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State‐space Models with Finite Dimensional Dependence
Author(s) -
Gourieroux Christian,
Jasiak Joann
Publication year - 2001
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00247
Subject(s) - mathematics , smoothing , observable , state space , nonlinear system , space (punctuation) , extension (predicate logic) , markov chain , state (computer science) , function (biology) , markov process , type (biology) , pure mathematics , algorithm , statistics , computer science , ecology , physics , quantum mechanics , evolutionary biology , biology , programming language , operating system
We consider nonlinear state‐space models, where the state variable (ζ t ) is Markov, stationary and features finite dimensional dependence (FDD), i.e. admits a transition function of the type: π(ζ t |ζ t −1 ) =π(ζ t ) a ′(ζ t ) b (ζ t −1 ), where π(ζ t ) denotes the marginal distribution of ζ t , with a finite number of cross‐effects between the present and past values. We discuss various characterizations of the FDD condition in terms of the predictor space and nonlinear canonical decomposition. The FDD models are shown to admit explicit recursive formulas for filtering and smoothing of the observable process, that arise as an extension of the Kitagawa approach. The filtering and smoothing algorithms are given in the paper. JEL. C4.