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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Efficient Estimation of Conditional Asset-Pricing

Models

Douglas J Hodgson & Keith P Vorkink

To cite this article: Douglas J Hodgson & Keith P Vorkink (2003) Efficient Estimation of

Conditional Asset-Pricing Models, Journal of Business & Economic Statistics, 21:2, 269-283, DOI: 10.1198/073500103288618954

To link to this article: http://dx.doi.org/10.1198/073500103288618954

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Published online: 24 Jan 2012.

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Ef’cient Estimation of Conditional

Asset-Pricing Models

Douglas J.

Hodgson

Département des Sciences Économiques, Université du Québec, Montréal, Québec, Canada H3C 3P8 (hodgson.douglas-james@uqam.ca)

Keith P.

Vorkink

Marriott School of Management, Brigham Young University, Provo, UT 84602 (kpv3@email.byu.edu)

A semiparametric efŽcient estimation procedure is developed for the parameters of multivariate gen-eralized autoregressive conditional heteroscedasticity-in-mean models when the disturbances have a conditional distribution assumed to be elliptically symmetric but otherwise unrestricted. Under high-level assumptions, the resulting estimator achieves the asymptotic semiparametric efŽciency bound. The elliptical symmetry assumption allows us to avert the curse of dimensionality problem that would oth-erwise arise in estimating the unknown error distribution. This framework is suitable for the estimation and testing of conditional asset-pricing models, such as the conditional capital asset-pricing model. We apply our procedure in an empirical study of stock prices, with Monte Carlo simulation results also reported.

KEY WORDS: Capital asset-pricing model; Multivariate autoregressive conditional heteroscedastic-ity; Semiparametric efŽciency.

1. INTRODUCTION

Modeling expected returns has permeated much of Žnancial research over the past three decades. The payoffs from a cor-rect relationship between risk and expected return are abun-dant and include applications to capital budgeting, portfolio performance, event studies, and others. The conditional mean-variance model of the risk–return relationship was initially implemented empirically for multivariate time series data by Bollerslev, Engle, and Wooldridge (1988), who developed a conditional capital asset-pricing model (C-CAPM) and associ-ated generalized autoregressive conditional heteroscedasticity-in-mean (GARCH-M) econometric model. A large empirical literature has subsequently developed in this area, gener-ally estimating the models with Gaussian quasi-maximum likelihood estimation (Q-MLE) techniques. Although such techniques typically retain their consistency and asymptotic normality properties in the presence of nonnormal data (Bollerslev and Wooldridge 1992), asymptotic inefŽciency and imprecise parameter estimates can occur due to the presence of thick tails in the distributions of Žnancial data. We propose a new estimation methodology for the multivariate GARCH-in-mean model designed to account for excess tail thickness by adopting a exible distributional assumption of conditional elliptical symmetry. The estimator will achieve the asymptotic semiparametric efŽciency bound in the presence of general elliptical symmetry in the data-generating process. We apply our estimator to a dataset of stock returns and perform asset-pricing tests of the C-CAPM.

It has been well documented that stock returns are not inde-pendent and identically distributed (iid) normal. In particular, they tend to exhibit substantial kurtosis and have moments that vary over time (see, e.g., Mandelbrot 1963; Fama 1965; Engle 1982; Bollerslev, Chou, and Kroner 1992). These phenomena are not unrelated. It is well known that time-varying volatil-ity implies a thick-tailed unconditional distribution. However, as shown by Bollerslev (1987), conditional volatility cannot

completely account for the tail behavior of the unconditional distribution in Žnancial returns (see also Diebold 1988; Nelson 1991; Vorkink 2001). Accurate description of return distribu-tions should include modeling of both of these properties.

We propose a semiparametric efŽcient estimator that attempts to improve on the inefŽciencies that may occur in the Gaussian Q-MLE when thick tails are present in the dis-tribution of the standardized innovations to the GARCH-M model. To do so, we assume the distribution of returns to be a member of the class of elliptically symmetric distribu-tions. This class includes those having conditional dependence among higher moments, inŽnite variance (Cauchy), Studentt, and others. (For further discussion of elliptical distributions, see Fang, Kotz, and Ng 1990; Muirhead 1982; the Appendix of this article.) We derive the asymptotic semiparametric efŽ-ciency bound for the estimation of our model’s parameters in the presence of an unknown elliptically symmetric innovation density, then propose a semiparametric estimator that achieves the bound. This estimator uses a nonparametric kernel estima-tor of the unknown innovation density.

This assumption of elliptical symmetry plays an integral role in our estimation methodology, particularly in the esti-mation of the residual density. We can think of two extreme methods of obtaining an estimate of a leptokurtokic residual density. One method is to Žt a fully parametric nonnormal distribution to the residuals. Alternatively, the density could be estimated in a fully nonparametric fashion. For example, Drost and Klaassen (1997) proposed a semiparametric efŽ-cient estimation method for univariate GARCH models that involves nonparametric kernel estimation of the innovation

©2003 American Statistical Association Journal of Business & Economic Statistics April 2003, Vol. 21, No. 2 DOI 10.1198/073500103288618954

269

density. However, their method is difŽcult to extend to a mul-tivariate setting, due to the “curse of dimensionality” prob-lem, in which the convergence rate of a nonparametric density estimate diminishes rapidly as the dimension of the density increases.

Elliptical symmetry provides a middle ground between a fully speciŽed Q-MLE approach and a fully nonparametric approach. Whereas the density is nonparametrically estimated within the elliptically symmetric class, this restriction allows us to do so without falling prey to the “curse of dimensional-ity.” SpeciŽcally, we are able to transform the nonparametric density estimator to one that is always one-dimensional.

This estimator’s roots lie in Bickel’s (1982) adaptive esti-mator. Assuming iid data, Bickel considered the problem of adaptively estimating mean and covariance parameters in ellip-tically symmetric location models. He found that under the assumption of elliptical symmetry, the mean could be tively estimated and the covariance parameters could be adap-tively estimated up to a scale. Linton (1993) showed that slope parameters can be adaptively estimated in a regression model with ARCH errors when the innovation density is symmetric. In both cases, the innovation density is otherwise unrestricted and is estimated using nonparametric kernel methods. Hodg-son, Linton, and Vorkink (in press) have derived adaptive esti-mators in time series models under the assumption of elliptical symmetry using a nonparametric estimate of the joint innova-tion density.

Hodgson et al. (in press) developed an estimator of lin-ear unconditional asset pricing models under elliptical sym-metry. Their estimator is fully asymptotically efŽcient and places no assumptions on the family of return distributions other than that this family is elliptically symmetric. They found that the more general estimator leads to substantially different estimates and conclusions when testing unconditional asset-pricing models. However, the treatment of conditional heteroscedasticity is ad hoc, which results in potential inef-Žciencies. This article extends this work by parameterizing the conditional heteroscedasticity in the form of a multivari-ate GARCH-M model with conditionally elliptically symmet-ric innovation distributions.

