Root- consistent density estimation in GARCH models

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• 作者：Aurore Delaigle^a ; ^{A.Delaigle@ms.unimelb.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Alexander Meister^b ; ^{alexander.meister@uni-rostock.de" class="auth_mail" title="E-mail the corresponding author} ; Jeroen Rombouts^c ; ^{rombouts@essec.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：C14 ; C22 ; C32
• 刊名：Journal of Econometrics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：192
• 期：1
• 页码：55-63
• 全文大小：455 K

文摘

We consider a new nonparametric estimator of the stationary density of the logarithm of the volatility of the GARCHclass="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407615002614&_mathId=si18.gif&_user=111111111&_pii=S0304407615002614&_rdoc=1&_issn=03044076&md5=f7c999eda34ac1cc84adb6a7ceeb2b2f" title="Click to view the MathML source">(1,1)class="mathContainer hidden">class="mathCode">$(1,1)$ model. This problem is particularly challenging since this density is still unknown, even in cases where the model parameters are given. Although the volatility variables are only observed with multiplicative independent innovation errors with unknown density, we manage to construct a nonparametric procedure which estimates the log volatility density consistently. By carefully exploiting the specific GARCH dependence structure of the data, our iterative procedure even attains the striking parametric root-class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407615002614&_mathId=si17.gif&_user=111111111&_pii=S0304407615002614&_rdoc=1&_issn=03044076&md5=a1d483c46cb5af4fdfd41a6f21f474ef" title="Click to view the MathML source">Tclass="mathContainer hidden">class="mathCode">$T$ convergence rate. As a by-product of our main results, we also derive new smoothness properties of the stationary density. Using numerical simulations, we illustrate the performance of our estimator, and we provide an application to financial data.