Specification tests of parametric dynamic conditional quantiles

Abstract

This article proposes omnibus speci…cation tests of parametric dynamic quantile regression models. Contrary to the existing procedures, we allow for a ‡exible and general speci…cation framework where a possibly continuum of quantiles are simultaneously speci…ed. This is the case for many econometric applications for both time series and cross section data which require a global diagnostic tool. We study the asymptotic distribution of the test statistics under fairly weak conditions on the serial dependence in the underlying data generating process. It turns out that the asymptotic null distribution depends on the data generating process and the hypothesized model. We propose a subsampling procedure for approximating the asymptotic critical values of the tests. An appealing property of the proposed tests is that they do not require estimation of the non-parametric (conditional) sparsity function. A Monte Carlo study compares the proposed tests and shows that the asymptotic results provide good approximations for small sample sizes. Finally, an application to some European stock indexes provides evidence that our methodology is a powerful and ‡exible alternative to standard backtesting procedures in evaluating market risk by using information from a range of quantiles in the lower tail of returns.

Testing for Granger-causality in quantiles

“…Theorem 2 demonstrates that under a local alternative H Z Y A,T of (12) the asymptotic distribution of S T has an extra shi function implying consistency against √ T-alternatives. Theorem 2 follows from Theorem 4 of Escanciano and Velasco (2010) and the continuous mapping theorem.…”

“…Assumption 5 provides a su cient condition for the estimator θ T to be consistent under a xed alternative hypothesis, see Angrist et al (2006) for conditions to satisfy Assumption 5(a); Assumption 5(b) holds if I t is bounded. Theorem 1 below is an application of Theorems 1-3 of Escanciano and Velasco (2010) and the continuous mapping theorem. Theorem 1.…”

“…Assumption 6 is analogous to Assumption A6 of Escanciano and Velasco (2010), but we consider just the restricted information set I Y t . To ensure nontrivial local power of our proposed test, we need to impose an assumption on the estimator θ T under a local alternative as given in (12).…”

“…Since it is an application of Theorem 2 of Whang (2006) and Theorem 5 of Escanciano and Velasco (2010), we omit the proof. (7),…”

“…Our test is an application of the method proposed by Escanciano and Velasco (2010) in the context of Granger-causality. To our knowledge, testing for Granger-causality in quantiles by parametric methods in a exible speci cation setting has not been analyzed in the literature before.…”