Semiparametric Conditional Quantile Estimation Through Copula-Based Multivariate Models

Abstract

We consider a new approach in quantile regression modeling based on the copula function that defines the dependence structure between the variables of interest. The key idea of this approach is to rewrite the characterization of a regression quantile in terms of a copula and marginal distributions. After the copula and the marginal distributions are estimated, the new estimator is obtained as the weighted quantile of the response variable in the model. The proposed conditional estimator has three main advantages: it applies to both iid and time series data, it is automatically monotonic across quantiles, and, unlike other copula-based methods, it can be directly applied to the multiple covariates case without introducing any extra complications. We show the asymptotic properties of our estimator when the copula is estimated by maximizing the pseudo log-likelihood and the margins are estimated nonparametrically including the case where the copula family is misspecified. We also present the finite sample performance of the estimator and illustrate the usefulness of our proposal by an application to the historical volatilities of Google and Yahoo companies.

Semiparametric copula quantile regression for complete or censored data

“…Assumption (C1) is standard in the context of quantile regression estimation. As for condition (C2), this is similar to assumption (C3)-(i) in Noh et al (2015) for the simplified case with no censoring, with an additional requirement on the conditional censoring probability that is resulting from the initial transformation of synthetic observations. Assumption (C3) is likewise emanating from the handling of censoring through these observations, and is rather usual in survival analysis.…”

“…, n, from (T, X). In this context, following the definition of a copula function, Noh et al (2015) noted that the conditional quantile function of T Figure 1: Copula-based quantile regression estimates of the minimalistic example. The Gaussian, Gumbel and Frank copulas are used for parametric copula estimation in (a), while (b) depicts the regression fit resulting from a nonparametric estimation of the copula density using the procedure of Geenens et al (2014) (more details given below).…”

“…• DGP A: (F T (T ), F 1 (X 1 ), F 2 (X 2 )) ∼ Gaussian copula with parameters (ρ T 1 , ρ T 2 , ρ 12 ) = (0.3, 0.9, 0.5). Given standard uniform marginal distributions for all three variables, the resulting true quantile regression may be calculated as m τ (x) = Φp −0.2Φ −1 (x 1 ) + Φ −1 (x 2 ) + 0.4Φ −1 (τ )q (see Noh et al (2015)). To include censoring, we introduce the variable C ∼ U [0, M ], where the parameter M is computed in order to obtain the desired average censoring proportion (M = 5/3 for 30% and M = 1 for 50%).…”

“…By taking advantage of copula modelling, we intend to provide a new class of estimators that would allow practitioners to analyse, in a flexible way, multidimensional survival data. Actually, our methodology is, in essence, an extension of the recent work of Noh et al (2013) and Noh et al (2015), as the central idea is to express the conditional quantile function in terms of an appropriate copula density and marginal distributions. In their original paper, Noh et al (2013) suggested subsequently to leave the marginal distributions unspecified while assuming a parametric model for the copula.…”

“…In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al (2013) andNoh et al (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable.…”