Asset-pricing theory exists that is consistent with the speci-Žcation of elliptical symmetry, at least for the case of the one-period unconditional CAPM, although the conditions under which these results would extend to a multiperiod condi-tional model are not known. Merton (1973) mentioned the restrictions required in a multiperiod model to generate one-period-ahead mean-variance pricing. It may be possible to show that Merton’s (1973) conditions, along with conditional elliptical symmetry, yield such pricing, but we know of no formal results to this effect. In the one-period CAPM, the assumption of normally distributed returns is sufŽcient, but not necessary, for a mean-variance result. Chamberlain (1983), Owen and Rabinovitch (1983), and Berk (1997) have obtained mean-variance pricing under the assumption that returns are elliptically symmetric. In fact, Berk (1997) found that ellip-tical symmetry is the most general distributional assumption that is consistent with mean-variance maximization when con-sumers are assumed to have concave utility functions. These exact-pricing models are more general and consistent with

the empirical regularities than their normal distribution coun-terparts. However, although these theoretical results can be obtained with more general distributional assumptions, estima-tion of the general model has not been feasible until recently. Our estimator is speciŽcally designed to be more efŽcient than the Gaussian Q-MLE in the presence of thick tails in the standardized innovations to the GARCH-M model. As shown by Bollerslev and Wooldridge (1992), the Q-MLE has the virtue of being consistent and asymptotically normal for a sub-stantial range of deviations of the innovations from iid normal-ity. We have not been able to derive similar properties for the semiparametric estimator developed in this article, and know its properties only when the assumption of iid elliptical sym-metry on the innovations holds. For data where such deviations from this assumption as conditional or unconditional skewness may be present, we can currently only conjecture as to the behavior of our estimator. Furthermore, empirical and simula-tion evidence reported herein suggests that the efŽciency gains of our estimator vis-à-vis the Q-MLE are quite modest for estimation of conditional mean parameters, although the evi-dence suggests that there may be potential gains in estimating conditional covariance parameters and conditional betas.

The article is organized as follows. Section 2 introduces the conditional CAPM model that we estimate and test. Section 3 presents our multivariate GARCH-M econometric model. Section 4 contains our derivation of the semiparametric efŽ-ciency bound for our model and describes a method of feasibly computing an estimator that will achieve the bound. Sections 5 and 6 report empirical and simulation results, and Section 7 concludes. The Appendix contains a discussion of elliptically symmetric densities and discusses some computational details relating to our estimator.

2. CONDITIONAL RETURN MODELS

It has been shown that the assumption of constant return distributions is not necessary to obtain equilibrium pricing equations. Merton (1973) derived an intertemporal CAPM that showed how investors would react to changing invest-ment opportunity sets. In an empirical setting, Bollerslev et al. (1988) estimated C-CAPM covariances assuming that the covariance matrix of returns followed a GARCH-M41115 process. They found that under this model’s parameterization, beta and the market price of risk are time-varying. They also showed that both returns and volatility are predictable and time-varying, in fact, they are able to predict a larger portion of the variability in returns than the unconditional counterpart (see also Harvey 1991; Buse, Korkie, and Turtle 1994; Braun, Nelson, and Sunier 1995; Jagannathan and Wang 1996). We suggest that a natural extension would be to estimate a CAPM where the residual distribution is assumed to be thick-tailed relative to the normal distribution and allow some exibility in the form of the conditional distribution.

We now introduce the C-CAPM return relationship. Our discussion closely follows that of Bollerslev et al. (1988), with some variations as suggested by, for example, DeSantis and Gerard (1997). The following equation demonstrates the main relationship of the C-CAPM, stating that the excess return on

asseti is linear in its covariance with the market portfolio, E_tƒ16Ri1 t7ƒrftD‹covtƒ14Rm1 t1 Ri1 t50 (1) We assume that there are n assets in the market,R_{i1 t} is the return on asseti in periodt,R_{m1 t} is the return on the market portfolio,r_ft is the return to a risk-free asset, and the sub-scripts on expectations and covariances indicate conditional moments. Note that it would be a straightforward matter to extend the model to allow for multiple factors to inuence returns. From (1), we can see that the expected return on the market portfolio is

E_tƒ16Rm1 t7ƒrftD‹vartƒ14Rm1 t51

so that the parameter ‹ can be interpreted as the market price on risk. Following DeSantis and Gerard (1997), we may treat this parameter either as a constant or as time-varying. In the latter case, it can be modeled as being dependent on an `-vector of state variables vt, and (1) can be generalized by writing‹Dexp4ƒ₀üCvT_tÃ15. In the model with a constant price of risk, we have ‹Dexp4ƒ₀ü5. We can also write our expected return relationship as

E_tƒ16R_{i1 t}7ƒr_ftDE_tƒ16R_{m1 t}ƒr_ft7‚_{i1 t}1 where‚_{i1 t}Dcovtƒ14Rm1 t1Ri1 t5

var_tƒ14Rm1 t5 is the conditional “beta” for asseti

in periodt.

DeŽne the n-vectorr_tDR_tƒr_ft‰_n, whereR_t is the vector of returns on the individual assets andÉ_n is ann-vector of 1s. Following Bollerslev et al. (1988), let×_tƒ1be then-vector of weights assigned to the assets in computing the “market,” so thatR_{m1 t}D×T

tƒ1Rt. Allowing for a possibly time-varying mar-ket price of risk, we may then write our CAPM relationship at timetfor our cross-section of assets as

E_tƒ1rtDexp

¡

ƒ₀üCvT_tÃ1

¢

èt×tƒ11 (2)

whereè_t is the covariance matrix of asset returns conditional on information available up to periodtƒ1. Note that our vec-tor of conditional betas is given by

Â_tD èt×tƒ1 ×T

tƒ1èt×tƒ1

0 (3)

Estimation of our model depends on the speciŽcation of a model for our conditional covariance matrix.

Testing the C-CAPM typically involves estimating the model

r_tDÁCexp¡ƒü 0Cv

T tÃ1

¢

è_t×_tƒ1Cut1 (4)

where an intercept is included to capture persistent variation in rtnot captured by variation in the market return. One common test of the asset-pricing model takes the form

H₀2 _iD01 iD11 : : : 1 n1 (5) which implies that no signiŽcant excess returns are present in each portfolio’s return that cannot be explained by variation

in the market portfolio return. This hypothesis can be tested by construction of a standard Wald test,

JD QÁ0_dvar4ÁQ5ÁQ1 (6) where ÁQ is an estimator anddvar4ÁQ5estimates its asymptotic covariance matrix. If this statistic deviates signiŽcantly from 0, then we conclude that the C-CAPM does not fully explain the deviations in returns.

It is also interesting to look at the time series of the implied betas, ‚_{i1 t}, to see whether the conditional variance parame-terization leads to substantial time variation in the covariation between the asset’s return and the market return. For exam-ple, Bollerslev et al. (1988) found substantial variation in the implied betas of their estimation of the U.S. stock and bond market, whereas Braun et al. (1995), using a modeling frame-work that differs from ours in a number of respects, found little variation in conditional betas.

3. THE ECONOMETRIC MODEL

The regression model that we estimate is given in (4). To arrive at a completely speciŽed econometric model, we must characterize our conditional covariance matrixè_t and our dis-turbance process8u_t9. There is no theory predicting a GARCH model of volatility; however, a relatively parsimonious model of time-varying second moments has been quite successful in capturing the time series behavior of volatility. Our general model of conditional volatility is a simpliŽed version of the multivariate GARCH model described by Engle and Kroner (1995),

è_tDexp4Š5H_t1 (7)

where

HtDCC T_C

Au_tƒ1u T

tƒ1ACDHtƒ1D1

CD

2 6 6 6 6 4

1 0 ¢ ¢ ¢ 0

00

0 0 00 000 00

0 0 00 0 c_{n1 ¢ ¢ ¢ ¢ ¢ ¢} c_nn

3 7 7 7 7 51

(8)

and the matrices A andD are diagonal. This model is less general than that developed by Engle and Kroner (1995), in which

H_tDCTCC n

X

jD1

A_ju_tƒ1uT_tƒ1A_jC n

X

jD1

D_jH_tƒ1D_j0

We adopt this simpliŽcation for computational purposes. Our model still has the generality to allow for systematically time-varying conditional variances and covariances. Other empirical articles, such as those by Bollerslev et al. (1988) and DeSantis and Gerard (1997), use simpliŽed GARCH-M models. This speciŽcation is more general than those of Bollerslev (1987) and Jeantheau (1998) in that it allows for time-varying condi-tional covariances. Note that under our assumptions onAand D, our restriction of the leading term of C to be unity does

not entail any further loss of generality. To see this, note that the conditional variance of the Žrst element ofut is

var_tƒ14u1t5D‘111 tDexp4Š5

¡

1Ca2 1u

2 11 tƒ1Cd

2 1h111 tƒ1

¢

1 which is the usual expression for conditional variance in a univariate GARCH model.

