Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T18:03:33.807Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZATIONS OF MULTINORMALITY AND CORRESPONDING TESTS OF FIT, INCLUDING FOR GARCH MODELS

Published online by Cambridge University Press:  22 May 2018

Norbert Henze ,
M. Dolores Jiménez–Gamero  and
Simos G. Meintanis
Norbert Henze
Affiliation:
Institute of Stochastics, Karlsruhe Institute of Technology
M. Dolores Jiménez–Gamero
Affiliation:
University of Seville
Simos G. Meintanis*
Affiliation:
National and Kapodistrian University of Athens and North–West University
*
*Address correspondence to Simos G. Meintanis, Department of Economics, National and Kapodistrian University of Athens, Athens, Greece; e-mail: [email protected] and Unit for Business Mathematics and Informatics, North–West University, Potchefstroom, South Africa.
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted L2-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 3 , June 2019 , pp. 510 - 546
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors thank the anonymous reviewers for their constructive comments. M.D. Jiménez-Gamero was partially supported by MTM2017–89422–P of the Spanish Ministry of Economy, Industry and Competitiveness/ERDF. Simos Meintanis was partially supported by grant Nr. 11699 of the Special Account for Research Grants (EΛKE) of the National and Kapodistrian University of Athens.

References

REFERENCES

Arcones, M. (2007) Two tests for multivariate normality based on the characteristic function. Mathematical Methods of Statistics 16, 177–201.CrossRefGoogle Scholar
Bai, J. & Chen, Z. (2008) Testing multivariate distributions in GARCH models. Journal of Econometrics 143, 19–36.CrossRefGoogle Scholar
Bardet, J.M. & Wintenberger, O. (2009) Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 2730–2759.CrossRefGoogle Scholar
Baringhaus, L. & Henze, N. (1988) A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35, 339–348.CrossRefGoogle Scholar
Baringhaus, L., Ebner, B., & Henze, N. (2017) The limit distribution of weighted L 2-goodness-of fit statistics under fixed alternatives, with applications. Annals of the Institute of Statistical Mathematics 69, 969–995.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1963) On the limit behaviour of extreme order statistics. The Annals of Mathematical Statistics 34, 992–1012.CrossRefGoogle Scholar
Bollerslev, T. (1990) Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505.CrossRefGoogle Scholar
Bonato, M. (2012) Modeling fat tails in stock returns: A multivariate stable-GARCH approach. Computational Statistics 27, 499–521.CrossRefGoogle Scholar
Bosq, D. (2000) Linear Processes in Function Spaces. Springer.CrossRefGoogle Scholar
Chen, Q., Gerlach, R., & Lu, Z. (2012) Bayesian value-at-risk and expected shortfall forecasting via the asymmetric Laplace distribution. Computational Statistics and Data Analysis 56, 3498–3516.CrossRefGoogle Scholar
Comte, F. & Lieberman, O. (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.CrossRefGoogle Scholar
Csörgő, S. (1986) Testing for normality in arbitrary dimension. The Annals of Statistics 14, 708–723.CrossRefGoogle Scholar
Csörgő, S. (1989) Consistency of some tests for multivariate normality. Metrika 36, 107–116.CrossRefGoogle Scholar
Csörgő, S. & Welsh, A.H. (1989) Testing for exponential and Marshall–Olkin distributions. Journal of Statistical Planning and Inference 23, 287–300.CrossRefGoogle Scholar
Dalla, V., Meintanis, S.G., & Bassiakos, Y. (2017) Characteristic function-based inference for GARCH models with heavy-tailed innovations. Communications in Statistics–Simulation and Computation 46, 2733–2755.CrossRefGoogle Scholar
