07350015%2E2014%2E887015
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Qiwei Yao
To cite this article: Qiwei Yao (2014) Comment, Journal of Business & Economic Statistics, 32:2,
201-201, DOI: 10.1080/07350015.2014.887015
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887015
Published online: 16 May 2014.
Submit your article to this journal
Article views: 57
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http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RA JA ALI HA JI
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Yao: Comment
201
xt
and σt (θxt0 )η . Thus, the difference
where yt is between σt (θ)η
|ST (ω, α, η) − ST (ω0 , α0 , η)| can be made asymptotically arbitrarily small, for α sufficiently close to α0 . In view of the
consistency of αˆ T , the estimator ηˆ f should thus have the same
asymptotic behavior as
T
1
ǫt
∗
− log η.
ηˆ f = arg max LT (θ 0 , η) = arg max
log f
T
η
η>0
η>0
t=1
Thus, the inconsistency of ωˆ T 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 condition γ < 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,” Econometric Theory, 20, 1203–1226. [200]
Comment
Qiwei YAO
Department 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 statistical 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 maximum likelihood estimation is based on non-Gaussian innovation
distributions (such as Student’s t). It provides more efficient estimation 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 likelihood 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 that f = g in practice, should a Gaussian likelihood be used in the event that the estimated value of ηt is 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 standard 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]
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887015
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Comment
Qiwei Yao
To cite this article: Qiwei Yao (2014) Comment, Journal of Business & Economic Statistics, 32:2,
201-201, DOI: 10.1080/07350015.2014.887015
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887015
Published online: 16 May 2014.
Submit your article to this journal
Article views: 57
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RA JA ALI HA JI
TANJUNGPINANG, KEPULAUAN RIAU]
Date: 11 January 2016, At: 20:45
Downloaded by [Universitas Maritim Raja Ali Haji], [UNIVERSITAS MARITIM RAJA ALI HAJI TANJUNGPINANG, KEPULAUAN RIAU] at 20:45 11 January 2016
Yao: Comment
201
xt
and σt (θxt0 )η . Thus, the difference
where yt is between σt (θ)η
|ST (ω, α, η) − ST (ω0 , α0 , η)| can be made asymptotically arbitrarily small, for α sufficiently close to α0 . In view of the
consistency of αˆ T , the estimator ηˆ f should thus have the same
asymptotic behavior as
T
1
ǫt
∗
− log η.
ηˆ f = arg max LT (θ 0 , η) = arg max
log f
T
η
η>0
η>0
t=1
Thus, the inconsistency of ωˆ T 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 condition γ < 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,” Econometric Theory, 20, 1203–1226. [200]
Comment
Qiwei YAO
Department 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 statistical 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 maximum likelihood estimation is based on non-Gaussian innovation
distributions (such as Student’s t). It provides more efficient estimation 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 likelihood 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 that f = g in practice, should a Gaussian likelihood be used in the event that the estimated value of ηt is 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 standard 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]
© 2014 American Statistical Association
Journal of Business & Economic Statistics
April 2014, Vol. 32, No. 2
DOI: 10.1080/07350015.2014.887015