPM.

of parameters

true parameter

ESTIMATION

testing. We

notation. {jti}"=1 denotes a sequence
the numberTHEestimator, which

hypotheses

estimate the parametersON9o) ‑i JV(O, (G'Q‑lG)‑1).

matrix ^‑1

is often unknown.

Therefore,

we calculate the first‑

step estimatorWof 0, §is, by using a known
weighting matrix (for example, choosing§is we can obtain an estimate ofas the identity matrix /). Then, using
the optimal

weightingmatrix Q"1

as
1

Q

:=
n

The feasibletwo‑step GMM

En g(xi,§is)g(xiJis)(2S‑GMM)

02S‑gmm(4)

is defined as■=argmin ng(9)''Q
9eR

lg(6)

(5)

72

KEIO

ECONOMICSTUDIESIf Q is a consistent estimator of Q, Qis‑gmm
and its asymptotic

is an asymptoticallyV^(02S‑gmm‑ 60) ‑i A/"(0,(G'Q‑lG)‑1).

In order to analyze the specification of the estimated(1982)

J test of overidentifying restrictions whichnull hypothesis

model,

we

1g02S‑GMM)‑> Xm‑pIn the next subsection, we present the Hansen's(1982)

parametersrestriction models.

of the standard CCAPM

Momentin moment

Restriction Model

GMM

■

(6)

estimator to estimate theEuler equation in the standard CCAPM,

in whichthat the utilityfunction is in the constant relative risk aversion (CRRA)
H

Q+i
J
c7

E'
where

Hansen's

property under thefor CCAPM

Let us consider the following
assumecan adopt

has the followingthat equation (1) is true:
J := ng(§2S‑GMMY^

2.2.

efficient estimator,2

distribution is given by

a + Ri,t+i)‑l＼=O,

weclass:(7)i = ＼,2,...,NRt,t+＼is the real return of the zth asset at time t + 1, which is defined as
Pi,t+i + dift+i
Ri,t+i

1.2

‑1

(8)

N

Put

where

the subscript /indicates time, Ct is real per capita consumption,pu

is the priceof the zth asset, d[j is the dividend of the zth asset, N is the numberof assets, (1 is the

subjective time discount factor, y is the relative risk aversion (RRA),

and Et [･]is the ex‑pectation operator conditional on the information

available. In most empirical research,equation (7) is used to estimate the subjective time discount factor and the degree ofrelative risk aversion (or its reciprocal, the intertemporalto test the restrictionsimposedWe

derive unconditional

error vectors uf+i(#)
ments, zt be a K
momentelasticityof substitution) andby equation (7).

momentdependingrestrictions for the standard CCAPM

on the underlying

x 1 vector of instruments

knownparameterwith eachN

vector 9 and the instru‑

at time t. We

define a m(―

N ･ K)

indicator vector gt(9) as
gt(0)=ut+i(0)RZt‑(9)

The error vector is defined asK
ut+i(6)
where

Rf+i

=

(Rht+i,Givenequations(7) and

R2,t+i,
(10),

Q+i

＼

■

we

..., RN,t+iY,
can

1

=

(1, 1,...,

must employ1)' and0=(fi,y)'.

derive￡*[ii*+i(0)]=O,
2 We

(10)

‑(l + Rf+i)l‑la heteroskedasticityand auto‑correlationconsistent(HAC)(11)
estimate of Q like the one

proposed by Newey and West (1987) when we use time‑series data. This estimator depends on a kernel anditsbandwidth that can be chosen using the procedure suggested by Andrews (1991).

NODA:where

A SURVEYONTHE

0 = (0, 0, ..., 0/. Let zt be an K

information

ESTIMATION

OF73

CCAPMS

vector of instruments

set available at time t,It, and define the NK(―which

is a subset of them) vector, gt(0), as(12)gt(0) =u,+i(0)<g>z,.
Then the Euler equation implies
E[gt(0)] = E[E[gt(9)＼It]]= E[E[ut+iRzt＼It]]= E[E[ut+i＼It]Rzt]= ￡[0]= 0.
where￡[･]is the unconditional(13)

expectation

operator. We

call equation (13) a moment

restriction model.
Generally, let {yt }fr=1,denote observations
usually assumed
hand

on a finitedimensional

to be stationary and strongly mixing

side of equation (12), gt(9), is called a moment

the parameters,
Whenbut also depends

our attention is focused(see Smith

process, which(2011)).

indicator, which

is a function of

on the data, yt, and potentially the instruments,on the parametersis

The right‑

zt.

to be estimated, yt and zt are usuallyomitted.
3.

TWO

Since Mehra

PUZZLESAND

and Prescott (1985)

are puzzles, the equity premium
marketwhen

EARLIEREMPIRICAL

and Weil (1989), many

IN JAPAN

studies indicate that there

puzzle and the risk‑free rate puzzle, in the financial

the standard CCAPM

framework

is used.

puzzles in the financial market: the equity premium
Second,STUDIES

we present a brief review

of JapaneseFirst, wefirstintroduce

twopuzzle and the risk‑free rate puzzle.

earlier empirical studies in the context ofthe standard CCAPM.
3.1.

The Equity Premium

PuzzleIn order to analyze the equity premium,Hansenand Singleton (1983)

assumethat thejoint conditional distribution of asset returns and the stochastic discount factor is log‑
normaland homoskedastic.

equity premium

These

is determined.

assumptions

When

a random

simplify the later discussion of how

thevariable X is conditionally lognormally

distributed,
log EtX
where

Varf logX

moskedastic,:=

Et[(＼ogX=

Et log X +‑Wait― Et logX)2].

