My dissertation consists of six essays which contribute new theoretical results

to two econometrics frontiers: nonparametrics and finite sample econometrics. Chapters 2 to 3 discuss the estimation and inference of the nonparametric and semiparametric models. In chapter 2 an efficient two-step estimator is developed in single nonparametric regression model with a general parametric error covariance. By fully utilizing the information incorporated in the error covariance into estimation, the newly developed method is more efficient compared to the conventional local linear estimator (LLLS) and some other two-step estimator. The corresponding asymptotic theorems are derived. Monte Carlo study shows the relative efficiency gain of the newly proposed estimator. Chapter 3 systematically develops a new set of results for seemingly unrelated regression (SUR) analysis within nonparametric and semiparametric framework. We study the properties of LLLS and local linear weighted least squares (LLWLS) estimators, provide an efficient two-step estimation for the system and establish the asymptotic theorems under both unconditional and conditional error variance-covariance cases. The procedures of estimation for various nonparametric and semiparametric SUR models are proposed. In addition, two nonparametric goodness-of-fit measures for the system are given. Chapter 4 applies the estimation method developed in chapter 2 and 3 to an empirical analysis on return to public capital in U.S.

Chapters 5 to 6 study the finite sample properties of the mean reversion parameter estimator in continuous time models. In chapter 5 we approximate the bias of the estimator for the Levy-based Ornstein-Uhlenbeck (OU) process, and propose bias corrected estimators. In chapter 6 the exact distribution of the MLE is investigated under different scenarios: known or unknown drift term, fixed or random start-up value, and zero or positive . The numerical calculations demonstrate the remarkably reliable performance of the proposed exact approach.

In chapter 7 we study the efficiency of the coefficient of determination based on

final prediction error and compare it with conventional goodness-of-fit measures

in linear regression models with both normal and non-normal disturbances. The

efficiency results show that R2 based on

final prediction error has practical use in empirical analysis, for examples,

panel data analysis and time series analysis.

Nonparametric approaches have widely been used due to their advancement in not making assumptions on the distribution of the data. Even with their extensive development, nonparametric hypothesis testing has not been developed as much as a nonparametric estimation even though it is one of the key components of the econometric analysis. This dissertation has mainly two parts. I first explore the systematic development of the current nonparametric tests and provide results on testing linearity as an illustration. Then I develop new nonparametric tests for detecting endogeneity in cross-sectional data and panel data respectively.

Elaborating each test's performance can be meaningful in that we can decide which test to use depending on the hypothesis and even construct a new test based on such a relationship. Under the hypotheses for linearity, Chapter 2 will categorize the types of nonparametric tests and discuss the analytical relationship of those tests. By imposing some conditions, I can compare the local power of each test asymptotically. While examining the analytical relationship, I develop a nonparametric Rao-Score test and show it to be equivalent to the Su and Ullah (2013) test.

Once analyzing the analytical relationship of the current nonparametric tests, I focus on developing a new nonparametric test for endogeneity. Since endogeneity is commonly observed in many economic contexts, detecting its presence is a preliminary step for choosing an estimation strategy. In Chapter 3, I construct a test using the control function approach under a triangular simultaneous equations model. My test can be summarized as being simple to implement as a test and being able to capture the locally nonlinear correlation with kernel weighting. Furthermore, I will apply these tests to the empirical analyses and show the contradicting results with the parametric test.

Not only in triangular simulation equations model, but also is endogeneity important model specification issue in panel data setting. The estimation strategy differs depending on the presence of endogeneity between the individual specific effects and the variable. I propose a new estimation method for the nonparametric panel random effects model and construct a new test for endogeneity using the residuals from the proposed estimation method. By obtaining the individual specific effects in the random effects model, I construct a test over the i index instead of the i index and time. With a large T, the test performs well in terms of size and power.

Nowadays, with advanced technology, it is easier to obtain data like never before. With more available data, comes new information that economists can extract to uncover relationships between economic variables. By using new state of the art machine learning algorithms and techniques that can handle data efficiently and can identify trends and patterns easily, we can help solve economic problems, theoretically and empirically. The primary goal of this dissertation is to help bridge the gap between machine learning and econometrics. With powerful machine learning models that exhibit great predictive ability, it would be useful to further explore machine learning methods and add them to our econometrics toolbox. In addition, we wish to extend these models to incorporate problems often faced in econometric models, including partial effects estimation using first derivatives, evaluating concavity of various economic functions using second derivatives, and allowing for heteroskedastic and autocorrelated errors in an econometric model. These issues are clearly often faced in economics, but not so much in machine learning. To incorporate machine learning techniques, machine learning estimation of nonparametric models and marginal effects are established throughout the dissertation. A derivative estimation procedure of smoothing weighted difference quotients based on random forest is proposed. The procedure of smoothing weighted difference quotients based on random forest is then used to estimate second derivatives. Lastly, a generalized framework for Kernel Regularized Least Squares that incorporates information in the error covariance when estimating the regression function is proposed.

