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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:45
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.
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Yao: Comment 201
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
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