(14)

If, in addition X, is conditionally ho‑

we obtain

VarflogX= E[(logX‑

EtlogX)2]

Thus, with joint conditional lognormality
consumption,log X,= Var(log X ‑ Et log X).

and the homoskedasticity

we can take logs of equation (7) to obtain(15)of asset returns and

74

KEIO

0

where

Et[n,t+i]+＼ogP

ct = log(Q),return innovations
consumption

ECONOMICSTUDIES‑ yEt[Act+i]

af + Y2°c

+

^Yaic

ri;f = log(l + Ri,t), of denotes the unconditional
Var(ri;f+i ― Et Ov+i)),

innovations

Var(cf+i

variance of log<tc2denotes the unconditional― Et(ct+i)),

variance of log‑return and log‑consumption

(16)

2

variance of log

and O[C denotes the unconditionalinnovations

Cou(ri;f+i― Et Ov+i),

co‑

ct+i ―

Et(ct+i)).
We

consider firstan asset with a risk‑free real return rinnovation

variance o＼ and the covariance(16) the following

expression
rf+1 =

Equation+l. For this asset the returna/c are both zero, so we obtain from equation

for the risk‑free real return:‑log￡+ yEt[Act+l]

2 2
?‑￡‑

‑

.

(17)

(17) indicates that the risk‑free real return is linear in expectedconsumption

growth, with a slope coefficient equal to the degree of relative risk aversion. Subtracting
equation (17) fromequation (16) yields an expression

for the expected excess return on

risky assets over the return on the risk‑free asset:

Et[ri,t+i ‑ r/+1] +Equation(18) showsthat is, when
Mehrawith consumptionof consumption

and Prescott (1985)

served difference between

‑you

■

(18)

by the degree

growth.

of relative risk aver‑This indicates that an asset withtends to have low returns whenthe marginal utilityof consumption

cial markets.

=that risk premia is determinedsion times the covariancea high covariance

y

show

consumption

is low,

is high.

that the standard CCAPM

fails to explain the ob‑

the average returns of risky and risk‑free assets in U.S. finan‑This puzzle, called the equity premium(18) describing the intertemporalpuzzle, comesfrom

an equation

rational behavior of participants in financial markets.

This puzzle we can easily verify using several statisticscalculated fromfinancial data

together with estimates of the subjective discount factor and the degree of relative risk
aversion.

Unless investors are extremely

too high to be consistent with observed
3.2.

risk averse, the observed

consumption

equity premium

The Risk‑free Rate Puzzle

Weil (1989)

indicates that even if weresolve the equity premium

puzzle by usinglarger values for the degree of relative risk aversion, y, this leads to a second
Equation (17) implies that the unconditionalmean

puzzle.

risk‑free return is given by
y V2

2

(19)

2

where

is

behavior.

g is the mean

growth rate of consumption.

puzzle called the "risk‑free rate puzzle" which
too high), then with powerintertemporally

utilitythey must(＼[ris too small). Given

Weil (1989)

showsthat there is anotheris that if investors are risk‑averse (y isalso be extremely

positive averagerisk‑free rate and a high rate of time preference,

unwilling

consumption

such investors

would

to substitute
growth,
have

a low

a strong

NODA:A SURVEY

Table

ON

1.

THE

ESTIMATION

Estimation

OF

75

CCAPMS

results in earlier studies

p

Y

AT

CONS

Hamori (1992)

0.997

0.216

96

Hamori (1994)

0.997

0.153

240

Tanigawa (1994)

0.998

0

Hori (1996)

0.997

0.049

Nakano

0.983

0

0.996

0.721

M
M
M
M
SA
Q

and Saito (1998)

Baba(2000)

HJD

108

RejectRejectRejectAccept179
65
76

Puzzles
No
No
Yes
Yes
Yes
No

Notes:(1)"P"

denotes

the estimate

the relative risk aversion,
bound

(2)

test, and"CONS"of the subjective"HJD""A/"" denotes

denotes

discount

rate, "y"the result of Hansen

the number

denotes

the estimate

and Jagannathan'sof degree(1991)

ofvolatilityof observations.

denotes the frequency of the gross per capita consumption

growth:"M"

for monthly,"SA" for semi‑annual, and "Q" for quarterly.

desire to borrowfromthe future to reduce

their average

consumption

low risk‑free rate is possible in equilibrium if investors have a low
time preference
If the degree

that reduces―y2crc2/2 in equation (19) dominates

down

again. The quadratic term reflects precautionary
consumption

streamsagainst their desire to borrow.

havea precautionary

But a reasonable

desire to save, whichsort to resolve these two puzzles.

same

several approaches

in earlier

Studies in Japan

there is a consensus

that the puzzles

exists in the U.S. stock market, thestatement is not true when

we use Japanesedata. In other words, several earlier

studies attempted

to resolve the puzzles in the Japanese

yet a general consensus.
that the puzzles

For example,

does not exist, while

Saito (1998)

show

brief review

of JapaneseHamori(1992,

Tanigawa

financial markets, there is not1994) and Baba

(1994),

Hori (1996)

that there exists the puzzles in Japan. Therefore,
earlier empirical studies belowthat (1) the estimates of parameters

to 0.2 except with Baba
and Jagannathan'svolatilityboundconcludeand Nakano

and

we only present aestimator in earlier studies. Itof the standard CCAPM(2000), (2) there is no consensus(1991)(2000)

to avoid confusion.

