February 2022 Adaptiveness of the empirical distribution of residuals in semi-parametric conditional location scale models
Christian Francq, Jean-Michel Zakoïan
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Bernoulli 28(1): 548-578 (February 2022). DOI: 10.3150/21-BEJ1357


This paper addresses the problem of deriving the asymptotic distribution of the empirical distribution function Fˆn of the residuals in a general class of time series models, including conditional mean and conditional heteroscedaticity, whose independent and identically distributed errors have unknown distribution F. We show that, for a large class of time series models (including the standard ARMA-GARCH with symmetric innovations), the asymptotic distribution of n{Fˆn(·)F(·)} is impacted by the estimation but does not depend on the model parameters. It is thus neither asymptotically estimation free, as is the case for purely linear models, nor asymptotically model dependent, as is the case for some nonlinear models. The asymptotic stochastic equicontinuity is also established. We consider an application to the estimation of the conditional Value-at-Risk.


The authors would like to thank the Editor, Associate Editor and two reviewers for valuable comments and suggestions. We are grateful to the Agence Nationale de la Recherche (ANR), France, which supported this work via the Project MultiRisk (ANR CE26 2016-CR), and the Labex ECODEC, France. The second author gratefully thanks the IDR ACP “Régulation et risques systémiques”, France for financial support.


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Christian Francq. Jean-Michel Zakoïan. "Adaptiveness of the empirical distribution of residuals in semi-parametric conditional location scale models." Bernoulli 28 (1) 548 - 578, February 2022. https://doi.org/10.3150/21-BEJ1357


Received: 1 June 2020; Revised: 1 December 2020; Published: February 2022
First available in Project Euclid: 10 November 2021

MathSciNet: MR4337716
zbMATH: 07467732
Digital Object Identifier: 10.3150/21-BEJ1357

Keywords: adaptive estimation , asymptotic distribution of quantiles , conditional VaR , empirical distribution of residuals , GARCH , stochastic equicontinuity

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 1 • February 2022
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