Chapter 1, 3 and 4 focus on the analysis of interval-valued data (joint with Professor González-Rivera). In Chapter 1, we propose a constrained regression model that preserves the natural order of the interval in all instances. Within the framework of interval time series, we specify a general dynamic bivariate system for the upper and lower bounds of the intervals, and propose a (modified) two-step estimator. Monte Carlo simulations show good finite sample properties of the proposed estimators. We model the daily interval of low/high SP500 returns before and after 2007, and find that truncation is very severe during and after the financial crisis of 2008, so that a modified two-step procedure should be implemented. In Chapters 3 and 4, we adopt an alternative modelling approach for interval-valued data that exploits the extreme property of lower/upper bounds of interval, which is ignored in the existing literature. Specifically, Chapter 3 and 4 propose two different models and estimation strategies (ML and semiparametric estimation) that combines the knowledge of order statistics and extreme value theory with interval-valued data respectively.;As a separate strand of research, in Chapter 2 (joint with Professor Ullah), we propose an adaptive spline estimator based on Friedman (1991)'s multivariate adaptive regression splines. The model takes the form of an expansion in the cross product spline bases, where the numbers of spline functions, the degree of tensor product and knot locations are automatically selected adaptively by using generalized cross validation. Our estimator is more tractable not only in computational implementations but in theoretical deductions as well. We establish the asymptotic normality of our adaptive estimator, and obtain the optimal convergence rate that it can possibly achieve. The optimal convergence rate depends on the order ratio of the number of selected spline basis functions to the total potential ones. The Monte Carlo simulation, comparing the adaptive estimator with classical regression splines given various DGP settings, shows that our estimator has more significant improvement upon classical regression splines by producing smaller AMSE given the DGP with multivariate covariates. We also apply our adaptive estimator to the study of the effect of public capital stock on the gross state product using the pooled panel data set in Baltagi and Pinnoi (1995).
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