Table 1 presents the estimation results using the GMM
shows

can workrate of time preference is obtained only

case. In the next section, we introduce

Earlier Empirical

the risk‑free ratesavings; risk‑averse agents withstudies that have

Althoughthen the negative quadratic

the linear term and pushes

as a knife‑edge

3.3.

rate. A

their desire to borrow.

of relative risk aversion y is high enoughterm

uncertain

growthor negative rate ofis almost fromzero

about the results of Hansen

test,and (3) the sample

sizes are allless than

300 in these papers.
In particular, the estimates by Hamori
do not exist, while Nakano
exists. We

can confirm

to the conclusion.

and Saito (1998)

that the Hamori's

However,cally in small sample(1992) lead to the conclusion

report quite opposite results: the puzzles(1992)

2S‑GMM

we argue that his 2S‑GMM

case as will be shownthat the puzzles

estimates of fi and y lead
estimates is unreliable empiri‑in next section. In turn, Nakano

and Saito

76

KEIO

(1998)

assert that their 2S‑GMM

ECONOMIC

estimates of p and y by a single asset CCAPM

stock data lead to contradiction among
real estate, and call money,

HamoriIn the context of the previous(1994)and Zin (1989,

1991). Bakshi(1996)and Fusecannot produce

and Wada

the standard

model(2006)

than 300.

use the GMM

the exception

However,

itis obvious

serious small sample

estimator in more

Small

Samplesmall sample
Economic
Generalizedof the vari‑

sizes are allless

sizes used in these papers is "small"GMM

estimator

ESTIMATORS

of GMM

the GMMestimator,

estimator and show

estimator and the GEL

the

ones.

Estimator

point out that the GMM

estimator has somethe special issue of the Journal

1996 is devotedof Moments").

performance

properties of the GMM

estimator. In particular, we present

empirical likelihood (GEL)

Statistics Vol.14 No.3

We

biases of the GMM

2. Over‑rejectionof the null hypothesis

properties of GMM‑based

to the "Small‑Sample

can classify these problemsof the GMM

1. Small sample

3. Poor power

the puzzles us‑

Survey.

et al.(2008), the sample

poor small sample

generalizedstudies which

Methodto examineand Expenditure

alternatives to the GMM

properties (for example,

poor finite sample

both a limited

and a persistent income

As described in next section, the GMM

ALTERNATIVE

Properties

are many

the international risk
employestimator to estimate the parameters

difference of the higher order bias between

Thereconsider trading costs,
cointegrating approach.

properties. Thus, we will discuss those properties of the

In this section, we firstintroduce

the recently developed

use
and

detailin next section.
4.

some

For ex‑by Epstein(1999)

et al. (2008)

and Duffie (1996)

that the sample

to im‑

applies Jagannathan(2002)

canonical

and Zeldes (1991)

of Kubota

et al. (1996)).

proposedtaking into accountFamily Income

CCAPM

and Tokunaga(2001)and Sonoda

and Kubota

by Constantinides

Withand then introduce

Maki

by Mankiw

the Japanese(see Hansen

and Hamori

and Reinhart's (1998)

suggestedfrom(1997)

estimate a CCAPM

All of these Japanese
ous CCAPMs.

and Naka

the U.S. and Japan,

proposeding micro‑data

has somea contradic‑

financial markets:

utility modelwith habit formation. Iwaisako

uses Ogaki

participation model

sampleNakanoon the puzzles, other earlier studies
modifythe non‑expectedcross‑section approach.

(2004)

sharing between

4.1.

work

or extremely

employsan asset pricing model

GMM

empirical

additional assumptions

HamoriWang'sstock,

Their estimates are as unreliable as those of

their ability to explain the puzzles in the Japanese

ample,

shock

the existence of the puzzles. However,drawbacks.

within three markets:several financial markets to lead to the puzzles.

introduce

Basu

moments(1992), and estimates of a multiple assets CCAPM

tion amongprove

the sample

suggesting

and Saito's analysis has two

STUDIESconcerning

estimator.
GMM‑based

tests.

&

Properties of

estimator as follows.

whenseriousof Businesstests are used.

theNODA:

The firstproblem,
estimation

A SURVEY

ON

the "small sample

errors, manytification. Altonji and(possibly

Segal (1996)

Windmeijer

bias. Han
(2009)and tests become

show

all the other parameters

as an example

The second

problemtests are used."The

concludeWald(1996)

prop‑of the standard

estimation.
when

GMM‑based

test and the overidentifying

concludethat there is sometests to overreject the null hypothesis.

when

the 2S‑GMM

The third issue concerns

HansenAhn

and iterative GMM

the "poor power

and Gadarowski(2004)(IT‑GMM)tendency

et al.(1996)

also

that the Hansen(1997)

size is small.

can un‑derstand that the volatility boundtest of Hansen

and Jagannathan

becausein a large part on GMM

Thus, we knowthat there are manystudies attempt to improvetor. Hall (2000)

reliable methodsof statisticalinferencetification problems.
and Windmeijer
Taylor expansionHan(2009)

when

and Phillips (2006),
proposesome

to avoid the many

modified
momentreport that the performance

properties of the GMM

4.2.

Empirical

Generalized

properties than the GMM
observations(2005)

proposeby weakiden‑(2007), and Newey

distributions using usual

problem.

of the bias‑corrected

However,

GMM

Donaldestimator is far

alternative estimators to overcome

estimator.

