Finite-sample multivariate tests of asset pricing models with coskewness ∗ Marie-Claude Beaulieu † Université Laval Jean-Marie Dufour ‡ McGill University Lynda Khalaf § Carleton University First version: July 2002 Revised: July 2002, April 2004, April 2005, September 2007 This version: February 2008 Compiled: March 16, 2008, 9:44pm This paper is forthcoming in the Computational Statistics and Data Analysis. * The authors thank the Editor David Belsley, two anonymous referees, Raja Velu, Craig MacKinlay, participants at the 2005 Finite Sample Inference in Finance II, the 2006 European Meetings of the Econometric Society, the 2006 New York Econometrics Camp, and the 2007 International workshop on Computational and Financial Econometrics conferences, for several useful comments. This work was supported by the William Dow Chair in Political Economy (McGill University), the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Bank of Canada (Research Fellowship), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), 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 Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fonds de recherche sur la société et la culture (Québec), the Fonds de recherche sur la nature et les technologies (Québec), the Chaire RBC en innovations financières (Université Laval), and NATECH (Government of Québec). † RBC Chair in Financial Innovations and Département de finance et assurance, Université Laval, CIRANO, and Centre Interuniversitaire sur le risque, les politiques economiques et l’emploi (CIRPEE). Mailing address: Département de finance et assurance, Pavillon Palasis-Prince, Université Laval, Québec, Québec G1K 7P4, Canada. TEL: 1 (418) 656-2926, FAX: 1 (418) 656-2624; e-mail: [email protected]‡ William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Centre interuniversitaire de recherche en économie quantitative (CIREQ). Mailing address: Department of Economics, McGill University, Leacock Building, Room 519, 855 Sherbrooke Street West, Montréal, Québec H3A 2T7, Canada. TEL: (1) 514 398 8879; FAX: (1) 514 398 4938; e-mail: [email protected] . Web page: http://www.jeanmariedufour.com § Canada Research Chair in Environmental and Financial Econometric Analysis (Université Laval), Economics De- partment, Carleton University, CIREQ, and Groupe de recherche en économie de l’énergie, de l’environnement et des ressources naturelles (GREEN), Université Laval. Mailing address: Economics Department, Carleton University, Loeb Building 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada. TEL: 1 (613) 520 2600 ext. 8697; FAX: 1 (613) 520 3906; e-mail: [email protected]
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Finite-sample multivariate tests of asset pricing models with
coskewness∗
Marie-Claude Beaulieu†
Université Laval
Jean-Marie Dufour‡
McGill University
Lynda Khalaf§
Carleton University
First version: July 2002
Revised: July 2002, April 2004, April 2005, September 2007
This version: February 2008
Compiled: March 16, 2008, 9:44pm
This paper is forthcoming in theComputational Statistics and Data Analysis.
∗ The authors thank the Editor David Belsley, two anonymous referees, Raja Velu, Craig MacKinlay, participantsat the 2005 Finite Sample Inference in Finance II, the 2006 European Meetings of the Econometric Society, the 2006New York Econometrics Camp, and the 2007 International workshop on Computational and Financial Econometricsconferences, for several useful comments. This work was supported by the William Dow Chair in Political Economy(McGill University), the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Bankof Canada (Research Fellowship), a Guggenheim Fellowship,a Konrad-Adenauer Fellowship (Alexander-von-HumboldtFoundation, Germany), the Institut de finance mathématiquede Montréal (IFM2), the Canadian Network of Centres ofExcellence [program onMathematics of Information Technology and Complex Systems(MITACS)], the Natural Sciencesand Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fondsde recherche sur la société et la culture (Québec), the Fondsde recherche sur la nature et les technologies (Québec), theChaire RBC en innovations financières (Université Laval), and NATECH (Government of Québec).
