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Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Christian Francq & Jean-Michel Zakoïan
To cite this article: Christian Francq & Jean-Michel Zakoïan (2014) Comment, Journal of Business & Economic Statistics, 32:2, 198-201, DOI: 10.1080/07350015.2013.879829
To link to this article: http://dx.doi.org/10.1080/07350015.2013.879829
Published online: 16 May 2014.
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Comment
Christian F
RANCQand Jean-Michel Z
AKO¨
IANCREST, 92245 Malakoff Cedex, France; Universit ´e Lille 3, 59653 Villeneuve d’Ascq Cedex, France (christian.francq@univ-lille3.fr; zakoian@ensae.fr)
In this interesting article, the authors developed and studied a non-Gaussian quasi-maximum likelihood (NGQML) method for estimating GARCH(p, q) models. The standard Gaussian QML estimator for generalized autoregressive conditional het-eroscedasticity (GARCH) is well known to be consistent and asymptotically Gaussian under mild conditions but it may lack accuracy when the underlying innovations density is far from the Gaussian. This article proposes to use a three-step method in which the unknown density is approximated via a rescaling of a given likelihood density. The GQML is used in a first step to provide approximations of the innovations, which are used in a second step to appropriately fit the scaling factor. The third step involves a new QML optimization, now using the density emerging from the second step.
In this comment, we would like to discuss two points, one concerning the practical implementation of the method and the comparison with the two-step quasi-maximum likelihood es-timate (hereafter 2QMLE) proposed by Francq, Lepage, and Zako¨ıan (2011), and the other one concerning the behavior of the method when strict stationarity does not hold.
1. PRACTICAL IMPLEMENTATION AND A COMPARISON
To explain the difference between the 2QMLE and the three-step NGQMLE that is proposed here, let us assume a GARCH(1,1) modelxt =vtεt, with
v2t(θ0)=c0+a0xt2−1+b0vt2−1(θ0), θ0=(c0, a0, b0)′. (1)
The 2QMLE is based on a density belonging to the class of the generalized Gaussian (GG) distributions
fβ(x)∝e−|x|β/β, β >0.
The two steps of this method are as follows:
1. Compute the estimator
θ∗ three steps are as follows:
1. Compute the GQML estimator
θT =(cT,aT,bT)=arg min
3. Compute the estimator
θT =arg min
As shown by the authors of the present article, the two esti-matorsθ∗T andθT have the same asymptotic distribution. The
2QMLE is, however, a little bit simpler because it involves only one numerical optimization. Moreover, the finite-sample dis-tributions of the two estimators may be significantly different. To see this, we generated 1000 independent replications of a GARCH(1,1) model (1) in which εt follows the Student dis-tribution with parameterν=4.5 (rescaled in such a way that
Eε21 =1), and we applied the NGQML and 2QML withβ=1.
Table 1shows that for the large sample sizeT =10,000, the 2QMLE and NGQMLE are equivalent, and they are both more accurate than the GQMLE. For small or moderate sample sizes (i.e.,T =500 orT =2000), the 2QMLE and NGQMLE are equivalent for the estimation ofa0, but the root mean square error (RMSE) of estimation ofc0andb0 is clearly smaller for
© 2014American Statistical Association Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2013.879829
Color versions of one or more of the figures in the article can be found online atwww.tandfonline.com/r/jbes.
Table 1. Comparison of three estimators of a GARCH(1,1) model (1), withc0=1,a0=0.05, andb0=0.85 and Student errors
c0 a0 b0
Mean RMSE Mean RMSE Mean RMSE
T =500
GQMLE 1.639 1.808 0.062 0.083 0.773 0.219
NGQMLE 1.681 1.788 0.055 0.042 0.771 0.209
2QMLE 1.625 1.643 0.055 0.043 0.778 0.193
T =2000
GQMLE 1.311 0.945 0.057 0.038 0.813 0.111
NGQMLE 1.212 0.694 0.053 0.019 0.826 0.080
2QMLE 1.197 0.580 0.053 0.019 0.828 0.068
T =10,000
GQMLE 1.066 0.365 0.052 0.014 0.842 0.045
NGQMLE 1.036 0.198 0.051 0.008 0.846 0.025
2QMLE 1.036 0.199 0.051 0.008 0.846 0.025
the 2QMLE than for the NGQMLE. This better finite-sample behavior of the 2QMLE with respect to the NGQMLE can be explained by the fact that the NGQMLE requires an additional optimization. Note in passing that the fact that the NGQMLE can be written as a generalized method of moments (GMM) estimator of the form (12) does not seem to have any practi-cal usefulness since it is obviously much harder to solve the high-dimensional optimization (12) than the two optimizations of Steps 1 and 3. Another potential explanation is that, contrary
to the 2QMLE, the NGQMLE uses the inefficient GQMLE in the first step.
