jagannathan-skoulakis-wang

Generalized Method of MomentsApplications in Finance

Ravi Jagannathan Georgios SkoulakisKellogg School of Management Northwestern University Evanston IL 60208 (rjagannanwuedu)

Zhenyu WangColumbia University Business School New York NY 10027

We provide a brief overview of applications of generalized method of moments in nance The modelsexamined in the empirical nance literature especially in the asset pricing area often imply momentconditions that can be used in a straight forward way to estimate the model parameters without makingstrong assumptions regarding the stochastic properties of variables observed by the econometricianTypically the number of moment conditions available to the econometrician would exceed the numberof model parameters This gives rise to overidentifying restrictions that can be used to test the validityof the model speci cations These advantages have led to the widespread use of the generalized methodof moments in the empirical nance literature

KEY WORDS Asset pricing models Bid-ask spreads CAPM Estimating scalar diffusions Gen-eralized method of moments Linear beta pricing models Market microstructureMean-variance ef ciency Mean-variance spanning Stochastic discount factor Termstructure of interest rates

1 INTRODUCTION

The development of the generalized method of moments(GMM) by Hansen (1982) has had a major impact on empiri-cal research in nance especially in the area of asset pricingGMM has made econometric evaluation of asset-pricing mod-els possible under more realistic assumptions regarding thenature of the stochastic process governing the temporal evolu-tion of exogenous variables

A substantial amount of research in nance is directedtoward understanding why different nancial assets earn dif-ferent returns on average and why a given asset may beexpected to earn different returns at different points in timeVarious asset pricing models that explain how prices of nan-cial claims are determined in nancial market have been pro-posed in the literature to address these issues These modelsdiffer from one another due to the nature of the assumptionsthat they make regarding investor characteristics that is pref-erences endowments and information sets the stochastic pro-cess governing the arrival of information in nancial marketsand the nature of the transactions technology for exchanging nancial and real claims among different agents in the econ-omy Each asset-pricing model speci es what the expectedreturn on a nancial asset should be in terms of observablevariables and model parameters at each point in time

Although these models have differences they also have sim-ilarities Most of the models start by studying the rst-orderconditions to the optimal consumption investment and port-folio choice problem faced by a model investor That leadsto the stochastic discount factor representation of these mod-els The price assigned by a model to a nancial asset equalsthe conditional expectation of its future payoff multiplied bya model-speci c stochastic discount factor In an information-ally ef cient market where the econometrician has less infor-mation than the model investor it should not be possible toexplain the difference between the market price of a securityand the price assigned to it by a model based on information

available to the econometrician GMM can be used to estimatethe model parameters and test the set of moment conditionsthat arise in this natural way For some models it would notbe convenient to work with the stochastic discount factor rep-resentation We provide examples where even in those casesit would be natural to use the GMM

The rst two important applications of GMM in nanceare those of Hansen and Hodrick (1980) and Hansen andSingleton (1982) Subsequent developments in time serieseconometric methods as well as re nements and extensions toGMM have made it a reliable and robust econometric method-ology for studying dynamic asset-pricing models allowingasset returns and the stochastic discount factor to be seri-ally correlated leptokurtic and conditionally heteroscedasticThe works by Newey and West (1987) Andrews (1991) andAndrews and Monahan (1992) on estimating covariance matri-ces in the presence of autocorrelation and heteroscedasticityare probably the most signi cant among these developments

Before the advent of GMM the primary econometric tool inthe asset-pricing area in nance was the maximum likelihood(ML) method which is often implemented using linear or non-linear regression methods The ML method has several limita-tions First for each asset-pricing model researchers have toderive a test for examining model misspeci cation which isnot always easy or possible Second linear approximation isoften necessary when studying nonlinear asset-pricing modelsThird researchers must make strong distributional assump-tions To make the estimation problem tractable the assumeddistributions often must be serially uncorrelated and condition-ally homoscedastic When the distributional assumptions arenot satis ed the estimated model parameters may be biased

copy 2002 American Statistical AssociationJournal of Business amp Economic Statistics

October 2002 Vol 20 No 4DOI 101198073500102288618612

470

Jagannathan Skoulakis and Wang Generalized Method of Moments in Finance 471

even in large samples These limitations severely restrict thescope of the empirical investigations of dynamic asset-pricingmodels GMM enables the econometrician to overcome theselimitations The econometrician does not have to make strongdistributional assumptionsmdashthe variables of interest can beserially correlated and conditionally heteroscedastic Furthernonlinear asset-pricing models can be examined without lin-earization The convenience and the generality of GMM arethe two main reasons why GMM has become so popular inthe nance literature

Despite its advantages GMM has a potential shortcom-ing when compared to the ML method When the distribu-tional assumptions are valid the ML method provides the mostef cient estimates of model parameters whereas the GMMmethod may not To apply GMM the econometrician typicallyuses the moment conditions generated by the stochastic dis-count factor (SDF) representation of an asset-pricing modelThe moment conditions representing the implications of anasset-pricing model chosen by the econometrician may notlead to the most ef cient estimation of model parameters Itis therefore important to understand whether GMM as com-monly applied in practice to examine asset-pricing modelsis less ef cient than the ML method in the situations wherethe ML method can be applied It is well known that GMMhas the same estimation ef ciency as the ML method whenapplied to the moment conditions generated by the classicalbeta representation of linear asset-pricing models Jagannathanand Wang (2002) demonstrated that GMM is as ef cient asthe ML method when applied to the moment conditions gen-erated by the SDF representation of linear asset-pricing mod-els This reinforces the importance and advantage of GMM inempirical asset-pricing applications

In what follows we review the use of GMM in nancewith emphasis on asset-pricing applications In Section 2 wediscuss using GMM for empirically evaluating the standardconsumption-based asset-pricing model and some of its exten-sions In Section 3 we discuss using GMM in examining fac-tor models In Section 4 we discuss the advantages of usingalternative weighting matrices while applying GMM We dis-cuss the ef ciency of GMM when applied to the SDF repre-sentation of asset-pricing models in Section 5 We examinethe use of GMM to estimate the stochastic process for short-term interest rates in Section 6 and discuss using GMM in themarket microstructure literature in Section 7 We summarizein Section 8

2 NONLINEAR RATIONAL EXPECTATIONS MODELS

In most asset-pricing models expectations about the futurerepresent an important factor in the decision making processIn practice however it is the actions of the agents ratherthan their expectations that we observe Empirical work onasset-pricing models relies on the hypothesis of rational expec-tations which in turn implies that the errors made by theagentsmdashnamely the differences between observed realizationsand expectationsmdashare uncorrelated with information on whichthe expectations are conditioned

21 Consumption-Based Capital Asset-Pricing Model

One of the rst applications of GMM appeared in the con-text of estimation of a consumption-based nonlinear ratio-nal expectations asset-pricing model of Hansen and Singleton(1982) Consider an expected utility-maximizing agent withpreferences on current and future consumption streams char-acterized by a time-additive utility function Let time rsquo referto the present let ctCrsquo be the agentrsquos consumption level dur-ing period t C rsquo let Irsquo be the information set available to theagent at rsquo let sbquo 2 40115 be the time-preference parameter andlet U be an increasing concave utility function The agent isthen assumed to choose consumption streams to maximize

X

tD0

sbquotE6U 4ctCrsquo 5mdashIrsquo 70 (1)

The agent can invest in any of N available securities withgross returns (1 C the rates of return) at time t denoted byRi1t1 i D 11 1 N Solving the agentrsquos intertemporal con-sumption and portfolio choice implies the following equationwhich relates security returns to the intertemporal marginalrate of substitution sbquo

U 04ctC15

U 04ct5

E6sbquo6U 04ctC15=U 04ct57Ri1tC1mdashIt7 D 11 i D 11 1 N 0 (2)

A standard assumption frequently used in the asset-pricingmacroeconomics literature is that the utility function U

belongs to the constant relative risk-aversion (CRRA) familyThe utility functions U in the CRRA family can be describedby a single parameter ƒ gt 0 referred to as the coef cient ofrelative risk aversion as

U4c5D c1ƒƒ

1ƒƒif ƒ 6D1 and U4c5D log4c5 if ƒ D10

(3)

For such utility function U (2) translates to

E6sbquo4ctC1=ct5ƒƒRi1tC1mdashIt7 D 11 i D 11 1 N 1 (4)

which by the law of iterated expectations implies that

E661ƒ sbquo4ctC1=ct5ƒƒRi1tC17zt7 D 01 i D 11 1N (5)

for any random vector zt measurable with respect to the infor-mation set It Assume that zt has dimension K 1 For infer-ence purposes zt is chosen to be observable by the econome-trician Typically zt would include lagged rates of return andlagged consumption growth rates

The parameter vector to be estimated is ˆ D 4sbquo1ƒ50 Let 1N

denote the N 1 vector of 1s let 0M denote the M 1 vec-tor of 0s where M D NK let Rt denote the vector of assetreturns at time t and let ytC1

D 4R0t1 ctC1=ct1 z0

t50 denote the

data available to the econometrician at time t C1 The orthog-onality conditions in (5) can be written more compactly as

E6h4ytC11 ˆ57 D 0M 1 where h4ytC11 ˆ5

D 61Nƒ sbquo41 C RtC154ctC1=ct5

ƒƒ 7 dagger zt1 (6)

472 Journal of Business amp Economic Statistics October 2002

with dagger denoting the Kronecker product The sample averageof h4ytC11 ˆ5 is

gT 4ˆ5 D 1T

TX

tD1

h4ytC11 ˆ51 (7)

and the GMM estimate OT is obtained by minimizing the

quadratic function

JT 4ˆ5 D gT 4ˆ50Sƒ1T gT 4ˆ51 (8)

where ST is a consistent estimate of the covariance matrixS04ˆ5 D E6h4ytC11 ˆ5h4ytC11 ˆ507 Note that according to (4)lags of the time series h4ytC11 ˆ5 are uncorrelated and thusthe spectral density matrix reduces to the covariance matrixS04ˆ5 A consistent estimate of S04ˆ5 is

STD 1

T

TX

tD1

h4ytC11O05h4ytC11

O05

01 (9)

where O0 is an initial consistent estimate of ˆ Typically

O0 is obtained by using the identity matrix as the weight-

ing matrix in the quadratic form that is by minimizingJ 0

T 4ˆ5 D gT 4ˆ50gT 4ˆ5 The test of the overidentifying restric-tions is based on the statistic