Likelihood

In this subsection, we introduce
ized empirical likelihood (GEL)and Swanson

asymptotic

estima‑the estimation error ofand Kleibergenconditions

from rosy. Therefore, recent several studies propose
poor small sample

estimation. Recently,

researchers are confronted
Chaolacks powerproperties of the GMM

matrix to modify

Q. Stock and Wright (2000)(1991)We

estimates.

with GMM‑based

small sample

applies a centered covarianceassociated with estimating

et al. (2009)

problems

on poortests." For ex‑

and Jagannathan's

the sample

their test statisticdepend

estimators used.

properties of GMM‑based

show

specification error test is quite unreliable whenmanynormalitythat tests of the overidentifying restrictionslead to overrejections of the modelin small samples

ample,

and asymptotic

identification in non‑linear GMM

and Eichenbaum

andestimator

is weakly identified,point to the parameters

tests refer to the GMM‑based

Wald

and Newey

are used to estimate parame‑is the "over‑rejection of the null hypothesis

restrictions test.Burnsidefor GMM‑based

momentsiden‑

the weight

for the GMM

that even if only one parameter

estimator.3 Theyof weakweakleads to a serious(2007)

approximations

there are many

show

and

momentsand Swanson

do not satisfy the consistency

erties of the usual 2S‑GMM

estimator," is due to: weight
instruments,

estimation and the sample

when

77

CCAPMS

indicate that the correlation between

that the asymptotic

impreciseOFor uninformed)

and Phillips (2006), Chaoters. Stock and Wright (2000)

CCAPM

ESTIMATION

biases of the GMM

weakestimation errors of the 2S‑GMM
small sampleTHE

one alternative to the GMM

estimator. The GEL

estimator. Let us define someof the data, and p denotes the number

3 Earlier studies of the problems associated with many
Nelson and Startz(1990), Bekker (1994), Boundestimator, the general‑

estimator has many
notation. {xi}^lof parameters

moredesirable

denotes i.i.d.

to be estimated. We

weak instruments and weak identificationinclude

et al.(1995), and Staiger and Stock (1997) which focus on

theinstrumental variable and two‑stage least square estimators(see also Stock et al.(2002) for details).

78

KEIO

sometimes

ECONOMIC

write g(‑) as a m‑dimensional

eters. We

assumethat mfollowing

momentcondition:

STUDIESvector of functions of the data and the param‑> p, and that the model

has a true parameter#o satisfying thef
E[g(Xi,90)]
wherethe moment

specified. We
random=g(x, 60)dF(x)function <?(･)is known,

the log‑likelihood

is basedprobability function pi =on the data {xj}"^.

Then,

P{x

=

xi} of theusing pi we can express

l°g Pi If there is no information■
is given byfunction as X^=i

likelihood function maximization(20)

but the distribution function F of xi is notconsider the multinominal

variable x which= 0,

about {xi }"=1, theproblemn￡log Pi ,

max
{Pi)U

n

y^

s.t. o < pi < i,To solve this restricted maximization

problem,npi = i.

we define the Lagrangian/function asnc

Pi ‑1)
v j=l

and obtain the following

first‑orderconditions,
8C

̲

1

dpi

0

pi

1
0

‑X
Pn

dc

n

E

1 ‑

Pi=O

Then we also obtain
1

1

1

k =

= ･ ･･ =
PI

P2

O

pi =

pi = ■■■= Pn ,Pnand derive the following maximum

likelihood estimator(MLE)

of pi

1

PI + P2 H
Therefore, the MLEtion (EDF),

and wedicator function. We

＼‑
Pn

npt

= n

1 o Pil(21)nof the distribution function F is the empirical distribution func‑
can showthat F(x)incorporate:― n

the momentl Y%=iH*icondition model<x) where1{‑} is an in‑(equation (20)) into thelikelihood function. Then,

we can write the moment

by using the multinominal

probability pi , and solve the likelihood function maximiza‑tion problem

associated with pi given as follows:

condition as Y11=i Pi9(xi , #o) = 0

NODA:A SURVEY

ON

max

THE

ESTIMATION

￡=
where

we

En log pi ‑ nX

multiply the second

calculations. Then,

Pi = lEn

PiQ(xi,9)

7

v j=l

term of right‑hand0

Pi ‑1)

PiQ(xi,6)

we also obtain the followingdC

79

CCAPMS

En log PinE

s.t. 0 < pi < 1

OF

(22)

of equation (22) by n to simplify the
first‑order conditions:

1‑nX'g(xi,e)+ y = 0

PI

(23)
1
dpn

nk'g(xn,O) + y = 0,

Pn￡n Pi9(xi,0)Multiplying

all the elements
PI

(24)

0

of equation (23) by pi gives,

1 ―nk'pig(xi,O)

dpi

0

+ YP＼

(25)
= 1 ‑ nX'png(xn, 9) ‑＼‑yPn=0

Pn

The equationsin (25) can be rewrittenconcisely as:r)C
Pi ― = 1 ‑ nk'ptgixi, 0) + ypi = 0 ,dpi
Summingover all elements

￡n Pi

(26)

of equation (25), we also derive that

n

dpi

Vi

n ― nX1 ￡ Pi9(xi,0) + y

n

E

n + y

Pi

Substituting this expression into equation (26), we also obtain the MLE

0

(27)

of pi

1
1

npi(l+Xf(e)g(xi,e))^so that pi is explicitly determined(28) by g(zt, 9) and summingpin(l+Xf(0)g(xi,9))

if X(9) is given. Multiplying

'

(28)

both sides of equation

over i gives^l+kf(O)g(xi,O)

(29)

80

KEIO

Given

g(xj, 6), we

ECONOMICSTUDIEScan derive the X whichempirical likelihood ratio (ELR)

satisfiesequation (29). We

and the empirical likelihood (EL)can define theestimator of 9, 9el,

as follows:nn

ELR(9)=J2^gPi‑J2lo^Pi
i

1=1

1
n
En ＼ogn{＼+X'(e)g(xi,e)) + E logrc
It

En lognn

E ＼og(l+Xf(9)g(Xi,0)) + ￡ logra―

En log (I+X'(9)§EL
ELR{6)(30)

g(Xi,0))(31)＼―argmax ELR(6),can also be obtained using the dual problem of equation (22)n=min‑^log(l+A/0(jc,‑,0)).

