Double asymptotics for explosive continuous time models

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• 作者：Xiaohu Wang^a ; ^{xiaohu.wang@cuhk.edu.hk" class="auth_mail" title="E-mail the corresponding author} ; Jun Yu^b ; ^{yujun@smu.edu.sg" class="auth_mail" title="E-mail the corresponding author}
• 关键词：C13 ; C22 ; G13
• 刊名：Journal of Econometrics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：193
• 期：1
• 页码：35-53
• 全文大小：542 K

文摘

This paper establishes a double asymptotic theory for explosive continuous time Lévy-driven processes and the corresponding exact discrete time models. The double asymptotic theory assumes the sample size diverges because the sampling interval (pan id="mmlsi62" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si62.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=8f986b8ad793af5812ed1734a8b21684" title="Click to view the MathML source">hpan>pan class="mathContainer hidden">pan class="mathCode">$h$pan>pan>pan>) shrinks to zero and the time span (pan id="mmlsi63" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si63.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=9fe3f2b93bc980ae7781b23a67e4968e" title="Click to view the MathML source">Npan>pan class="mathContainer hidden">pan class="mathCode">$N$pan>pan>pan>) diverges. Both the simultaneous and sequential double asymptotic distributions are derived. In contrast to the long-time-span asymptotics (pan id="mmlsi64" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si64.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=792aed9bbc3c7e57178b1503cb7ceb88" title="Click to view the MathML source">N→∞pan>pan class="mathContainer hidden">pan class="mathCode">$N\to \infty$pan>pan>pan> with fixed pan id="mmlsi62" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si62.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=8f986b8ad793af5812ed1734a8b21684" title="Click to view the MathML source">hpan>pan class="mathContainer hidden">pan class="mathCode">$h$pan>pan>pan>) where no invariance principle applies, the double asymptotic distribution is derived without assuming Gaussian errors, so an invariance principle applies, as the asymptotic theory for the mildly explosive process developed by Phillips and Magdalinos (2007). Like the in-fill asymptotics (pan id="mmlsi66" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si66.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=c9cd552b930a8ae8141a4fc38a81d6fc" title="Click to view the MathML source">h→0pan>pan class="mathContainer hidden">pan class="mathCode">$h\to 0$pan>pan>pan> with fixed pan id="mmlsi63" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304407616300173&_mathId=si63.gif&_user=111111111&_pii=S0304407616300173&_rdoc=1&_issn=03044076&md5=9fe3f2b93bc980ae7781b23a67e4968e" title="Click to view the MathML source">Npan>pan class="mathContainer hidden">pan class="mathCode">$N$pan>pan>pan>) of Perron (1991), the double asymptotic distribution explicitly depends on the initial condition. The convergence rate of the double asymptotics partially bridges that of the long-time-span asymptotics and that of the in-fill asymptotics. Monte Carlo evidence shows that the double asymptotic distribution works well in practically realistic situations and better approximates the finite sample distribution than the asymptotic distribution that is independent of the initial condition. Empirical applications to real Nasdaq prices highlight the difference between the new theory and the theory without taking the initial condition into account.