Testing three-moments based asset pricing models: an ... - CiteSeerX

Testing three-moments based asset pricing models: an exact non-Gaussian multivariate regression approach∗ Marie-Claude Beaulieu † Université Laval

Jean-Marie Dufour ‡ Université de Montréal

Lynda Khalaf § Université Laval

First version: April 2004 Revised: December 2004, April 2005 This version: April 2005 Compiled: April 27, 2005, 10:47am

∗ This work was supported by the Institut de Finance Mathématique de Montréal [IFM2], the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Canada Research Chair Program, the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fond Banque Royale en Finance and the Fonds FQRSC and NATECH (Government of Québec). † Centre Interuniversitaire de Recherche sur les Politiques Économique et de l’Emploi (CIRPÉE), Département de finance et assurance, Pavillon Palasis-Prince, local 3632, Université Laval, Québec, Canada G1K 7P4. TEL: (418) 6562131-2926, FAX: (418) 656-2624. email: [email protected] ‡ Canada Research Chair Holder (Econometrics). Centre interuniversitaire de recherche en économie quantitative (CIREQ), Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Département de sciences économiques, Université de Montréal. Mailing address: Département de sciences économiques, Université de Montréal, C.P. 6128 succursale Centre-ville, Montréal, Québec, Canada H3C 3J7. TEL: 1 514 343 2400; FAX: 1 514 343 5831; e-mail: [email protected]. Web page: http://www.fas.umontreal.ca/SCECO/Dufour . § Canada Research Chair Holder (Environmental and Financial Econometric Analysis). Centre interuniversitaire de recherche en économie quantitative (CIREQ), and Groupe de recherche en économie de l’environnement et des ressources naturelles (GREEN), Université Laval, Pavillon J.-A. De Sève, St. Foy, Québec, Canada, G1K 7P4. TEL: (418) 656 2131-2409; FAX: (418) 656 7412. email: [email protected]

ABSTRACT

We propose exact inference methods for asset pricing models with unobservable risk-free rates and coskewness, specifically, the quadratic market model (QMM) [Kraus and Litzenberger (1976)] which incorporates the effect of asymmetry of return distribution on asset valuation. In this context, exact tests are appealing given: (i) the increasing popularity of such models in finance, (ii) the fact that traditional market models (which assume that asset returns move proportionally to the market) have not fared well in empirical tests, (iii) available related studies [e.g., Dittmar (2002), Harvey and Siddique (2000), Chen, Hong and Stein (2000), Simaan (1993), Barone-Adesi (1985), BaroneAdesi, Gagliardini and Urga (2004a, 2004b)] are only asymptotic so there is no guarantee that results are non-spurious; exact QMM tests are unavailable even with Gaussian errors. We consider the empirical models of Barone-Adesi (1985) and Barone-Adesi, Gagliardini and Urga (2004a, 2004b) where the procedure to assess the significance of coskewness preference is LR-based, and relates to the statistical and econometric literature on dimensionality tests which are interesting in their own right. We obtain exact versions of these tests, allowing for non-normality of fundamentals. Our results show that the model fares quite well with well known data sets.

i

Contents 1.

Introduction

1

2.

Framework

2

3.

Constrained estimation and test statistics 3.1. The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 5

4.

Exact distributional results 4.1. Tests on QMM parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Tests for QMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. A bootstrap-type LR test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 8 11 15

5.

The case of error distributions with unknown parameters

16

6.

A less restricted model

17

7.

Empirical analysis

17

8.

Conclusion

21

A.

B.

Monte Carlo tests A.1. General method . . . . . . . . . . . . . . . . . . . A.2. MC skewness and kurtosis tests . . . . . . . . . . . A.2.1. Estimating expected skewness and kurtosis . A.2.2. Individual excess skewness and kurtosis tests A.2.3. Combined excess skewness and kurtosis test

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Multivariate diagnostic checks

23 23 24 25 25 26 26

List of Assumptions, Propositions and Theorems 4.1 4.2 4.3 4.4

Theorem : Distribution of tests for uniform Linear hypotheses . . . . . . . . . . . . Theorem : Distribution of Linear QMM test statistics . . . . . . . . . . . . . . . . . Theorem : A general bound on the null distribution of Wald and LR non-linear test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem : A general bound on the null distributions of QMM non-linear test statistics

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List of Tables 1 2 3 4 5

QMM test: Value-Weighted Portfolios . . . QMM test: Equally Weighted Portfolios . . GQMM test: Value-Weighted Portfolios . . GQMM test: Equally-Weighted Portfolios . Multivariate diagnostics . . . . . . . . . .

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18 19 20 21 22

1. Introduction The traditional market model which supposes that asset returns move proportionally to the market has not fared well in empirical tests; see e.g. Campbell (2003). These failures have spurred studies on possible nonlinearities and asymmetries in the dependence between asset and market returns. In particular, the Quadratic Market Model (QMM) was proposed to extend the standard CAPM framework to incorporate the effect of asymmetry of return distribution on asset valuation; see Kraus and Litzenberger (1976). A number of well known empirical studies [e.g. Dittmar (2002), Harvey and Siddique (2000), Chen et al. (2000), Simaan (1993), Barone-Adesi (1985), BaroneAdesi, Gagliardini and Urga (2004a, 2004b)] have attempted to assess such asset pricing models; yet these studies are only asymptotically justified so there is no guarantee that results are nonspurious. Exact QMM tests seem lacking even with Gaussian errors; on problems with asymptotic tests in this context, see Shanken (1996), Campbell, Lo and MacKinlay (1997), Beaulieu, Dufour and Khalaf (2003, 2004, 2005). We consider empirical settings as in Barone-Adesi (1985) and Barone-Adesi, Gagliardini and Urga (2004a, 2004b) where the procedure to assess the significance of coskewness preference is LR-based. There are at least two reasons for focusing our attention on likelihood ratio [LR] type criteria. First, as demonstrated in Dufour (1997), size correction techniques have a much better chance of success with LR statistics [relative to instrumental variables based Wald-type criteria]. Secondly, of course, the power advantages of LR approaches are, in general, well-established. The model we assess (based on the Arbitrage Pricing Theoretical setting of Barone-Adesi (1985)) places nonlinear cross-equation constraints on a Multivariate Linear Regression (MLR) based QMM, which includes the expected return in excess of the zero-beta portfolio (assumed unknown and denoted γ) and an extra regressor: the expected excess returns on a portfolio perfectly correlated with the squared market returns; the latter parameter (denoted θ) is also unobservable and should be estimated in empirical applications. This formulation is interesting because it nests Black’s fundamental model [Black (1972), Gibbons (1982), Shanken (1996)] theoretically and statistically. To obtain an exact version of Barone-Adesi’s test, we formulate the hypothesis in the form of a dimensionality test on the parameters of the MLR-based QMM, where estimates and LR statistics are obtained through eigenvalue problems. Exact dimensionality tests are available under Gaussian fundamentals [see, for example, Zhou (1991), Zhou (1995) and Velu and Zhou (1999) who point out references to the statistical literature], yet these procedures have not been applied to the QMM framework. Of course, empirical evidence on non-normalities of financial returns [see Richardson and Smith (1993)] may lead one to question the usefulness of purely Gaussian based exact tests. The results of Beaulieu, Dufour and Khalaf (2003, 2004, 2005) related to the standard CAPM confirm this issue: despite relaxing normality (using provably exact non-Gaussian tests), the linear CAPM is still rejected for several sub-periods. We thus propose to extend the statistical framework of Beaulieu, Dufour and Khalaf (2003) to the QMM case. In general MLR-based tests, discrepancies between asymptotic and finite sample distributions are usually attributed to the curse of dimensionality: as the number of equations increases, the number of error cross-correlations grows rapidly which leads to reductions in degrees of freedom and test size distortions [see Shanken (1996), Campbell et al. (1997), and Dufour and Khalaf (2002b)].