ELR{6)
Therefore,

we

(32)

can consider that the empirical likelihood is an expansion

function using a semi‑parametricapproach.4

Newey

and Smith (2004)

of likelihood
generalize theduality between

equation (22) and equation (32), and define the generalized

likelihood (GEL)

estimator as

empiricalnOgel:=argmin￡(#)
6e&:= argminsupV^p(k'g(xj, 9))
6e&
1 z―'
i―l

(33)

This has a dual problem as follows
Ogel

= are

n log h( pi),EmmeeR,{Pi%=inByspecifying

s.t. 0 < pi < 1

E

Pi = lEn Pig(xi,O)=O.

and

(33)

and (34),

we

p

･

Empirical･

Exponential

p(v)

p(v)
･

=

h in equations

likelihood
log(l

‑

―

v) or h(p)or h(p)updating―(1canderivefollowing

estimators.

(EL):
=

‑ log(p).

tilting (ET):― ― exp(i>)

Continuous
p(v)

(34)

+

v)2/2

GMM

― p log(p),proposedby Imbens

et al. (1998).

(CU‑GMM):or h(p)― [(np)2― l]/2n,proposedbyHansenet al.

(1996).
4 The empirical likelihoodincorporates the moment

condition using the multinominal probability o,‑.

NODA:We

A SURVEYON

can thus confirm that the GEL

contains the GMM

THE

ESTIMATION

OF

estimator is more

81

CCAPMSa general formulation

estimator as a special case. The asymptotic

because it

distribution of the GEL

estimator is given as^(OgelThat is, the GEL‑ 0o) ‑^ A/"(0,(G'Q‑lG)‑1).

estimator is asymptotically(35)

equivalent to the optimal GMM

estimator,

and is thus asymptoticallyefficient. Similar to the J‑statisticused

estimated

estimator, we can use a J‑statisticto test the overidentifying

using the GMM

restrictions when
for the GEL

the parameters

are estimated using the GEL

estimator is computed

Section 4 in Newey

‑In
Under

the null hypothesis

modelsareestimator. The J‑statistic

using the kernel‑smoothed

and Smith (2004)

when

momentindicator (seefor details).

min (0) = ‑In6≫e01(6Gel)■(36)that equation (20) is true, this statistici s asymptoticallydis‑tributed as xi‑p‑4.3.

HigherOrder Biases ofGMM

In section 4.2, we
the GMMconfirmedthat the GEL

estimator. However,in the asymptotic

expansionand GELestimator is asymptoticallyfor both estimators. In the case of small samples, we must

analyze the properties of the estimators in more
Newey

and Smith (2004)

and GELestimators.

They

decomposite

equivalent tothis theoretical result is confined to the first‑orderterm

detail using higher order expansions.

apply the stochastic expansionof Nagarthe higher order bias of the 2S‑GMM(1959)

to the GMM

estimator, 02s‑gmm,

asfollows:

Bias(6GMM)= Bj + BG + Bn + Bw ,Bi = H(‑a+ E[GHg(Xi, 0)])/n,(37)

BG =‑XE[GfPg(Xi,e)]/n,
Bn = HE[g(Xi, 0)g(xi,eYPg(xi,e)]/np
BW=‑HJ2^Oj (%‑ H)'ej/n= ￡[G(0o)]are both

,

7= 1
where

G(0)

=

dg(xi,O)/d0,E[g(xi, 9)g(xi, 0)'] is a m

and G

x m positive definite matrix. Now￡ =(G/fi"1G)"1,// = EG/^"1,
Hw

The matrix W
W

used

P

that there exists a W

functions ￡(jej).L et

Q=

we define= fi"1‑n"1GEG/fi"1,

estimator.

Assumption

and t‑(x) such that W

depends

on the initialweighting

4 in Newey=W

on the preliminary

and Smith+ J21=i HM/npositive definite, ^[^(jci)] = 0, and ￡[￨￨￡(Jti)￨￨6]< co. Weorder bias dependsAnd= (G'W"^)"^'^"1.

appearing in the definition of Hw

for the GMM

p x m matrices.

+matrix(2004)
Op{nstatesl),W

isshould note that the higher

estimator § only throughthe limit of W

and the82

KEIO

ECONOMICSTUDIES

Qdj = E[d(g(Xi,eo)g(xi,eoy)/dej],
and let abeamgj(xi, 9) is the jth elementj = i,2,...,P,i =＼,2,...,n,j = l,2,...,m,As Newey

and Smith (2004)

of g{xi, 0), and ej is a unit vector.

explain, the terms of Bias(0GMM)

interpretation. Bj denotes the asymptotic
known

1,2,...,≪,

vector such thataj =tYCZE[d2gj(xi,6o)/d6d6f])/2,
where

i =

Q and G, Be

have

biases for an optimal

GMM

denotes the estimation biases associated with estimating

denotes the estimation biases associated with estimating the second
Bw

the following
estimator

momentwith

G, Bq,

matrix, anddenotes the estimation biases associated with estimating the preliminary

weightingmatrix.
In contrast, Newey
Ogel,

and Smith (2004)

show

the higher order bias of the GEL

is given by

Bias(9GEL)
where

Pj(v)

P3 =

―2 (in the EL== B! + {＼ +3‑/p(y)/3u‑/ and pj=

That is, we= Bias(6EL)

estimator because

mationbiases associated with estimating G (Be)In particular, the small sample

squaredmomenterror (MSE,that of the GMM

Donaldpreferable comparedAN

In this section, we
dard CCAPM

estimator is less than the higher
estimator is free fromand Bq(2001))

how

and Anatolyev

dependheavily on the numberthat the higher order meanof the GEL

estimator is smaller than(2005)and GEL

Last, we show

the nonexistence

estimator isfor details).