T gT 4 OT 50Sƒ1

T gT 4 OT 51 (10)

which asymptotically follows a chi-squared distribution withM ƒ 2 degrees of freedom because we have M orthogonalityconditions and two parameters to be estimated

22 Assessment of Asset Pricing Models Usingthe Stochastic Discount Factor Representation

The method given in the preceding section can be used toevaluate an arbitrary asset-pricing model For this purpose werewrite the asset-pricing model given in (2) as

E6mtC1Ri1 tC1mdashIt7 D 11 i D 11 1N 1 (11)

where mtC1D sbquoU 04ctC15=U 04ct5 Any random variable mtC1

that satis es (11) is referred to as a SDF In general a num-ber of random variables satisfying (11) exist and hence thereis more than one SDF An asset-pricing model designates aparticular random variable as a SDF GMM can be appliedin exactly the same way as described earlier to estimate theasset-pricing model parameters and test the overidentifyingrestrictions implied by the asset-pricing model using its SDFrepresentation of an asset-pricing model In what follows weprovide some examples of SDFs We also give an example toshow that the GMM is helpful in the econometric analysis ofasset-pricing models even when the SDF representation of amodel is not used

23 More General Utility Functions

Several researchers have modeled consumption expendi-tures as generating consumption services over a period oftime Notable among these are Dunn and Singleton (1986)and Eichenbaum Hansen and Singleton (1988) Sundaresan(1989) and Constantinides (1990) made the case for the impor-tance of allowing for habit formation in preferences wherehabit depends on an agentrsquos own past sequence of consump-tion Abel (1990) and Campbell and Cochrane (1999) modeledhabit in such a way that it is affected not by an agentrsquos owndecisions but rather by the decisions of other agents in theeconomy Ferson and Constantinides (1991) examined the casein which the utility function exhibits both habit persistenceand durability of consumption expenditures In these cases theutility function is no longer time separable because the owof consumption services in a given time period depends onconsumption expenditures in the past

Campbell Lo and MacKinlay (1997) considered a repre-sentative agent who maximizes the in nite-horizon utility

X

jD0

sbquoj E6U 4ctCj5mdashIt71 where U4ctCj5 D4ctCj=xtCj5

1ƒƒ ƒ 1

1 ƒ ƒ1

(12)

ct denotes consumption and xt is the variable summarizingthe current state of habit in period t When habit is externaland equals the aggregate consumption ct during the previousperiod the SDF is given by

mtC1 D sbquo

sup3ct

ctƒ1

ƒƒ1sup3ctC1

ct

ƒƒ

0 (13)

When habit is internal the SDF representation is not conve-nient for empirically examining the model Instead it is moreconvenient to apply GMM to the following relation betweenthe expected return on any asset and fundamentals that mustbe satis ed when the model holds

Et

poundc

ƒƒ1tƒ1 cƒƒ

tƒ sbquocƒƒ1

t cƒƒtC14ctC1=ct5

curren

D Et

poundRtC14c

ƒƒ1t c

ƒƒ

tC1ƒ sbquoc

ƒƒ1tC1 c

ƒƒ

tC24ctC2=ctC155curren0 (14)

Campbell and Cochrane (1999) developed a model inwhich the habit variable xt enters the in nite-horizon utilityPˆ

jD0 sbquojE6U 4ctCj 5mdashIt7 additively in the following way

U 4ctCj 5 D 4ctCjƒ xtCj5

1ƒƒ ƒ 1

1 ƒ ƒ0 (15)

They assumed that the habit variable xt is external that isit is determined by the consumption paths of all other agentsin the economy In particular xt is a weighted average of pastaggregate per capita consumptions Campbell and Cochraneshowed that under some additional assumptions the SDF isgiven by mtC1 D sbquo4

stC1

st

ctC1

ct5ƒ where st

D ct ƒxt

ctis the surplus

consumption ratio These examples illustrate the versatility ofGMM in examining a variety of asset-pricing models

Epstein and Zin (1991) and Weil (1989) developed a utilityfunction that breaks the tight link between the coef cient of


relative risk aversion ƒ and the elasticity of intertemporal sub-stitution ndash in the time-separable CRRA case For this class ofpreferences the SDF is given by

mtC1 Dmicro

sbquo

sup3ctC1

ct

ƒ1=ndashparaˆmicro 1Rm1tC1

para1ƒˆ

1 where ˆ D 1ƒƒ

1ƒ1=ndash0

(16)

When the period utility function is logarithmic the foregoingSDF reduces to the one identi ed by Rubinstein (1976) (Formore examples see Campbell et al 1997 and Cochrane 2001)

3 FACTOR MODELS

31 Stochastic Discount Factor Representation

Linear and nonlinear factor pricing models are popular in nance In this section we discuss the estimation of such mod-els by applying GMM to their SDF representation We rstconsider the standard capital asset-pricing model (CAPM)which can be thought of as a linear single-factor pricingmodel for econometric evaluation

311 Standard Capital Asset-Pricing Model Considerthe standard CAPM the most standard and widely used equi-librium model Consider an economy with N risky assets andlet Rit denote the gross return on the ith asset during periodt Further let Rmt denote the gross return for the market port-folio of all assets in the economy over the same period tThen the CAPM states that in equilibrium the expected grossreturns are given by

E6Ri7 D ƒ C lsaquosbquoi1 where sbquoiD cov6Ri1Rm7

var6Rm71 (17)

lsaquo is the market risk premium and ƒ assumed to be nonzerois the return of the zero-beta asset This is equivalent to

E6mRi7 D 11 where m D ˆ0 C ˆ1Rm1 (18)

for some ˆ0 and ˆ1 It follows that m is the SDF corre-sponding to the CAPM The foregoing equivalence betweenthe beta and SDF representations of the CAPM was rstpointed out by Dybvig and Ingersoll (1982) and Hansen andRichard (1987) Once again we can apply GMM to estimatethe CAPM parameters and test the overidentifying restrictionsas discussed earlier

312 Linear Factor Models Ross (1976) and Connor(1984) showed that when there are only K economy-wide per-vasive factors denoted by f11 1 fK the expected return onany nancial asset will be a linear function of the K-factorbetas

E6Ri7 D ƒ CKX

kD1

lsaquoksbquoik1 (19)

where 4sbquoi11 1 sbquoiK 50 sup2 sbquoiD 4var6f 75ƒ1E64Ri

ƒ E6Ri754f ƒE6f 757 and f sup2 4f11 1 fK 50 Using algebraic manipulationssimilar to those in the CAPM it can be shown that the fore-going linear K-factor beta pricing model can be representedas E6mRi7 D 1 where the SDF m is given by

m D ˆ0 CKX

kD1

ˆkfk (20)

for some parameters ˆ01 1 ˆK GMM can be used to esti-mate these parameters and to test the model

313 NonlinearFactorModels Bansal and Viswanathan(1993) derived a nonlinear factor pricing model by assum-ing that the intertemporal marginal rate of substitution of themarginal investor is a constant function of a nite number ofeconomy-wide pervasive factors They suggested approximat-ing the function using polynomials In the single-factor casethis gives rise to the SDF

m D sbquo0C sbquo1rf

CX

jD112100K

sbquojmr jm1 (21)

where K is the order of the polynomial used and rf and rm

denote the risk-free rate and the excess return on the mar-ket index portfolio Bansal Hsieh and Viswanathan (1993)suggested approximating the function using neural nets Thisgives rise to the following SDF when the excess return on themarket index portfolio is the single pervasive factor

m D sbquo0 C sbquo1rfC

X

jD112100K

sbquojmg4ƒojC ƒ1jrm51 (22)

where g4cent5 is the logistic function given by g4cent5 D exp4cent51Cexp4cent5

Again the GMM is a natural way to estimate the modelparameters and test the overidentifying restrictions

314 Mean-Variance Spanning Whether the returns ona subset of a given collection of nancial assets are suf cientto span the unconditional mean-variance frontier of returnshas received wide attention in the literature For example aninvestor may be interested in examining whether it would bepossible to construct a xed-weight portfolio of some bench-mark assets that dominate a given managed portfolio in themean-variance space If the answer is af rmative then theinvestor may not have to include the managed portfolio inthe menu of opportunities This can be examined by checkingwhether the set of benchmark assets spans the mean-variancefrontier of returns generated by the primitive assets and themanaged portfolios taken together

Suppose that we are interested in examining whether a setof K benchmark assets spans the mean-variance ef cient fron-tier generated by N primitive assets plus the K benchmarkassets for some value for the risk-free return Huberman andKandel (1987) showed how to construct the statistical tests forthis purpose when asset returns have an iid joint normal distri-bution Note that when the mean-variance frontier of returns isspanned by K benchmark returns for some value of the risk-free return the SDF will be an af ne function only of thoseK benchmark returns Bekaert and Urias (1996) showed howthis property can be used to test for mean-variance spanningusing GMM The advantage to using GMM is that it allows forconditional heteroscedasticity exhibited by returns on nancialassets

32 Beta Representation

Applications of GMM in nance are not restricted to exam-ining asset-pricing models using the SDF framework In thissection we present an application of GMM to empirical eval-uation of linear beta pricing models using the beta represen-tation instead of the implied SDF representation


321 Unconditional Linear Beta Pricing Models Fornotational simplicity we assume here that there is only oneeconomy-wide pervasive factor The results generalize in astraightforward manner to the case of multiple factors Forexpositional convenience we assume that the single factor isthe return on the market portfolio and consider the standardCAPM Following MacKinlay and Richardson (1991) assumean economy with a risk-free asset and N risky assets withexcess returns over period t denoted by Rit for i D 11 1N Further let Rpt denote the excess return on a portfolio p overperiod t Mean-variance ef ciency of the portfolio p impliesthat

E6Rit7 D sbquoiE6Rpt71

where sbquoiD cov6Rit1Rpt7

var6Rpt71 i D 11 1N 0 (23)

Using vector notation will ease the exposition Let RtD

4R1t1 1 RNt50 denote the column vector of excess returns

over period t let sbquo D 4sbquo11 1sbquoN 50 denote the column vectorof betas and de ne ut

D Rtƒ E6Rt7 ƒ 4Rpt

ƒ E6Rpt75sbquo Thisleads to the regression representation

RtD C Rptsbquo C ut1 with E6ut7 D 01

E6Rptut7 D 01 and D E6Rt7 ƒ E6Rpt7sbquo1 (24)

as a consequence of the de nition of sbquo Mean-variance ef -ciency of p implies D 0 which imposes testable restrictionson the data

