Abstract
ABSTRACT We propose a simple yet powerful method to construct strictly stationary Markovian models with given but arbitrary invariant distributions. The idea is based on a Poisson-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.
Highlights
Stationarity and other stability properties represent a crucial component in the theory and application of stochastic processes
Part of the dependence in the model is induced by the choice of marginal density, f, which in turn can be selected by the nature of the phenomenon or data under study
A related, but different, idea is the one proposed by Nakajima et al (2012), where innovations of the underlying latent process follow a Type I generalized extreme value (GEV) distribution and induce an observation response again characterized by a Type I GEV distribution
Summary
Stationarity and other stability properties represent a crucial component in the theory and application of stochastic processes. Defining Markov models with prescribed invariant distributions poses a tradeoff between marginal and conditional properties, as one can have several models with different dependence structure while retaining the same stationary distribution This issue can be handled, to some extent, with a particular context in mind, for example, fulfilling certain continuity or dependency requirements. Within a similar setup, Pitt, Chatfield, and Walker (2002) exploited the reversibility property characterizing Gibbs sampler Markov chains to build strictly stationary AR(1)-type models with any choice of marginal distribution Their approach is very general and requires to make choices of dependence to accommodate the specific modeling needs. In the continuous-time setup, we use the gamma distribution as basic building block and obtain, via suitable transformations, a richer class of diffusion processes with known transition density This includes, for instance, diffusions with generalized extreme value (GEV) invariant distributions that, to the best of our knowledge, have not been derived before.
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