Graduate Thesis Or Dissertation
 

Estimation of the order of an autoregressive time series : a Bayesian approach

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  • Finite order autoregressive models for time series are often used for prediction and other inferences. Given the order of the model, the parameters of the models can be estimated by least squares, maximum likelihood, or the Yule-Walker method. The basic problem is estimating the order of the model. A number of statisticians have examined this problem. The most recent and widely accepted method was proposed by Akaike (1969, 1970, 1974), which has been shown to give quite accurate estimates for simulated data. In this dissertation, the problem of autoregressive order estimation is placed in a Bayesian framework. This is done with the intent of illustrating how the Bayesian approach brings the numerous aspects of the problem together into a coherent structure which is both complementary to presently used methods and intuitively satisfying. A joint prior probability density is proposed for the order, the partial autocorrelation coefficients and the variance, and the marginal posterior probability distribution for the order, given the data, is obtained. It is noted that the value with maximum posterior probability is the Bayes estimate of the order with respect to a particular loss function. The asymptotic posterior distribution of the order is also given. In conclusion, Wolfer's sunspot data as well as simulated data corresponding to several autoregressive models are analyzed according to Akaike's method and the Bayesian method proposed in this dissertation. Both methods are observed to perform quite well, although the Bayesian method was clearly superior in most cases.
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