The traditional approach to testing mean-variance ef ciencyrelies on the assumption that excess returns are temporally iidand multivariate normal This assumption allows use of theWald statistic for testing the null hypothesis D 0 As Gib-bons Ross and Shanken (1989) showed the Wald statisticfollows an F4N 1 N ƒ T ƒ 15 distribution in nite samples andan asymptotic chi-squared distribution with N degrees of free-dom under the null hypothesis

As MacKinlay and Richardson (1991) pointed out it isimportant to relax the iid normal assumption This is becausethere is overwhelming empirical evidence suggesting thatreturns are heteroscedastic and have fatter tails than thoseimplied by the normal distribution In contrast with the tra-ditional approach GMM provides an econometric frameworkthat allows for testing mean-variance ef ciency without theneed to make strong distributional assumptions It is onlyassumed that a riskless rate of return exists and that the excessasset returns are stationary and ergodic time series with nitefourth moments Let 84R1t1 1RNt1Rpt5 2 t D 11 1 T 9 bea sample of T time series observations Using (24) de neuit4 i1sbquoi5 D Rit

ƒ iƒ sbquoiRpt for all i D 11 1N and t D

11 1 T Let bdquo D 411 sbquo11 1 N 1 sbquoN 50 denote the 2N 1parameter vector and de ne the following 2N 1 vectors

ht4bdquo5 D 6u1t411 sbquo15

u1t411sbquo15Rpt uNt4N 1sbquoN 5

uNt4N 1sbquoN 5Rpt70 (25)

and gT 4bdquo5 D 1T

PTtD1 ht4bdquo5 Note that (24) implies the moment

condition E6ht4bdquo57 D 0 which we exploit for estimation andtesting using the GMM methodology One approach is to esti-mate the unrestricted system and then test the hypothesis D 0using the unrestricted estimates The alternative approach isto impose the restriction D 0 estimate the restricted systemand then test for overidentifying restrictions Next we brie ydescribe these two approaches

Unrestricted Case The sample analog of E6ht4bdquo57 D 0 isthe system of equations

1T

TX

tD1

uit4 i1 sbquoi5 D 1T

TX

tD1

uit4 i1 sbquoi5RptD 01 i D 11 1N 0

(26)

Hence there are 2N equations and 2N unknown parametersand the system is identi ed The foregoing equations coincidewith the normal equations from ordinary least squares (OLS)and thus this version of GMM is equivalent to OLS regres-sion for each i It follows from the general theory of GMMas developed by Hansen (1982) that the GMM estimator Obdquois asymptotically normal with mean bdquo and covariance matrix4D0

0Sƒ10 D05

ƒ1 where

D0D E

hiexclht

iexclbdquo0 4bdquo5i

and S0D

CX

rDƒˆE6ht4bdquo54htƒr 4bdquo55070

(27)

In practice consistent estimates DT and ST are used instead ofthe unknown population quantities D0 and S0 Let rdquo1 denotethe test statistic for the hypothesis D 0 Then under the nullhypothesis

rdquo1D T O 06P6D0

T Sƒ1T DT 7ƒ1P07ƒ1 O (28)

has an asymptotic chi-squared distribution with N degrees offreedom where P D IN

dagger 61 07 and O D P ObdquoRestricted Case Imposing the restriction D 0 yields

gT 401sbquo5 D 1T

TX

tD1

ht401sbquo50 (29)

Hence there are 2N restrictions but only N unknown param-eters the system is overidenti ed and not all of the sam-ple moments can be set equal to 0 The GMM estimateof sbquo say Osbquo is obtained by minimizing the quadratic formgT 401sbquo50Sƒ1

T gT 401sbquo5 where ST is a consistent estimator ofS04sbquo5 D PCˆ

rDƒˆ E6ht401sbquo54htƒr 401sbquo5507 It follows then fromthe general GMM theory that Osbquo has an asymptotically normaldistribution with mean sbquo and covariance matrix

D04sbquo504S04sbquo55ƒ1D04sbquo5centƒ1

1 where D04sbquo5DE

microiexclht

iexclsbquo0 401sbquo5

para0

(30)

Finally a test of the N overidentifying restrictions can bebased on the test statistic

rdquo2DTgT 4 Osbquo50Sƒ1

T gT 4 Osbquo51 (31)


which under the null hypothesis asymptotically follows a chi-squared distribution with 2N ƒN DN degrees of freedom

MacKinlay and Richardson (1991) demonstrated the dan-ger of incorrect inference under the iid multivariate normalassumption when the true return distribution is multivariatet that exhibits contemporaneous conditional heteroscedastic-ity given the market return The remedy to the misspeci ca-tion problem is provided by the robust GMM procedure underwhich the test statistic has an asymptotic chi-squared distribu-tion regardless of the true underlying data-generating process

322 Conditional Linear Beta Pricing Models The lackof empirical support for linear factor models in general andthe CAPM in particular coupled with mounting evidence ofpredictable time variations in the joint distribution of secu-rity returns led to the empirical evaluation of conditional lin-ear factor pricing models Early work examining conditionalasset-pricing models includes that of Hansen and Hodrick(1983) Gibbons and Ferson (1985) and Campbell (1987)More recently Harvey (1989) Ferson and Harvey (1993) Fer-son and Korajczyk (1995) and Ghysels (1998) among oth-ers have examined variations of the conditional version of thelinear beta pricing model given in (19) allowing E6Ri7 ƒlsaquokrsquos and sbquoikrsquos to vary over time For example Harvey (1989)examined the following conditional version of the CAPM

E6ri1tC1mdashIt7DsbquoitE4rm1tC1

mdashIt51 (32)

where ri1tC1 and rm1tC1 denote the date tC1 excess returnson security i and the market sbquoit

D cov4ri1tC1 1rm1tC1mdashIt 5

var4rm1tC1mdashIt 5 and E4centmdashIt5

cov4centmdashIt5 and var4centmdashIt5 denote conditional expectation con-ditional covariance and conditional variance operators basedon the information set It Harvey assumed that

E4rm1tC1 mdashIt 5

var4rm1tC1 mdashIt 5Dƒ

With this assumption and the de nitions of sbquoit and conditionalcovariance cov4centmdashIt5 (32) becomes

E4ri1tC1mdashIt5DƒE6ri1tC14rm1tC1

ƒE4rm1tC1mdashIt5mdashIt70 (33)

Suppose that E4rm1tC1mdashIt5Dbdquo0

mzm1t where zm1t denotes a Lm-dimensional vector of variables in the information set It atdate t Substituting this expression into (33) yields

E6ri1tC1mdashIt7DƒE6ri1tC14rm1tC1 ƒbdquo0mzm1t5mdashIt70 (34)

De ne

ui1tC1D ri1tC1

ƒƒ6ri1tC14rm1tC1ƒbdquo0

mzm1t570 (35)

Let zi1t denote the Li-dimensional vector of variables in theinformation set It at date t This gives the following momentconditions for each security i

E6ui1tC1zi1t7D00 (36)

GMM can be applied to these moment conditions to estimateand test the conditional CAPM Harvey (1991) empiricallyexamined the cross-section of stock index returns across sev-eral countries using the world CAPM and taking a relatedapproach Ferson and Harvey (1993) empirically examined amultifactor extension of (32) using GMM They generatedthe moment conditions by assuming that conditional betas

are xed af ne functions of variables in the date t informa-tion set It Ghysels and Hall (1990) showed that the standardGMM tests for overidentifying restrictions tend to not rejectthe model speci cations even when the assumptions regardingbeta dynamics are wrong Ghysels (1998) tested for parameterstability in conditional factor models using the sup-LM testproposed by Andrews (1993) and found evidence in favor ofmisspeci ed beta dynamics

4 ALTERNATIVE WEIGHTING MATRICES

An asset-pricing model typically implies a number ofmoment conditions The number of model parameters in gen-eral will be much less than the number of moment conditionsGMM chooses a subset of the possible linear combinationsof the moment conditions and picks the parameter values thatmake them hold exactly in the sample The moment condi-tions are chosen so as to maximize asymptotic estimation ef -ciency However in some cases the econometrician may havesome a priori information on which moment conditions con-tain relatively more information than others In those casesit may be advisable to bring in the prior information avail-able to the econometrician through an appropriate choice ofthe weighting matrix For example Eichenbaum et al (1988)suggested prespecifying the subset of moment conditions tobe used for estimation and testing the model using the addi-tional moment conditions This would correspond to choosingthe weighting matrix with 0s and 1s as entries to pick outthe moment conditions that should be used in the estimationprocess

There is another reason for prespecifying the weightingmatrix used When making comparisons across models it isoften tempting to compare the minimized value of the criterionfunction T gT 4 O

T 50Sƒ1T gT 4 O

T 5 across the models One modelmay do better than another not because the vector of averagepricing errors gT associated with it are smaller but ratherbecause the inverse of the optimal weighting matrix ST asso-ciated with it is larger To overcome this dif culty Hansen andJagannathan (1997) suggested examining the pricing error ofthe most mispriced portfolio after normalizing for the ldquosizerdquoof the portfolio This corresponds to using the inverse of thesecond moment matrix of returns AD 4E6RtR

0t75

ƒ1 sup2Gƒ1 asthe weighting matrix under the assumption that G is pos-itive de nite Cochrane (1996) suggested using the identitymatrix as the weighting matrix Hansen and Jagannathan(1997) showed that dist4ˆ5D

pE6wt4ˆ507Gƒ1E6wt4ˆ57 equals

the least-squares distance between the candidate SDF and theset of all valid discount factors Further they showed thatdist4ˆ5 is equal to the maximum pricing error per unit normon any portfolio of the N assets

We now illustrate use of the nonoptimal matrix Gƒ1 in thecontext of SDFs that are linear in a number of factors follow-ing Jagannathan and Wang (1996) Suppose that the candidateSDF is of the form mc

t 4ˆ5DF0tˆ where ˆ D 4ˆ01 1ˆKƒ15

0

and FtD 411F11t1 1FKƒ11t5

0 is the time t factor realization


vector Let

DTD 1

T

TX

tD1

RtF0t

PƒE6RtF0t7sup2D and

GTD 1

T

TX

tD1

RtR0t

PƒE6RtR0t7sup2G0 (37)