To complete our speciŽcation of the model, we assume that our regression disturbances8u_t9have the elliptically symmet-ric conditional density

p_tƒ14ut5D —èt—ƒ

1 2gQ¡_uT

tèƒ 1 t ut

¢

0 (9)

Our objective in this article is to obtain a semiparametric efŽ-cient estimator of the parameters of our model, treating the functional form ofgQ as an unknown inŽnite-dimensional nui-sance parameter. The function g4Q ¢5 has only a scalar as its

argument, which plays an important role in the nonparametric estimation of the density. We also deŽneìp1 t to be the condi-tional information matrix ofp_tƒ14ut5; it is proportional to the inverse of H_t and è_t. We have ì_{p1 t}Dconst¢èƒt1, with the constant greater than or equal to 1. (It equals 1 ifp_tƒ14ut5is Gaussian.) Mitchell (1989) computed the value of the constant for various elliptically symmetric densities.

Note that because we are treatinggQ as being of unknown functional form, we can also write the density as

p_tƒ14u_t5D —H_t—ƒ1 2g¡uT

tHƒ 1 t ut

¢

1 (10)

where the constant of proportionality relatingHt and èt has now been absorbed into the functiong. This speciŽcation fol-lows the example of Linton (1993), who did not consider efŽcient estimation of Š. Note that g4Q _¢5 as deŽned in (9) is the density function of the iid spherically symmetrically dis-tributed random variableèƒ1=2

t ut with unit covariance matrix. As deŽned in (10),g4_¢5 is still the density of an iid spheri-cally distributed random variable, but without the restriction of a unit covariance matrix. We also do not concern ourselves with efŽcient estimation ofŠ, and we rewrite our regression model as

r_tDÁCexp¡

ƒ₀CvT_tÃ₁¢

H_t×_tƒ1Cu_t1 (11) where ƒ0 DŠCƒ0ü. We do not consider semiparameteric efŽcient estimation of the parameters Š and ƒü

0 separately (although in principle it would be possible to do so), but con-sider only their sum,ƒ0. We justify this parameterization in our case because our parameters of primary interest are the intercept parameterÁand the conditional beta vectorÂ_t. Note that the latter depends only on the parameters of the function H_t as deŽned in (7), because

Â_tD èt×tƒ1 ×T

tƒ1èt×tƒ1 D exp4Š5Ht×tƒ1

×T

tƒ1exp4Š5Ht×tƒ1 D Ht×tƒ1

×T

tƒ1Ht×tƒ1 0

Let 411cT5T _{Dvech4C5, so that c is the} n4nC15ƒ2 2 -vector of unknown elements of C, aDdiag4A5, dDdiag4D5, and È₂ D8cT_1aT_1dT₉T _{is the vector of unknown parameters in} the conditional covariance function. Note that there are h2D

n4nC55ƒ2

2 parameters in È2. Likewise, let the vector of param-eters in the conditional mean function be given by È₁ D 8ÁT_1ÃT₉T_{, where ÃD4ƒ}

01Ã T 15

T_{, so that È}

1 is of dimension h₁DnC`C1. Our h4Dh₁Ch₂5-dimensional full parame-ter vector is Ȳ4ÈT

11È T 25

T_{, which is usually estimated using} Q-MLE procedures resulting from a speciŽcation of iid nor-mality for the normalized disturbance processØtD8Hƒ

1=2 t ut9. Although few analytical results are available, Bollerslev and Wooldridge (1992) have shown, under high-level assump-tions, that the Gaussian Q-MLE will be pT-consistent and asymptotically normal, even under distributional misspeciŽca-tion. We derive estimators that are asymptotically semipara-metrically efŽcient under our elliptical symmetry assumption (along with high-level assumptions similar to those of Boller-slev and Wooldridge 1992), but without placing additional restrictions on the return distribution. We use a semiparametric Newton–Raphson-type estimator following the basic approach of Bickel (1982).

4. EFFICIENT ESTIMATION

In this section we give our derivation of a semiparamet-ric efŽciency bound for the aforementioned model. Follow-ing the literature in the area of multivariate GARCH mod-els, we derive our estimation theory under a set of high-level assumptions. The restrictions that such assumptions imply for the parameters of our model are not known and presumably could be obtained only with great difŽculty. This is an endemic problem in multivariate GARCH modeling. Jeantheau (1998) provided a recent example of a consistency result for a mul-tivariate GARCH model that does not rely on such high-level assumptions, but at the cost of using a very restric-tive parameterization. We assume that our data are stationary and ergodic, that conditional variances are always Žnite and bounded away from 0, and that the score function has Žnite variance. Any expectation or derivative taken in the following sections is assumed to exist, and conditions for the consistency and asymptotic normality of the estimators used are assumed to hold. We can apply a result of Brown and Hodgson (2002) to obtain a semiparametric efŽciency bound for our model, which necessitates that we make the further assumptions that g4w5 is three times differentiable with bounded third deriva-tive, wherewD˜T_{˜, that ln—H}

t4È5ƒ1=2— is three times differ-entiable with respect to Èwith bounded third derivative, and thatØt4È5is three times differentiable with respect toÈ.

We now turn to the issue of semiparametric efŽcient esti-mation of the parameter vector È. (For a fuller discussion of semiparametric efŽciency bounds and the related concepts used here, see Newey 1990.) We must derive an expression for theefŽcient scoreforÈ, the orthogonal complement of the projection of the score forÈonto thetangent space, which is, loosely speaking, the space spanned by all scores for param-eterizations Ò of the unknown density g4_¢5 that include the true model of g4¢5as special cases. Such a parameterization,

which we write as g4uT t4È5Hƒ

1

t 4È5ut4È51Ò5, is known as a

parametric submodel. A semiparametric efŽciency bound for our model can be obtained by applying theorem 1 of Brown and Hodgson (2002), which applies to a class of nonlinear models with elliptical distributions that contains our model. We now give an heuristic derivation of the bound.

The log-likelihood for the aforementioned parametric sub-model for a sample of size T, where we follow the usual practice of conditioning on initial observations whose uncon-ditional density is assumed to have an asymptotically negligi-ble effect on analysis of the likelihood, is

ln¬

T4È1Ò5D ƒ 1 2

T

X

tD1

ln—H_t4È5—C T

X

tD1 g¡

uT_t4È5Hƒ1

t 4È5ut4È51Ò

¢

D ƒ1 2

T

X

tD1

ln—Ht4È5—C T

X

tD1 g¡

ØT

t4È5Øt4È51Ò

¢

D ƒ1 2

T

X

tD1

ln—Ht4È5—C T

X

tD1

g4wt4È51Ò51

where w_t4È5DØT

t4È5Øt4È5. The score of the tth observation with respect to the nuisance parameterÒ is

`_t’4È1Ò5Dg24wt4È51Ò5 g4wt4È51Ò5

1

where g_j4_¢1_¢5denotes the partial derivative ofg with respect to its jth argument, for jD112. Note that because 8wt4È59 is assumed to be an iid sequence, so is8`t’4È1Ò59. Similarly, becausew_t4È5is independent of

