The second-order least squares (SLS) method in regression model proposed by Wang (2003, 2004) is based on the first two conditional moments of the response variable given the observed predictor variables. Wang and Leblanc (2008) show that the SLS estimator (SLSE) is asymptotically more efficient than the ordinary least squares estimator (OLSE) if the third moment of the random error is nonzero. We apply the SLS method to variable selection problems and propose the adaptively weighted L1 regularized SLSE (L1-SLSE). The L1-SLSE is robust against the shape of error distributions in variable selection problems. Finite sample simulation studies show that the L1-SLSE is more efficient than L1-OLSE in the case of asymmetric error distributions. A real data application with L1-SLSE is presented to demonstrate the usage of this method. / October 2015

In this dissertation we proposed two generalizations of the Second-Order Least Squares (SLS) approach in two popular dynamic econometrics models. The first one is the regression model with time varying nonlinear mean function and autoregressive conditionally heteroskedastic (ARCH) disturbances. The second one is a linear dynamic panel data model. We used a semiparametric framework in both models where the SLS approach is based only on the first two conditional moments of response variable given the explanatory variables. There is no need to specify the distribution of the error components in both models. For the ARCH model under the assumption of strong-mixing process with finite moments of some order, we established the strong consistency and asymptotic normality of the SLS estimator. It is shown that the optimal SLS estimator, which makes use of the additional information inherent in the conditional skewness and kurtosis of the process, is superior to the commonly used quasi-MLE, and the efficiency gain is significant when the underlying distribution is asymmetric. Moreover, our large scale simulation studies showed that the optimal SLSE behaves better than the corresponding estimating function estimator in finite sample situation. The practical usefulness of the optimal SLSE was tested by an empirical example on the U.K. Inflation. For the linear dynamic panel data model, we showed that the SLS estimator is consistent and asymptotically normal for large N and finite T under fairly general regularity conditions. Moreover, we showed that the optimal SLS estimator reaches a semiparametric efficiency bound. A specification test was developed for the first time to be used whenever the SLS is applied to real data. Our Monte Carlo simulations showed that the optimal SLS estimator performs satisfactorily in finite sample situations compared to the first-differenced GMM and the random effects pseudo ML estimators. The results apply under stationary/nonstationary process and wih/out exogenous regressors. The performance of the optimal SLS is robust under near-unit root case. Finally, the practical usefulness of the optimal SLSE was examined by an empirical study on the U.S. airfares.

We study the consistency, robustness and efficiency of parameter estimation in different but related models via semiparametric approach. First, we revisit the second- order least squares estimator proposed in Wang and Leblanc (2008) and show that the estimator reaches the semiparametric efficiency. We further extend the method to the heteroscedastic error models and propose a semiparametric efficient estimator in this more general setting. Second, we study a class of semiparametric skewed distributions arising when the sample selection process causes sampling bias for the observations. We begin by assuming the anti-symmetric property to the skewing function. Taking into account the symmetric nature of the population distribution, we propose consistent estimators for the center of the symmetric population. These estimators are robust to model misspecification and reach the minimum possible estimation variance. Next, we extend the model to permit a more flexible skewing structure. Without assuming a particular form of the skewing function, we propose both consistent and efficient estimators for the center of the symmetric population using a semiparametric method. We also analyze the asymptotic properties and derive the corresponding inference procedures. Numerical results are provided to support the results and illustrate the finite sample performance of the proposed estimators.

In this thesis, we first review the current development of optimal regression designs under the second-order least squares estimator in the literature. The criteria include A- and D-optimality. We then introduce a new formulation of A-optimality criterion so the result can be extended to c-optimality which has not been studied before. Following Kiefer's equivalence results, we derive the optimality conditions for A-, c- and D-optimal designs under the second-order least squares estimator. In addition, we study the number of support points for various regression models including Peleg models, trigonometric models, regular and fractional polynomial models. A generalized scale invariance property for D-optimal designs is also explored. Furthermore, we discuss one computing algorithm to find optimal designs numerically. Several interesting applications are presented and related MATLAB code are provided in the thesis. / Graduate