## Bernoulli

• Bernoulli
• Volume 19, Number 2 (2013), 462-491.

### Parameter estimation for pair-copula constructions

Ingrid Hobæk Haff

#### Abstract

We explore various estimators for the parameters of a pair-copula construction (PCC), among those the stepwise semiparametric (SSP) estimator, designed for this dependence structure. We present its asymptotic properties, as well as the estimation algorithm for the two most common types of PCCs. Compared to the considered alternatives, that is, maximum likelihood, inference functions for margins and semiparametric estimation, SSP is in general asymptotically less efficient. As we show in a few examples, this loss of efficiency may however be rather low. Furthermore, SSP is semiparametrically efficient for the Gaussian copula. More importantly, it is computationally tractable even in high dimensions, as opposed to its competitors. In any case, SSP may provide start values, required by the other estimators. It is also well suited for selecting the pair-copulae of a PCC for a given data set.

#### Article information

Source
Bernoulli, Volume 19, Number 2 (2013), 462-491.

Dates
First available in Project Euclid: 13 March 2013

https://projecteuclid.org/euclid.bj/1363192035

Digital Object Identifier
doi:10.3150/12-BEJ413

Mathematical Reviews number (MathSciNet)
MR3037161

Zentralblatt MATH identifier
06168760

#### Citation

Hobæk Haff, Ingrid. Parameter estimation for pair-copula constructions. Bernoulli 19 (2013), no. 2, 462--491. doi:10.3150/12-BEJ413. https://projecteuclid.org/euclid.bj/1363192035

#### References

• [1] Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance Math. Econom. 44 182–198.
• [2] Bedford, T. and Cooke, R.M. (2001). Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32 245–268.
• [3] Bedford, T. and Cooke, R.M. (2002). Vines – a new graphical model for dependent random variables. Ann. Statist. 30 1031–1068.
• [4] Berg, D. and Aas, K. (2009). Models for construction of multivariate dependence. European Journal of Finance 15 639–659.
• [5] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. J. Econometrics 130 307–335.
• [6] Chollete, L., Heinen, A. and Valdesogo, A. (2009). Modeling international financial returns with a multivariate regime switching copula. Journal of Financial Econometrics 7 437–480.
• [7] Claeskens, G. and Hjort, N.L. (2008). Model Selection and Model Averaging. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge Univ. Press.
• [8] Clayton, D.G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65 141–151.
• [9] Czado, C. and Min, A. (2010). Bayesian inference for multivariate copulas using pair-copula constructions. Journal of Financial Econometrics 8 511–546.
• [10] Czado, C., Min, A., Baumann, T. and Dakovic, R. (2009). Pair-copula constructions for modeling exchange rate dependence. Unpublished manuscript.
• [11] Czado, C., Schepsmeier, U. and Min, A. (2010). Maximum likelihood estimation of mixed c-vines with application to exchange rates. Unpublished manuscript.
• [12] Dissmann, J., Brechmann, E., Czado, C. and Kurowicka, K. (2011). Selecting and estimating regular vine copulae and application to financial returns. Unpublished manuscript.
• [13] Fischer, M., Köck, C., Schlüter, S. and Weigert, F. (2007). Multivariate copula models at work: Outperforming the “desert island copula”? Technical Report 79, Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie.
• [14] Genest, C. (1987). Frank’s family of bivariate distributions. Biometrika 74 549–555.
• [15] Genest, C., Gerber, H.U., Goovaerts, M.J. and Laeven, R.J.A. (2009). Editorial to the special issue on modeling and measurement of multivariate risk in insurance and finance. Insurance Math. Econom. 44 143–145.
• [16] Genest, C., Ghoudi, K. and Rivest, L.P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82 543–552.
• [17] Genest, C. and Rivest, L.P. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88 1034–1043.
• [18] Hall, P. and Yao, Q. (2005). Approximating conditional distribution functions using dimension reduction. Ann. Statist. 33 1404–1421.
• [19] Heinen, A. and Valdesogo, A. (2009). Asymmetric capm dependence for large dimensions: the canonical vine autoregressive model. CORE Discussion Paper (2009/69).
• [20] Hobæk Haff, I. (2012). Supplement to “Parameter estimation for pair-copula constructions.” DOI:10.3150/12-BEJ413SUPPA, DOI:10.3150/12-BEJ413SUPPB.
• [21] Hobæk Haff, I., Aas, K. and Frigessi, A. (2010). On the simplified pair-copula construction – simply useful or too simplistic? J. Multivariate Anal. 101 1296–1310.
• [22] Joe, H. (1996). Distributions with Fixed Marginals and Related Topics, Chapter Families of M-Variate Distributions with Given Margins and $m(m-1)/2$ Dependence Parameters. Hayward, CA: IMS.
• [23] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. London: Chapman & Hall.
• [24] Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. J. Multivariate Anal. 94 401–419.
• [25] Joe, H., Li, H. and Nikoloulopoulos, A.K. (2010). Tail dependence functions and vine copulas. J. Multivariate Anal. 101 252–270.
• [26] Joe, H. and Xu, J. (1996). The estimation method of inference functions for margins for multivariate models. Technical Report 166, Univ. British Columbia, Dept. Statistics.
• [27] Kim, G., Silvapulle, M.J. and Silvapulle, P. (2007). Comparison of semiparametric and parametric methods for estimating copulas. Comput. Statist. Data Anal. 51 2836–2850.
• [28] Klaassen, C.A.J. and Wellner, J.A. (1997). Efficient estimation in the bivariate normal copula model: Normal margins are least favourable. Bernoulli 3 55–77.
• [29] Kolbjørnsen, O. and Stien, M. (2008). D-vine creation of non-Gaussian random field. In Procedings of the Eight International Geostatistics Congress 399–408. Santiago, Chile: GECAMIN Ltd.
• [30] Kurowicka, D. and Cooke, R. (2006). Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley Series in Probability and Statistics. Chichester: Wiley.
• [31] Lehmann, E.L. (2004). Elements of Large-Sample Theory. Springer Texts in Statistics. New York: Springer.
• [32] McNeil, A.J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Series in Finance. Princeton, NJ: Princeton Univ. Press.
• [33] Min, A. and Czado, C. (2010). Scomdy models based on pair-copula constructions with application to exchange rates. Unpublished manuscript.
• [34] Oakes, D. (1982). A model for association in bivariate survival data. J. Roy. Statist. Soc. Ser. B 44 414–422.
• [35] Schirmacher, D. and Schirmacher, E. (2008). Multivariate dependence modeling using pair-copulas. Presented at The 2008 ERM Symposium, Chicago.
• [36] Shih, J.H. and Louis, T.A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51 1384–1399.
• [37] Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229–231.
• [38] Stacy, E.W. (1962). A generalization of the gamma distribution. Ann. Math. Statist. 33 1187–1192.
• [39] Stute, W. (1986). On almost sure convergence of conditional empirical distribution functions. Ann. Probab. 14 891–901.
• [40] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357–375.
• [41] Tsukahara, H. (2011). Personal communication.