4r_tƒ11rtƒ21 : : : 3Ht4È51Htƒ14È51 : : : 3vt1vtƒ11 : : : 51

so is`t’4È1Ò5. Furthermore, we have

E6`<sub>t’</sub>4È1Ò57DE6`_t’4È1Ò5—ë_tƒ17D01 where we deŽne the‘-Želd

ë_tƒ1D‘ 4rtƒ11rtƒ21 : : : 3Ht4È51Htƒ14È51 : : : 3vt1vtƒ11 : : : 50 The tangent space´ _{is the inŽnite-dimensional Hilbert space} spanned by all functions having the deŽning characteristics of `_t’4È1Ò5, namely that it is a function only ofØT_{Øand that it} has zero mean,

´ _D8s4ØT_{Ø5 2 E6s4Ø}T_Ø57D090

The projection of an arbitrary function

R4rt1rtƒ11rtƒ21 : : : 3Ht4È51Htƒ14È51 : : : 3

vt1vtƒ11 : : : 3Øt1Øtƒ11 : : : 5DR4yt5 on the tangent space can be shown to be

Pr6R4yt5—´7DE6R4yt5—ØTØ70

In calculating the efŽcient score forÈ, we Žrst consider the score forÈ, which for observationt can be written as

`_tˆ4È1Ò5D ƒ1 2

¡ln—Ht— ¡È C2

¡uT t ¡È H

ƒ1 t ut

g14wt1Ò5 g4w_t1Ò5 C¡4vecH

ƒ1 t 5T

¡È vec4utu

T t5

g₁4w_t1Ò5 g4wt1Ò5

1 (12)

where we have suppressed dependence of è_t, w_t, and u_t on È to prevent cluttered notation. In considering the projection of`<sub>tˆ</sub>4È1Ò5onto the tangent space, Žrst note that the Žrst two components of`_tˆ4È1Ò5are orthogonal to the nuisance scores `t’4È1Ò5 for any parametric submodel and hence are orthog-onal to the tangent space. Considering the Žrst component on the right side of (12), we have

E

µ

¡ln—Ht—

¡È `t’4È1Ò5

¶

DE

µ

¡ln—Ht— ¡È

¶

E6`_t’4È1Ò57D01

becauseE6`t’4È1Ò57D0. Considering now the second com-ponent, note that ¡uTt

¡È andHt are both measurable with respect toë_tƒ1, yielding

E

µ

¡uT t ¡È H

ƒ1 t ut

g14wt1Ò5

g4w_t1Ò5`t’4È1Ò5

¶

DE

µ

¡uT t ¡È H

ƒ1=2 t

¶

E

µ

˜t

g₁4w_t1Ò5 g4wt1Ò5

`t’4È1Ò5

¶

D01

where E6˜_tg14wt1Ò5

g4wt1Ò5`t’4È1Ò57D0 by symmetry. It remains to

consider the projection of the third component of the right side of (12) onto the tangent space, which is given by

E

µ

¡4vecHƒ1 t 5

T

¡È

¡

H1=2 t †H

1=2 t

¢

vec4Ø_tØT t5

g0_4w t5 g4wt5

­ ­ ­ ­ w_t ¶ DE µ

¡4vecHƒ1 t 5T ¡È

¡

H_t1=2†H1_t=2¢

¶

E£

vec¡ØtØTt

¢

—wt

¤g04wt5

g4w_t51 with the equality holding because Ht is independent of Øt. Note thatE6vec4Ø_tØT

t5—wt76DE6vec4ØtØTt57. Here and in what follows, we drop the nuisance parameterÒ from our notation, because the notion of a parametric submodel has served its purpose and we now concern ourselves with the semiparamet-ric model. The derivative ofg4_¢5is now denoted byg0₄

¢5.

The projection of the periodtscore for Èonto the tangent space is, therefore,

Pr6`_tˆ4È5—´_7DE

µ

¡4vecHƒ1 t 5

T

¡È

¡

H1=2 t †H

1=2 t

¢ ¶

€E£

vec¡ØtØTt

¢

—wt

¤g04wt5

g4w_t51

and the periodtefŽcient score forÈis ã_{t1 T}4È5D`<sub>tˆ</sub>4È5ƒPr6`_tˆ4È5—´₇

D ƒ1 2

¡ln—H_t— ¡È C À 2¡u T t ¡È H

ƒ1 t utC

¡4vecHƒ1 t 5

T

¡È

¡

H1=2 t †H

1=2 t

¢

€vec¡Ø_tØT t

¢

ƒE

µ

¡4vecHƒ1 t 5

T

¡È

¡

H1=2 t †H

1=2 t

¢ ¶

€E£

vec¡

ØtØTt

¢

—wt

¤

Á

g0_4w t5 g4w_t5 D ƒ1

2

¡ln—Ht4È5—

¡È Cât4È5

g0_4w t4ˆ55

g4w_t4È551 (13) where

â_t4È5D2¡u T t4È5 ¡È H

ƒ1

t 4È5ut4È5C

¡4vecHƒ1 t 4È55T ¡È €¡

H1=2 t 4È5†H

1=2 t 4È5

¢

vec¡

Ø_t4È5ØT t4È5

¢

ƒE

µ

¡4vecHƒ1 t 4È55

T

¡È

¡

H1=2 t 4È5†H

1=2 t 4È5

¢ ¶

€E£

vec4ØØT¢

—wt4È5

¤

0

Our efŽcient score function for the sample of sizeT is then

ãT4È5D T

X

tD1

ãt1 T4È51 (14)

with the semiparametric efŽcient estimator ÈQüü

T being that value,ˆ2ä, that sets the efŽcient score equal to 0, that is, such that

ãT

¡Q

Èüü_T¢

D T

X

tD1 ãt1 T

¡Q

Èüü_T¢

D00

Under the high-level assumptions outlined at the start of this section, the semiparametric efŽcient estimator will have the asymptotic distribution

p n¡Q

Èüü_T ƒÈ0

¢ d

!N 401¢_{51 (15)} where the semiparametric efŽciency bound¢ _{is given by}

¢_D©

E£

ã_{t1 T}4È₀5ãT t1 T4È05

¤ªƒ1

0

Note that under our assumptions, an information matrix equal-ity will hold here, so that

E£

ãt1 T4È05ã T t1 T4È05

¤

D ƒE

µ

¡ã_{t1 T}4È₀5 ¡ÈT

¶

0

Note that under any misspeciŽcation (such as, e.g., the fail-ure of either our iid assumption or our elliptical symmetry assumption on the errors) this equality will fail to hold, so the possibility exists of a White (1982)-style speciŽcation test. However, we do not explore this possibility here.

If we had available a pT-consistent preliminary estimator O

ÈT, the Gaussian Q-MLE for example, and if we furthermore

knew the functional form of the density g4¢5 and the

expec-tations E6¡4vecHtƒ15T

¡È 4H 1=2 t †H

1=2

t 57 and E6vec4ØtØTt5—wt7, then we could compute the following one-step iterative estimator, which would be asymptotically equivalent to the semiparamet-ric efŽcient estimatorÈQüü

T :

Q

Èü_T D OÈ_TC " _T

X

tD1

ã_{t1 T}4ÈO_T5ãT t1 T4ÈOT5

#ƒ1

ã_T4ÈO_T51 (16) with the asymptotic covariance matrix estimated by

"

Tƒ1 T

X

tD1

ãt1 T4ÈOT5ã T t1T4ÈOT5

#ƒ1 0

As an alternative information estimator in (16) and in the com-putation of standard errors, we could use

" ƒTƒ1

T

X

tD1

¡ã_{t1 T}4OÈ_T5 ¡ÈT

#ƒ1 0

Of course, it is infeasible to compute ÈQü

T, because the afore-mentioned density and expectation functions are unknown. We must therefore replace these quantities with nonparamet-ric estimates, for which purpose we draw on existing results of Brown and Hodgson (2002) and Hodgson et al. (in press). To estimateE6¡4vecHƒt15T