Diamantopoulos, K. & Vrontos, I.D. (2010) A Student-𝑡 full factor multivariate GARCH model. Computational Economics 35, 63–83.CrossRefGoogle Scholar
Eaton, M.L. & Perlman, M.D. (1973) The non-singularity of generalized sample covariance matrices. The Annals of Statistics 1, 710–717.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, Th. (1997) Modelling Extremal Events. Springer.CrossRefGoogle Scholar
Epps, T.W. (1999) Limiting behavior of the ICF test for normality under Gram–Charlier alternatives. Statistics and Probability Letters 42, 175–184.CrossRefGoogle Scholar
Epps, T.W. & Pulley, L.B. (1983) A test for normality based on the empirical characteristic function. Biometrika 70, 723–726.CrossRefGoogle Scholar
Escanciano, J. (2009) On the lack of power of omnibus specification tests. Econometric Theory 25, 162–194.CrossRefGoogle Scholar
Fama, E. (1965) The behavior of stock market prices. Journal of Business 38, 34–105.CrossRefGoogle Scholar
Fang, K.-T., Li, R.-Z., & Liang, J.-J. (1998) A multivariate version of Ghosh’s T 3-plot to detect non-multinormality. Computational Statistics and Data Analysis 28, 371–386.CrossRefGoogle Scholar
Francq, C., Jiménez-Gamero, M.D., & Meintanis, S.G. (2017) Tests for sphericity in multivariate GARCH models. Journal of Econometrics 196, 305–319.CrossRefGoogle Scholar
Francq, C. & Meintanis, S.G. (2016) Fourier-type estimation of the power GARCH model with stable Paretian innovations. Metrika 79, 389–424.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.M. (2010) GARCH Models: Structure, Statistical Inference and Applications. Wiley.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.M. (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory 28, 179–206.CrossRefGoogle Scholar
Ghoudi, K. & Rémillard, B. (2014) Comparison of specification tests for GARCH models. Computational Statistics and Data Analysis 76, 291–300.CrossRefGoogle Scholar
Ghosh, S. (1996) A new graphical tool to detect non-normality. Journal of the Royal Statistical Society Series B 58, 691–702.Google Scholar
Giacomini, R., Politis, D.N., & White, H. (2013) A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29, 567–589.CrossRefGoogle Scholar
Goodman, I.R. & Kotz, S. (1973) Multivariate θ-generalized normal distributions. Journal of Multivariate Analysis 3, 204–219.CrossRefGoogle Scholar
Henze, N. (1997) Extreme smoothing and testing for multivariate normality. Statistics and Probability Letters 35, 203–213.CrossRefGoogle Scholar
Henze, N. (2002) Invariant tests for multivariate normality: A critical review. Statistical Papers 43, 467–506.CrossRefGoogle Scholar
Henze, N. & Koch, S. (2017) On a test of normality based on the empirical moment generating function. Statistical Papers. Available at https://doi.org/10.1007/s00362-017-0923-7.CrossRefGoogle Scholar
Henze, N. & Wagner, T. (1997) A new approach to the BHEP tests for multivariate normality. Journal of Multivariate Analysis 62, 1–23.CrossRefGoogle Scholar
Henze, N. & Zirkler, B. (1990) A class of invariant consistent tests for multivariate normality. Communications in Statistics– Theory and Methods 19, 3595–3617.CrossRefGoogle Scholar
Janssen, A. (2000) Global power functions of goodness of fit tests. The Annals of Statistics 28, 239–253.CrossRefGoogle Scholar
Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.CrossRefGoogle Scholar
Jiménez-Gamero, M.D. (2014) On the empirical characteristic function process of the residuals in GARCH models and applications. Test 23, 409–432.CrossRefGoogle Scholar
Jiménez-Gamero, M.D. & Pardo-Fernández, J.C. (2017) Empirical characteristic function tests for GARCH innovation distribution using multipliers. Journal of Statistical Computation and Simulation 87, 2069–2093.CrossRefGoogle Scholar
