Is a Normal Copula the Right Copula?

ABSTRACT

We derive computationally simple and intuitive expressions for score tests of Gaussian copulas against generalized hyperbolic alternatives, including symmetric and asymmetric Student t, and many other examples. We decompose our tests into third and fourth moment components, and obtain one-sided Likelihood Ratio analogs, whose standard asymptotic distribution we provide. Our Monte Carlo exercises confirm the reliable size of parametric bootstrap versions of our tests, and their substantial power gains over alternative procedures. In an empirical application to CRSP stocks, we find that short-term reversals and momentum effects are better captured by non-Gaussian copulas, whose parameters we estimate by indirect inference. Supplementary materials for this article are available online.

KEY WORDS:

• Indirect inference
• Kuhn–Tucker test
• Momentum strategies
• Nonlinear dependence
• Short-term reversals
• Supremum test.

SUPPLEMENTARY MATERIALS

The Supplemental Appendix contains detailed expressions for the score, Hessian and information matrix of the econometric model, together with a useful reparametrization of the GH distribution, local power comparisons, computational details, and additional Monte Carlo and empirical results.

ACKNOWLEDGMENTS

The authors thank Manuel Arellano, Yanqin Fan, Pedro García Ares, Pascal Lavergne, Javier Mencía, Andrew Patton, and Javier Perote for their comments, advice, and suggestions. The authors have also benefitted from the feedback provided by seminar participants at Alicante, CEMFI, Duke, Erasmus, ESSEC, Exeter, Indiana, Liverpool, Murcia, Surrey, Torcuato Di Tella, UCL, UPF and Yale, as well as audiences at the TSE Financial Econometrics Conference (May 2013), the XXI Finance Forum (Segovia, November 2013), the XXXVIII Symposium on Economic Analysis (Santander, December 2013), the LATAM Econometrics Workshop (São Paulo, December 2013), the IEEM Meeting (Montevideo, December 2013), the 68th ESEM (Toulouse, August 2014), the 11th World Congress of the Econometric Society (Montreal, August 2015, the New Methods for the Empirical Analysis of Financial Markets SanFI conference (Comillas, June 2017), and the EcoSta (Hong Kong, June 2017). The authors thank Difang Huang and Zaici Li for their help in collecting the data and especially Xinyue Bei and Julio Crego for their excellent research assistance. The input of a co-editor and two anonymous referees has also greatly improved the article. Of course, the usual caveat applies.

Funding

Financial support from the Spanish Ministries of Science & Innovation and Economy & Competitiveness through grants ECO 2011-26342 and 2014-59262, respectively, and the Santander - CEMFI Research Chair on Finance is gratefully acknowledged.