1

Further, in the QMM case, the model involves nonlinear restrictions whose identification raises serious non-regularities: γ and θ are not identified over the whole parameter space, which can strongly affect the distributions of estimators and test statistics, leading to the failure of standard asymptotics; see, for example, Dufour (1997, 2003), Staiger and Stock (1997), Wang and Zivot (1998), Zivot, Startz and Nelson (1998), Dufour and Jasiak (2001), Kleibergen (2002), Stock, Wright and Yogo (2002) and Moreira (2003). We propose exact test procedures immune to such difficulties. The paper is organized as follows. Section 2 sets the framework. Section 3 presents model estimates and test statistics. Our distributional results are discussed in Section 4. In Section 5, we deal with the unknown distributional parameters and explain the procedures we use to solve this problem. In Section 6, we consider the homogeneity hypothesis. Section 7 reports our empirical results.

2. Framework ˜ Mt the returns Let Rit , i = 1, . . . , n, be returns on n securities for period t, t = 1, . . . , T and R on a market portfolio under consideration. Kraus and Litzenberger (1976)’s quadratic market model (QMM) takes the following form ˜ Mt + ci R ˜ 2Mt + uit , Rit = ai + bi R

i = 1, . . . , n, t = 1, . . . , T,

(2.1)

where uit are random disturbances. Our main statistical results require that the vectors (u1t , . . . , unt )0 denoted Vt , t = 1, . . . , T , satisfy the following distributional assumption: Vt = JWt , t = 1, . . . , T ,

(2.2)

where J is an unknown, non-singular matrix and the distribution of the vector w vec(W1 , . . . , WT ) : (i) is fully specified and denoted F(W ) where

=

W = [W1 , . . . , WT ]0 , or (ii) is specified up to an unknown nuisance-parameter, and denoted F(W ; ν) where ν refers to the nuisance parameter in question, w ∼ F(W ; ν) . (2.3) We will treat first the case where ν is known; the case of unknown ν is discussed in Section 5. As emphasized by Barone-Adesi (1985) and Barone-Adesi, Gagliardini and Urga (2004a, 2004b), (2.1) is not an equilibrium model. However, using Ross’s Arbitrage Pricing Theory, BaroneAdesi (1985) derived a model of equilibrium, which entails the following restrictions on (2.1): HQ : ai + γ(bi − 1) + ci θ = 0 ,

i = 1, . . . , n, for some γ and θ.

(2.4)

The coefficients γ and θ are unknown, so that HQ is a nonlinear restriction. The scalar γ may be interpreted as the return on the zero-beta portfolio (ci = 0 , i = 1, . . . , n), whereas θ is the expected value of the difference between the square of the returns on the market portfolio and

2

the expected excess returns on a portfolio perfectly correlated with it.1 Clearly, γ may be weakly identified if the bi coefficients are close to 1, and θ may be weakly identified if the ci coefficients are close to 0. For any specified values γ 0 and θ0 , the hypothesis HQ (γ 0 , θ0 ) : ai + γ 0 (bi − 1) + ci θ0 = 0 , i = 1, . . . , n,

(2.5)

is linear. This observation underlies our exact test procedure; we can also use pivotal statistics associated with the latter hypothesis as in Beaulieu et al. (2003) to obtain a joint confidence set for γ and θ. To avoid confusion, all through this paper HQ will refer to the QMM hypothesis (2.4) for an unknown γ and θ, whereas HQ (γ 0 , θ0 ) means that the hypothesis sets γ and θ to known scalars. Our notation for the test statistics under consideration follows conformably: LRQ and WQ will denote the LR and Wald statistics associated with HQ , where γ and θ are unknown, whereas LR(γ 0 , θ0 ) and W (γ 0 , θ0 ) will denote the LR and Wald statistics associated with HQ (γ 0 , θ0 ) which treats γ and θ as known scalars. To simplify the presentation, let us transform the model as follows: ˜ Mt = ai + (bi − 1)R ˜ Mt + ci R ˜ 2 + uit , t = 1, . . . , T, i = 1, . . . , n. Rit − R Mt

(2.6)

The above model is a special case of the following MLR Y = XB + U

(2.7)

where Y = [Y1 , ... , Yn ] is T × n, X is T × k with rank k and is assumed fixed, and U = [U1 , . . . , Un ] = [V1 , . . . , VT ]0 is the T × n matrix of error terms. In most cases of interest, we also have n ≥ k. Clearly, (2.6) corresponds to the case where: ˜ M , ... , Rn − R ˜ M ] , X = [ιT , R ˜M, R ˜ 2M , ], Y = [R1 − R 2 ˜ M = (R ˜ 1M , ... , R ˜ T M )0 , R ˜ 2M = (R ˜ 1M ˜ T2 M )0 , Ri = (R1i , ... , RT i )0 , R , ... , R   a1 ··· an  B = b1 − 1 · · · bn − 1  . c1 ··· cn

(2.8)

In this context, the (Gaussian) quasi likelihood ratio (QLR) criterion for testing (2.4) is: ˆQ |/|Σ|) ˆ LRQ = T ln(|Σ

(2.9)

ˆQ is the (Gaussian) quasi maximum likelihood estimator (QMLE) of Σ under HQ and Σ ˆ where Σ is the unrestricted QMLE of Σ, i.e. ˆ = U ˆ 0U ˆ /T = Y 0 M Y /T , Σ 1

ˆ = Y − XB ˆ, U

(2.10)

Note that Barone-Adesi, Gagliardini and Urga (2004b) treat a special case of the latter problem imposing an observable risk-free rate. The restriction they test (on a MLR model expressed in excess returns) remains non-linear.

3

ˆ = (X 0 X)−1 X 0 Y , B

M = I − X(X 0 X)−1 X 0 .

(2.11)

Barone-Adesi (1985) implements a linearization procedure as in Gibbons (1982) to approximate the latter statistic; in a related framework which corresponds to a known γ, Barone-Adesi et al. (2004b) obtains a test statistic of this form using iterative numerical maximization. In what follows, we propose [based on Gouriéroux, Monfort and Renault (1993, 1995) and Dufour and Khalaf (2002b)]: (i) simple eigenvalue-based non-iterative formula to derive (2.9), (ii) exact bounds on p-values and, alternative (iii) bootstrap-type cut-off p-values. Rejections based on the former and non–rejections based on the latter p-values are compelling (may be interpreted from an exact test perspective). Our formulation of the problem reveals similarities with the statistical foundations of MLRbased tests of Black’s version of the CAPM. Indeed, if ci = 0, the above model nests Black’s model. The statistical results we provide here thus extend Shanken (1986), Zhou (1991) and Velu and Zhou (1999) to the three-moments CAPM case. We note some results from Zhou (1995) are also relevant for the Gaussian case, although Zhou (1995) did not study the three-moments CAPM. Alternative formulations of the three moments CAPM allow for a less restrictive nonlinear hypothesis [see Barone-Adesi, Gagliardini and Urga (2004a)] as follows: ¯ Q : ai + γ(bi − 1) + ci θ = φ, i = 1, . . . , n, H

for some γ, θ and φ,

(2.12)

where φ is the same across portfolios. We will consider this hypothesis in Section 6.