¡È 4H 1=2 t †H

1=2

t 57, we can use

b

E

µ

¡4vecHƒ1 t 5

T

¡È

¡

H1_t=2†H_t1=2¢

¶

DTƒ1 T

X

tD1 ¡¡

vecHƒ1 t 4ÈOT5

¢T

¡È

¡

H1_t=24OÈT5†H 1=2 t 4ÈOT5

¢

0 (17)

Note that the derivative ¡4vecHƒt15T

¡È is difŽcult to calculate, as are the derivatives ¡uTt

¡È and ¡ln—Ht—

¡È , which also appear in our expression for the score. These difŽculties are discussed in the Appendix. To estimate the conditional expectation func-tionE6vec4Ø_tØT

t5—wt7, we make use of the fact that for ellip-tically symmetric distributions, the random n-vectorØ has a distribution (conditional on wDØT_{Ø) that is uniform on the} 4nƒ15-dimensional hypersphere with radiuspw. As Brown and Hodgson (2002) observed, the desired conditional expec-tation can be estimated to an arbitrarily high degree of preci-sion by

b

E£

vec¡

Ø_tØT t

¢

—wt

¤

DMƒ1 M

X

iD1

vec¡

Øü iØü

T i

¢

1 (18) where Øü

i,iD11 : : : 1 s are iid draws from the uniform distri-bution on a hypersphere with radiuspwtandMis chosen suf-Žciently large to achieve the desired degree of precision. The Øü

i are easily computed, as pointed out by Werner Ploberger. Draw an iid sequence8ØQi9MiD1from then-dimensional standard normal, then compute

Øü_i D s

wt4OÈT5 Q ØT

iØQi Q Ø_i0

We now consider the problem of deriving nonparamet-ric estimates of the functions g4_¢5 and g0₄

¢5. We closely

follow the discussion of Hodgson et al. (in press). Using our preliminary estimator OÈ_T, we compute the standardized residuals8Ø_t4ÈO_T59T

tD1 and the sequence of scalars wt4ÈOT5D

ØT

t4ÈOT5Øt4ÈOT5for everytD11 : : : 1 T. Next, compute the trans-formationztD’4wt5, where the transformation ’ 4¢5 belongs

to the Box–Cox (1964) family, ’4w_t5Dw

† tƒ1

† 0

We now compute kernel estimates of the density functionƒ4z5 of the transformed random variable z, and of its derivative ƒ0_{4z5, and use these estimates to indirectly obtain estimates} of the ratio g_g4w504w5. DeŽne the kernelK_‘T4¢5, with a bandwidth

parameter‘T, and use the kernel to compute the following estimates:

O

ƒ_t4z5D4Tƒ15ƒ1 T

X

sD1 s6Dt

K_‘T4zƒz_s4OÈ_T55

and

O

ƒ_t04z5D4Tƒ15ƒ1 T

X

sD1 s6Dt

K_‘0

T4zƒzs4OÈT550

We can then use the estimatesƒO_t4z5andƒO0

t4z5to estimate the ratio g_g4w504w5 as follows:

O g0

t O g_t4wt5D

8 > < > :

s4wt5C’04wt5O ƒt0

b

ƒt4zt5

if trimming conditions hold, 0 otherwise

(19)

where s4w5D41ƒn=25wƒ1_ƒJ’0

J’8’ 4w59’

0_{4w5 and J} ’4z5D —¡’ƒ1_4z5

¡z —. The trimming conditions referred to in (19) will depend on the kernel used. For certain kernels, such as the quartic or the logistic, trimming will not be required. In the Appendix, we provide expressions for the trimming conditions in the case where a Gaussian kernel is used. Even in this case, very little trimming (i.e., less than 1% of the observations) has been shown, in another context (Hodgson 1998), to yield semiparametric estimators that work well in Monte Carlo sim-ulations.

Finally, we have our semiparametric estimator for the period tscore,

b

ãt1 T4ÈOT5D ƒ 1 2

¡ln—H_t4OÈ_T5—

¡È Cbât4ÈOT5 O g0

t O gt

4wt4OÈT551 (20) where the expectation and score estimators are as deŽned in (17), (18), and (19) and where

b

ât4OÈT5D2 ¡uT

t4OÈT5 ¡È H

ƒ1

t 4ÈOT5ut4OÈT5C

¡4vecHƒ1 t 4OÈT55T ¡È €¡

H1_t=24ÈOT5†H 1=2 t 4ÈOT5

¢

vec¡Øt4ÈOT5ØTt4ÈOT5

¢

ƒEb

µ

¡4vecHƒ1 t 4OÈT55T ¡È

¡

H1=2

t 4OÈT5†H 1=2 t 4OÈT5

¢ ¶

€E6bvec4ØØT_5—w t4OÈT570

We then have our semiparametric score estimator for the sam-ple of sizeT,

b

ãT4OÈT5D T

X

tD1 b ãt1 T4ÈOT50

Our last step in deriving a semiparametric efŽcient estimator ofÈis to come up with a consistent semiparametric estimator of the expected outer product of the score. To this end, note that

E£

ã_{t1 T}4È05ãt1 T4È05 T¤

D1 4E

µ

¡ln—Ht4È05— ¡È

¡ln—Ht4È05— ¡ÈT

¶

CE

µ

ât4È05ât4È05 T

³

g0

g4wt4È055

´2¶

0 (21) To establish (21), note that

E

µ

¡ln—H_t4È05— ¡È ât4È05

g0

g4wt4È055

¶

DE

µ

¡ln—Ht4È05— ¡È

¡uT t4È05 ¡È H

ƒ1=2 t 4È05

¶

€E

µ

Øt4È05 g0

g4wt4È055

¶

CE

µ

¡4vecHƒ1 t 4È055T ¡È

¡

H1=2

t 4È05†H 1=2 t 4È05

¢ ¶

¢E

µ

4vec4Ø_t4È05Ø T

t4È055ƒE6vec4ØØ T_5—w

t4È0575

€g0

g4wt4È055

¶

0 Now we have

E

µ

Øt4È05 g0

g4wt4È055

¶

DE

µ

g0

g4wt4È055E6Øt4È05—wt4È057

¶

D01 because E6Øt4È05—wt4È057D0. Equation (21) then follows because

E

µ ¡

vec¡

Ø_t4È₀5ØT t4È05

¢

ƒE6vec4ØØT

5—wt4È057

¢g0

g4wt4È055

¶

DE

µ

E64vec4Ø_t4È₀5ØT t4È055

ƒE6vec4ØØT_5—w

t4È0575—wt4È057 g0

g4wt4È055

¶

DE

µ

E64vec4Øt4È05Ø T

t4È05—wt4È055

ƒE6vec4ØØT_5—w t4È05757

g0

g4wt4È055

¶

D00 In place of the unknown expectation,

E

µ

â_t4È₀5â_t4È₀5T

³

g0

g4wt4È055

´2¶

1

Table 5. Semiparametric Ef’cient Estimation of the C-CAPM

Parameter Size 1 Size 2 Size 3

 0145 0027 ƒ0269

(0022) (0018) (0036)

ƒ0 ƒ20665

(0133)

C1 C2 C3

C1 1

C2 10319 0459

(0030) (0033)

C3 10127 0266 0005

(0042) (0061) (0018)

A 0507 0423 0263

(0030) (0021) (0018)

D 0720 0811 0920

(0015) (0015) (0004)

NOTE: Data are from the CRSP dataset of stocks listed on the NYSE, AMEX, and NASDAQ. Value-weighted returns are calculated daily from January 1995–December 1997. Three size portfolios are created according to the previous day’s market value of equity. The previous day’s NYSE size quartiles are used as the cutoffs for the size portfolios, as well as for con-struction of weights. The ’rst two quartiles are grouped into the ’rst size portfolio, with the remaining two quartiles each representing the other two portfolios. The C-CAPM takes on the parameterizationrtDÁCexp(ƒ0)Ht×tƒ1Cut, with the scaled conditional variance

parameter-izationHtDCCTCAutƒ1uT

tƒ1ACDHtƒ1D. Point estimates of the parameters are reported, as are standard errors, which are reported below the point estimates in parentheses. Estimates are obtained using the SE procedure. We report estimates in this table using the bi-quartic kernel and the transformation function’(wt)D(wt†ƒ1)=†.