STANDARD

CCAPM5

estimators to estimate the parameters

the higher

ing the GMM

matrixalso show

can consider that the GELTO THE

apply the GEL

Second,weightingestimator in terms of its higher order properties

APPLICATION

to computethe esti‑estimator is serious whenand calculate the higher order bias on the estimations

the standard CCAPM.

order bias of the GMM

we estimate the parametersof the stan‑. In particular, weand GELestimatesof the standard CCAPM

for
us‑

estimators and calculate the higher order bias of our estimates.
of the well‑known

financial markets by using the GEL

This

Be

As a result, wewith the 2S‑GMM

and Smith (2004)
5.

firstshow

and Newey(39)

and the preliminary

and Smith (2004)

estimator.

(38)

#/ .

biases of the 2S‑GMM

conditions because

conditions. Newey(see Newey=

the GEL(Bw).

there are many

BQ

simplifies to:find that the higher order bias of the GEL

order bias of the 2S‑GMM

of moment^jp/(0) for each j. In the special case wherecase), the bias expression
Bias(§GEL)

5

estimator,

section relies on Ito and Noda

estimates.

(2012a).

risk‑free rate puzzle in the JapaneseNODA:

5.7.

HigherA SURVEY

ON

THE

Order Bias on the Standard

ESTIMATION

83

CCAPMS

CCAPM

Let us apply the higher order bias formulathe GELOFof the GMM

estimator in equation (37) andestimator in equation (38) to the standard CCAPM.

The

momentrestrictions

can be written as≫( Ct+i9(xt,9)j･ (l + R,+i)‑ll

0zt

ct
Then,

we derive the following

gradient of g(xt, 9) as
Y

)

(

■(1 +

dg(xt,6)
G(9) =89

We'(

R Zf .

r

Q+i

R/+i)

(

･ (1 +

Rf+i)

Ct+i

■log

)

define()

yo･ (1 + R*+i)(‑A><8>z*,

＼‑yo

Q+i

)

･ (1 + R,+i) ･ log

Q

ctwhere #ois the true value of 9.
n = E[g(xt,8o)g(xt,8oy]y0‑(l^[[(/*°(^f)+ Rf+i)‑l)Rzf]
y0‑(1X[(A)(^1)
￡2denotes the covariance+ R^+l)‑1)RZf]]

matrix of "the residuals" of the model.

Wesubstitute Q, andG into the right‑handsides of definitions of ￡, H,

P, and Hw.

denotesmatrixreflects the marginal

the covariancethe parameters
the marginal
secondary

preliminarychangesof the parameters,Hestimated affecting the covarianceof the parametersof the model,

and Hw

In this context, ￡changesofmatrix Q, P also reflects
correspondsHto theestimates.

Q8j

‑fe(*

(H

Ct+i
ct

Q+i

)

･(l + R,+i)‑lJ<g>z,J)･(l + Rf+i)‑lJ<g>zfJct

Q$j denotes the marginal change of the jth parameter 6j independently affectingthe
covariance matrix Q.

84

KEIO

ECONOMICSTUDIES‑(2*,p(I+R,+l),0E(%,)dk=‑(^)‑"≫.a+R>+,,bE(^)",0(^i)‑r≫.a+R>+,,j,o￡(%i)￨2̲x <g>zjt,,＼ /2,
ak denotes that the second
kth momentWe

can compute(2004)
weorder effect of the marginal

restriction g^. k runs 1 throughthe higher

changeof the parameterm is the number

Following

the section 5 in Newey

distribution to estimate the above

and Smith's

and Smithbias formula.

on theof asset returns.

order biases for specifications using Newey

theoretical framework.

use the empiricalm, where(2004),

Specifically, we

replace the expectation operators by the averages.
5.2.

Data

In this paper, quarterly data fromsumption
2000)"

is computed

1980Q3as "Nondurable

goods

to 2009Q4

are used. The per capita con‑plus service consumption(benchmark

divided by the estimates of the total population reported in the Annual

National Accounts
using the X‑12

in Japan.

ARIMA

The

per capita consumption

procedure.

Fama‑French's

The returns on short‑term instruments

marketondata are seasonally adjusted

as the return on the risk‑free asset and these are obtained fromligence. The

year

ReportNikko

are employed
Financial Intel‑

portfolio returns are treated as the returns on therisky asset and these are obtained fromTo deflate all series,the "Nondurable

Nikkei

Portfolio Master.6

plus service consumption"on National Accounts

is used.7 Lagged

deflator published in

the Annual

Reporttion growthrate, the real return on the risk‑free asset, and the real return on the market

portfolio are used as instruments.

For the GMM

appear in the moment

should be stationary. To checkconditions

satisfy the stationarity condition, we use the ADF
ble 2 provides somevariables, the ADF
at conventional

and GELvalues of the real consump‑
estimator, all variables thattest of Dickey

whether

descriptive statisticsand the results of the ADF
test rejects the null hypothesis

the variables

and Fuller (1981). Ta‑
tests. For all thethat the variable contains a unit root

significance levels.

5.3. GMM
and GEL Estimates
We estimate the two basic parameters in the standard CCAPM.
To confirm the accu‑racy of our estimates,we compare GMM
and GEL estimatesfor the standard CCAPM.
Table 3 shows the empirical resultswith GMM
estimators(2S‑GMM
and CU‑GMM).
In GMMestimations,we employ an appropriate HAC covariance matrix of Andrews(1991) to reduce estimation biases, which is the asymptotically optimal lag trunca‑tion/bandwidth for the quadraticspectralkernel estimator we used.

6 Fama‑French's
7 The

market factorsin Japan are calculated by following Kubota and Takehara (2007).

"Nondurable

plus service consumption"

deflatoris a weighted inflationrate using "Nondurable

goods" and "Service" deflatorsthat are also published in the Annual Report on National Accounts.

NODA:A SURVEY

Table

2.