This dissertation details my exploration of two compounds: LaCrGe3 and LaCrSb3. While their names differ by only two letters, and therefore their compositions by only one element, they are quite different. The relevant question to ask about these two ferromagnetic systems is: “what happens when their ferromagnetism is suppressed?" Although the magnetic phase diagram of LaCrGe3 under pressure has previously been charted, there are aspects of its rich temperature-pressure-magnetic field phase space that are contested. In particular, through a meticulous study of its magnetic domain behavior, I provide evidence in favor of the existence of multiple ferromagnetic states in LaCrGe3. We find that investigating domain behavior can lead to a more accurate way of characterizing new materials and perhaps a method of probing crossovers between ferromagnetic states. On the other hand, the ferromagnetism of LaCrSb3 is relatively robust to pressure. Therefore, I use Fe substitution to suppress its Curie temperature and discover the first reported magnetic phase diagram of its kind—one with an avoided quantum tricritical point. The LaCr1-xFexSb3 system has a temperature-chemical substitution-magnetic field phase diagram that is ripe with magnetic features for closer examination. Further study of both materials will likely lead to additional discoveries in the world of magnetism.

This dissertation consists of four chapters that study estimation and inference in systemof equations and panel data under model uncertainty. In Chapter 2, I consider model uncertainty in a panel data model, and introduce a Stein-like shrinkage estimator that is a weighted average of an unrestricted estimator and a restricted estimator. The restricted estimator represents a belief about where the parameters of the model are likely to be close. Chapter 3 considers the estimation uncertainty from choosing different number of lagged dependent variables as instruments in dynamic panel data models. Generalized method of moments (GMM), the typical estimation method, can produce efficient estimators when all lagged dependent variables are used as instruments. However, estimation using all instruments can cause substantial bias. Conversely, the GMM estimators that use one lag as instrument are asymptotically unbiased under forward demeaning transformation, but at the cost of losing efficiency. Therefore, I introduce an averaging estimator which is a weighted average of the two GMM estimators where the averaging weight is proportional to a quadratic loss function that minimizes the asymptotic risk. In Chapter 4, I consider simultaneous equations models (SEM), and develop an estimator to deal with the model uncertainty about the magnitude of endogeneity. Ordinary least squares (OLS) estimators are the most efficient estimators, however, may suffer from substantial bias when the degree of endogeneity is substantial. On the contrary, two-stage least squares (2SLS) estimators are consistent but not efficient. Therefore, I consider a Stein-like shrinkage estimator which is a weighted average of the OLS and 2SLS estimators, where the weight is inversely related to a Wu-Hausman statistic that measures the magnitude of the endogeneity. Chapter 5 considers latent group structures to model uncertainty resulting from unobserved heterogeneity in panel data models. Basically, I consider a panel data model where the slope parameters are heterogenous across groups but homogenous within a group, and the group identity is unknown. I provide a framework for estimation and identification of the latent group structure using a pairwise fusion penalized approach.

This dissertation is composed with 4 essays. They explore modelling uncertainty following two major directions. The former 2 contains topics on ordinary and general ridge-type shrinkage estimation developed from model averaging and kernel density estimation. The third one critically reviews recent literature in the areas of model averaging and model selection both parametrically and nonparametrically and proposes topics for future work. The last one focuses on nonparametric panel data estimation with random effects. In chapter 2, ordinary ridge-type shrinkage estimation is extensively studied, where a class of well-behaved ordinary ridge-type semiparametric estimators is proposed. Monte Carlo simulations, theoretical derivations, as well as empirical out-of-sample forecasts are all investigated to prove their usefulness in reducing mean squared errors, i.e. risks. Chapter 3 develops the works in Chapter 2 to the general ridge regressions. By connecting general ridge regression with kernel density estimation, an asymptotically optimal semiparametric ridge-type estimator is built. By connecting general ridge regression with model averaging, a class of model averaging ridge-type estimators are obtained. These estimators are observed to have different improvements upon the feasible general ridge estimators when model uncertainties, i.e., the error variances are different. To encourage better understanding on model averaging and model selection, Chapter 4 gives a comprehensive literature review and analysis on these topics from a frequentist's point of view. Parametric and nonparametric procedures in the recent developments are explored. Chapter 5 starts investigating panel data estimation by introducing nonparametrics in the picture. The proposed two-stage estimator shows good behaviors in Monte Carlo simulation. In addition, illustrative empirical examples in health economics and environmental economics are also introduced.

Economic and Financial phenomena convey enormous information about the underlying structure of economic and policy interest. The first objective of the thesis is mainly concerned with how to make use of information efficiently, specifically, (1) how to separate noises from useful information in the presence of large dimensional data, (2) how to incorporate prior information (economic constraint), and (3) how to employ model structure, to conduct more informed inference, and thus to understand the economic structure wisely and draw sound policy conclusions.

The second dimension of information refers to the recent developments in the information theory that measure how much information content the observed data contains. The formalism of Maximum Entropy provides an information-theoretic approach to tackle economic problems, especially those with data observed in aggregate terms. Thus, the second objective of the thesis is to make use of this line of research and develop a new estimation method to measure quantities of economic interest when researchers are faced with model uncertainty.