3.

Constrained estimation and test statistics

In this section, we provide convenient non-iterative formulae for computing QMLE-based test statistics associated with HQ (γ 0 , θ0 ) and HQ .

3.1.

The linear case

Let us first consider results relevant to the hypothesis HQ (γ 0 , θ0 ) in (2.5), which sets the basis for the nonlinear formulation. HQ (γ 0 , θ0 ) is a special case of H(C0 ) : C0 B = 0

(3.1)

where C0 ∈ M(k − r, k), where 0 ≤ k − r ≤ k and M(m1 , m2 ) denotes the set of full-rank m1 × m2 matrices with real elements. In this case, the constrained QMLE estimates are ¡ ¢ £ ¡ ¢ ¤ ˆ 0) = B ˆ − X 0 X −1 C 0 C0 X 0 X −1 C 0 −1 C0 B ˆ, B(C (3.2) 0 0 £ ¡ ¢ ¤ ˆ 0) = Σ ˆ +B ˆ 0 C00 C0 X 0 X −1 C00 −1 C0 B ˆ. Σ(C (3.3) Furthermore, the standard LR and Wald statistics for testing H(C0 ) – denoted respectively LR(C0 ) and W (C0 ) – can be expressed in terms of the eigenvalues µ ˆ 1 (C0 ) ≥ µ ˆ 2 (C0 ) ≥ · · · ≥ µ ˆ n (C0 ) of ˆ 0 )−1 [Σ(C ˆ 0 ) − Σ]. ˆ Note that 1 − µ the matrix Σ(C ˆ 1 (C0 ), . . . , 1 − µ ˆ n (C0 ) are the eigenvalues of ˆ 0 )−1 Σ. ˆ In view of (3.3), the rank of Σ(C ˆ 0 )−1 [Σ(C ˆ 0 ) − Σ] ˆ is equal to l ≡ min{k − r, n} (with Σ(C

4

probabili-1ty one), and 0 ≤ µ ˆ i (C0 ) ≤ 1, i = 1, . . . , n.2 We have: ¡ ¢ ¡ ¢ ˆ 0 )|/|Σ| ˆ = −T ln |I − Σ(C ˆ 0 )−1 [Σ(C ˆ 0 ) − Σ]| ˆ LR(C0 ) = T ln |Σ(C = −T

n X

ln[1 − µ ˆ i (C0 )] = −T

i=1

l X

ln[1 − µ ˆ i (C0 )] ,

(3.4)

i=1

n X ¢ ¡ −1 ˆ [Σ(C ˆ 0 ) − Σ] ˆ =T W (C0 ) = T trace Σ i=1

l X µ ˆ i (C0 ) µ ˆ i (C0 ) =T . (3.5) 1−µ ˆ i (C0 ) 1−µ ˆ i (C0 ) i=1

In particular if k ≤ n, we have l = k − r . Applying these expressions to test HQ (γ 0 , θ0 ), we obtain the following expression for the constrained QMLE estimates: £ ¡ ¢ ¤ ¡ ¢ ˆ, ˆ 0 , θ0 ) = {I − X 0 X −1 C 0 C0 X 0 X −1 C 0 −1 C0 }B B(γ (3.6) 0 0 £ ¡ ¢ ¤ ˆ 0 , θ0 ) = Σ ˆ +B ˆ 0 C00 C0 X 0 X −1 C00 −1 C0 B ˆ = Y 0 M (γ 0 , θ0 )Y , Σ(γ (3.7) where X and Y are defined in (2.8), C0 = [1, γ 0 , θ0 ] and £ ¤−1 M (γ 0 , θ0 ) = M + X(X 0 X)−1 C00 C0 (X 0 X)−1 C00 C0 (X 0 X)−1 X 0 .

(3.8)

In this case, we have l = rank(C0 ) = k − r = 1, so that the matrix ˆ 0 , θ0 )−1 [Σ(γ ˆ 0 , θ0 ) − Σ] ˆ Σ(γ has only one non-zero root which we denote µ ˆ (γ 0 , θ0 ), and the test statistics are: LR(γ 0 , θ0 ) = −T ln[1 − µ ˆ (γ 0 , θ0 )] , µ ¶ µ ˆ (γ 0 , θ0 ) W (γ 0 , θ0 ) = T . 1−µ ˆ (γ 0 , θ0 )

(3.9) (3.10)

In this case, the statistics LR(γ 0 , θ0 ) and W (γ 0 , θ0 ) are strictly increasing functions of a single eigenvalue µ ˆ (γ 0 , θ0 ), hence strictly increasing transformations of each other.

3.2.

The nonlinear case

In the general framework of rank-tests, Gouriéroux, Monfort and Renault (1993, 1995) provide the following formulae for testing in the context of the general MLR model (2.7), hypotheses of the form HN L : CB = 0 , for some C ∈ M(k − r, k). (3.11) 2 The bounds on the µ ˆ i follow from standard results on the ordering of positive semidefinite matrices; see Horn and Johnson (1985, Theorems 7.6.3 and 7.7.3).

5

The LR and Wald statistics may then be written: ½ ¾ ¡ ¢ ¡ ¢ ˆ ˆ LRN L = T min ln |Σ(C0 )| − ln |Σ| = C∈M(k−r, k)

= −T

k X

min

C∈M(k−r, k)

ˆ i ), ln(1 − λ

{LR(C)} (3.12)

i=r+1

WN L

=

min

C∈M(k−r, k)

{W (C)} = T

k X

ˆi λ

ˆ i=r+1 (1 − λi )

,

(3.13)

ˆ1 ≥ λ ˆ2 ≥ · · · ≥ λ ˆ k are the eigenvalues of where LR(C) and W (C) are defined in (3.4)-(3.5) and λ RXY = (X 0 X)−1 X 0 Y (Y 0 Y )−1 Y 0 X. ˆ i ≤ 1, i = 1, . . . , n.3 Let eˆ1 , eˆ2 , . . . , eˆk denote the eigenvectors associated with Note that 0 ≤ λ ˆ1, λ ˆ2, . . . , λ ˆ k , normalized so that λ 0

eˆi X 0 X eˆj

= 1,

for i = j ,

= 0,

for i 6= j .

(3.14)

Then the QMLE estimate for C can be obtained as follows: CˆN L = [ˆ er+1 , . . . , eˆk ]0 (X 0 X) . It is also useful (to simplify the presentation of our test procedures in what follows) to consider the following reformulation of the hypothesis under test. We can also observe that HN L is equivalent to HN L : B = δβ for some δ

(3.15)

where β is r × n and δ is a k × r matrix of rank r which are linked to C through the orthogonality condition Cδ = 0 . (3.16) In the same vein, the hypothesis

˜ Q (δ 0 ) : B = δ 0 β H

(3.17)

where δ 0 is a known k × r matrix of rank r, is linear and equivalent to H(C0 ) : C0 B = 0, with C0 orthogonal to δ 0 . Gouriéroux, Monfort and Renault (1993, 1995) show that QMLE estimates for δ can be obtained as: ˆδ = [ˆ e1 , . . . , eˆr ] (3.18) ˆ ) can be obtained by applying OLS to the ˆN L and β and QMLE estimates of β and Σ (denoted Σ NL 3ˆ

ˆ k are also eigenvalues of RY X = (Y 0 Y )−1 Y 0 X(X 0 X)−1 X 0 Y . λ1 , . . . , λ

6

transformed regression

Y = (X ˆδ)β + U.