Standard errors tend to fall somewhat when using the semiparametric efŽcient estimator rather than the Gaussian Q-MLE, and the point estimates ofusing the semiparamet-ric estimator tend to be smaller than for their Q-MLE coun-terparts. The Wald test statistics of the validity of the CAPM, formed from the  estimates, are given in Table 6. For the unconditional CAPM, we Žnd that OLS leads to a marginal rejection of the CAPM at the 5% level. When we look at the tests of the C-CAPM, both estimation methods lead to strong rejections of the model withp values<.01.

Although the inferences regarding  are quite similar for the two methodologies, we Žnd some potentially interesting differences in the estimated systematic risk as measured by beta 4Â_t5. These differences are seen in lists of parameter estimates Tables 4 and 5, or perhaps more easily observed in Figures 1–3. These Žgures plot the conditional betas for the three portfolios showing the Q-MLE beta as well as the

Table 6. Mean-Variance Ef’ciency Tests

J(pvalue)

Unconditional CAPM

OLS 7095 (005)

C-CAPM

Q-MLE 264011 (<001)

SE 543033 (<001)

NOTE: The test statistics are constructed using the intercepts from estimating the C-CAPM. Mean-variance ef’ciency implies that the intercepts are jointly equal to 0:H02 iD0iD

11 : : : 1n. The table lists the tests that result from estimating the unconditional CAPM model via OLS, the C-CAPM using Q-MLE techniques, as well as the ef’cient procedure using the bi-quartic kernel. Under the null that the C-CAPM is the true model,Jis distributed asymptotically chi-squared (3).

Figure 1. Conditional Beta for the Size 1 Portfolio (- - - Q-MLE; —– SE).

semiparametrically estimated beta using the bi-quartic kernel. We observe that the‚11 t (size 1 portfolio) tends to be higher for the Q-MLE relative to the semiparametric estimates. How-ever, for the other two portfolios, the estimatedÂ_t is greater for the semiparametric estimator than for the Q-MLE. We also Žnd that the variability of Â_t is greater for the Q-MLE than for the semiparametric estimates. This is true for all of the portfolios, but especially for the size 1 and 2 portfolios. For these, the standard deviation of Â_t is 48% smaller for the semiparametric estimate then for the Q-MLE estimate. On the other hand, the standard deviation of ‚_{31 t} is only 2% smaller for the semiparametric estimate.

We also provide graphs of conditional expected returns for the three portfolios. These graphs are found in Figures 4–6. We deŽne the conditional expected return as

E_tƒ1rtD OÁCexp4ƒO05bHt×tƒ10

These graphs incorporate both the intercepts and the condi-tional betas and give a net effect on the parameters of inter-est for our semiparametric efŽcient method relative to Q-MLE methods. In general, we Žnd that the semiparametric efŽcient method leads to estimates of conditional expected return that are greater than those obtained by Q-MLE methods. These differences are small for the size 1 portfolio but increase as

Figure 2. Conditional Beta for the Size 2 Portfolio (- - - Q-MLE; —– SE).

Figure 3. Conditional Beta for the Size 3 Portfolio (- - - Q-MLE; —– SE).

we move to the larger Žrm portfolios. We observe that the dif-ferences in the estimates of the scaled conditional covariance matrix4H_t5tend to dominate differences in the intercept4Á5. In general, the semiparametric method estimates a larger por-tion of return due to systematic risk and a smaller porpor-tion of return coming from unexplained effects relative to Q-MLE.

6. SIMULATIONS

We simulate series of multivariate GARCH41115time series using the data-generating process

r_tDÁCexp4ƒ05Ht×tƒ1Cut1 (24) with

HtDCC

T_C

Autƒ1u

T tƒ1A0

We setnD2,T D759, and use the parameterizations

CD µ

1 0

0 1015 ¶

1 AD

µ

05 0

0 025 ¶

1

and ƒ₀D ƒ2075. This simulation setup is a simpliŽcation of our empirical model, adopted for the purpose of reduc-ing the computer time required to run the simulations. We

Figure 4. Conditional Expected Return for the Size 1 Portfolio

( Q-MLE; —– SE).

Figure 5. Conditional Expected Return for the Size 2 Portfolio

( Q-MLE; —— SE).

use the same ×_tƒ1 from our empirical analysis but reduce the dimension to a 2€1 vector by combining the smaller two quantile weights into one weight. We also simulate data using two differentÁ vectors,ÁD80109 under the null and ÁD8ƒ0151 0159under the alternative.

We add a randomly selected residual4ut5from some pre-speciŽed distribution. We consider a normal, a mixture of normals, a Studentt with 3 degrees of freedom, and a chi-squared with 5 degrees of freedom. The Žrst three distri-butions are elliptical, whereas the third is asymmetric and is included as a check on the robustness of our estima-tor to misspeciŽcation. To compute the mixture of normals, we Žrst deŽne the uniform random variable, U 260117. If

U < 41ƒ…5, then letu_tDpŠ1uMt, where uMt¹N 40115.

Oth-erwise, we let u_t DpŠ2uMt. The resulting ut will follow a mixed normal distribution. We set …D08 andŠ1D065 in the

simulations, and for all distributions the errors are scaled to have unit variances. We use the same residual in construct-ing both the alternative and null series. For each simulation, we estimate (24) using Q-MLE and the semiparametric efŽ-cient estimator. We replicate each simulation 2,000 times for each distribution and report the results of the simulations in Tables 7–9.

Figure 6. Conditional Expected Return for the Size 3 Portfolio

( Q-MLE; —– SE).

Table 7. Parameter Estimate Results for the Simulation Study

MN t3 •2

5 Normal

Parameter Bias Std. Dev. MSE Bias Std. Dev. MSE Bias Std. Dev. MSE Bias Std. Dev. MSE

0 Q-MLE ƒ0004 0053 0003 ƒ0011 0081 0007 0133 0083 0025 ƒ0015 0050 0003 SE 0005 0052 0003 ƒ0020 0078 0006 0160 0081 0032 0025 0089 0009

1 Q-MLE ƒ0007 0059 0004 ƒ0024 0043 0002 ƒ0162 0072 0031 ƒ0019 0079 0007 SE ƒ0004 0058 0003 ƒ0019 0044 0002 ƒ0160 0074 0031 ƒ0010 0121 0015

ƒ0 Q-MLE ƒ0229 10006 10064 ƒ0431 10169 10552 ƒ0013 10053 10109 ƒ0435 0906 10010 SE ƒ0398 10731 30154 ƒ0497 20404 60026 ƒ0196 10781 30210 ƒ10207 20075 50762 C2 Q-MLE ƒ0017 0235 0056 ƒ0106 0286 0093 ƒ0030 0238 0058 ƒ0011 0187 0035 SE 0015 0223 0050 ƒ0097 0279 0087 0006 0252 0064 0068 0208 0047 A1 Q-MLE ƒ0170 0161 0055 ƒ0106 0259 0078 ƒ0173 0173 0060 ƒ0186 0145 0056 SE ƒ0074 0157 0030 ƒ0076 0251 0068 ƒ0011 0180 0033 ƒ0280 0162 0105 A2 Q-MLE ƒ0129 0119 0031 0086 0216 0054 ƒ0123 0173 0045 ƒ0142 0122 0035 SE ƒ0111 0105 0023 0151 0208 0066 ƒ0082 0167 0034 ƒ0117 0164 0041

NOTE: This table lists the estimation results from a Monte Carlo study. Four different simulations were performed where the residuals were drawn from three different distributions: mixed normal (MN),t3,•₅2, and normal as indicated on the top row. The following parameter values were used in the simulation study:0D1D0,ƒ0D ƒ2050,C2D1015,A1D05, andA2D025. Each series had a length of 759, with 2,000 replications performed for each of the four distributions. Q-MLE and SE methods are used to estimate the model. The bias (average estimated valueƒtrue value), standard deviation of the parameter estimates, and MSE are reported.