The choice β=1 is arbitrary and efficiency gains can be expected by an adaptive choice of β, or by choosing another densityf in the case of the NGQMLE. Indeed, when the instru-mental densityfβ is used, the asymptotic variance of the two estimators is an increasing function of
Aβ =
μ2β−μ2β β2μ2
β ,
whereμs =E|ε1|sfors >0. A natural method, which is pro-posed in the two articles, consists in choosingβminimizing
ˆ
Aβ= μˆ2β,T −μˆ
2
β,T β2μˆ2
β,T
, Aˆ∗β =μˆ ∗
2β,T −( ˆμ∗β,T)
2
β2( ˆμ∗ β,T)2
for the NGQMLE and 2QMLE, respectively. Here we would like to insist on the fact that the optimization has to be performed on a bounded set. Indeed, Lemma 3.1 in Francq, Lepage, and Zako¨ıan (2011) shows that, almost surely,
ˆ
A∗β→0 asβ → ∞,
and we similarly have ˆAβ →0.Figure 1illustrates this point. For different sample sizes, we computed ˆAβ using Gaussian
innovations for which the optimal asymptotic value isβ =2.If the minimization is performed for a too large set of valuesβ, then the estimated optimal value is always arbitrarily far from
β =2. Thus, if formula (24) had to be used in practice, we
0 10 20 30 40
12345678
β
Aβ T=300
T=500 T=700
Figure 1. Behavior of the empirical estimator ofAβ.
200 Journal of Business & Economic Statistics, April 2014
suggest performing the optimization over a compact set for the values ofβ(and probably also forν).
To conclude our comment on this point, a few lessons can be drawn from our experiments:
• The two estimators clearly outperform the GQML in pres-ence of heavy-tailed errors.
• Within the class of GG QML, obtained by choosing the densities fβ, the 2QMLE only requires one
optimiza-tion and might provide better efficiency in finite sample. Asymptotically, the two methods are equivalent.
• The adaptive choice of β in practice is not obvious and requires a bounded set of possible values.
• The NGQML allows a priori for greater flexibility than the 2QML by permitting the choice of QML densities not belonging to the GG class. Further research, how-ever, is required to show empirically the efficiency gains that can be expected from widening the class of QML densities.
2. RELAXING STRICT STATIONARITY
The assumptions required in this article to establish the asymptotic results are comparable to those used for the theory of the standard GQMLE. One maintained assumption, however, is the strict stationarity of the observed process.
Several authors have studied the asymptotic behavior of the QML when the strict stationarity condition is violated. The seminal results on this topic were obtained by Jensen and Rahbek (2004a,b) and recently completed by Francq and Zako¨ıan (2012), hereafter FZ, and Francq and Zako¨ıan (2013). In financial applications, this situation is obviously not frequent but FZ gave several examples of stock series for which the strict stationarity can be questioned. In the GARCH framework, nonstationarity, in the strict sense, means explosiveness. For the standard GARCH(1,1) in the strictly explosive case, the asymp-totic results can be summarized as follows: (i) the estimators of the parameters α andβ, in the standard parameterization, are strongly consistent and asymptotically Gaussian, with the usual rate of convergence but a modified asymptotic covariance matrix and (ii) the estimator of the intercept is inconsistent (at least under mild additional assumptions).
Whether or not such results extend to the method of this article is an interesting issue that we now want to discuss. In this informal comment, we will focus on the main ideas rather than on detailed proofs. For simplicity, the discussion is limited to the ARCH(1) model. First note that with the parameterization used in this article, namely,
'
xt =σ0vtǫt, t=1,2, . . .
v2t =1+a0xt2−1,
the GQML estimators ofσ0 anda0are both inconsistent when the process is explosive (i.e., whenElog(a0σ02ǫ
2
t)>0). Only a0σ02can be consistently estimated. It is thus more convenient
to consider the standard parameterization
' xt =
√
htǫt, t =1,2, . . . ht =ω0+α0xt2−1,
under the assumption of explosiveness, that is,
γ =Elogα0ǫ2t
the article, the true innovation density is denoted bygand the likelihood function is denoted byf. In this simple model, the three steps of the proposed estimation method are as follows:
1. Define the GQML ofθ0=(ω0, α0) by
with an initial value forǫ2 0.