Of course, these estimates can also be obtained on replacing C0 by Cˆ in (3.6)-(3.7). Formally, convenient expressions for the QMLE estimates of the parameters of (2.7) under (3.11) may be obtained as

ˆN L U

0 ˆ, Σ ˆN L = ˆδ β ˆN L = U ˆN ˆ B L UN L /T, ˆ, β ˆ = (X ˆβ ˆ 0 X) ˆ −1 X ˆ 0 Y, X ˆ = X ˆδ =Y −X

where ˆδ is as defined in (3.18). Under standard regularity assumptions, Gouriéroux, Monfort and Renault (1993, 1995) show that ¡ ¢ ¡ ¢ ∞ ∞ LRN L ∼ χ2 (n − r)(k − r) , WN L ∼ χ2 (n − r)(k − r) . The QMM restrictions under test here are a special case of (3.11) with C = [1, γ, θ] where γ and θ are unknown. Since k = 3 and k − r = 1, we have here r = 2. The fact that one element of C is equal to one implies a normalization that does not alter the formulae for the test statistic. Then we can obtain an analytical (non-iterative) expression for the associated LR and ˆQ ≥ λ ˆQ ≥ λ ˆ Q be the eigenvalues of Wald test statistics as follows. Let λ 1 2 3 RXY = (X 0 X)−1 X 0 Y (Y 0 Y )−1 Y 0 X

(3.19)

where X and Y are as defined in (2.8); there are k = 3 non-zero roots here, then LRQ = WQ =

Q

ˆ ), inf {LR(γ, θ)} = −T ln(1 − λ 3 γ, θ

© ª inf W(γ,θ) = T γ, θ

ˆQ λ 3 , ˆQ) (1 − λ

(3.20) (3.21)

3

where LRQ and WQ are as defined in (3.9) and (3.10). As for LR(γ 0 , θ0 ) and W (γ 0 , θ0 ), the statistics LRQ and WQ are monotonic functions of a single eigenvalue, hence monotonic transformations of each other. ˆQ ˆQ ˆQ For further reference, let eˆQ ˆQ ˆQ 1, e 2, e 3 denote the eigenvectors associated with λ1 ≥ λ2 ≥ λ3 [normalized as in (3.14)]. Convenient expressions for the QMLE estimates of the parameters of (2.7) under HN L may be obtained as: 0 ˆ ˆ , Σ ˆQ = ˆδ Q β ˆQ = U ˆQ B UQ /T , Q ˆ , ˆQ = Y − X ˆQβ U Q 0 ˆ −1 ˆ 0 ˆ ˆ ˆ Q = X ˆδ Q , β Q = (XQ XQ ) XQ Y , X

7

(3.22) (3.23) (3.24)

h i ˆδ Q = eˆQ eˆQ . 1 2

(3.25)

Consistent estimates for γ and θ [denoted γˆ QM LE and ˆθQM LE ] may be obtained from (3.25) using the orthogonality condition (3.16). To the best of our knowledge, the latter formulae for BaroneAdesi (1985)’s model have not been applied to date. The original application in Barone-Adesi (1985) relied on a linearized version of the model and a pre-set estimator for θ which is not the QMLE; while Barone-Adesi et al. (2004a, 2004b) do tackle the nonlinear problem, they follow a standard iterative based numerical MLE. Under standard regularity assumptions, the results of Gouriéroux, Monfort and Renault (1993, 1995) yield: ∞ ∞ LRQ ∼ χ2 (n − 2) , WQ ∼ χ2 (n − 2) . (3.26) However, Dufour and Khalaf (2002b) (and the references cited therein) show that the latter approximations perform poorly in finite samples, particularly if n (the number of portfolios in this case) is large relative the sample size. In addition, the underlying asymptotics are not corrected for weak-identification and may lead to severe size distortions. Note finally again that the statistics ˆ Q , hence strictly increasing LRQ and WQ are strictly increasing functions of a single eigenvalue λ 3 transformations of each other.

4. Exact distributional results In Dufour and Khalaf (2002b), we have recently derived several exact distributional results regarding the QLR criteria discussed in the previous section for linear and nonlinear hypotheses. The following results are relevant to the problem under consideration.

4.1.

Tests on QMM parameters

Let us first consider results relevant to the hypothesis (2.5). This result is a special case of Theorem 3.1 in Dufour and Khalaf (2002b). Theorem 4.1 (2.7) and

D ISTRIBUTION

OF TESTS FOR UNIFORM

L INEAR

HYPOTHESES .

Under (2.2),

H(C0 ) : C0 B = 0 where C0 ∈ M(k − r, k), the statistics ¡ ¢ ˆ 0 )−1 [Σ(C ˆ 0 ) − Σ]| ˆ LR(C0 ) = −T ln |In − Σ(C = −T

l X

ln[1 − µ ˆ i (C0 )] ,

(4.1)

i=1 l X ¡ −1 ¢ ˆ [Σ(C ˆ 0 ) − Σ] ˆ =T W (C0 ) = T trace Σ i=1

8

µ ˆ i (C0 ) , 1−µ ˆ i (C0 )

(4.2)

are respectively distributed like LR(C0 ) = −T

l X

ln[1 − µi (C0 )] ,

(4.3)

i=1

¯ (C0 ) = T W

l X i=1

µi (C0 ) , 1 − µi (C0 )

(4.4)

where l ≡ min{k − r, n}, µ ˆ 1 (C0 ) ≥ µ ˆ 2 (C0 ) ≥ · · · ≥ µ ˆ n (C0 ) are the largest eigenvalues of the ˆ 0 )−1 [Σ(C ˆ 0 ) − Σ], ˆ matrix Σ(C ˆ = Y 0M Y , Σ

ˆ 0 ) = Y 0 M (C0 )Y , Σ(C

M = I − X(X 0 X)−1 X ,

M (C0 ) = M + X(X 0 X)−1 C00 [C0 (X 0 X)−1 C00 ]−1 C0 (X 0 X)−1 X 0 , while µ1 (C0 ) ≥ µ2 (C0 ) ≥ · · · ≥ µl (C0 ) are the eigenvalues of the matrix F¯L (C0 ) ≡ [W 0 M (C0 )W ]−1 [W 0 M (C0 )W − W 0 M W ] and W = [W1 , . . . , WT ]0 is defined by (2.2). The hypothesis H(C0 ) is equivalent to H(δ 0 ) : B = δ 0 β

(4.5)

where δ 0 is a known k × r matrix of rank r and β is r × n matrix such that C0 δ = 0. Thus the constrained projection matrix can also be calculated as ˆ 0 (X ˆ 00 X ˆ 0 )−1 X ˆ0 , M (C0 ) = M (δ 0 ) = I − X

ˆ 0 = Xδ 0 . X

(4.6)

For certain values of k − r and normal errors, the null distribution in question reduces to the F distribution. For instance, if k − r = 1, then ¢ ¤ ¡ ¢ T − (k − 1) − n £¡ ˆ ˆ − 1 ∼ F n, T − (k − 1) − n . |Σ(C0 )|/|Σ| n

(4.7)

