Strong orthogonal decompositions and non‐linear impulse response functions for infinite‐variance processes

Abstract

Abstract:The author proves that Wold-type decompositions with strong orthogonal prediction innovations exist in smooth, reflexive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. His theory includes as special cases all previous Wold-type decompositions of discrete time processes, completely characterizes when non-linear heavy-tailed processes obtain a strong-orthogonal moving average representation, and easily promotes a theory of non-linear impulse response functions for infinite-variance processes. The author exemplifies his theory by developing a non-linear impulse response function for smooth transition threshold processes, and discusses how to test decomposition innovations for strong orthogonality and whether the proposed model represents the best predictor. A data set on currency exchange rates allows him to illustrate his methodology. Decompositions orthogonales fortes et fonctions reponses a impulsion non lineaire pour des processus a variance infinieR6ssumi : L'auteur ddmontre que pour qu'un espace de Banach dflexif lisse de processus ti temps discret admette une dkomposition de type Wold 21 innovations prkvisionnelles fortement oahogonales, il faut et il suffit que I'op6rateur projectif engendrant les innovations posshle la propriktk d'itdrations. Son rdsultat. qui englobe comme cas particuliers toutes les ddcompositions de type Wold obtenues antdrieurement pour des processus ti temps discret, caractkrise complktement les situations oil les processus non lindaires ti queues lourdes possMent une reprdsentation en moyenne mobile fortement orthogonale, en plus de conduire naturellement ti une thdorie des fonctions rdponses ti impulsion non linkaire pour les processus ti variance infinie. Il se sert de cette thdorie pour ddvelopper une fonction ti impulsion non lindaire pour des processus ti seuil de transition lisse et montre comment tester I'orthogonalitk forte des innovations de la dkcomposition et vkrifier si le modkle proposk est be1 et bien le meilleur pr6dicteur. Un jeu de donnks sur des taux de change de devises lui permet d'illustrer son propos.

Infinite-Variance, Alpha-Stable Shocks in Monetary SVAR: Final Working Paper Version

“…The work by Hill (2006) cited in section 1 and other, similar efforts may offer some hope for an alternative approach when standard SVAR analysis is precluded by problems with infinite-variance. The empirical generality of the findings presented here is not yet known.…”

“…In a statistics journal, Hill (2006) has drawn attention to a broader array of problems with the use of VARs on data possessing fat-tailed distributions. Zarepour and Roknossadati (2008) have studied a VAR with infinite-variance non-Gaussian shocks.…”

Infinite-variance, alpha-stable shocks in monetary SVAR

“…The work by Hill (2006) RESULTS FROM GARCH(1,1) MODEL .7858 .5580 GARCH (-1) .5879 .0312 Notes: *Presample variances computed using backcasting parameter = 0.7 **S.E. = Bollerslev-Wooldridge (1992) robust standard error …”

“…In a statistics journal, J.B. Hill (2006: 3) mentions the possibility that VAR shocks could have infinite variance, leading to problems with orthogonalization for SVAR analysis.…”

Mixed‐Norm Spaces and Prediction of SαS Moving Averages

“…The resulting MAs, however, need not have independent increments, but rather increments satisfying a kind of orthogonality condition. See Hill () for necessary and sufficient conditions for a sequence in a Banach space to have strong orthogonal innovations. In the Hilbert space case, the Wold decomposition assures that a stationary Gaussian process can be written as an independent sum of an MA process and its completely deterministic part.…”