† RBC Chair in Financial Innovations and Département de finance et assurance, Université Laval, CIRANO, andCentre Interuniversitaire sur le risque, les politiques economiques et l’emploi (CIRPEE). Mailing address: Départementde finance et assurance, Pavillon Palasis-Prince, Université Laval, Québec, Québec G1K 7P4, Canada. TEL: 1 (418)656-2926, FAX: 1 (418) 656-2624; e-mail: [email protected]
‡ William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse desorganisations (CIRANO), and Centre interuniversitaire derecherche en économie quantitative (CIREQ). Mailing address:Department of Economics, McGill University, Leacock Building, Room 519, 855 Sherbrooke Street West, Montréal,Québec H3A 2T7, Canada. TEL: (1) 514 398 8879; FAX: (1) 514 3984938; e-mail: [email protected] . Webpage: http://www.jeanmariedufour.com
§ Canada Research Chair in Environmental and Financial Econometric Analysis (Université Laval), Economics De-partment, Carleton University, CIREQ, and Groupe de recherche en économie de l’énergie, de l’environnement et desressources naturelles (GREEN), Université Laval. Mailingaddress: Economics Department, Carleton University, LoebBuilding 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada. TEL: 1 (613) 520 2600 ext. 8697; FAX: 1 (613)520 3906; e-mail: [email protected]
ABSTRACT
Finite-sample inference methods are proposed for asset pricing models with unobservable risk-free
rates and coskewness, specifically, for thequadratic market model(QMM) which incorporates the
effect of asymmetry of return distribution on asset valuation. In this context, exact tests are ap-
pealing for several reasons: (i) the increasing popularityof such models in finance, (ii) the fact that
traditional market models (which assume that asset returnsmove proportionally to the market) have
not fared well in empirical tests, (iii) finite-sample testsfor the QMM are unavailable even with
Gaussian errors. Empirical models are considered where theprocedure 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. Exact versions of these tests are ob-
tained, allowing for non-normality of fundamentals. A simulation study documents the size and
power properties of asymptotic and finite-sample tests. Empirical results with well known data sets
reveal temporal instabilities over the full sampling period, namely 1961-2000, though tests fail to
reject the QMM restrictions over 5-year subperiods.
Note – Numbers shown are empirical rejections for proposed tests ofHQ [hypothesis (2.5) in the context of (2.1)]. Thestatistic considered is the quasi-LR statistic (3.19); associatedp-values rely on, respectively, the asymptoticχ2(n − 2)distribution and the LMCp-values assuming the error distribution is known, which corresponds to a parametric bootstrap.Columns (1) - (8) refer to the choices for the parametersγ andθ underlying the various simulation designs considered.These parameters are fixed, in turn, to their QMLE counterparts based on the data set analyzed in section 8, over eachof the eight 5-year subperiods under study, so column (1) refers to parameters estimated using the 1961-65 subsample,column (2) to the 1966-70 subsample, etc.
null hypothesis. Bootstrapping reduces over-rejections so tests based on LMCp-values deviate only
moderately from their nominal size: rejection probabilities range from 2.9% to 8.7% with normal
errors and from 2.9% to 10.9% with Student-t errors.
Results of the power study reported in Table 2 show that the tests have a good power perfor-
mance. Observe that empirical rejections associated with∆ 6= 0 in columns (1) and (4) of Table
2 convey a misleading assessment of power, since the underlying χ2-based tests are severely over-
sized. In such cases, a size-correction scheme is required;for instance, one may compute an artificial
size-correct cut-off point from the quantiles of the simulated statistics conditional on each design.
Since the bootstrap-type correction seems to work, at leastlocally, for the design under study, we
prefer to analyze the bootstrap version of the tests rather than resorting to another artificial size
correction. Indeed, whereas bootstrap-based size corrections are empirically applicable, a local cor-
rection corresponds to a practically infeasible test. Though simulation results generally depend on
the designs considered, the following findings summarized next are worth noting.
1. Estimation costs for the degrees-of-freedom parameter with Student-t errors are unnoticeable.
Indeed, the empirical rejections based onBMC p-values are identical whetherκ is treated
as a known or as an unknown scalar. The bootstrap typeLMC tests are affected albeit
Note – Numbers shown are empirical rejections for proposed tests ofHQ [hypothesis (2.5) in the context of (2.1)]. Thenull hypothesis corresponds to∆ = 0. The statistic considered is the quasi-LR statistics [refer to (3.19)]; associatedp-values rely on, respectively, the asymptoticχ2(n − 2) distribution [columns (1) and (4)], the BMC and LMCp-valuesassuming the error distribution is known [columns (2,3) and(5,6)], and the BMC and LMCp-values imposing multivariatet(κ) errors with unknownκ [columns (7,8)]; the latterp-values are the largest over the degrees-of-freedom parameter κ
within the specified search set.
2. Size-correct tests seem to perform well for the considered alternative [see (7.2)], which fo-
cuses on homogenous deviations fromHQ. Such alternatives may be harder to detect relative
to the nonhomogeneous case; our findings thus provide a realistic appraisal of power given a
possibly less favorable (though empirically relevant) scenario.