Table 7 reports bias, standard deviation, and mean squared error (MSE) for the estimators for the four different distributions. For the nonnormal elliptical densities, the semi-parametric estimator (SE) yields only slight improvements in estimation of the intercepts, with larger improvements found in the estimation of the conditional variance parameters. This is consistent with our empirical study, in which we found that the SE point estimates had greater impact on conditional covariances than on intercepts. Estimation of the risk aversion parameterƒ0 deteriorates when we move from the Q-MLE to

the SE, but neither estimator accurately estimates this param-eter. We should point out that for the purposes of this article, ƒ0 is not of substantive interest, because we have focused our

attention on testing for zero intercepts and estimating betas, both problems in which ƒ0 can be thought of as a nuisance

parameter. Note that in the case of asymmetric errors, the SE provides reasonably good estimates of most of the parameters.

Table 8. Analysis of‚Performance From the Simulation Study

Distribution Performance measure ‚011 ‚012 ‚Q-MLE11 ‚Q-MLE12 ‚SE11 ‚SE12

MN ‚N_t 10055 0929 1004 0941 10066 0931

‘‚Nt 0135 0166 0127 0152 0132 0162

Averageabs(‚_iƒ‚0) 0108 0132 0099 0104

Average(‚iƒ‚0)2 0027 0040 0025 0036

t3 ‚Nt 10068 0914 0993 10005 10004 0983

‘‚N_t 0086 0099 0060 0068 0068 0079

Averageabs(‚_iƒ‚0) 0084 0102 0078 0098

Average(‚iƒ‚0)2 0014 0021 0011 0018

•2

5 ‚Nt 10056 0925 10051 0935 10059 0930

‘‚N_t 0123 0147 0118 0133 0128 0152

Averageabs(‚_iƒ‚0) 0108 0130 0103 0125

Average(‚_iƒ‚0)2 0028 0042 0028 0039

Normal ‚N_t 10066 0915 10046 0940 10040 0937

‘‚N_t 0004 0005 0114 0139 0118 0142

Averageabs(‚_iƒ‚0) 0092 0112 0109 0136

Average(‚_iƒ‚0)2 0019 0028 0023 0039

NOTE: This table lists various measures of‚performance of the Q-MLE and the SE estimator. The measure‚N_tis constructed by taking the sample average of‚t for each simulation and then averaging over the simulations. The measure‘‚N_t is constructed by taking the standard deviation of the average‚_t’s over the simulations.‚0corresponds with the “true” beta as opposed to estimated beta. The measure

averageabs(‚_iƒ‚0)is constructed by taking the average absolute difference between the estimated‚_tand‚0and then averaging over

the simulations. Average(‚_iƒ‚0)2reports where squared differences are taken as opposed to absolute differences.

This is important because, recalling our earlier comments, we have no theoretical results on the behavior of the SE under asymmetry, but the simulation results suggest that the SE may be robust to asymmetry. As we would expect, for the case of normal errors we see some deterioration in the performance of the SE conditional mean estimates relative to their Q-MLE counterparts, because the latter are fully efŽcient maximum likelihood estimates in this case. However, the SE estimates of conditional variance parameters are quite close to their MLE counterparts.

Table 8 compares the simulation results in the estimation of betas. For each simulation, we use the true parameter val-ues and the simulated residuals to construct a time series of “true” betas, Âi

0, where i indexes the simulation. We

com-pute the average values ofÂi

0 for each portfolio over the time

series as well as the standard deviations of Âi

0, 4‘Âi

01 j5 for each simulation. We then deŽne ‚01 j D 210001

P21000

iD1 Â

i

01 j and

Table 9. Conditional Mean-Variance Tests From the Simulation Study

Size Power (size-corrected)

Distribution Q-MLE SE Q-MLE SE

MN 0055 0058 071 073

t3 0054 0057 066 066

•52 0071 0078 062 064

Normal 0044 0039 078 058

NOTE: The table lists results of conditional mean-variance tests as shown in (6). Both size and power performance at the .05 level are listed. The power reports listed here have been adjusted for any size problems.

‘‚01 j D

1 21000

P21000

iD1 ‘Âi

01 j forjD112. We construct this same measure using the parameter and residual estimates from the two estimation methods. The Žnal two measures reported in Table 8 are constructed by looking at absolute and squared differences between the estimated and “true” conditional betas for each simulation and then averaging over the simulations. One apparent advantage of the semiparametric estimator is that estimated volatilities of the conditional betas are closer to the “true” beta volatilities than for the Q-MLE estimates. Note that in our empirical application, the SE produceslessvolatile estimated betas than the Q-MLE, whereas the reverse is true in the simulations. We are not sure why this is the case, although the simulation setup is different in a couple of important ways from the empirical model, which may explain the difference in results. The important point to take note of, in our view, is that the volatility of the SE betas is closer to the true beta volatility than that of the Q-MLE betas. We also note that the performance of the SE-estimated betas for the simulation with normal residuals indicates that there is a loss relative to the MLE, with the losses for this case of approximately the same order as the gains in the nonnormal simulations.

Table 9 considers the Wald tests of the zero-intercept null hypothesis. We calculate the empirical size and power of the test statistics for the two estimation methods as discussed by Davidson and MacKinnon (1998), using the p values from each test statistic. The power results are adjusted for any biases in size. The two methods lead to quite similar size and power properties in the asset-pricing tests for all of the cases other than normality, with the SE being slightly more oversized but having slightly higher size-corrected power. Both methods lead to reasonably sized tests for the elliptical distributions but overreject for the asymmetric •2

5 distribution. The MLE

method has substantially greater power than the SE method for the case of normality.

7. CONCLUSION

We propose a new estimation methodology that captures the nonnormalities of return distributions arising from tail thick-ness by using a multivariate GARCH-in-mean model with the exible distributional assumption of conditional elliptical sym-metry. Under high-level assumptions, this framework should lead to more efŽcient estimates than Q-MLE and should yield more powerful asset-pricing tests. We Žnd that in empiri-cal and simulation analysis our estimator does not improve signiŽcantly over the Gaussian Q-MLE in the estimation of

conditional mean parameters, but the semiparametric efŽcient estimates of the conditional betas do improve on the Q-MLE estimates in the presence of nonnormality to a degree that may be of potential interest to applied workers.

Further work on the properties of our estimator in the pres-ence of speciŽcation failure is suggested. In particular, the work of Harvey and Siddique (1999), among others, suggests that a derivation of the semiparametric efŽciency bounds of the GARCH-in-mean model with conditional densities that are not required to be symmetric would be a useful extension to this research. We have seen in our simulations that the estima-tor in this article does not misbehave too badly in the presence of asymmetric errors, but it would be desirable to have an estimator that explicitly accounts for the possibility of asym-metry. Such an estimator would have to take into account the fact that the conditional location is unidentiŽed in the pres-ence of asymmetric errors of unknown distributional form, and would require the use of multivariate generalizations of the approaches taken by Newey and Steigerwald (1997) and Drost and Klaassen (1997), both of which analyze univariate GARCH models with possibly asymmetric conditional densi-ties of unknown functional form.