We first apply Theorem 4.1 to inference on the parameters γ and θ associated with model (2.1). These parameters may be interpreted as “structural parameters” which are defined by the relationship HQ in (2.4), where n ≥ 2. For that purpose, it will be convenient to study first the problem testing hypotheses of the form HQ (γ 0 , θ0 ) that set γ and θ. The exact null distribution of the relevant test statistics is established in the following theorem. Theorem 4.2 D ISTRIBUTION OF L INEAR QMM TEST STATISTICS. Under (2.2), (2.6), (2.7), (2.8), and HQ (γ 0 , θ0 ) : ai + γ 0 (bi − 1) + ci θ0 = 0, i = 1, . . . , n,

9

where γ 0 and θ0 are known, the statistics ¡ ¢ ˆ 0 , θ0 )−1 [Σ(γ ˆ 0 , θ0 ) − Σ)]| ˆ LR(γ 0 , θ0 ) = −T ln |I − Σ(γ = −T ln[1 − µ ˆ (γ 0 , θ0 )] , ¡ −1 ¢ ˆ [Σ(γ ˆ 0 , θ0 ) − Σ] ˆ =T W (γ 0 , θ0 ) = T trace Σ

(4.8) µ ˆ (γ 0 , θ0 ) , 1−µ ˆ (γ 0 , θ0 )

(4.9)

are respectively distributed like LR(γ 0 , θ0 ) = −T ln[1 − µ(γ 0 , θ0 )], ¯ (γ 0 , θ0 ) = T µ(γ 0 , θ0 ) W 1 − µ(γ 0 , θ0 )

(4.10) (4.11)

ˆ 0 , θ0 )−1 [Σ(γ ˆ 0 , θ0 ) − Σ], ˆ where µ ˆ (γ 0 , θ0 ) is the non-zero eigenvalue of the matrix Σ(γ ˆ = Y 0M Y , Σ

ˆ 0 , θ0 ) = Y 0 M (γ 0 , θ0 )Y Σ(γ

M (γ 0 , θ0 ) = M + X(X 0 X)−1 C00 [C0 (X 0 X)−1 C00 ]−1 C0 (X 0 X)−1 X 0 ,

M = I − X(X 0 X)−1 X ,

µ(γ 0 , θ0 ) is the non-zero eigenvalue of the matrix F¯L (γ 0 , θ0 ) ≡ [W 0 M (γ 0 , θ0 )W ]−1 [W 0 M (γ 0 , θ0 )W − W 0 M W ]

(4.12)

and W = [W1 , . . . , WT ]0 is defined by (2.2). In this case, the tests based on LR(γ 0 , θ0 ) and W (γ 0 , θ0 ) are equivalent because they are functions of a single eigenvalue µ ˆ (γ 0 , θ0 ), hence monotonic transformations of each other. The latter theorem shows that the exact distributions of LR(γ 0 , θ0 ) and W (γ 0 , θ0 ) do not depend on any unknown nuisance parameter as soon as the distribution of W is completely specified. In particular, the values of B and J are irrelevant, though it depends in general upon X, γ 0 and θ0 . Although possibly non standard, the relevant distributions may easily be simulated if taking draws from the distribution of W1 , . . . , WT . We explain in Appendix A.1 how MC exact p-values [Dufour (2002), Dufour and Khalaf (2002b)] may be applied based on the pivotal quantity (4.12). The method may be summarized as follows. Conditional on ν, generate, according to (2.2), N i.i.d. draws from the distribution of W1 , . . . , WT ; using (4.10)-(4.12), these yield N simulated values of the test statistic. The exact MC p-value is then calculated (respectively) from the rank of the observed LR(γ 0 , θ0 ) and W (γ 0 , θ0 ) relative to the simulated ones. To compare our results with published work in this field, and in view of the equivalence between tests based on LR(γ 0 , θ0 ) and W (γ 0 , θ0 ), let us we focus on LR(γ 0 , θ0 ). For further reference, we will call the MC p-value so obtained pˆPNM C [LR(γ 0 , θ0 )|ν]

10

(4.13)

where: (i) N is the number of MC replications, (ii) the superscript P M C emphasizes the fact that the simulated statistic is a proper pivot and (ii) ν refers to the parameter of the error distribution [see (2.2))] which is considered known for now. Since the p-values just described do allow one to test any hypothesis of the form HQ (γ 0 , θ0 ), which sets the values of both γ and θ, we can build a confidence set for (γ, θ), by considering all pairs (γ 0 , θ0 ) that are not rejected, i.e. such that pˆPNM C [LR(γ 0 , θ0 )|ν] > α : C(α; γ, θ) = {(γ 0 , θ0 ) : pˆPNM C [LR(γ 0 , θ0 )|ν] > α} .

(4.14)

By construction, we have: P[(γ, θ) ∈ C(α; γ, θ)] ≥ 1 − α .

(4.15)

In the special case where the errors are i.i.d. Gaussian, i.e., W1 , . . . , WT ∼ N [0, In ]

(4.16)

the null distribution of LR(γ 0 , θ0 ) takes a simpler form. Since rank(C0 ) = k − r = 1, we have: ¢ ¤ (T − 2 − n) £¡ ˆ ˆ − 1 ∼ F (n, T − 2 − n ) . |Σ(γ 0 , θ0 )|/|Σ| n

(4.17)

In this case the distribution of LR(γ 0 , θ0 ) does not depend on X, γ 0 or θ0 . There is no need to use the technique of Monte Carlo tests to implement the procedures. This convenient feature is, however, specific to the Gaussian error distribution (though we cannot exclude the possibility that it holds for other distributions).

4.2. Tests for QMM Let us now consider the problem of testing the QMM restrictions as expressed by HQ . The latter constitute nonlinear restrictions for which a finite-sample distributional theory may be difficult to obtain. We propose here a number of possible avenues in order to deal with the nonlinear nature of the problem. For that purpose, we will start from the confidence set C(α; γ, θ) for (γ, θ). Under the assumption HQ , C(α; γ, θ) contains the true parameter vector (γ, θ) with probability at least 1 − α, in which case the set C(α; γ, θ) is obviously not empty. By contrast, if HQ does not hold, no vector (γ 0 , θ0 ) does satisfy HQ : if the test is powerful enough, all proposed values should be rejected, in which case C(α; γ, θ) is empty. The size condition (4.15) entails the following property: under HQ , P[C(α; γ, θ) = ∅] ≤ α . (4.18) Thus, by checking whether we have C(α; γ, θ) = ∅, i.e. {(γ 0 , θ0 ) : pˆPNM C [LR(γ 0 , θ0 )|ν] > α} = ∅ , we get a critical region with level α for HQ .

11

(4.19)

In view of the fact that the set C(α; γ, θ) only involves two parameters, it can be established fairly easily by numerical methods. But it would be useful if the condition C(α; γ, θ) = ∅ might be checked in an even simpler way. To do this, we shall derive bounds on the distribution of LRN L and WN L . This is done in the two following theorems. In the first one of these, we consider tests for general (nonlinear) restrictions of the form HN L in (3.11). Theorem 4.3 A

GENERAL BOUND ON THE NULL DISTRIBUTION OF

LINEAR TEST STATISTICS .