3. Tests based onLMC p-values outperform bound tests based onBMC cutoffs. We observe
power differences averaging around 20% with normal errors,38% with Student-t errors and
known degrees-of-freedom, and 37% with Student-t errors and unknown degrees-of-freedom.
TheBMC test is however not utterly conservative. Taken collectively, size and power rank-
ings emerging from this study illustrate the reliability ofour proposed test strategy which
relies on theBMC in conjunction with theLMC p-values.
4. Controlling for nuisance parameter effects, it seems that kurtosis in the data reduces power.
Two issues need to be raised in this regard. First, recall that both LMC and BMC p-
values are obtained using the Gaussian QMLE estimates ofγ and θ and bothLMC and
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BMC procedures are not invariant to the latter. In addition, thetest statistic is based on
Gaussian-QMLE, although we have corrected its critical region for departures from normal-
ity. Gaussian-QMLE coincides with least squares in this model, and least-squares-based sta-
tistics are valid (although possibly not optimal) in non-normal settings, at least in principle.
We view our results as a motivation for research on finite-sample robust test procedures in
MLR models.
8. Empirical analysis
For our empirical analysis, we use Fama and French’s data base. We produce results for monthly
returns of 25 value-weighted portfolios from 1961-2000. The portfolios which 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 years are the New
York Stock Exchange (NYSE) market equity quintiles at the end of June of years. The ratio of
book equity to market equity for June of years is the book equity for the last fiscal year end ins−1
divided by market equity for December of years− 1. The ratio of book equity to market equity are
NYSE quintiles. The portfolios for July of years to June of years + 1 include all NYSE, AMEX,
NASDAQ stocks for which we have market equity data for December of years−1 and June of year
s, and (positive) book equity data fors − 1. All MC tests where applied with 999 replications, and
multivariate normal and multivariate Student-t errors; formally, as in (2.3) and (2.4) respectively.
Table 3 reports tests ofHQ [hypothesis (2.5) in the context of (2.1)] and ofHGQMM [hypothesis
(2.14) in the context of (2.1)], over intervals of 5 years andover the whole sample. Subperiod
analysis is usual in this literature [seee.g. the surveys of Black (1993) or Fama and French (2004)],
and is mainly motivated by structural stability arguments.Our previous and ongoing work on related
asset pricing applications [Beaulieu, Dufour and Khalaf (2005, 2006, 2007), Dufour, Khalaf and
Beaulieu (2008)] have revealed significant temporal instabilities which support subperiod analysis
even in conditional models which allow for time varyingbetas.
To validate our statistical setting, companion diagnostictests are run and are reported in Table
4. These include: (i) goodness of fit tests associated with distributional hypotheses (2.3) and (2.4);
Note – Columns (1)-(5) pertain to tests ofHQ [hypothesis (2.5) in the context of (2.1)]; columns (6)-(10) pertain to tests ofHGQMM [hypothesis (2.14) in the context of (2.1)]. Numbers shown arep-values, associated with the quasi-LR statistics[refer to (3.19)], relying on, respectively, the asymptotic χ2(n−2) distribution [columns (1) and (6)], the Gaussian basedBMC and LMC p-values [columns (2)-(3) and (7)-(8)], and the BMC and LMCp-values imposing multivariatet(κ)errors [columns (4)-(5) and (9)-(10)]. MCp-values fort(κ) errors are the largest over the degrees-of-freedom parameterκ within the specified confidence sets; the latter is reported in column 6 of Table 4. January and October 1987 returns areexcluded from the dataset.
(ii) tests for departure from the maintained errori.i.d. hypothesis, and (iii) tests for exogeneity of
the market factors.
The goodness-of-fit tests rely on the multivariate skewness-kurtosis criteria described in section
5. For the normal distribution, we apply the pivotal MC procedure to the omnibus statistic:
MN =T
6SK+
T [KU−n(n + 2)]2
8n(n + 2). (8.1)
For the Student-t distribution, we report the confidence set for the degrees-of-freedom parameter
which inverts the combined skewness-kurtosis statistic (5.7).
