Dynamic copula quantile regressions and tail area dynamic dependence in Forex markets

Abstract

We introduce a general approach to nonlinear quantile regression modelling based on the copula function that defines the dependency structure between the variables of interest. Hence, we extend Koenker and Bassett's (1978. Regression quantiles. Econometrica, 46, no. 1: 33–50.) original statement of the quantile regression problem by determining a distribution for the dependent variable Y conditional on the regressors X, and hence the specification of the quantile regression functions. The approach exploits the fact that the joint distribution function can be split into two parts: the marginals and the dependence function (or copula). We then deduce the form of the (invariably nonlinear) conditional quantile relationship implied by the copula. This can be achieved with arbitrary distributions assumed for the marginals. Some properties of the copula-based quantiles or c-quantiles are derived. Finally, we examine the conditional quantile dependency in the foreign exchange market and compare our quantile approach with standard tail area dependency measures.

Keywords:

• Copula
• Quantile
• Regression
• dependence
• foreign exchange markets

Acknowledgements

First version January 2001, we would like to thank participants at the NSF/NBER Time Series Conference 2002 at Penn, the CIRANO Workshop in Financial Econometrics in Montreal, October 2002 and seminars at Nuffield College, Oxford, University of Melbourne, University of Western Australia, Australian National University, Cambridge University, Warwick University and the Bank of England for their comments on an earlier version of this paper; in particular Andrew Patton, Neil Shephard and Adrian Pagan.

Notes

Koenker and Bassett discuss properties of their estimator, especially through the following theorem:

Theorem 1 Let . Then, the following properties hold:

1.
2.	for ,
3.	with Γ non-singular (k×k) matrix.

Note that C ₁ has to be partially invertible in its second argument. If it is not analytically invertible, a numerical root-finding procedure can be used.

Again for simlicity, we start by examining the dependence between the level of the exchange rates while recognizing that on the basis of some statistical criteria exchange rate levels may appear to be nonstationary. However, it is in principle clear that exchange rates cannot follow a stochastic process with an infinite variance which can also take negative values.

It is natural that we find contemporaneous dependence between the exchange rates given the presence of triangular arbitrage relationships in the market.

The Joe–Clayton copula was preferred by the data in AIC comparisons with several alternative copulae including the Gaussian copula.

We are grateful to Jan Heffernan for making the SPLUS code for computing these figures publically available.