Despite strong recent economic growth, gender inequality remains a major concern for India. This dissertation examines the effectiveness of public policy in improving some important human development outcomes, with a focus on gender issues. The national Pre-Conception and Pre-Natal Diagnostics Techniques (PNDT) Act of 1994, implemented in 1996, banned sex-selective abortions in the Indian states which hitherto had not legislated such a policy. Using village-level and town-level longitudinal data from the 1991 and 2001 censuses, along with household survey data from other sources, the first essay finds a significantly positive impact of the PNDT Act on the female-to-male juvenile sex ratio (number of females per 1000 males below the age of 6 years). Although researchers frequently mention the futility of the Act, this study is among the first to use a treatment-effect type analysis of the pre-ban and post-ban periods to show that the law hindered any further worsening of the gender imbalance in India. I find that in the possible absence of the PNDT Act, juvenile sex ratio would have declined by another 13-20 points on average. A second study evaluates the `unintended consequences' of the PNDT Act on child quality. Using household survey data from two time periods, and exploiting a natural experiment framework originating from the timing of the PNDT Act, I find a mixed impact of the law on gender-relative child quality outcomes. Since the PNDT Act partially improved the sex ratio but did not uniformly worsen the nutritional and immunization status of girls, it could be regarded as a truly welfare enhancing public policy. Finally, a third study examines the effectiveness of the Indian school feeding program in improving the nutritional and learning outcomes of children. Using a household fixed-effect and a propensity score matching framework, the outcomes of children receiving school meals are compared with that of similar children who are not covered under the program. The results show that the school meal program generally does not have any significant effect on the child nutrition nor learning outcomes, neither does it have any impact on the relative outcomes of the girl children.

The first chapter establishes a way of inferring risk aversion in a first-price auction (FPA) model when an entry decision is endogenous. Bidders' risk aversion is captured by a parameter in constant relative risk aversion utility functions and the parameter is then partially identified in a set under a "monotonicity" condition. The recovery of

the partially identified risk aversion parameter is concluded in a confidence set (CS). The CS is constructed by inverting a test dealing with many inequality restrictions of quantiles. In the spirit of Andrews and Shi (2013), we implement quantile selection to address possible slackness. Asymptotic results show desired properties of size and power against fixed (and some local) alternatives. Confidence sets perform fairly well in finite samples and the comparison of results highlights the necessity of quantile selection. The inference is illustrated by using US Forest Service timber auction data and detects considerable risk aversion.

Exogenous entry is a convenient assumption to make for identification and inference in auction models. The second chapter examines this assumption and develops a test against endogenous entry in first-price auctions with risk aversion. The approach also takes auction observed heterogeneity into account. The desirable property of asymptotic size and consistency of the test is proven, and Monte Carlo simulations approve the test at finite samples. The application to US Forest Service timber auctions brings in interesting implication: entry is exogenous with lower entry level but is endogenous with higher entry level. The result also suggests a relatively stronger risk aversion attitude than what is obtained in the literature.

The third chapter shows nonparametric identification and estimation of private value distribution and density functions in first-price auctions with endogenous entry. In the model, symmetric bidders face a nontrivial entry cost and a binding reserve price. We identify latent structures by solving a two stage game, and estimate density

functions (point-wisely) by using and comparing two different methods. Monte Carlo experiments show good performance of our estimators.

This dissertation covers topics in classification with high-dimensional data, variable selection in sparse semiparametric single-index models and statistical inference under heteroskedasticity. In particular, Chapter 1 provides the motivation and background of the dissertation.

Chapter 2 provides a summary of boosting methods for classification, namely Discreet AdaBoost, Real AdaBoost, P-AdaBoost, Gentle AdaBoost and LogitBoost. We compare these methods with alternative machine learning classification tools such as Deep Neural Network and demonstrate the empirical applications in economics, such as prediction of business cycle turning points and directional prediction of stock price indexes.

Chapter 3 generalizes the Discreet AdaBoost shown in Chapter 2 for binary classification problem with state-dependent loss functions. We introduce Asymmetric AdaBoost that solves the asymmetric maximum score problem with high-dimensional data. Asymmetric AdaBoost produces a nonparametric classifier via minimizing the ``asymmetric exponential risk'' which is a convex surrogate of the traditional non-convex score risk or 0-1 risk. The convex risk function gives huge computation advantage over non-convex risk functions, e.g. Maximum Score (Manski, 1975, 1985), especially when the data is high-dimensional.

Chapter 4 considers the "Regularization of Derivative Expectation

Operator" (RODEO) of Lafferty and Wasserman (2008) and propose a modified RODEO algorithm for sparse semiparametric single-index models which we call the SIM-RODEO. The SIM-RODEO method is able to distinguish relevant explanatory variables from irrelevant variables and gives a competitive estimator for the model.

In addition, the algorithm finishes in a reasonable period of time. In addition, the algorithm finishes in a reasonable period of time.

Chapter 5 investigates the methods for statistical inference under the presence of heteroskedasticity of unknown form in the disturbances of linear regression models. We propose an F-type test statistic for testing regression parameters under the heteroskedasticity of unknown form. The accuracies of the test statistic are confirmed by extensive Monte Carlo experiments. And Chapter 6 concludes.