WALD

AND

LR

NON -

Under (2.2), (2.7), and

HN L : CB = 0, for some C ∈ M(k − r, k) , the distribution of the Gaussian QMLE statistics LRN L = − T

k X

ˆ i ), ln(1 − λ

WN L = − T

i=r+1

k X

ˆi) , ln(1 − λ

i=r+1

ˆ1 ≥ λ ˆ2 ≥ · · · ≥ λ ˆ k are the eigenvalues of where λ RXY = (X 0 X)−1 X 0 Y (Y 0 Y )−1 Y 0 X may be bounded as follows: for all (B, J), P(B, J) [LRN L ≥ x] ≤ P(B, J) [WN L ≥ x] ≤

£

sup (B, J)∈H(C0 )

sup (B, J)∈H(C0 )

P(B, J) − T £ P(B, J) T

l X

¤ ln[1 − µi (C0 )] ≥ x , ∀x, (4.20)

i=1 l X i=1

¤ µi (C0 ) ≥ x , ∀x, 1 − µi (C0 )

(4.21)

where P(B, J) represents the distribution of Y when the parameters are (B, J), H(C0 ) = {(B, J) : C0 B = 0}, C0 ∈ M(k − r, k), l = min{k − r, n}, and µ1 (C0 ) ≥ µ2 (C0 ) ≥ · · · ≥ µn (C0 ) are the eigenvalues of the matrix F¯L (C0 ) = [W 0 M (C0 )W ]−1 [W 0 M (C0 )W − W 0 M W ] ,

(4.22)

with M = I − X(X 0 X)−1 X 0 and M (C0 ) = M + X(X 0 X)−1 C00 [C0 (X 0 X)−1 C00 ]−1 C0 (X 0 X)−1 X 0 .

(4.23)

P ROOF . Let HN L = {(B, J) : CB = 0 for some C ∈ M(k − r, k)}. It is clear that HN L =

∪

C0 ∈M(k−r, k)

12

H(C0 ) .

(4.24)

From Gouriéroux, Monfort and Renault (1993, 1995), we have: LRN L

=

WN L

=

inf

{LR(C0 )} ,

inf

{W (C)} ,

C0 ∈M(k−r, k) C0 ∈M(k−r, k)

which implies that LRN L ≤ LR(C0 ) and WN L ≤ W (C0 ), for any C0 ∈ M(k − r, k). This entails that, for each C0 ∈ M(k − r, k) and for all (B, J) ∈ H(C0 ), £ ¤ P(B, J) [LRN L ≥ x] ≤ P(B, J) LR(C0 ) ≥ x , ∀x, (4.25) £ ¤ P(B, J) [WN L ≥ x] ≤ P(B, J) W (C0 ) ≥ x , ∀x. (4.26) Furthermore, from Theorem 4.1, we see that, under the null hypothesis, LR(C0 ) and W (C0 ) are P P µi (C0 ) distributed (respectively) like the variables −T li=1 ln(1 − µi (C0 )) and T li=1 1−µ where i (C0 ) µ1 (C0 ) ≥ µ2 (C0 ) ≥ · · · ≥ µn (C0 ) are the eigenvalues underlying (4.22) and l = min{k − r, n}. From there on, (4.20)-(4.21) follow straightforwardly.

Applying these distributional results to the problem at hand leads to the following. Theorem 4.4 A GENERAL BOUND ON THE NULL DISTRIBUTIONS OF QMM NON - LINEAR TEST STATISTICS . Under (2.2), (2.6), (2.7), (2.8) and HQ : ai + γ(bi − 1) + ci θ = 0, i = 1, . . . , n, for some γ and θ, the null distribution of the Gaussian QMLE statistics LRQ = WQ Q

Q

=

ˆ Q ), − T ln(1 − λ 3 ˆ Q ), − T ln(1 − λ 3

Q

ˆ ≥λ ˆ ≥λ ˆ are the non-zero roots of where λ 1 2 3 RXY = (X 0 X)−1 X 0 Y (Y 0 Y )−1 Y 0 X may be bounded as follows: for all (B, J), P(B, J) [LRQ ≥ x] ≤ P(B, J) [WQ ≥ x] ≤

sup (B, J)∈H(γ 0 , θ0 )

sup (B, J)∈H(γ 0 , θ0 )

£ ¤ P(B, J) − T ln[1 − µ(γ 0 , θ0 )] ≥ x , ∀x, · P(B, J)

¸ µ(γ 0 , θ0 ) ≥ x , ∀x, T 1 − µ(γ 0 , θ0 )

(4.27) (4.28)

where P(B, J) represents the distribution of Y when the parameters are (B, J), H(γ 0 , θ0 ) =

13

{(B, J) : [1, γ 0 , θ0 ] B = 0}, where µ(γ 0 , θ0 ) is the non-zero eigenvalue of the matrix F¯L (γ 0 , θ0 ) = [W 0 M (γ 0 , θ0 )W ]−1 [W 0 M (γ 0 , θ0 )W − W 0 M W ] ,

(4.29)

M (γ 0 , θ0 ) = M + X(X 0 X)−1 C00 [C0 (X 0 X)−1 C00 ]−1 C0 (X 0 X)−1 X 0 ,

(4.30)

C0 = [1, γ 0 , θ0 ] , γ 0 and θ0 are any given scalars, M = I − X(X 0 X)−1 X 0 , and W = [W1 , . . . , Wp ] is defined by (2.2). The latter theorem shows that the distribution of LRN L can be bounded by bounding the distributions of the test statistics LR(γ 0 , θ0 ) – each one under the corresponding null hypothesis HQ (γ 0 , θ0 ) – over the set of possible values (γ 0 , θ0 ), and similarly for WN L using W (γ 0 , θ0 ). In particular, if the distribution of LR(γ 0 , θ0 ) is the same for all (γ 0 , θ0 ), this yields a single bounding distribution. The latter situation obtains in the Gaussian case (4.16), where an F -distribution can be used. For HN L and k − r = 1, Theorem 4.4 and (4.7) yield the following probability inequality: £ ¤ P ([T − (k − 1) − n]/n) (ΛQ − 1) ≥ x ≤ P[F (n, T − (k − 1) − n) ≥ x], ∀x. (4.31) where ΛQ ≡ exp(LRQ /T ). Further, for HQ , we further have k = 3, hence £ ¤ P ([T − 2 − n]/n) (ΛQ − 1) ≥ x ≤ P[F (n, T − 2 − n) ≥ x], ∀x.

(4.32)

For non-Gaussian error distributions, the situation is more complex. Again, if the bounding test statistics LR(γ 0 , θ0 ) have the same distribution [under HQ (γ 0 , θ0 )] irrespective of (γ 0 , θ0 ), we can make MC bound test by simulating the distribution of the bounding test statistic LR(γ 0 , θ0 ). We explain in Appendix A.1 how MC bound p-values [Dufour (2002), Dufour and Khalaf (2002b)] may be obtained. The method may be summarized as follows given (2.2). Conditional on ν, generate, imposing (2.2), N i.i.d. draws from the distribution of W1 , . . . , WT ; these yield N simulated values of the bounding test statistic LR(γ 0 , θ0 ). The exact MC bound p-value is then calculated from the rank of the observed LRQ and WQ relative to the simulated ones. Of course, both tests will be equivalent in this case; to compare our results with published work in this field, we focus on LRQ . As explained in Appendix A.1, we will call the bound MC p-value C pˆBM (LRQ |γ, θ, ν) N

(4.33)

where: (i) the superscript BM C [rather than P M C as was used in (4.13)] indicates that the MC p-value is bound-based, (ii) ν refers to the parameter of the error distribution (see Appendix A.2.3) which is for the moment considered known, and (iii) conditioning on γ and θ emphasizes the specific choice for γ and θ underlying the bound. If invariance of the bounding distributions generated by LR(γ 0 , θ0 ) does not hold, numerical methods may be needed to evaluate the bounds in (4.27)-(4.28). In this case, it is not clear that this is any simpler than drawing the set C(α; γ, θ) using numerical methods. However, even in this case, a useful bound may still be obtained by considering the distribution the simulated distribution

14

obtained on replacing (γ 0 , θ0 ) by their QMLE estimates in LR(γ 0 , θ0 ), where the estimates are treated as fixed known parameters. This yields the following bound p-value: C pˆBM (LRQ |ˆ γ QM LE , ˆθQM LE , ν). N

(4.34)

Then, we have the following implication: C pˆBM (LRQ |ˆ γ QM LE , ˆθQM LE , ν) > α ⇒ pˆPNM C [LR(γ 0 , θ0 )|ν] > α , for some (γ 0 , θ0 ). (4.35) N C (LR |ˆ ˆ In other words, if pˆBM Q γ QM LE , θ QM LE , ν) > α, we can be sure that C(α; γ, θ) is not N empty so that HQ is not rejected by the test in (4.18). The proof is similar to the one in Beaulieu et al. (2003).

