## Bayesian analysis for mixtures of autoregressive components with application to financial market volatility

### Sampietro, Stefano ; Arbia, Giuseppe (Dir.)

### Thèse de doctorat : Università della Svizzera italiana, 2004 ; 2004ECO004.

This thesis presents a Bayesian analysis of a non-linear time series model. In particular, we deal with a mixture of normal distributions whose means are linear functions of the past values of the observed variable. Since the component densities of the mixture can be viewed as the conditional distributions of different Gaussian autoregressive models, the model is referred as mixture of... Plus

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- This thesis presents a Bayesian analysis of a non-linear time series model. In particular, we deal with a mixture of normal distributions whose means are linear functions of the past values of the observed variable. Since the component densities of the mixture can be viewed as the conditional distributions of different Gaussian autoregressive models, the model is referred as mixture of autoregressive components. Bayesian perspective implies some advantages, especially in terms of model determination. First of all, it has been recognized that usual criterions like AIC and BIC are not satisfactory for the mixture of autoregressive components. In addition, these standard approaches do not take into account model uncertainty because they select a single model and then make inference based on this model. On the contrary, our Bayesian approach maintains consideration of several models, with the input of each into the analysis weighted by the model posterior probability. Both parameter estimation and model selection do not lend themselves to analytic solutions and we use Markov Chain Monte Carlo (or MCMC) approximation methods, which have had a real explosion over the last years, especially in Bayesian statistics. Our work takes into account the stationarity conditions of the autoregressive coefficients of the mixture components through a reparametrization in terms of partial autocorrelations. Finally, this thesis addresses the important task of modelling and forecasting return volatility. Several stylized facts about volatility have been recognized and they are captured by the mixture of autoregressive components.