Then wt4ˆ5Dmct 4ˆ5Rt

ƒ1D 4RtF0t5ˆƒ1 and so NwT 4ˆ5D

DT ˆƒ1 The GMM estimate OT of ˆ is the solution to

minˆNwT 4ˆ50Gƒ1

TNwT 4ˆ5 The corresponding rst-order con-

dition is D0T Gƒ1

TNwT 4 O

T 5D0 from which we obtain OT

D4D0

T Gƒ1T DT 5ƒ1D0

T Gƒ1T 1 Because the weighting matrix Gƒ1

T dif-fers from the optimal choice in the sense of Hansen (1982)the asymptotic distribution of T NwT 4 O

T 50Gƒ1T

NwT 4 OT 5 will not

be a chi-squared distribution Suppose that for some ˆ0 we

havep

T NwT 4ˆ05currenƒ N401S05 where S0 is a positive de nite

matrix and also that the N K matrix DDE6RtF0t7 has rank

K and the matrix GDE6RtR0t7 is positive de nite Let

ADS1=2Gƒ1=2poundIN

ƒ4Gƒ1=25D4D0Gƒ1D5ƒ1D0Gƒ1=2curren

4Gƒ1=2504S1=2501 (38)

where S1=2 and G1=2 are the upper triangular matrices fromthe Cholesky decomposition of S and G and IN is the N N

identity matrix Jagannathan and Wang (1996) showed that Ahas exactly N ƒK nonzero eigenvalues that are positive anddenoted by lsaquo11 1lsaquoN ƒK and the asymptotic distribution ofdist4ˆ5 is given by

T NwT 4 OT 50Gƒ1

TNwT 4 O

T 5currenƒ

N ƒKX

jD1

lsaquojvj as T ˆ1 (39)

where v11 1vN ƒK are independent chi-squared random vari-ables with one degree of freedom

Using GMM with the weighting matrix suggested byHansen and Jagannathan (1997) and the sampling theoryderived by Jagannathan and Wang (1996) Hodrick and Zhang(2001) evaluated the speci cation errors of several empiricalasset-pricing models that have been developed as potentialimprovements on the CAPM On a common set of assets theyshowed that all the recently proposed multifactor models canbe statistically rejected

5 EFFICIENCY OF GENERALIZED METHODOF MOMENTS FOR STOCHASTIC DISCOUNT

FACTOR MODELS

As discussed in the preceding sections GMM is usefulbecause it can be applied to the Euler equations of dynamicasset-pricing models which are SDF representations A SDFhas the property that the value of a nancial asset equals theexpected value of the product of the payoff on the asset and theSDF An asset-pricing model identi es a particular SDF that isa function of observable variables and model parameters Forexample a linear factor pricing model identi es a speci c lin-ear function of the factors as a SDF The GMM-SDF methodinvolves using the GMM to estimate the SDF representationsof asset-pricing models The GMM-SDF method has become

common in the recent empirical nance literature It is suf -ciently general so it can be used for analysis of both linearand nonlinear asset-pricing models including pricing modelsfor derivative securities

Despite its wide use there have been concerns that com-pared to the classical ML-beta the generality of the GMM-SDF method comes at the cost of ef ciency in parameterestimation and power in speci cation tests For this reasonresearchers compare the GMM-SDF method with the GMM-beta method for linear factor pricing models For such modelsthe GMM-beta method is equivalent to the ML-beta methodunder suitable assumptions about the statistical properties ofreturns and factors On the one hand if the GMM-SDFmethod turns out to be inef cient relative to the GMM-betamethod for linear models then some variation of the ML-betamethod may well dominate the GMM-SDF method for non-linear models as well in terms of estimation ef ciency Onthe other hand if the GMM-SDF method is as ef cient as thebeta method then it is the preferred method because of itsgenerality

Kan and Zhou (1999) made the rst attempt to comparethe GMM-SDF and GMM-beta methods for estimating theparameters related to the factor risk premium Unfortunatelytheir comparison is inappropriate They ignored the fact thatthe risk premium parameters in the SDF and beta representa-tions are not identical and directly compared the asymptoticvariances of the two estimators by assuming that the risk pre-mium parameters in the two methods take special and equalvalues For that purpose they made the simplifying assump-tion that the economy-wide pervasive factor can be standard-ized to have zero mean and unit variance Based on their spe-cial assumption they argued that the GMM-SDF method is farless ef cient than the GMM-beta method The sampling errorin the GMM-SDF method is about 40 times as large as that inthe GMM-beta method They also concluded that the GMM-SDF method is less powerful than the GMM-beta method inspeci cation tests

Kan and Zhoursquos (1999) comparison as well as their con-clusion about the relative inef ciency of the SDF method isinappropriate for two reasons First it is incorrect to ignorethe fact that the risk premium measures in the two methodsare not identical even though they are equal at certain param-eter values Second the assumption that the factor can bestandardized to have zero mean and unit variance is equiva-lent to the assumption that the factor mean and variance areknown or predetermined by the econometricians By makingthat assumption Kan and Zhou give an informational advan-tage to the GMM-beta method but not to the GMM-SDFmethod

For a proper comparison of the two methods it is necessaryto incorporate explicitly the transformation between the riskpremium parameters in the two methods and the informationabout the mean and variance of the factor while estimatingthe risk premium When this is done the GMM-SDF methodis asymptotically as ef cient as the GMM-beta method asestablished by Jagannathan and Wang (2002) These authorsfound that asymptotically the GMM-SDF method provides asprecise an estimate of the risk premium as the GMM-betamethod Using Monte Carlo simulations they demonstrated


that the two methods provide equally precise estimates in nite samples as well The sampling errors in the two meth-ods are similar even under nonnormal distribution assump-tions which allow conditional heteroscedasticity Thereforelinearizing nonlinear asset-pricing models and estimating riskpremiums using the GMM-beta or ML-beta method will notlead to increased estimation ef ciency

Jagannathan and Wang (2002) also examined the speci ca-tion tests associated with the two methods An intuitive testfor model misspeci cation is to examine whether the modelassigns the correct expected return to every asset that iswhether the vector of pricing errors for the model is 0 Theyshow that the sampling analog of pricing errors has smallerasymptotic variance in the GMM-beta method However thisadvantage of the beta method does not appear in the speci ca-tion tests based on Hansenrsquos (1982) J statistics Jagannathanand Wang demonstrate that the GMM-SDF method has thesame power as the GMM-beta method

6 STOCHASTIC PROCESS FOR SHORT-TERMINTEREST RATES

Continuous-time models are widely used in nance espe-cially for valuing contingent claims In such models the con-tingent claim function that gives the value of the claim asa function of its characteristics depends on the parametersdescribing the stochastic process driving the prices of under-lying securities In view of this a vast literature on estimatingcontinuous-time models has evolved refer to for example thework by Aumlt-Sahalia Hansen and Scheinkman (2001) for adiscussion of traditional methods Gallant and Tauchen (2001)for a discussion of the ef cient method of moments and Gar-cia Ghysels and Renault (2001) for a discussion of the liter-ature in the context of contingent claim valuation AlthoughGMM may not be the preferred method for estimating modelparameters it is easy to implement and provides estimates thatcan be used as starting values in other methods In what fol-lows we provide a brief introduction to estimating continuous-time model parameters using GMM

61 Discrete Time Approximations ofContinuous-Time Models

Here we present an application of GMM to testing and com-paring alternative continuous-time models for the short-rateinterest rate as given by Chan Karolyi Longstaff and Sanders(1992) Other authors who have used GMM in empiricaltests of interest rate models include Gibbons and Ramaswamy(1993) who tested the Cox-Ingersoll-Ross (CIR) model Har-vey (1988) and Longstaff (1989)

The dynamics for the short-term riskless rate denoted byrt as implied by a number of continuous-time term structuremodels can be described by the stochastic differential equa-tion

drtD 4 Csbquort5dtClsquo rƒ

t dWt1 (40)

where Wt is standard Brownian motion process The forego-ing speci cation nests some of the most widely used mod-els of the short-term rate The case where sbquoDƒ D0 (ie

Brownian motion with drift) is the model used by Merton(1973) to derive discount bond prices The case where ƒ D0(the so-called OrnsteinndashUhlenbeck process) corresponds to themodel used by Vasicek (1977) to derive an equilibrium modelof discount bond prices The case where ƒ D1=2 (the so-calledFeller or square-root process) is the speci cation that CoxIngersoll and Ross (1985) used to build a single-factor gen-eral equilibrium term structure model Other special cases of(40) as short-term rate speci cations have been given by forinstance Dothan (1978) Brennan and Schwartz (1980) Coxet al (1980) Courtadon (1982) and Marsh and Rosenfeld(1983)

Following Brennan and Schwartz (1982) and Sanders andUnal (1988) Chan et al (1992) used the following discrete-time approximation of (40) to facilitate the estimation of theparameters of the continuous-time model

rtC1 ƒrtD Csbquort

C˜tC11 with Et6˜tC17D0

and Et6˜2tC17Dlsquo 2r 2ƒ

t 1 (41)

where Et6cent7 denotes the conditional expectation given theinformation set at time t It should be emphasized that GMMnaturally provides a suitable econometric framework for test-ing the validity of the model (41) Application of ML tech-niques in this setting would be problematic because the dis-tribution of the increments critically depends on the value ofƒ For example if ƒ D0 then the changes are normally dis-tributed whereas if ƒ D1=2 then they follow a gamma distri-bution A brief description of the GMM test follows

Let ˆ denote the parameter vector comprising 1sbquo1lsquo 2 andƒ The structure of the approximating discrete-time modelin (41) implies that under the null hypothesis the momentrestrictions E6ft4ˆ57D0 hold where

ft4ˆ5D

2666664

˜tC1

˜tC1rt

˜2tC1

ƒlsquo 2r 2ƒt

4˜2tC1

ƒlsquo 2r2ƒt 5rt

3777775

0 (42)

Given a dataset 8rt 2 t D11 1T C19 of T C1 observa-tions following the standard GMM methodology let gT 4ˆ5D1T

PTtD1 ft4ˆ5 The GMM estimate ˆ is then obtained by min-

imizing the quadratic form JT 4ˆ5DgT 4ˆ50WgT 4ˆ5 where Wis a positive de nite symmetric weighting matrix or equiva-lently by solving the system of equations D04ˆ50WgT 4ˆ5D0where D04ˆ5D iexclgT

iexclˆ0 4ˆ5 The optimal choice of the weight-ing matrix in terms of minimizing the asymptotic covari-ance matrix of the estimator is WD 4S04ˆ55ƒ1 where S04ˆ5DPCˆ

rDƒˆ E6ft4ˆ54ftƒr 4ˆ5507 The asymptotic distribution of theGMM estimator O is then normal with mean ˆ and covariancematrix consistently estimated by 4D04