ON

THE

Descriptive

ESTIMATION

statistics andOF

85

CCAPMS

unit root tests

CGtM

1.00370.00910.97701.0312‑10.5587

10.00500.0063‑0.01430.0207‑5.73821

Rf

0.01210.1041‑0.33350.2331‑7.22081

118

Notes:(1) "CGf"

denotes the gross real per capita consumption

on risk‑freeasset,and "R(2) "ADF"

growth, "/?/" denotes the real return" denotes the real return on market portfolio.

denotes the Augmented

Dickey‑Fuller (ADF)

test statistics," Lag" denotes the lagorder selected by the Bayesian information criterion,and "A/"" denotes the numberof obser‑vations.
(3) In computing

the ADF

values at the 1%

test,a model with a time trend and a constant is assumed. The criticalsignificancelevel for the ADF

Table

3.

Empirical

testis "―3.99".

results

GEL

GMM
2S‑GMM

EL

ET

0.99720.9984

0.99850.99780.9981[0.0007][0.0008][0.0008][0.0008][0.0008]0.56010.86850.85020.75260.8026[0.1836][0.2343][0.2084][0.2067][0.2087]

0.91790.3947

0.33520.58900.48456

6

6

CU‑GMM

CUE

Y

PJ
DF

6

6

Notes:(1) "P" denotes the estimate of the subjective discount rate, and "y" denotes the estimate ofdegree of the relativerisk aversion.
(2) "pj"

denotes the p‑value for Hansen's J, and "DF"denotes the degrees of freedom

for theHansen's J test.
(3) The Andrews(1991) adjusted standard errors for each of the estimates are reported in brack‑ets.
(4) R version 2.15.1 was used to computeset equal to B =the estimates, the startingvalues of the parameters are

1 and y = 1.

All estimates of /? and y are statisticallysignificant at conventional levels. The

esti‑mates of /3 range from 0.9972 to 0.9984, which is plausible, but the estimates of y range
from

0.5601

to 0.8685, which implies lack of robustness.

test are large enoughTable 3 also showsthe empirical results with GEL

estimations, we choose(1997)and Smith (1997)

moment8 We

Anatolyev(2005)

the smoothedsuggests that the kernel smoothing

GEL

the momentdemonstrates
GEL

J

conditions hold.

estimators (CUE,the truncated kernel proposedto smooth

function, the smoothed

employp‑values for Hansen'sthat we cannot reject the null that the moment

In GELcase) because

The

by Kitamura

EL, and ET).
and Stutzer

function (that is, equation (33) in ourthat,in the presenceof correlation in theestimator is efficient.8In addition, we employ

anestimator, but the optimal kernel weights do not exceed one.

This

has no effect.

86

KEIO

appropriate HAC

covariance

ECONOMICSTUDIESmatrix of Andrews(1991) to reduce estimation biases. The

estimates of ft and y are statisticallysignificant at conventional levels. Theof p range

from

0.9978

to 0.9985;

In contrast to the GMM
for the Hansen's

the estimatesestimators, the GEL

J test are large enoughof y range

fromestimates

0.7526

to 0.8502.

estimates are very stable. The

that wep‑values

cannot reject the null that the moment

conditions hold.
In addition, we calculate the higher order biases of Newey
tigate the asymptotic

not so large that we cannot rely on the GMMof the estimates. Table 4 showsWelarger than those of the CU‑GMM

and the GEL

ticular,biases of the preliminary

weightingestimates (CUE,in case of CU‑GMM

of higher order biases, the GMM
result corresponds

estimatesto that of Noda

in Nodaand Sugiyama(2010)
Table

Estimates

P2S‑GMMY2S‑GMMPCU‑GMMYCU‑GMM
YCUE

of poten‑of these estimates are also more

and the GEL

estimates.

Weshouldis zero, verifying the independence
matrixW.

For the sameother than CU‑GMM

and Sugiyama(2010), whofor 2S‑GMM

and CU‑GMM

objective functions to be minimized

than 20 times

EL, and ET). In par‑estimates are unreliable because

also find that the higher order MSEs

estimate of the covariancefor each estimate.

estimates is are morethan 20 times larger than those of the CU‑GMM

the preliminary

size ismatrix estimator are huge in the 2S‑GMM

estimates. This suggests that both the GMM

note that the value of Bw

to inves‑

the sample

estimates, we should suspect the reliability

higher order biases and MSEs

find that higher order biases of the 2S‑GMM

tial biases. We

and Smith (2004)

higher order properties of our estimates.9 Whenreasonof

as in caseare unreliable. This
comparethe shapes of(see Figures 3 and 4

for details).
4.

Higherorder

biases for eachestimates

MSE

Bi

bg

Bg

0.0001‑0.00020.00010.00440.00440.0258‑0.0501

0.00751.14761.13070.0002‑0.00030.00030.0001

0.0463‑0.08390.04060.00290.00020.00030.00050.04180.04270.0845Pel

0.00010.0001Yel

0.03410.0341Pet

0.00010.00030.0003

Yet

0.03810.03680.05640.05490.04390.0467

Notes:(1)

Bj

denotes the total higher order biases, and MSE

denotes the higher order meansquarederror for each estimation.
(2) To computethe estimates,R version 2.15.1 was used.

9 In the case of stationary time series,we should employ
because there are no kernel‑smoothing(2004).

the formula in Anatolyev (2005).

However,

effectsin our estimations, we employ the formula in Neweyand SmithNODA:

Therefore,we

A SURVEY

conclude

ON

THE

ESTIMATION

that the CU‑GMM

estimates in asymptotic

ple size is small.

