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Stability of nonlinear AR‐GARCH models
Author(s) -
Meitz Mika,
Saikkonen Pentti
Publication year - 2008
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/j.1467-9892.2007.00562.x
Subject(s) - conditional variance , mathematics , autoregressive model , autoregressive conditional heteroskedasticity , heteroscedasticity , markov chain , nonlinear system , conditional probability distribution , conditional expectation , nonlinear autoregressive exogenous model , moment (physics) , econometrics , star model , stability (learning theory) , setar , volatility (finance) , statistics , autoregressive integrated moving average , time series , computer science , physics , classical mechanics , quantum mechanics , machine learning
. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR( p )] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β ‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.