4.3. A bootstrap-type LR test With N draws from the distribution of W1 , . . . , WT , N simulated samples can be constructed conformably with (2.6)-(2.8), (2.2) under the null hypothesis (2.4), given any value for {B, Σ} ∈ ΨQ where ΨQ refers to the parameter space compatible with the null hypothesis HQ . Applying (3.20) to each of these samples yields N simulated statistics from the null data generating process. ˆ N (.) in Count the number of simulated statistics ≥ {observed LRQ } and replace this number for G (A.1) of Appendix A; this provides a MC p-value conditional on the choice for B and Σ which we denote C pˆLM (LRQ |B, Σ, ν) (4.36) N a Local MC (LMC) p-value to emphasize that it is estimated given a specific nuisance parameter ˆQ and Σ ˆQ defined in (3.22)value. In particular, it is natural to consider the QMLE estimates B (3.24), 0 ˆ ˆ , Σ ˆQ = ˆδ Q β ˆQ = U ˆQ B UQ /T , Q 0 0 ˆ , β ˆ = (X ˆQ = Y − X ˆQβ ˆQX ˆ Q )−1 X ˆQ U Y, Q Q h i ˆ Q = X ˆδ Q , ˆδ Q = eˆQ , eˆQ X r 1

which yields

C ˆQ , Σ ˆQ , ν). pˆLM (LRQ |B N

(4.37)

This bootstrap-type p-value differs from the bounds-based p-value (4.34) BM C ˆ pˆN (LRQ |ˆ γ QM LE , θQM LE , ν), although both use QMLE estimates. Indeed, as a consequence of Theorem 4.4, it is straightforward to see that C C ˆQ , Σ ˆQ , ν). pˆBM (LRQ |ˆ γ QM LE , ˆθQM LE , ν) ≥ pˆLM (LRQ |B N N

15

(4.38)

C satisfies the test’s level control by construction, nothing guarantees that While relying on pˆBM N C will yield a level correct test in finite samples. However, since relying on pˆLM N C ˆQ , Σ ˆQ , ν) > α ⇒ pˆLM (LRQ |B N

sup {B,Σ}∈ΨQ

C pˆLM (LRBCAP M |B, Σ, ν) > α N

where B, Σ refer to the parameter space compatible with the null hypothesis. It follows that a C (LR |B ˆ ˆ non-rejection based on pˆLM Q Q , ΣQ , ν) can be considered conclusive from an exact test N perspective.

5. The case of error distributions with unknown parameters We will now extend the above results to the unknown distributional parameter case for the error families of particular interest, namely (2.2). For ν given, the distributional hypothesis underlying (2.2) satisfies (2.2). Thus if errors follow (2.2), Theorems 4.3-4.4 yield exact bounds (invariant to B and Σ) only when ν is specified. Here we show how to take this parameter explicitly into consideration. We consider a solution originally proposed by Dufour and Kiviet (1996) which we also applied in Beaulieu et al. (2003, 2004, 2005). The method requires two steps: (i) an exact confidence set with levels (1 − α1 ) is derived for ν, and (ii) the MC p-values (presented above) maximized over all values of ν in the latter confidence set are referred to the level α2 . This yields an exact test with overall level α = α1 + α2 . In our empirical application, we used α1 = α2 = α/2. Empirically, maximizing over a set estimate for ν (rather than overall the nuisance parameter set) ensures that the error distributions retained are consistent with our data. Recall that the null hypothesis imposes distributional constraints jointly with the restrictions on the regression coefficients. In view of this, retaining empirically relevant values for ν helps to interpret the QMM test rejections. For any (1 − α1 ) confidence set for ν which we will denote C(Y ) where Y refers to the return data (as in e.g. (2.7)), step 2 proceeds as follows. Let C QBM (γ, θ) = U C QBM (ˆ γ QM LE , ˆθQM LE ) = U C ˆ ˆQ ) = QLM (BQ , Σ U

C sup pˆBM (LRQ |γ, θ, ν), N

(5.1)

C sup pˆBM (LRQ |ˆ γ QM LE , ˆθQM LE , ν), N

(5.2)

C ˆQ , Σ ˆQ , ν), sup pˆLM (LRQ |B N

(5.3)

ν ∈ C(Y ) ν ∈ C(Y ) ν∈ C(Y )

C (LR |γ, θ, ν) [defined in (4.33)] is the bound associated with Theorem 4.4, where: pˆBM Q N BM C pˆN (LRQ |ˆ γ QM LE , ˆθQM LE , ν) [defined in (4.34)] is the specific bound based on QMLE estiC (LR |B ˆ ˆ mates of γ and θ and pˆLM Q Q , ΣQ , ν) [defined in (4.37)] is the LMC bootstrap-type pN value. Then the critical regions C QBM (γ, θ) ≤ α2 , U

C QBM (ˆ γ QM LE , ˆθQM LE ) ≤ α2 , U

16

(5.4)

C (B ˆQ , Σ ˆQ ) is not necessarily are exactly of level α1 + α2 respectively.4 The critical region QLM U level correct in finite samples. However, it can be considered exact if non-significant. Let us now present the set estimation method we propose to obtain C(Y ). Given the recent literature documenting the poor performance of asymptotic Wald-type confidence intervals [see Dufour (1997) and the references cited therein], we prefer to build a confidence set by ”inverting” a test for the null hypotheses (2.2) where ν = ν 0 for known ν 0 . Inverting an α1 level test denoted T (Y ) formally implies the following. Let T0 (Y ) denote the value of the statistic computed from the observed sample. Obtain the p-value pˆ(T0 (Y )|ν 0 ) conforming with (2.2). The confidence set for ν corresponds to the values of ν 0 , for which pˆ(T0 (Y )|ν 0 ) > α1 . The test statistic we consider in our empirical analysis is presented in Appendix A.2 (specifically: equation (A.11)).

6. A less restricted model We now turn to the general hypothesis (2.12). Let us first observe that the hypothesis may be rewritten in matrix form HGQM M : [1, γ, θ] B = φι0n where ιn is n-dimensional vector of ones. In addition, if φ was known, then it is possible to use the above derived eigenvalue test procedure to derive its associated LR statistic; one may consider the following MLR: ˜ Mt − φ = ai + (bi − 1)R ˜ Mt + ci R ˜ 2Mt + uit , t = 1, . . . , T, i = 1, . . . , n. Rit − R

(6.5)

In this context, testing HGQM M corresponds to assessing ai + γ(bi − 1) + ci θ = 0, i = 1, . . . , n,

(6.6)

which leads to the above framework. This also suggest a simple procedure to derive the LR statistic to test HGQM M and the associated MLE estimate of φ. Indeed, one may minimize, over φ, the eigenvalue based criterion associated with (6.5)-(6.6); this may be conducted numerically (yet since the argument of the underlying minimization is scalar, this approach is numerically much more tractable than the iterative maximization applied by Barone-Adesi et al. (2004b)). The statistical solutions we derived above also hold in this framework: indeed, bounding cut-off points can be obtained using the null distribution of the test statistic which fixes φ to a known value. Note however that we have shown in Dufour and Khalaf (2002b) that the latter distribution does not depend on φ, which will lead one to the same bound whether φ is zero or not. Tighter bounds can be obtained under normality (see Velu and Zhou (1999) or Zhou (1995)).