We

obtain the economically

CCAPM

employ5.4.

wethe CU‑GMM

A Solution to the Risk‑Free

Although
Japaneseestimates are indisputably

higher order properties whenrealistic parameters

estimator and GEL

estimators.

to resolve the risk‑free rate puzzle in thefinancial markets, there is not yet a general consensus.
(1992) lead to the conclusion

exist, while Nakano

and Saito (1998)

as does the equity premium

the sam‑of the standard

Rate Puzzle in Japan

several earlier studies attempted

timates by Hamori

87

CCAPMSand the GEL

better than the 2S‑GMM
when

OF

puzzle.

For example,

the es‑that the risk‑free rate puzzle does notreport quite opposite results: the puzzle
We

confirmthat the Hamori's(1992)

exists,2S‑GMMestimates of fiand y lead to the conclusion

when

Kandelwe argue that his estimates by the 2S‑GMM

and Stambaugh(1991).

However,

estimator is unreliable as we show
is quite large for samples
the problem
Sugiyama

of weak(2010)

the Japanesethat the CU‑GMM

and Wright(2000)

estate, and call money,(2010)

suggesting

point out; Noda

a contradiction among

there is the risk‑free rate puzzle in the Japaneseand GELs

estimates. Underand homoskedasticity

deliver a convenient

0 = Et[Rift+i]the assump‑of asset returns, Hansen

equation:+ log￡ ‑ yEt[ACt+l]

+

‑(a?

+ y1o1c ‑ lyaic),

07 and oq are the standard deviations of the zth asset and consumption,

tively, and (Tjcis the covarianceequation shownby Kandel

betweenthem.

and StambaughThis equation implies

y 2V

E[RJ

growthrate of real

] is the unconditional
consumption,

estimatesof fi andy

GMM

the GELestimates

andon

expectation
andthe standard(CUE,

10 The equation of Kandel and Stambaughof interestrates derived by Breeden (1986).

+ yg ‑

respec‑the following

CCAPM
and

2

of the risk‑free

into
ET),

we

(41)

2

a^ is the variance

EL,

(40)

(1991):] = ‑logB
where

with

in three markets: stock, realcannot produce

we use the CU‑GMM

tion of joint conditional lognormality
and Singleton (1983)

and Saito

to lead to the puzzle.

we investigate whether
whenapplied todoes not (seeTheir estimates are as unreliable as those of Hamori(1992), and estimates of a single asset CCAPM

financial markets

moments

andthe existence of the risk‑free rate puzzle. However,their analysis has two drawbacks.

several financial markets

estimatesfor details). In turn, Nakano

the sampleofhe fails to avoid

estimates of ft and y by a single asset CCAPM

stock data lead to contradiction among

where

Furthermore,estimate of the standard CCAPM

and Sugiyama

assert that their 2S‑GMM

Therefore,

100.

financial data successfully identifies while that of 2S‑GMM

Figures 3 and 4 in Noda
(1998)

that the higher order biases of the 2S‑GMM

with the size of around

identification as Stockshow

we substitute them into the formulainterest rate, g is the meanof p.10When

this equation
derive

E[r{]

we

and

substituteemployin the range

the

the
CU‑

0.0047(1991) is a special case of the "mean‑variance" representation

88

KEIO

to 0.0050,
(see

which

Table

in Japanis closeto 0.0050,

2 for details). Thus,
when

if oneadopts

we

CCAPM

we

provideand some

STUDIESthe sample

meanconcludeempirical

REMARKSa brief review

of estimationof its serious empiricalproblems,

there is a consensus

explain movements
for Japanese

problems.
CCAPM

Last, wemeasuresolving the puzzles.

performs

dard CCAPM.

shown

We

on poorestimator, to overcomeof the risk‑free rate puzzle in themay

timator to the random
issue and has been
for the 2S‑GMM

properties of GMM

the asymptotic

estimator in the stan‑higher order biases in the GMM

changing

parameter

regression

framework

model

Therefore,of Ito (2007).

properties. We

weassumees‑

This is a very recent
using a rolling method

indicates that the degree of risk aversion has
given poor

estimator, itis likely that a rolling 2S‑GMM

of the standard CCAPM

weby applying the GEL

subject to very littleresearch. For example,(2009)

when

instability of the standard

the failure of the standard CCAPM.
in the standard CCAPM

during the post‑war period. However,poor small sample

the two estimators as

to exist evencan consider that the parameter

estimator, Kim

ties of the 2S‑GMM

the standard

puzzle is stillfoundarise fromparameters(2012a)).

to investigate whether

the equity premium

becausetime‑varying

thoseorder bias on the standard

instability of the stan‑estimator. We

webe madeproperties of GMM

verify the parameter

use the GEL

beensmall sample

adequately

cannotshoulddard CCAPM
CCAPM

statement

estimators to empirically reveal the differences between

in section 5.3. Second,puz‑

studies. In particular,

financial markets. First,we should use the GEL

small sample

can then measurepuzzles in the

restriction mod‑estimator (see also Ito and Noda

two alternative methods

exist

ones and suggest a possibility forthe nonexistencewell in the Japaneseestimator to improveand GELestimator and the GEL
we show

we suggestpoor

estimators, the GEL

financial markets using the GEL

Therefore,
CCAPM

And

the twoof the standard CCAPM

the difference of the higher

the GMM

not

asset

for the standard

via moment

earlier empiricalthat the estimatesfinancial data. Third, we expound

betweenJapanese

methods

in the U.S. financial market, the same

estimators and the alternative GMM

doesof the two puzzles, the equity premium

zle and the risk‑free rate puzzle, and Japanesealthoughrate puzzle

namely,the standard CCAPM

we present a brief review

on the risk‑freemethod.

CONCLUDING

financial market. First, we introduce
els. Second,of the returns

that the risk‑free

the appropriate

6.
In this paper,

ECONOMIC

small sample
methodproper‑will also haveaddress these two issues is relation to the estimatesin future research (see Ito and Noda

(2012b)for details).

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