The independence process in conditional quantile location-scale models and an application to testing for monotonicity

Abstract

In this paper the nonparametric quantile regression model is considered in a locationscale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is investigated. For the test for validity of the location-scale model as well as for the monotonicity test smooth residual bootstrap versions of Kolmogorov-Smirnov and Cramér-von Mises type test statistics are suggested. We give rigorous proofs for bootstrap versions of the weak convergence results. The performance of the tests is demonstrated in a simulation study. The authors would like to thank two anonymous referees and the associate editor for careful reading and for very constructive suggestions to improve the paper. Our special thanks go to one of the referees for several very careful readings of the manuscript and insightful comments. Part of this work was conducted while Stanislav Volgushev was postdoctoral fellow at the Ruhr University Bochum, Germany. During that time Stanislav Volgushev was supported by the Sonderforschungsbereich "Statistical modelling of nonlinear dynamic processes" (SFB 823), Teilprojekt (C1), of the Deutsche Forschungsgemeinschaft. arXiv:1609.07696v1 [math.ST]

Estimation and hypotheses testing in boundary regression models

“…Wilson (2003) points out the analogy to tests for independence between errors and covariates in regression models, but no asymptotic distributions are derived. Tests for independence in nonparametric mean and quantile regression models that are similar to the test we will consider are suggested by Einmahl and Van Keilegom (2008) and Birke et al (2016+).…”

“…We are not aware of hypothesis tests for monotonicity or other shape constraints in the context of boundary regression, but would like to mention Gijbels' (2005) review on testing for monotonicity in mean regression. Tests similar in spirit to the one we are suggesting here were considered by Birke and Neumeyer (2013) and Birke et al (2016+) for mean and quantile regression models, respectively.…”

Specification testing in nonparametric AR‐ARCH models

“…Furthermore, note that a nonsmooth residual bootstrap, that is, drawing with replacement from (standardized) residuals, might result in a time series that is not mixing. Note that Neumeyer () and Birke, Neumeyer, and Volgushev () proved the validity of smooth residual bootstrap procedures in the context of residual processes for regression models with independent data. Similar methods could be applied in our context, but a rigorous proof is beyond the scope of the paper.…”

A Smooth Nonparametric, Multivariate, Mixed-Data Location-Scale Test

“…A variety of tests have been proposed for assessing the appropriateness of the location-scale assumption that is often invoked in applied settings; see by way of illustration Akritas & Van Keilegom (2001) and Li & Racine (forthcoming), who adopt the location-scale framework, see Einmahl & Van Keilegom (2008), Birke, Neumeyer & Volgushev (2017) and Neumeyer, Noh & Van Keilegom (2016) for various approaches that have been proposed to test the location-scale assumption in a range of settings, and see Neumeyer (2009) for a bootstrap procedure for the error distribution in these models. These approaches employ test statistics that are based on conditional mean models, in particular, the difference between the joint distribution of the predictor and error and the product of the marginal distributions of the predictor and error, and include the Kolmogorov-Smirnov (Kolmogorov 1933, Smirnov 1948, Cramér-von-Mises (Cramér 1928, von Mises 1928 and Anderson-Darling (Anderson & Darling 1952) statistics, among others.…”