Klar, B., Lindner, F., & Meintanis, S.G. (2012) Specification tests for the error distribution in GARCH models. Computational Statistics and Data Analysis 56, 3587–3598.CrossRefGoogle Scholar
Kotz, S., Kozubowski, T.J., & Podgórski, K. (2001) The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhäuser.CrossRefGoogle Scholar
Lee, J., Lee, S., & Park, S. (2014) Maximum entropy test for GARCH models. Statistical Methodology 22, 8–16.CrossRefGoogle Scholar
Lindsay, B., Markatou, M., & Ray, S. (2014) Kernels, degrees of freedom, and power properties of quadratic distance goodness–of–fit statistics. Journal of the American Statistical Association 109, 395–410.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Lukacs, E. (1970) Characteristic Functions. Griffin.Google Scholar
Mandelbrot, B. (1963) The variation of certain speculative prices. Journal of Business 36, 394–419.CrossRefGoogle Scholar
Mardia, K.V. (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530.CrossRefGoogle Scholar
Meintanis, S.G. (2007) A Kolmogorov-Smirnov type test for skew normal distributions based on the empirical moment generating function. Journal of Statistical Planning and Inference 137, 2681–2688.CrossRefGoogle Scholar
Meintanis, S.G. & Hlávka, Z. (2010) Goodness-of-fit test for bivariate and multivariate skew-normal distributions. Scandinavian Journal of Statistics 37, 701–714.CrossRefGoogle Scholar
Móri, T.F., Rohatgi, V.K., & Székely, G.J. (1993) On multivariate skewness and kurtosis. Theory of Probability and its Applications 38, 547–551.CrossRefGoogle Scholar
Nikitin, Y. (1995) Asymptotic Efficiency of Nonparametric Tests. Cambridge University Press.CrossRefGoogle Scholar
Ogata, H. (2013) Estimation of multivariate stable distributions with generalized empirical likelihood. Journal of Econometrics 172, 248–254.CrossRefGoogle Scholar
Pudelko, J. (2005) On a new affine invariant and consistent test for multivariate normality. Probability and Mathematical Statistics 25, 43–54.Google Scholar
Rao, C.R. (1973) Linear Statistical Inference and its Applications. Wiley.CrossRefGoogle Scholar
Shorack, G.R. & Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley.Google Scholar
Spierdijk, L. (2016) Confidence intervals for ARMA-GARCH value-at-risk: The case of heavy tails and skewness. Computational Statistics and Data Analysis 100, 545–559.CrossRefGoogle Scholar
Tenreiro, C. (2009) On the choice of the smoothing parameter for the BHEP goodness–of–fit test. Computational Statistics and Data Analysis 53, 1038–1053.CrossRefGoogle Scholar
Tenreiro, C. (2011) An affine invariant multiple test procedure for assessing multivariate normality. Computational Statistics and Data Analysis 55, 1980–1992.CrossRefGoogle Scholar
Trindade, A.A. & Zhu, Y. (2007) Approximating the distributions of estimators of financial risk under an asymmetric Laplace law. Computational Statistics and Data Analysis 51, 3433–3447.CrossRefGoogle Scholar
Tsay, R.S. (2006) Multivariate volatility models. In: IMS Lecture Notes–Monograph Series, vol. 52, Time Series and Related Topics, pp. 210–222. Institute of Mathematic Statistics.Google Scholar
Ushakov, N.G. (1999) Selected Topics in Characteristic Functions. VSP.CrossRefGoogle Scholar
Volkmer, H. (2014) A characterization of the normal distribution. Journal of Statistical Theory and Applications 13, 83–85.CrossRefGoogle Scholar
Zghoul, A.A. (2010) A goodness-of-fit test for normality based on the empirical moment generating function. Communications in Statistics–Simulation and Computation 39, 1292–1304.CrossRefGoogle Scholar
Zhu, D. & Zinde-Walsh, V. (2009) Properties and estimation of asymmetric exponential power distribution. Journal of Econometrics 148, 86–99.CrossRefGoogle Scholar