Mixture Autoregressive Models: Asymptotic Properties and Application to Financial Risk

Abstract

This thesis extensively studies the class of Mixture autoregressive (MAR) models in terms of its asymptotic properties and applications to financial risk evaluation. We establish geometric ergodicity of the MAR models and by implication absolute regular and strong-mixing properties of the models. In addition, we also show the consistency and asymptotic normality of the maximum likelihood estimators of the MAR models. We compare the estimates of Value at Risk (VaR) and Expected Shortfall (ES) based on the MAR models to estimates based on a number of other methods, for individual stocks, exchange rates and stock indices. We find that the MAR models consistently perform better than the other models. In addition, tail density forecast performance of individual stocks, stock indices and exchange rate, based on some popular GARCH models are compared to tail forecasts based on MAR models with both Gaussian and Student-t innovations. The MAR models mostly outperform the other models. Confirming the claim that MAR models are better suited to capture the kind of data dynamics present in financial data.All the data analysis are implemented in R.The traditional residuals of the MAR model are computed as the difference between the observed values and their conditional means. We show that these residuals form a martingale difference sequence and that the unconditional variance of these residuals is strictly positive and bounded by the expected value of its conditional variance. We compare the class of MAR Models to the class of GARCH models and observed that both the GARCH type models andMAR models can be cast into the framework of random coefficient autoregressive models as well as generalized hidden markov models.We also show that for the MAR(2;1,1) model, the variance-covariance matrix ispositive definite and the same for both the conditional least square and maximum likelihood penalty functions.

Date of Award	1 Aug 2013
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Georgi Boshnakov (Supervisor)