t almost surely converges
to infinity. Noting that ˜θT =arg minθ∈Q˜T(θ),where ˜QT(θ)=
in the sense that
lim
We have, by a Taylor expansion,
|ST(ω, α, η)−ST(ω0, α0, η)|
where yt is between xt
σt(θ)η and
xt
σt(θ0)η. Thus, the difference
|ST(ω, α, η)−ST(ω0, α0, η)| can be made asymptotically ar-bitrarily small, for α sufficiently close to α0. In view of the consistency of ˆαT, the estimator ˆηf should thus have the same
asymptotic behavior as
ˆ
η∗f =arg max
η>0
LT(θ0, η)=arg max
η>0 1
T T
t=1 logf
ǫt
η
−logη.
Thus, the inconsistency of ˆωT should not impact the asymptotic properties of the second-step estimator. The same analysis can be conducted on the third-step estimator.
To conclude, in the three-step method of this article the strict stationarity conditionγ <0 can probably be removed, at least in the ARCH(1) case, to obtain consistency of the estimator (except for the intercept). An open issue is the asymptotic distribution of this estimator under (2).
3. CONCLUSION
The contribution of the authors is quite welcome, because it highlights the fact that the Gaussian QMLE should not be
routinely used in times series models, in particular when the errors are suspected to have fat tails. Non-Gaussian QMLEs are generally inconsistent, but the authors show how they can nevertheless be used to construct consistent estimators through the clever introduction of a scaling factor. Without diminishing in any sense their work, our remarks are an attempt to enhance its practical aspects and to make connections to related issues that have been worked out recently.
REFERENCES
Francq, C., Lepage, G., and Zako¨ıan, J.-M. (2011), “Two-Stage Non Gaussian QML Estimation of GARCH Models and Testing the Efficiency of the Gaussian QMLE,”Journal of Econometrics, 165, 246–257. [198,199] Francq, C., and Zako¨ıan, J. M. (2012), “Strict Stationarity Testing and Estimation
of Explosive and Stationary GARCH Models,”Econometrica, 80, 821–861. [200]
——— (2013), “Inference in Non Stationary Asymmetric GARCH Models,”
The Annals of Statistics, 41, 1970–1998. [200]
Jensen, S. T., and Rahbek, A. (2004a), “Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case,”Econometrica, 72, 641– 646. [200]
——— (2004b), “Asymptotic Inference for Nonstationary GARCH,” Econo-metric Theory, 20, 1203–1226. [200]
Comment
Qiwei Y
AODepartment of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK (q.yao@lse.ac.uk)
I congratulate the authors for tackling a challenging statis-tical problem with an important financial application, that is, estimating heavy-tailed GARCH models. The significance of the proposed three-step quasi maximum likelihood procedure is two-fold. It rectifies the inconsistency issue when quasi maxi-mum likelihood estimation is based on non-Gaussian innovation distributions (such as Student’st). It provides more efficient es-timation when the innovations are heavy-tailed. As heavy-tailed residuals are common place in empirical modeling for financial returns, one tends to use heavy-tailed distributions to form like-lihood functions. Hence, this article fills in an important gap in the literation on the estimation of GARCH models.
The key to success is the introduction of a scale parameter
ηt, which is cute. Can it be further developed into a “selector”? Since it is rare thatf =gin practice, should a Gaussian likeli-hood be used in the event that the estimated value ofηtis around 1? Perhaps some additional test is required. This could be a valid question as GARCH processes driven by Gaussian innovations can be very heavy-tailed. See, for example, Theorem 8.4.12 of Embrechts, Kl¨uppelberg, and Mikosch(1997).
Another advantage of the proposal is that the estimators for the heteroscedastic parameters enjoy the standard√T
conver-gence rate. Would this be enough to guarantee that the stan-dard parametric bootstrap method is valid for both the testing and the interval estimation for those parameters, avoiding the subsample-resampling method of Hall and Yao (2003)? The size of subsample is a tuning parameter causing extra difficulties in practice.
My final comment is on possible extension of the method to multivariate volatility models, which are practically more relevant and technically more challenging.
REFERENCE
Embrechts, P., Kl¨uppelberg, C., and Mikosch, T. (1997),Modelling Extremal Events, Berlin: Springer. [201]
© 2014American Statistical Association Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2 DOI:10.1080/07350015.2014.887015