O50Sƒ1T D04

O55ƒ1 whereST is a consistent estimator of S04ˆ5 In the case of the unre-stricted model we test for the signi cance of the individ-ual parameters using the foregoing asymptotic distribution Ifwe restrict any of the four parameters then we can test thevalidity of the model using the test statistic TgT 4 O50Sƒ1

T gT 4 O5which is asymptotically distributed as a chi-squared random


variable with 4ƒk degrees of freedom where k is the num-ber of parameters to be estimated Alternatively we can testthe restrictions imposed on the model by using the hypothesistests developed by Newey and West (1987) Suppose that thenull hypothesis representing the restrictions that we wish totest is of the form H0 2 a4ˆ5D0 where a4cent5 is a vector func-tion of dimension k Let JT 4 Q5 and JT 4 O5 denote the restrictedand unrestricted objective functions for the optimal GMMestimator The test statistic RDT 6JT 4 Q5ƒJT 4 O57 then followsasymptotically a chi-squared distribution with k degrees offreedom This test provides a convenient tool for making pair-wise comparisons among the several model candidates

62 The HansenndashScheinkman Test Function Method

Hansen and Scheinkman (1995) showed how to derivemoment conditions for estimating continuous-time diffu-sion processes without using discrete-time approximationsWhereas other methods such as the ef cient method ofmoments may be more ef cient and may be preferred toGMM they are computationally much more demanding thanGMM In view of this it may be advisable to obtain initialestimates using GMM and use them as starting values in othermore ef cient methods to save computational effort

In what follows based on work of Sun (1997) we illus-trate the HansenndashScheinkman method for estimating the modelparameters when the short rate evolves according to

drtDŒ4rt5dt Clsquo 4rt5dWt1 (43)

where Wt is the standard Brownian motion and Œ4cent5 and lsquo 4cent5are the drift and diffusion of the process assumed to satisfythe necessary regularity conditions to guarantee the existenceand uniqueness of a solution process Assume that the shortrate process described by (43) has a stationary marginal dis-tribution The stationarity of the short rate rt implies that forany time t E6rdquo4rt57Dc where rdquo is a smooth function andc is a constant Taking differences of the foregoing equationbetween two arbitrary points t and T gtt and applying Itorsquoslemma we obtain

0DE

microZ T

t

microŒ4rs5rdquo

04rs5C 1

2lsquo 4rs5rdquo

004rs5

parads

para

C E

microZ T

t

lsquo 4rs5rdquo04rs5dWs

para0 (44)

By the martingale property of the Ito integral the sec-ond expectation vanishes at 0 as long as f 4t5Dlsquo 4rt5rdquo

04rt5

is square integrable Applying Fubinirsquos theorem to the rstexpectation in (44) and then using the stationarity propertyyields

E

microŒ4rt5rdquo

04rt5C 1

2lsquo 4rt5rdquo

004rt5

paraD00 (45)

This is the rst class of HansenndashScheinkman moment con-ditions The space of test functions rdquo for which (45) iswell de ned can be given a precise description (see Hansen

Scheinkman and Touzi (1996)) To illustrate the use of (45)consider the CIR square root process

drtDa4b ƒrt5dt Clsquo

prt dWt1 agt01b gt00 (46)

Letting the test function rdquo equal xk1kD11213 we obtain from(45) that

E6a4b ƒrt57 D E62a4b ƒrt5rtClsquo 2rt7

D E62a4b ƒrt5r2t

Clsquo 2r 2t 7D01

implying that the rst three moments of rt are given by bb2 C lsquo 2b

2a and b3 C 3lsquo 2b2

2aC lsquo 4b

2a2 The rst class of moments conditions in (45) uses only

information contained in the stationary marginal distributionBecause the evolution of the short rate process is governed bythe conditional distribution (45) alone would not be expectedto produce ef cient estimates in nite samples especially ifthere is strong persistence in the data We now consider thesecond class of moment conditions which uses the informa-tion contained in both the conditional and the marginal distri-butions

Kent (1978) showed that a stationary scalar diffusion pro-cess is characterized by reversibility This says that if theshort rate process 8rt9 is modeled as a scalar stationarydiffusion process then the conditional density of rt givenr0 is the same as that of rƒt given r0 The reversibilityof rt implies that for any two points in time t and sE6ndash4rt1rs57DE6ndash4rs1rt57 where ndash 2 ograve2 ograve is a smooth func-tion in both arguments Using this equality twice and sub-tracting yields E6ndash4rtCatildet1rs57ƒE6ndash4rt1rs57DE6ndash4rs1rtCatildet57ƒE6ndash4rs1rt57 Now applying Itorsquos lemma to both sides of thelast equation using the martingale property of the Ito integraland applying Fubinirsquos theorem we have

E

microŒ4rt54ndash14rt1rs5ƒndash24rs1rt55

C 1

2lsquo 4rt54ndash114rt1rs5ƒndash224rs1rt55

paraD00 (47)

This is the second class of moment conditions with ndash asthe test function As an illustration we note that for thecase of the CIR square-root process and for ndash4x1y5 beingequal to 4xƒy53 (47) translates to E6a4b ƒrt54rt

ƒrs52 C

lsquo 2rt4rtƒrs57D0

Appropriate choices of test functions (45) and (47) gener-ate moment restrictions directly implied from a scalar station-ary diffusion The rst class (45) is a restriction on uncon-ditional moments whereas the second class (47) is a jointrestriction on conditional and unconditional moments Thetwo classes of moment conditions were derived by using onlystationarity reversibility and Itorsquos lemma without knowingany explicit forms of the conditional and unconditional dis-tributions Because times t and s in (45) and (47) are arbi-trary the two classes of moment conditions take into explicitaccount that we observe the short rate only at discrete inter-vals This makes the HansenndashScheinkman moment conditionsnatural choices in GMM estimation of the continuous-timeshort rate process using the discretely sampled data GMM


estimation requires that the underlying short rate process beergodic to approximate the time expectations with their sam-ple counterparts The central limit theorem is also required toassess the magnitude of these approximation errors Hansenand Scheinkman (1995) proved that under appropriate regular-ity conditions these GMM assumptions are indeed satis ed

Although (45) and (47) provide moment conditions oncetest functions rdquo and ndash have been speci ed the choice ofappropriate test functions remains a critical issue To illustratethe point we consider the linear drift and constant volatil-ity elasticity model of Chan et al (1992) drt

Da4bƒrt5dtClsquo r

ƒt dWt where a1b1lsquo and ƒ are the speed of adjustment

the long-run mean the volatility coef cient and the volatil-ity elasticity It is apparent that (45) and (47) can identifya and lsquo 2 only up to a common scale factor This prob-lem may be eliminated by making test functions dependon model parameters When estimating the model of Chanet al (1992) consider the test functions rdquo14y5D

R yxƒadx

and rdquo24y5DR y

xƒlsquo dx for the class (45) It can be shownthat these are indeed valid test functions in the appropri-ate domain In a related work Conley Hansen Luttmerand Scheinkman (1997) found that the score functions forthe implied stationary densitymdashthat is rdquo34y5D

R yxƒ2ƒdx

and rdquo44y5DR y

xƒ2ƒC1dxmdashare ef cient test functions for (45)They also used the cumulative function of the standard normaldistribution

ndash4rs1rt5DZ rt ƒrs

expplusmn

ƒ x2

2bdquo2

sup2dx (48)

as the test function in (47) where bdquo is a scaling constantThe combination of rdquo11rdquo21rdquo31rdquo4 and ndash yields the momentconditions E6fHS4t1ˆ57D0 where

fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1

a4b ƒrtC15rƒ2ƒC1tC1

ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1

ƒ 12lsquo 2r 2ƒƒaƒ1

tC1

a4b ƒrtC15rƒlsquotC1

ƒ 12lsquo 3r 2ƒƒlsquo ƒ1

tC1

pounda4b ƒrtC15ƒ rtC1ƒrt

2bdquo2 lsquo 2r 2ƒt

currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0

Then GMM estimation of the model of Chan et al (1992)proceeds in the standard way using these moment conditions

7 MARKET MICROSTRUCTURE

A substantial portion of the literature in this area focuseson understanding why security prices change and why trans-actions prices depend on the quantity traded Huang and Stoll(1994) developed a two-equation time series model of quoterevision and transactions returns and evaluated the relativeimportance of different theoretical microstructure models pro-posed in the literature using GMM Madhavan Richardsonand Roomans (1997) used GMM to estimate and test a struc-tural model of intraday price formation that allows for pub-lic information shocks and microstructure effect to under-stand why a U-shaped pattern is observed in intraday bid-askspreads and volatility Huang and Stoll (1997) estimated thedifferent components of the bid-ask spread for 20 stocks in

the major market index using transactions data by applyingGMM to a time series microstructure model

There are several other market microstructure applicationsof GMMmdashtoo many to discuss in a short article of this natureFor example Foster and Viswanathan (1993) conjectured thatadverse selection would be relatively more severe on Mondaysthan on other days of the week This implies that volumewould be relatively low on Mondays Because of the presenceof conditional heteroscedasticity and serial correlation in thedata these authors used GMM to examine this hypothesis Theinterested reader is referred to the surveys of this literature byBiais Glosten and Spatt (2002) and Madhavan (2000)

8 SUMMARY

GMM is one of the most widely used tools in nancialapplications especially in the asset-pricing area In this arti-cle we have provided several examples illustrating the use ofGMM in the empirical asset-pricing literature in nance

In most asset-pricing models the value of a nancial claimequals the expected discounted present value of its future pay-offs The models differ from one another in the stand thatthey take regarding which discount factor to use An econo-metrician typically has a strictly smaller information set thaninvestors who actively participate in nancial markets Hencethe value computed using a given asset-pricing model basedon the information available to the econometrician in generalwould not equal the market price of a nancial claim How-ever the difference between the observed market price and thevalue computed by the econometrician using an asset-pricingmodel should be uncorrelated with information available tothe econometrician when investors have rational expectations

By a judicious choice of instruments available in the econo-metricianrsquos information set we obtain a set of moment condi-tions that can be used to estimate the model parameters usingGMM The number of moment conditions in general would begreater than the number of model parameters The overidenti-fying restrictions provide a natural test for model misspeci -cation using the GMM J statistic

