Bootstrapping Stationary ARMA-GARCH Models by Kenichi Shimizu

By Kenichi Shimizu

Im Jahre 1979 hat Bradley Efron mit seiner Arbeit Bootstrap tools: one other examine the Jackknife das Tor zu einem in den vergangenen 30 Jahren intensiv bearbeiteten Forschungsgebiet aufgestossen. Die simulationsbasierte Methode des Bootstraps hat sich in den verschiedensten Bereichen als ein ausserordentlich - ?zientes Werkzeug zur Approximation der stochastischen Fluktuation eines Sch- zers um die zu schatzende Grosse erwiesen. Prazise Kenntnis dieser stochastischen Fluktuation ist zum Beispiel notwendig, um Kon?denzbereiche fur Schatzer an- geben, die die unbekannte interessierende Grosse mit einer vorgegebenen Wa- scheinlichkeit von, sagen wir, ninety five oder ninety nine% enthalten. In vielen Fallen und bei korrekter Anwendung ist das Bootstrapverfahren dabei der konkurrierenden und auf der Approximation durch eine Normalverteilung basierenden Methode ub- legen. Die Anzahl der Publikationen im Bereich des Bootstraps ist seit 1979 in einem atemberaubenden pace angestiegen. Die wesentliche und im Grunde e- fache Idee des Bootstraps ist die Erzeugung vieler (Pseudo-) Datensatze, die von ihrer wesentlichen stochastischen Struktur dem Ausgangsdatensatz moglichst a- lich sind. Die aktuellen Forschungsinteressen im Umfeld des Bootstraps bewegen sich zu einem grossen Teil im Bereich der stochastischen Prozesse. Hier stellt sich die zusatzliche Herausforderung, bei der Erzeugung die Abhangigkeitsstruktur der Ausgangsdaten adaquat zu imitieren. Dabei ist eine prazise examine der zugrunde liegenden state of affairs notwendig, um beurteilen zu konnen, welche Abhangigkei- aspekte fur das Verhalten der Schatzer wesentlich sind und welche nicht, um a- reichend komplexe, aber eben auch moglichst einfache Resamplingvorschlage fur die Erzeugung der Bootstrapdaten entwickeln zu ko

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01 (1, 10, 000), bold line). 5614]. 01 ulation. That is, the wild bootstrap technique underestimates the existing risk and could lead a financial institution to the false decision to take the too much risk that is not covered by her capital charge. 01 3 Parametric AR(p)-ARCH(q) Models In this chapter we consider the parametric AR(p)-ARCH(q) model based on ARCH regression models of Engle (1982). For simplicity, we estimate the parameters by the ordinary least squares (OLS) method and adopt the two-step estimation for the ARCH part, in which the parameters of the ARCH part are estimated based on the residuals of the AR part.

1 X XT T −1 X p+q−1 X p+q .. ... .. XT −2 ... ⎛ ⎛ a∗ = (X X)−1 X x∗ , b∗ = (E E)−1 E e∗ , ∗2 ∗2 ε p+q+2 ... εT∗2 . 4 ηt∗ has the following properties: ηt − μ σ T ∑ t=1 = 1 σ 1 T 1 T T ∑ ηt −μ t=1 =μ = 0, E∗ (ηt∗2 ) = T ∑ t=1 = ηt − μ σ 1 1 σ2 T T =σ 2 = 1, 2 1 T ∑ (ηt − μ )2 t=1 ⎞ + u∗ , ⎛ ∗ ⎞ a ⎞ ε p+q+1 Xq+1 ⎜ 0 ⎟ a 1 ⎜ ⎟ ⎜ ∗ ⎟ Xq+2 ⎟ ⎟ ⎜ ⎟ ⎜ ε p+q+2 ⎟ .. ⎟ ⎜ a2 ⎟ + ⎜ .. ⎟ . ⎟ ⎝ . ⎠ . ⎠⎜ ⎝ .. , XT . Analogously to the previous section, we observe a∗ = (X X)−1 X (Xa + u∗ ) = a + (X X)−1 X u∗ , that is, T − p − q(a∗ − a) = 1 XX T − p−q −1 1 √ X u∗ .

1 we already have 1 p XX− → A. T − p−q in probability. 50 3 Parametric AR(p)-ARCH(q) Models Together with the Slutsky theorem it suffices to show T 1 1 d √ X u∗ = √ ε ∗ xt → N 0, V ∑ T − p−q T − p − q t=p+q+1 t Now we observe 1 1 ε ∗ xt = √ y∗t := √ T − p−q t T − p−q ht ηt∗ xt , in probability. , T. , ηT∗ is independent. 1 = o p (1) p − →V. 2 = =o p (1). , c y∗T show T ∑ y∗t → − N 0, V d in probability. 1) satisfying assumptions (A1)(A4) and (B1)-(B4). e. T − p − q(b∗ − b) − → N 0, B−1 WB−1 d in probability.

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