7. Empirical analysis For our empirical analysis, we use Fama and French’s data base. We produce results for monthly returns of 25 value weighted and equally weighted portfolios from 1961-2000. The portfolios which 4

For proofs and further references, see Dufour and Kiviet (1996) and Dufour (2002).

17

Table 1. QMM test: Value-Weighted Portfolios Normality tests Sample 1961-65 1966-70 1971-75 1976-80 1981-85 1986-90 1991-95 1996-00

Tests of HQ

(1)

(2)

(3)

(4)

(5)

(6)

SK

KU

CSK

MN

LR

p∞

.034 .051 .017 .004 .001 .757 .318 .001

.007 .061 .016 .002 .001 .637 .081 .001

.006 .075 .032 .001 .001 .754 .091 .001

.026 .044 .015 .004 .001 .401 .176 .001

33.190 42.346 34.853 67.900 51.807 46.310 49.425 33.489

.078 .008 .054 .000 .001 .003 .001 .073

(7)

(8)

Normal

(9)

Student-t

C pˆBM N

C p˜LM N

C pˆBM t

C p˜LM t

.586 .246 .515 .006 .081 .163 .114 .543

.405 .134 .341 .002 .022 .074 .055 .386

.596 .255 .529 .015 .093 .193 .148 .591

.414 .143 .367 .006 .033 .091 .076 .420

CS(κ) ≥8 ≥8 ≥7 ≥6 ≥5 ≥ 12 ≥9 3 − 11

Note – Columns (1)-(4) report the exact normality tests presented in Appendix A.2, specifically: MC tests based on multivariate skewness (A.6), kurtosis (A.7), and the combined tests (A.11) and (A.8) respectively. Column (5) presents the quasi-LR statistic defined in (3.20) to test HQ [(2.4)]; the subsequent columns report the associated p-values using, respectively, the asymptotic χ2 (n − 2) distribution [column (6)], the Gaussian based BMC and LMC p-values [column (7)], and the BMC and LMC p-values imposing multivariate t(κ) errors [column (8)]; see Section 4 for a description of the underlying LMC and BMC tests. MC p-values for t(κ) errors are the largest over the degrees-of-freedom parameter κ within the specified confidence sets; the latter is reported in column (9) [see Section 5]. January and October 1987 returns are excluded from the dataset.

are constructed at the end of June, are the intersections of five portfolios formed on size (market equity) and five portfolios formed on the ratio of book equity to market equity. The size breakpoints for year s are the New York Stock Exchange (NYSE) market equity quintiles at the end of June of year s. The ratio of book equity to market equity for June of year s is the book equity for the last fiscal year end in s − 1 divided by market equity for December of year s − 1. The ratio of book equity to market equity are NYSE quintiles. The portfolios for July of year s to June of year s + 1 include all NYSE, AMEX, NASDAQ stocks for which we have market equity data for December of year s − 1 and June of year s, and (positive) book equity data for s − 1. All MC tests where applied with 999 replications, and multivariate normal and multivariate student t errors. Formally, these correspond to the following special case of assumption (2.2): Wt ∼ FN (Wt ) ⇔ Wt is multivariate normal (0, In )

(7.1)

Wt ∼ Ft (Wt ; κ) ⇔ Wt = Z1t /(Z2t /κ)1/2 ,

(7.2)

and where Z1t is multivariate normal (0, In ) and Z2t is a χ2 (κ) variate independent from Z1t . As is usual in this literature (see e.g. the surveys of Black (1993) or Fama and French (2004)), we estimate and test the model over 5 year subperiods.5 We also perform the exact normality tests 5

Our previous and ongoing work on related asset pricing applications [see Beaulieu et al. (2003, 2004, 2005), and

18

Table 2. QMM test: Equally Weighted Portfolios Normality tests Sample 1961-65 1966-70 1971-75 1976-80 1981-85 1986-90 1991-95 1996-00

Tests of HQ

(1)

(2)

(3)

(4)

(5)

(6)

SK

KU

CSK

MN

LR

p∞

.017 .002 .103 .010 .001 .366 .736 .001

.022 .001 .122 .002 .001 .369 .261 .001

.019 .001 .155 .009 .001 .456 .600 .001

.016 .002 .066 .001 .001 .200 .382 .001

33.392 29.829 34.045 74.203 42.337 43.215 37.354 30.287

.075 .154 .065 .000 .008 .007 .030 .141

(7)

(8)

Normal

(9)

Student-t

C pˆBM N

C p˜LM N

C pˆBM t

C p˜LM t

.562 .686 .535 .003 .248 .247 .418 .697

.361 .515 .355 .001 .122 .122 .261 .496

.576 .695 .549 .004 .273 .255 .420 .710

.363 .530 .368 .002 .144 .136 .282 .518

CS(κ) ≥8 ≥6 ≥9 ≥6 ≥6 ≥ 10 ≥ 11 4 − 19

Note – Columns (1)-(4) report the exact normality tests presented in Appendix A.2, specifically: MC tests based on multivariate skewness (A.6), kurtosis (A.7), and the combined tests (A.11) and (A.8) respectively. Column (5) presents the quasi-LR statistic defined in (3.20) to test HQ [(2.4)]; the subsequent columns report the associated p-values using, respectively, the asymptotic χ2 (n − 2) distribution [column (6)], the Gaussian based BMC and LMC p-values [column (7)], and the BMC and LMC p-values imposing multivariate t(κ) errors [column (8)]; see Section 4 for a description of the underlying LMC and BMC tests. MC p-values for t(κ) errors are the largest over the degrees-of-freedom parameter κ within the specified confidence sets; the latter is reported in column (9) [see Section 5]. January and October 1987 returns are excluded from the dataset.

reported in Appendix A.2, specifically: MC tests based on (A.6), (A.7), (A.8) and (A.11). For further reference on these exact tests, see Dufour, Khalaf and Beaulieu (2003). Our results can be summarized as follows. Multivariate normality is rejected in many subperiods and provides us with a reason to investigate whether test results shown under multivariate normality are still prevalent once we use Student-t distributions. Table 1 shows that the test restriction is rejected at the 5% level in five subperiods out of eight using asymptotic p-values. Using a finite sample tests under multivariate normality reveals that the restriction is rejected in only one sub-period, namely 1976-1980. The LMC p-value confirms all these non-rejections but one. Using the same approach under the multivariate Student-t distribution leads to the same conclusion. The results presented in Table 2 for equally weighted portfolios show that the restriction imposed by the QMM is only rejected in the 1976-1980 sub-period using finite sample normal and student-t distributions, although asymptotic p-values are significant for four subperiods. Tables 3 and 4 present the tests on GQMM. Using value-weighted portfolios, HGQM M is not rejected in any sub-period allowing for t-errors, although the normal LMC p-value is