Continuous-time models are used extensively in nanceespecially for valuing contingent claims These models valuea contingent claim using arbitrage arguments taking thestochastic process for the prices of the primitive assets asexogenous This gives the value of a contingent claim as afunction of the prices of underlying assets and the parametersof the stochastic process determining their evolution over timeHence estimating the parameters of continuous-time stochas-tic processes describing the dynamics of primitive asset priceshas received wide attention in the literature We discussed anexample showing how GMM can be used for that purposeThe GMM estimates thus obtained can be used as starting val-ues in more ef cient estimation methods

The economic models often examined in empirical studiesin nance imply moment conditions that can be used in a nat-ural way for estimation and testing the models using GMMThis combined with the fact that GMM does not requirestrong distributional assumptions has led to its widespreaduse in other areas of nance as well We cannot possibly dis-cuss all of the numerous interesting applications of GMM in nance here and new applications continue to appear


ACKNOWLEDGMENTS

The authors thank Eric Ghysels and Joel Hasbrouck for theirvaluable comments and suggestions

[Received May 2002 Revised June 2002]

REFERENCES

Abel A (1990) ldquoAsset Prices Under Habit Formation and Catching Up Withthe Jonesesrdquo American Economic Review 80 38ndash42

Aumlt-Sahalia Y Hansen L P and Scheinkman J (2003) ldquoDiscretely-Sampled Diffusionsrdquo in Handbook of Financial Econometrics eds Y Aumlt-Sahalia and L P Hansen forthcoming

Andrews D W K (1991) ldquoHeteroscedasticity and Autocorrelation Consis-tent Covariance Matrix Estimationrdquo Econometrica 59 817ndash858

(1993) ldquoTests for Parameter Instability and Structural Change WithUnknown Change Pointrdquo Econometrica 61 821ndash856

Andrews D W K and Monahan J C (1992) ldquoAn Improved Heteroscedas-ticity and Autocorrelation Consistent Covariance Matrix Estimatorrdquo Econo-metrica 60 953ndash966

Bansal R Hsieh D A and Viswanathan S (1993) ldquoA New Approach toInternational Arbitrage Pricingrdquo Journal of Finance 48 1719ndash1747

Bansal R and Viswanathan S (1993) ldquoNo Arbitrage and Arbitrage PricingA New Approachrdquo Journal of Finance 48 1231ndash1262

Bekaert G and Urias M S (1996) ldquoDiversi cation Integration and Emerg-ing Market Closed-End Fundsrdquo Journal of Finance 51 835ndash869

Biais B Glosten L R and Spatt C S (2002) ldquoThe Microstructure ofStock Marketsrdquo Discussion Paper 3288 CEPR

Brennan M J and Schwartz E S (1980) ldquoAnalyzing Convertible BondsrdquoJournal of Financial and Quantitative Analysis 15 907ndash929

(1982) ldquoAn Equilibrium Model of Bond Pricing and a Test of MarketEf ciencyrdquo Journal of Financial and Quantitative Analysis 17 75ndash100

Campbell J Y (1987) ldquoStock Returns and the Term Structurerdquo Journal ofFinancial Economics 18 373ndash399

Campbell J Y and Cochrane J H (1999) ldquoBy Force of Habit AConsumption-Based Explanation of Aggregate Stock Market BehaviorrdquoJournal of Political Economy 107 205ndash251

Campbell J Y Lo A W and MacKinlay A C (1997) The Econometricsof Financial Markets Princeton NJ Princeton University Press

Chan K C Karolyi G A Longstaff F A and Sanders A B (1992) ldquoAnEmpirical Comparison of Alternative Models of the Short-Term InterestRaterdquo Journal of Finance 1209ndash1227

Cochrane J H (2001) Asset Pricing Princeton NJ Princeton UniversityPress

Cochrane J H (1996) ldquoA Cross-Sectional Test of an Investment-Based AssetPricing Modelrdquo Journal of Political Economy 104 572ndash621

Conley T Hansen L P Luttmer E and Scheinkman J A (1997) ldquoShort-Term Interest Rates as Subordinated Diffusionsrdquo Review of Financial Stud-ies 10 525ndash577

Connor G (1984) ldquoA Uni ed Beta Pricing Theoryrdquo Journal of EconomicTheory 34 13ndash31

Constantinides G (1982) ldquoHabit Formation A Resolution of the Equity Pre-mium Puzzlerdquo Journal of Political Economy 98 519ndash543

Courtadon G (1982) ldquoThe Pricing of Options of Default-Free Bondsrdquo Jour-nal of Financial and Quantitative Analysis 17 75ndash100

Cox J C Ingersoll J E and Ross S A (1980) ldquoAn Analysis of VariableRate Loan Contractsrdquo Journal of Finance 35 389ndash403

(1985) ldquoA Theory of the Term Structure of Interest Ratesrdquo Econo-metrica 53 385ndash407

Dothan U L (1978) ldquoOn the Term Structure of Interest Ratesrdquo Journal ofFinancial Economics 6 59ndash69

Dunn K B and Singleton K J (1986) ldquoModelling the Term Structureof Interest Rates Under Nonseparable Utility and Durability of GoodsrdquoJournal of Financial Economics 17 27ndash55

Dybvig P H and Ingersoll J E (1982) ldquoMean-Variance Theory in Com-plete Marketsrdquo Journal of Business 55 233ndash252

Eichenbaum M Hansen L P and Singleton K J (1988) ldquoA Time-SeriesAnalysis of Representative Agent Models of Consumption and LeisureChoice Under Uncertaintyrdquo Quarterly Journal of Economics 103 51ndash78

Epstein L and Zin S (1991) ldquoSubstitution Risk Aversion and the TemporalBehavior of Consumption and Asset Returns An Empirical InvestigationrdquoJournal of Political Economy 99 263ndash286

Ferson W E and Constantinides G (1991) ldquoHabit Persistence and Dura-bility in Aggregate Consumption Empirical Testsrdquo Journal of FinancialEconomics 36 29ndash55

Ferson W E and Harvey C R (1993) ldquoThe Risk and Predictability ofInternational Equity Returnsrdquo Review of Financial Studies 6 527ndash566

Ferson W E and Korajczyk R A (1995) ldquoDo Arbitrage Pricing Mod-els Explain the Predictability of Stock Returnsrdquo Journal of Business 68309ndash349

Foster D and Viswanathan S (1993) ldquoVariations in Trading VolumesReturn Volatility and Trading Costs Evidence in Recent Price FormationModelsrdquo Journal of Finance 48 187ndash211

Gallant R and Tauchen G (2003) ldquoSimulated Score Methods and IndirectInference for Continuous-Time Modelsrdquo in Handbook of Financial Econo-metrics eds Y Aumlt-Sahalia and L P Hansen forthcoming

Garcia R Ghysels E and Renault E (2003) ldquoOption Pricing Modelsrdquo inHandbook of Financial Econometrics eds Y Aumlt-Sahalia and L P Hansenforthcoming

Ghysels E (1998) ldquoOn Stable Factor Structures in the Pricing of Risk DoTime-Varying Betas Help or Hurtrdquo Journal of Finance 53 549ndash574

Ghysels E and Hall A (1990) ldquoAre Consumption-Based IntertemporalCapital Asset Pricing Models Structuralrdquo Journal of Econometrics 45121ndash139

Gibbons M R and Ferson W E (1985) ldquoTesting Asset Pricing ModelsWith Changing Expectations and an Unobservable Market Portfoliordquo Jour-nal of Financial Economics 14 217ndash236

Gibbons M R and Ramaswamy K (1993) ldquoA Test of the Cox Ingersolland Ross Model of the Term Structurerdquo Review of Financial Studies 6619ndash658

Gibbons M Ross S A and Shanken J (1989) ldquoA Test of the Ef ciencyof a Given Portfoliordquo Econometrica 57 1121ndash1152

Hansen L P (1982) ldquoLarge Sample Properties of Generalized Method ofMoments Estimatorsrdquo Econometrica 50 1029ndash1054

Hansen L P and Hodrick R J (1980) ldquoForward Exchange Rates as OptimalPredictors of Future Spot Rates An Econometric Analysisrdquo Journal ofPolitical Economy 88 829ndash853

(1983) ldquoRisk Averse Speculation in the Forward Foreign ExchangeMarket An Econometric Analysis of Linear Modelsrdquo in Exchange Ratesand International Macroeconomics ed J Frenkel Chicago University ofChicago Press

Hansen L P and Jagannathan R (1997) ldquoAssessing Speci cation Errors inStochastic Discount Factor Modelsrdquo Journal of Finance 62 557ndash590

Hansen L P and Richard S F (1987) ldquoThe Role of Conditioning Informa-tion in Deducing Testable Restrictions Implied by Dynamic Asset PricingModelsrdquo Econometrica 55 587ndash613

Hansen L P and Scheinkman J (1995) ldquoBack to the Future GeneratingMoment Implications for Continuous-Time Markov Processesrdquo Economet-rica 63 767ndash804

Hansen L P Scheinkman J and Touzi N (1998) ldquoSpectral Methods forIdentifying Scalar Diffusionsrdquo Journal of Econometrics 86 1ndash32

Hansen L P and Singleton K J (1982) ldquoGeneralized Variables Esti-mation of Nonlinear Rational Expectations Modelsrdquo Econometrica 501269ndash1286

Harvey C R (1988) ldquoThe Real Term Structure and Consumption GrowthrdquoJournal of Financial Economics 22 305ndash333

(1989) ldquoTime-Varying Conditional Covariances in Tests of AssetPricing Modelsrdquo Journal of Financial Economics 24 289ndash318

(1991) ldquoThe World Price of Covariance Riskrdquo Journal of FinancialEconomics 46 111ndash157

Hodrick R and Zhang X (2001) ldquoAssessing Speci cation Errors inStochastic Discount Factor Modelsrdquo Journal of Financial Economics 62327ndash376

Huang R D and Stoll H R (1994) ldquoMarket Microstructure and StockReturn Predictionsrdquo Review of Financial Studies 7 179ndash213

(1997) ldquoThe Components of the Bid-Ask Spread A GeneralApproachrdquo Review of Financial Studies 10 995ndash1034

Huberman G and Kandel S (1987) ldquoMean-Variance Spanningrdquo Journal ofFinance 32 339ndash346

Jagannathan R and Wang Z (1996) ldquoThe Conditional CAPM and theCross-Section of Expected Returnsrdquo Journal of Finance 51 3ndash53

(2002) ldquoEmpirical Evaluation of Asset Pricing Models A Compari-son of the SDF and Beta Methodsrdquo Journal of Finance forthcoming

