Predictive Modelling of Extreme Values in Unbalanced Panel Data
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This thesis aims at predictive modelling of a two-sided, heavy-tailed data under a mixture model setting. A robust regression method is used to fit the main body, while the peaks-over-threshold method is employed to select the tails or the extreme events. Based on the extreme value theory, the tails are modelled with Pareto or exponential distributions. For the estimation of the tail distributions, the Bayesian maximum a posterior estimation (MAP) with conjugate priors is used to smooth the maximum likelihood estimates (MLEs). With regard to each of the two tail, the MAP approach leads to two optimization problems for the estimation of tail parameters: the tail decay rate and the tail quantile level. This filter tuning process provides stability and efficiency in computation and prediction. Several constrained, non-convex optimization problems have been converted to unconstrained, convex problems by quadratic approximation and variable changes. Newton’s iteration method is employed to solve the optimization problems numerically. This formulated methodology is applied to a large, multi-period, unbalanced data set of daily returns of global stocks, containing nearly 120,000 records. The out-of-sample prediction results show the out-performance of the smoothed estimates over the regular MLEs.