Serial dependence tests [from Dufour et al. (2008) and Beaulieu et al. (2007)] are summarized
here for convenience. In particular, we apply the LM-GARCH test statistic [Engle (1982)] and
the variance ratio statistic which assesses linear serial dependence [Lo and MacKinlay (1988)], to
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standardized residuals, namelyWit, the elements of the matrix
W = U S−1
U, (8.2)
whereSU is the Cholesky factor ofU ′U . So the modified GARCH test statistic for equationi,
denotedEi, is given byT× (the coefficient of determination in the regression of the equation’s
squared OLS residualsW 2it on a constant andW 2
(t−j),i , j = 1, . . . , q) whereq is the ARCH order
against which the test is designed. The modified variance ratio is given by:
˜V Ri = 1 + 2K
∑
j=1
(
1 −j
K
)
ρij , ρij =
∑Tt=j+1 WitWi,t−j
∑Tt=1 W 2
ti
. (8.3)
12 lags are used for both procedures. We combine inference across equation via the joint statistics:
E = 1 − min1≤i≤n
[
p(Ei)]
, ˜V R = 1 − min1≤i≤n
[
p( ˜V Ri)]
, (8.4)
wherep(Ei) andp( ˜V Ri) refer top-values, obtained using theχ2(q) andN[
1, 2(2K − 1)(K −
1)/(3K)]
respectively. In Dufour et al. (2008), we show that under (2.2), W has a distribution
which depends only onκ, so the MC test technique can be applied to obtain a size correct p-value
for E and ˜V R. To deal with an unknownκ, we apply an MMC test procedure following the same
technique proposed for tests onHQ. Specifically, we use the same confidence set forκ, of level
(1 − α1); we maximize thep-value function associated withE and ˜V R over all values ofκ in the
latter confidence set; we then refer the latter maximalp-value toα2 whereα = α1 + α2. Power
properties of these tests are analyzed in Dufour et al. (2008) and suggest a good performance for
sample sizes compatible with our subperiod analysis.
We also apply the Wu-Hausman test to assess the potential endogeneity of our regressors. It
consists in appending, to each equation, the residuals froma first stage regression of returns on a
constant and the instruments, and testing for the exclusionof these residuals using the usual OLS
basedF-statistic [see Hausman (1978), Dufour (1987)]. This test is run, in turn, for each equation,
with one lag ofRM , R2M andRi, i = 1, 25 as instruments. Numbers shown are the minimum
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p-values over all equations. The usualF-basedp-value is computed for the normal case; for the
Student-t, we compute MMCp-values, as follows. In each equation, and ignoring contemporaneous
correlation of the error term, theF-statistic in question is location-scale invariant and caneasily
be simulated to derive a MCp-value given draws from a Student-t distribution, conditional on
its degrees-of-freedom. We maximize thep-value so obtained overκ in the same confidence set
used for all other tests as described above; we then refer thelatter maximalp-value toα2 where
α = α1 + α2. For presentation clarity, we report the minimump-value in each case, over all
equations.
For all confidence set based MMC tests under the Student-t hypothesis, we considerα1 = 2.5%
so, in interpreting thep-values reported in following tables for the Student-t case,α1 must be
subtracted from the adopted significance level; for instance, to obtain a5% test, reportedp-values
should be referred to 2.5% as a cut-off.
From Table 3, we see that, when assessed using the whole sample, bothHQ andHGQMM are
soundly rejected, using asymptotic or MCp-values, the confidence sets on the degrees-of-freedom
parameter is quite tight and suggests high kurtosis, and normality is definitely rejected. Unfortu-
nately, the diagnostic tests (Table 4) reveal significant departures from the statistical foundations
underlying the latter tests (even when allowing for non-normal errors); temporal instabilities thus
cast doubt on the full sample analysis.
Results over subperiods can be summarized as follows. Multivariate normality is rejected in
many subperiods and provides us with a reason to investigatewhether test results shown under
multivariate normality are still prevalent once we use Student-t distributions. HQ is rejected at
the 5% level in five subperiods out of eight using asymptoticp-values. Using finite-sample tests
under multivariate normality reveals thatHQ is rejected at the5% level in only one subperiod,
namely 1976-1980. The LMCp-value confirms all these non-rejections but one. Using the same
approach under the multivariate Student-t distribution leads to the same conclusion.HGQMM is not
rejected in any subperiod allowing fort-errors, although the normal LMCp-value is less than5% for
1986-90, and asymptoticp-values are highly significant for three subperiods spanning 1981-1995.
Diagnostic tests allowing fort errors reveal significant (at the5% level) departures from thei.i.d.
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Table 4. Multivariate diagnostics
Time Dependence Goodness-of-fit Exogeneity
E ˜V R MN CS(κ) Wu-Hausman
(1) (2) (3) (4) (5) (6) (7) (8)
Sample Normal Student-t Normal Student-t Normal Student-t Normal Student-t