Kan R and Zhou G (1999) ldquoA Critique of the Stochastic Discount FactorMethodologyrdquo Journal of Finance 54 1221ndash1248

Kent J (1978) ldquoTime-Reversible Diffusionsrdquo Advances in Applied Probabil-ity 10 819ndash835

Longstaff F A (1989) ldquoA Nonlinear General Equilibrium Model of the TermStructure of Interest Ratesrdquo Journal of Financial Economics 23 195ndash224

MacKinlay A C and Richardson M P (1991) ldquoUsing Generalized Methodof Moments to Test Mean-Variance Ef ciencyrdquo Journal of Finance 46551ndash527

Madhavan A (2000) ldquoMarket Microstructure A Surveyrdquo Journal of Finan-cial Markets 3 205ndash258


Madhavan A Richardson M and Roomans M (1997) ldquoWhy Do SecurityPrices Change A Transaction-Level Analysis of NYSE Stocksrdquo Review ofFinancial Studies 10 1035ndash1064

Marsh T A and Rosenfeld E R (1983) ldquoStochastic Process for InterestRates and Equilibrium Bond Pricesrdquo Journal of Finance 38 635ndash646

Merton R C (1973) ldquoThe Theory of Rational Option Pricingrdquo Bell Journalof Economics and Management Science 4 141ndash183

Newey W and West K (1987) ldquoHypothesis Testing With Ef cient Methodof Moments Estimationrdquo International Economic Review 28 777ndash787

Ross S A (1976) ldquoThe Arbitrage Theory of Capital Asset Pricingrdquo Journalof Economic Theory 13 341ndash360

Rubinstein M (1976) ldquoThe Valuation of Uncertain Income Streams and thePricing of Optionsrdquo Bell Journal of Economics 7 407ndash425

Sanders A B and Unal H (1988) ldquoOn the Intertemporal Stability of theShort-Term Rate of Interestrdquo Journal of Financial and Quantitative Anal-ysis 23 417ndash423

Sun G (1997) ldquoThe Test Function Method for Estimating Continuous-TimeModels of Short-Term Interest Raterdquo working paper University of Min-nesota Carlson School of Management

Sundaresan S (1989) ldquoIntertemporally Dependent Preferences and theVolatility of Consumption and Wealthrdquo Review of Financial Studies 273ndash88

Vasicek O (1977) ldquoAn Equilibrium Characterization of the Term StructurerdquoJournal of Financial Economics 5 177ndash188

Weil P (1989) ldquoThe Equity Premium Puzzle and the Risk-Free Rate PuzzlerdquoJournal of Monetary Economics 24 401ndash421


even in large samples These limitations severely restrict thescope of the empirical investigations of dynamic asset-pricingmodels GMM enables the econometrician to overcome theselimitations The econometrician does not have to make strongdistributional assumptionsmdashthe variables of interest can beserially correlated and conditionally heteroscedastic Furthernonlinear asset-pricing models can be examined without lin-earization The convenience and the generality of GMM arethe two main reasons why GMM has become so popular inthe nance literature

Despite its advantages GMM has a potential shortcom-ing when compared to the ML method When the distribu-tional assumptions are valid the ML method provides the mostef cient estimates of model parameters whereas the GMMmethod may not To apply GMM the econometrician typicallyuses the moment conditions generated by the stochastic dis-count factor (SDF) representation of an asset-pricing modelThe moment conditions representing the implications of anasset-pricing model chosen by the econometrician may notlead to the most ef cient estimation of model parameters Itis therefore important to understand whether GMM as com-monly applied in practice to examine asset-pricing modelsis less ef cient than the ML method in the situations wherethe ML method can be applied It is well known that GMMhas the same estimation ef ciency as the ML method whenapplied to the moment conditions generated by the classicalbeta representation of linear asset-pricing models Jagannathanand Wang (2002) demonstrated that GMM is as ef cient asthe ML method when applied to the moment conditions gen-erated by the SDF representation of linear asset-pricing mod-els This reinforces the importance and advantage of GMM inempirical asset-pricing applications

In what follows we review the use of GMM in nancewith emphasis on asset-pricing applications In Section 2 wediscuss using GMM for empirically evaluating the standardconsumption-based asset-pricing model and some of its exten-sions In Section 3 we discuss using GMM in examining fac-tor models In Section 4 we discuss the advantages of usingalternative weighting matrices while applying GMM We dis-cuss the ef ciency of GMM when applied to the SDF repre-sentation of asset-pricing models in Section 5 We examinethe use of GMM to estimate the stochastic process for short-term interest rates in Section 6 and discuss using GMM in themarket microstructure literature in Section 7 We summarizein Section 8

2 NONLINEAR RATIONAL EXPECTATIONS MODELS

In most asset-pricing models expectations about the futurerepresent an important factor in the decision making processIn practice however it is the actions of the agents ratherthan their expectations that we observe Empirical work onasset-pricing models relies on the hypothesis of rational expec-tations which in turn implies that the errors made by theagentsmdashnamely the differences between observed realizationsand expectationsmdashare uncorrelated with information on whichthe expectations are conditioned

21 Consumption-Based Capital Asset-Pricing Model

One of the rst applications of GMM appeared in the con-text of estimation of a consumption-based nonlinear ratio-nal expectations asset-pricing model of Hansen and Singleton(1982) Consider an expected utility-maximizing agent withpreferences on current and future consumption streams char-acterized by a time-additive utility function Let time rsquo referto the present let ctCrsquo be the agentrsquos consumption level dur-ing period t C rsquo let Irsquo be the information set available to theagent at rsquo let sbquo 2 40115 be the time-preference parameter andlet U be an increasing concave utility function The agent isthen assumed to choose consumption streams to maximize

X

tD0

sbquotE6U 4ctCrsquo 5mdashIrsquo 70 (1)

The agent can invest in any of N available securities withgross returns (1 C the rates of return) at time t denoted byRi1t1 i D 11 1 N Solving the agentrsquos intertemporal con-sumption and portfolio choice implies the following equationwhich relates security returns to the intertemporal marginalrate of substitution sbquo

U 04ctC15

U 04ct5

E6sbquo6U 04ctC15=U 04ct57Ri1tC1mdashIt7 D 11 i D 11 1 N 0 (2)

A standard assumption frequently used in the asset-pricingmacroeconomics literature is that the utility function U

belongs to the constant relative risk-aversion (CRRA) familyThe utility functions U in the CRRA family can be describedby a single parameter ƒ gt 0 referred to as the coef cient ofrelative risk aversion as

U4c5D c1ƒƒ

1ƒƒif ƒ 6D1 and U4c5D log4c5 if ƒ D10

(3)

For such utility function U (2) translates to

E6sbquo4ctC1=ct5ƒƒRi1tC1mdashIt7 D 11 i D 11 1 N 1 (4)

which by the law of iterated expectations implies that

E661ƒ sbquo4ctC1=ct5ƒƒRi1tC17zt7 D 01 i D 11 1N (5)

for any random vector zt measurable with respect to the infor-mation set It Assume that zt has dimension K 1 For infer-ence purposes zt is chosen to be observable by the econome-trician Typically zt would include lagged rates of return andlagged consumption growth rates

The parameter vector to be estimated is ˆ D 4sbquo1ƒ50 Let 1N

denote the N 1 vector of 1s let 0M denote the M 1 vec-tor of 0s where M D NK let Rt denote the vector of assetreturns at time t and let ytC1

D 4R0t1 ctC1=ct1 z0

t50 denote the

data available to the econometrician at time t C1 The orthog-onality conditions in (5) can be written more compactly as

E6h4ytC11 ˆ57 D 0M 1 where h4ytC11 ˆ5

D 61Nƒ sbquo41 C RtC154ctC1=ct5

ƒƒ 7 dagger zt1 (6)



gT 4ˆ5 D 1T

TX

tD1

h4ytC11 ˆ51 (7)


quadratic function



STD 1

T

TX

tD1

h4ytC11O05h4ytC11

O05

01 (9)





T gT 4 OT 50Sƒ1

T gT 4 OT 51 (10)










X

jD0


1ƒƒ ƒ 1

1 ƒ ƒ1

(12)


mtC1 D sbquo

sup3ct

ctƒ1

ƒƒ1sup3ctC1

ct

ƒƒ

0 (13)


Et

poundc

ƒƒ1tƒ1 cƒƒ

tƒ sbquocƒƒ1

t cƒƒtC14ctC1=ct5

curren

D Et

poundRtC14c

ƒƒ1t c

ƒƒ

tC1ƒ sbquoc

ƒƒ1tC1 c

ƒƒ





1ƒƒ ƒ 1

1 ƒ ƒ0 (15)


stC1

st

ctC1

ct5ƒ where st

D ct ƒxt

ctis the surplus





mtC1 Dmicro

sbquo

sup3ctC1

ct


para1ƒˆ

1 where ˆ D 1ƒƒ

1ƒ1=ndash0

(16)


3 FACTOR MODELS





var6Rm71 (17)





E6Ri7 D ƒ CKX

kD1




m D ˆ0 CKX

kD1

ˆkfk (20)




CX

jD112100K

sbquojmr jm1 (21)




X

jD112100K












var6Rpt71 i D 11 1N 0 (23)
















and gT 4bdquo5 D 1T




1T

TX

tD1


TX

tD1


(26)


0Sƒ10 D05

ƒ1 where

D0D E

hiexclht


and S0D

CX


(27)


rdquo1D T O 06P6D0




gT 401sbquo5 D 1T

TX

tD1

ht401sbquo50 (29)





1 where D04sbquo5DE

microiexclht


para0

(30)









mdashIt51 (32)





E4rm1tC1 mdashIt 5








De ne

ui1tC1D ri1tC1


mzm1t570 (35)








T 50Sƒ1T gT 4 O


0t75






0




vector Let

DTD 1

T

TX

tD1

RtF0t


GTD 1

T

TX

tD1

RtR0t





minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

TNwT 4 O


D4D0

T Gƒ1T DT 5ƒ1D0



T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



T NwT 4 OT 50Gƒ1

TNwT 4 O

T 5currenƒ

N ƒKX

jD1





FACTOR MODELS
















t dWt1 (40)




rtC1 ƒrtD Csbquort



t 1 (41)



ft4ˆ5D

2666664

˜tC1

˜tC1rt

˜2tC1

ƒlsquo 2r 2ƒt

4˜2tC1

ƒlsquo 2r2ƒt 5rt

3777775

0 (42)






O50Sƒ1T D04










0DE

microZ T

t

microŒ4rs5rdquo

04rs5C 1

2lsquo 4rs5rdquo

004rs5

parads

para

C E

microZ T

t


para0 (44)


