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Nonlinearity and Temporal Dependence

Author

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  • Xiaohong Chen
  • Lars P. Hansen
  • Marine Carrasco

Abstract

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: β−mixing and ρ−mixing. Weshow that β−mixing and ρ−mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be ρ−mixing, we show that they are still β−mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence. Les non-linéarités dans les coefficients de mouvement et de diffusion ont une incidence sur la dépendance temporelle dans le cas des modèles de diffusion scalaire. Nous examinons ce lien en recourant à deux notions de dépendance temporelle : mélange β et mélange ρ. Nous démontrons que le mélange β et le mélange ρ avec dégradation exponentielle constituent des concepts fondamentalement équivalents en ce qui a trait aux diffusions scalaires. Pour ce qui est des diffusions stationnaires qui ne se classent pas dans le mélange ρ, nous démontrons qu'elles appartiennent quand même au mélange β, sauf que les taux de dégradation sont lents plutôt qu'exponentiels. Pour ce genre de processus, nous recourons à des transformations des états de Markov dont les variations sont finies, mais dont les densités spectrales sont infinies à la fréquence zéro. Certains états ont des densités spectrales qui divergent à la fréquence zéro de la même façon que dans le cas des processus stochastiques à mémoire longue. En terminant, nous indiquons la façon dont l'échantillonnage de Poisson qui est non linéaire et dépendant de l'état modifie la distribution inconditionnelle et la dépendance temporelle.

Suggested Citation

  • Xiaohong Chen & Lars P. Hansen & Marine Carrasco, 2009. "Nonlinearity and Temporal Dependence," CIRANO Working Papers 2009s-17, CIRANO.
  • Handle: RePEc:cir:cirwor:2009s-17
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    References listed on IDEAS

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    More about this item

    Keywords

    Mixing; Diffusion; Strong dependence; Long memory; Poisson sampling.; mélange; diffusion; forte dépendance; mémoire longue; échantillonnage de Poisson.;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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