ACKNOWLEDGMENTS

For their helpful comments, we are grateful to the refer-ees, to Jeffrey Wooldridge, Oliver Linton, Bill Brown, Rene Garcia, Werner Ploberger, Michael Brandt, and seminar par-ticipants at USC, UCSB, UQAM, Maryland, Rice, BYU, the 2000 American Winter Meetings of the Econometric Soci-ety, the 2000 EEA Summer Meetings, and the 2000 Canadian Econometrics Study Group. We acknowledge Žnancial sup-port from L’Institut de Finance Mathematique de Montréal and the National Science Foundation under CAREER grant SBR-9701959.

APPENDIX: ON ELLIPTICAL DENSITIES AND SOME COMPUTATIONAL ISSUES

A.1 Elliptical Densities

Ann-dimensional random vectoruis said to be elliptically distributed about the origin if its density can be written as

p4u5D4detè5ƒ1=2

g4u0èƒ1

u51

whereèis a positive deŽnite, symmetric matrix that is propor-tional to the covariance matrix ofu(when a Žnite covariance matrix exists) and is also proportional to the inverse of the information matrix ofp. The characteristic function ofuis

–4s5DE6exp4isT_u57D”4sT_ès5

for some function”4_¢5. The standardizedn-vector˜Dèƒ1=2_u

is said to be spherically symmetric, with density

p4˜5Dg4˜T_˜50

Note that the isoprobability contours of the density of the elliptical random variable u will be elliptical in shape, and those of the spherical random variable˜will be spherical (cir-cular in the case ofnD2).

Some examples of spherical densities are as follows: (a) Gaussian,

g4¢5Dconst¢exp

³ ƒ˜

T_˜ 2

´

(b) Studentt with’ degrees of freedom, g4_¢5Dconst_¢

³ 1C˜

T_˜ ’

´ƒ4nC’5=2

(c) Cauchy,

g4¢5Dconst¢41C˜T˜5ƒ4nC15=2 (d) logistic,

g4_¢5Dconst_¢exp4ƒ˜T_{˜5=61Cexp4ƒ˜}T_˜572

(e) scale mixed normal, g4¢5Dconst¢

Zˆ

0

sƒn=2

exp ³ ƒ˜ T_˜ 2s ´ dF 4s5

for some cdfF 4_¢5. Note that all of the non-Gaussian densities listed here feature thick tails, and some of them are popular candidates for modeling tail thickness in empirical work that takes a fully parametric tack.

Elliptical distributions have a few properties of interest. First, deŽne the norm˜˜˜ Dp˜T_{˜. The random variables} ˜

˜˜˜ and˜˜˜are independent of one another. Furthermore, the ran-dom variable _˜˜_˜˜ has a uniform distribution on the 4nƒ15 -dimensional unit hypersphere. These two features of elliptical distributions form the basis for Beran’s (1979) test for ellip-tical symmetry, whereas the latter fact plays a central role in our derivation of the semiparametric efŽciency bound for this model, following the results of Brown and Hodgson (2002).

DeŽne the nü_{€n matrix ê, of rank n}ü_{µn. Then the n}ü -dimensional random variableêu is elliptically symmetrically distributed with characteristic matrix of êèêT_{. DeŽne the} partitionuD4uT

11 u

T

25

T _{and partitionè conformably as}

µ

è11 è12

è21 è22

¶ 0

Then the marginal densities ofu1andu2are of the same form

as the joint density ofu, with respective characteristic matrices ofè11 andè22. The conditional mean can be written as

E6ui—uj7Dèijèƒ

1

jj uj0

Furthermore, the density of u_i conditional on u_j will be elliptically symmetric with a characteristic matrix of è_iiƒ è_ijèƒ1

jj èji.

Many of these characteristics of elliptical distributions are well known among economists to apply to the Gaussian den-sity. That they also apply to the more general elliptical family explains why the unconditional CAPM also holds in this case, a point discussed in more detail by Owen and Rabinovitch (1983). (See also Fang et al. 1990 and Owen and Rabinovitch 1983 for further discussion of elliptical distributions.)

A.2 Computation of Derivatives

We remark here on the difŽculty of obtaining expressions for the derivatives ¡ln—Ht—

¡È , ¡uTt

¡È , and ¡4vecHƒ1

t 5T

¡È . The basic prob-lem is that each of these derivatives involves an inŽnite recur-sion, because the expression for ¡uTt

¡È, for example, involves ¡4Ht5T

¡È , which in turn involves ¡uT_tƒ1

¡È , and so on. Our prac-tical approach is to construct the derivatives by assuming that ¡vec4H05

¡ˆ and

¡u0

¡ˆ take on their unconditional values, which allows us to obtain ¡vec4H15

¡ˆ and

¡u₁

¡ˆ. Given the derivatives for period one, we can construct the same for period two and con-tinue in a likewise manner to construct the derivatives for allT periods. We could also have assumed that ¡vec4H05

¡ˆ and

¡u0 ¡ˆ are 0, following Drost and Klaassen (1997), but have found in pre-liminary calculations that the empirical properties of the esti-mator were quite robust to the assumptions placed on ¡vec4H05

¡ˆ and ¡u0

¡ˆ. As stated earlier, the asymptotics of the estimator should not depend on the assumptions of the initial period. A.3 Kernels and Trimming

The two kernels that we consider are the bi-quartic,

K‘4u5D

15 16 ³

1ƒu

2

‘2

´2À­ ­ ­ ­ u ‘ ­ ­ ­ ­ µ1 Á 1 and the Gaussian,

K_‘4z5D 1

‘p2exp

³ ƒ z

2

2‘2

´ 0

The bi-quartic is applied without trimming. To establish con-sistency of the Gaussian kernel estimator, it is sufŽcient to apply the following trimming conditions, as shown by Hodg-son et al. (2002):

(a) ƒOt4z5¶dT, (b) —z—µe_T, (c) —‹4z5—µb_T, (d) —1=2_4z5ƒO0

t4z5—µcTƒOt4z5, where4z5Dw’04w5Jƒ1

’ 4w5[recall thatwD’ƒ14z5],J’4z5D —¡’ƒ1_4z5

¡z —, and 4d=dz5ƒ

1_1=2_{4z5. The constants‘}

T, dT, eT, bT, andc_T have the properties that, asT ! ˆ, we have‘_T !0,

cT ! ˆ, eT ! ˆ, bT ! ˆ, dT !0,‘TcT !0, eT‘Tƒ3D

o4T 5, andbT‘Tƒ3Do4T 5.

A.4 Schuster’s Correction

For most standard choices of symmetric kernel, the den-sity estimator fT4z5 typically performs poorly on the right neighborhood of 0. This bias arises because for points x_i in the right neighborhood of 0, the contribution of xi given by Tƒ1_K

‘T4xƒxi5 to fT4x5 extends to pointsxµ0 where

f 4x5D0. A similar bias arises in the multivariate density

estimates that impose the elliptical symmetry restriction. This bias creates a volcano-like contour in the density estimate. The overow in weights beyond the lower support of 0 can be corrected by using an estimator that incorporates this addi-tional support constraint information intofT4x5.

Schuster (1985) offered a correction that incorporates this overow to the regionz < c, for Žnitec, back into the region

z¶c by adding a mirror-image term Tƒ1_K

‘T4zƒ2cCzs5to

Tƒ1_K

‘T4zƒzs5. The resulting estimator forz¶c is given by

O

ƒt4z5D4Tƒ15ƒ1

T

X

sD1

s6Dt

6K‘T4zƒ Nzs5CK‘T4zƒ2cC Nzs570

In our case, cD0. Schuster (1985) also proved consistency and asymptotic normality results for this estimator.

[Received August 1999. Revised January 2002.]

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