04rt5


E

microŒ4rt5rdquo

04rt5C 1

2lsquo 4rt5rdquo

004rt5

paraD00 (45)







D E62a4b ƒrt5r2t

Clsquo 2r 2t 7D01



2aC lsquo 4b




E


C 1


paraD00 (47)


ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

and rdquo24y5DR y


R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1


tC1



tC1



currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0






8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES












































































gT 4ˆ5 D 1T

TX

tD1

h4ytC11 ˆ51 (7)


quadratic function



STD 1

T

TX

tD1

h4ytC11O05h4ytC11

O05

01 (9)





T gT 4 OT 50Sƒ1

T gT 4 OT 51 (10)










X

jD0


1ƒƒ ƒ 1

1 ƒ ƒ1

(12)


mtC1 D sbquo

sup3ct

ctƒ1

ƒƒ1sup3ctC1

ct

ƒƒ

0 (13)


Et

poundc

ƒƒ1tƒ1 cƒƒ

tƒ sbquocƒƒ1

t cƒƒtC14ctC1=ct5

curren

D Et

poundRtC14c

ƒƒ1t c

ƒƒ

tC1ƒ sbquoc

ƒƒ1tC1 c

ƒƒ





1ƒƒ ƒ 1

1 ƒ ƒ0 (15)


stC1

st

ctC1

ct5ƒ where st

D ct ƒxt

ctis the surplus





mtC1 Dmicro

sbquo

sup3ctC1

ct


para1ƒˆ

1 where ˆ D 1ƒƒ

1ƒ1=ndash0

(16)


3 FACTOR MODELS





var6Rm71 (17)





E6Ri7 D ƒ CKX

kD1




m D ˆ0 CKX

kD1

ˆkfk (20)




CX

jD112100K

sbquojmr jm1 (21)




X

jD112100K












var6Rpt71 i D 11 1N 0 (23)
















and gT 4bdquo5 D 1T




1T

TX

tD1


TX

tD1


(26)


0Sƒ10 D05

ƒ1 where

D0D E

hiexclht


and S0D

CX


(27)


rdquo1D T O 06P6D0




gT 401sbquo5 D 1T

TX

tD1

ht401sbquo50 (29)





1 where D04sbquo5DE

microiexclht


para0

(30)









mdashIt51 (32)





E4rm1tC1 mdashIt 5








De ne

ui1tC1D ri1tC1


mzm1t570 (35)








T 50Sƒ1T gT 4 O


0t75






0




vector Let

DTD 1

T

TX

tD1

RtF0t


GTD 1

T

TX

tD1

RtR0t





minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

TNwT 4 O


D4D0

T Gƒ1T DT 5ƒ1D0



T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



T NwT 4 OT 50Gƒ1

TNwT 4 O

T 5currenƒ

N ƒKX

jD1





FACTOR MODELS
















t dWt1 (40)




rtC1 ƒrtD Csbquort



t 1 (41)



ft4ˆ5D

2666664

˜tC1

˜tC1rt

˜2tC1

ƒlsquo 2r 2ƒt

4˜2tC1

ƒlsquo 2r2ƒt 5rt

3777775

0 (42)






O50Sƒ1T D04










0DE

microZ T

t

microŒ4rs5rdquo

04rs5C 1

2lsquo 4rs5rdquo

004rs5

parads

para

C E

microZ T

t


para0 (44)


04rt5


E

microŒ4rt5rdquo

04rt5C 1

2lsquo 4rt5rdquo

004rt5

paraD00 (45)







D E62a4b ƒrt5r2t

Clsquo 2r 2t 7D01



2aC lsquo 4b




E


C 1


paraD00 (47)


ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

and rdquo24y5DR y


R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1


tC1



tC1



currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0






8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES












































































mtC1 Dmicro

sbquo

sup3ctC1

ct


para1ƒˆ

1 where ˆ D 1ƒƒ

1ƒ1=ndash0

(16)


3 FACTOR MODELS





var6Rm71 (17)





E6Ri7 D ƒ CKX

kD1




m D ˆ0 CKX

kD1

ˆkfk (20)




CX

jD112100K

sbquojmr jm1 (21)




X

jD112100K












var6Rpt71 i D 11 1N 0 (23)
















and gT 4bdquo5 D 1T




1T

TX

tD1


TX

tD1


(26)


0Sƒ10 D05

ƒ1 where

D0D E

hiexclht


and S0D

CX


(27)


rdquo1D T O 06P6D0




gT 401sbquo5 D 1T

TX

tD1

ht401sbquo50 (29)





1 where D04sbquo5DE

microiexclht


para0

(30)









mdashIt51 (32)





E4rm1tC1 mdashIt 5








De ne

ui1tC1D ri1tC1


mzm1t570 (35)








T 50Sƒ1T gT 4 O


0t75






0




vector Let

DTD 1

T

TX

tD1

RtF0t


GTD 1

T

TX

tD1

RtR0t





minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

TNwT 4 O


D4D0

T Gƒ1T DT 5ƒ1D0



T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



T NwT 4 OT 50Gƒ1

TNwT 4 O

T 5currenƒ

N ƒKX

jD1





FACTOR MODELS
















t dWt1 (40)




rtC1 ƒrtD Csbquort



t 1 (41)



ft4ˆ5D

2666664

˜tC1

˜tC1rt

˜2tC1

ƒlsquo 2r 2ƒt

4˜2tC1

ƒlsquo 2r2ƒt 5rt

3777775

0 (42)






O50Sƒ1T D04










0DE

microZ T

t

microŒ4rs5rdquo

04rs5C 1

2lsquo 4rs5rdquo

004rs5

parads

para

C E

microZ T

t


para0 (44)


04rt5


E

microŒ4rt5rdquo

04rt5C 1

2lsquo 4rt5rdquo

004rt5

paraD00 (45)







D E62a4b ƒrt5r2t

Clsquo 2r 2t 7D01



2aC lsquo 4b




E


C 1


paraD00 (47)


ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

and rdquo24y5DR y


R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1


tC1



tC1



currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0






8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES














































































var6Rpt71 i D 11 1N 0 (23)
















and gT 4bdquo5 D 1T




1T

TX

tD1


TX

tD1


(26)


0Sƒ10 D05

ƒ1 where

D0D E

hiexclht


and S0D

CX


(27)


rdquo1D T O 06P6D0




gT 401sbquo5 D 1T

TX

tD1

ht401sbquo50 (29)





1 where D04sbquo5DE

microiexclht


para0

(30)









mdashIt51 (32)





E4rm1tC1 mdashIt 5








De ne

ui1tC1D ri1tC1


mzm1t570 (35)








T 50Sƒ1T gT 4 O


0t75






0




vector Let

DTD 1

T

TX

tD1

RtF0t


GTD 1

T

TX

tD1

RtR0t





minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

TNwT 4 O


D4D0

T Gƒ1T DT 5ƒ1D0



T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



T NwT 4 OT 50Gƒ1

TNwT 4 O

T 5currenƒ

N ƒKX

jD1





FACTOR MODELS
















t dWt1 (40)




rtC1 ƒrtD Csbquort



t 1 (41)



ft4ˆ5D

2666664

˜tC1

˜tC1rt

˜2tC1

ƒlsquo 2r 2ƒt

4˜2tC1

ƒlsquo 2r2ƒt 5rt

3777775

0 (42)






O50Sƒ1T D04










0DE

microZ T

t

microŒ4rs5rdquo

04rs5C 1

2lsquo 4rs5rdquo

004rs5

parads

para

C E

microZ T

t


para0 (44)


04rt5


E

microŒ4rt5rdquo

04rt5C 1

2lsquo 4rt5rdquo

004rt5

paraD00 (45)







D E62a4b ƒrt5r2t

Clsquo 2r 2t 7D01



2aC lsquo 4b




E


C 1


paraD00 (47)


ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

and rdquo24y5DR y


R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1


tC1



tC1



currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0






8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES















































































mdashIt51 (32)





E4rm1tC1 mdashIt 5








De ne

ui1tC1D ri1tC1


mzm1t570 (35)








T 50Sƒ1T gT 4 O


0t75






0




vector Let

DTD 1

T

TX

tD1

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T

TX

tD1

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minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

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T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



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rtC1 ƒrtD Csbquort



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4˜2tC1

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04rs5C 1

2lsquo 4rs5rdquo

004rs5

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para

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microZ T

t


para0 (44)


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microŒ4rt5rdquo

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2lsquo 4rt5rdquo

004rt5

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paraD00 (47)


ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

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expplusmn

ƒ x2

2bdquo2

sup2dx (48)


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tC1


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REFERENCES











































































vector Let

DTD 1

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TX

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GTD 1

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TX

tD1

RtR0t





minˆNwT 4ˆ50Gƒ1


dition is D0T Gƒ1

TNwT 4 O


D4D0

T Gƒ1T DT 5ƒ1D0



T 50Gƒ1T

NwT 4 OT 5 will not


havep




ADS1=2Gƒ1=2poundIN


4Gƒ1=2504S1=2501 (38)



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TNwT 4 O

T 5currenƒ

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t dWt1 (40)




rtC1 ƒrtD Csbquort



t 1 (41)



ft4ˆ5D

2666664

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REFERENCES



















































































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rtC1 ƒrtD Csbquort



t 1 (41)



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2666664

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Da4bƒrt5dtClsquo r



R yxƒadx

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and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

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tC1ƒƒlsquo 2rƒ1

tC1


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ACKNOWLEDGMENTS



REFERENCES

















































































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ƒrs52 C






Da4bƒrt5dtClsquo r



R yxƒadx

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R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


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8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES













































































Da4bƒrt5dtClsquo r



R yxƒadx

and rdquo24y5DR y


R yxƒ2ƒdx

and rdquo44y5DR y



expplusmn

ƒ x2

2bdquo2

sup2dx (48)


fHS4t1ˆ5D

26666666664

a4b ƒrtC15rƒ2ƒ

tC1ƒƒlsquo 2rƒ1

tC1


ƒ4ƒ ƒ 125lsquo 2

4bƒrtC15rƒatC1


tC1



tC1



currenexp

poundƒ 4rtC1ƒrt 5

2

2bdquo2

curren

37777777775

0






8 SUMMARY







ACKNOWLEDGMENTS



REFERENCES











































































ACKNOWLEDGMENTS



REFERENCES






















































































Documents

jagannathan-skoulakis-wang