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Multiple linear regression techniques are often employed in the statistical analysis of data. The most frequently used regression procedure is ordinary least squares. However, it is accepted that the method of least squares does not necessarily provide either accurate estimates of unknown regression coefficients or accurate predictions at future data points. Several classes of biased estimators have emerged as possible alternatives to ordinary least squares. We review the origins, definitions and properties of existing biased estimation procedures such as ridge, shrinkage, principal components and partial least squares regression. In addition, two new classes of estimator, multistage and multistep, are introduced. Simulation is the obvious means for assessing the relative merits of different estimation procedures. We review the design and results of previous simulation studies in which comparisons have been made between the performances of different estimation procedures. The designs of most previous studies are somewhat limited and unrealistic. Consequently, few clear guidelines have emerged regarding the circumstances in which individual procedures should either be applied or avoided. To provide some clarification, we conducted a series of simulation experiments that were designed to compare the performances of different regression procedures over a broad range of realistic situations.

Modern data analysis often involves a large number of variables, which gives rise to the problem of multicollinearity in regression models. It is well-known that in a linear model, when the design matrix X is nearly singular, then the ordinary least squares (OLS) estimator may perform poorly because of its numerical instability and large variance. To overcome this problem, many linear or nonlinear biased estimators are studied. In this work we consider a class of generalized shrunken least squares (GSLS) estimators that include many well-known linear biased estimators proposed in the literature. We compare these estimators under the mean square error and matrix mean square error criteria. Moreover, a simulation study and two numerical examples are used to illustrate some of the theoretical results.

Modern data analysis often involves a large number of variables, which gives rise to the problem of multicollinearity in regression models. It is well-known that in a linear model, when the design matrix X is nearly singular, then the ordinary least squares (OLS) estimator may perform poorly because of its numerical instability and large variance. To overcome this problem, many linear or nonlinear biased estimators are studied. In this work we consider a class of generalized shrunken least squares (GSLS) estimators that include many well-known linear biased estimators proposed in the literature. We compare these estimators under the mean square error and matrix mean square error criteria. Moreover, a simulation study and two numerical examples are used to illustrate some of the theoretical results.

In this thesis are presented the Tully-Fisher (TF) relations for a sample of 99 galaxies within the four nearby dusters; Coma, Abell 2199. Abell 2634 and Abell 194. Each cluster was compromised of two samples. The first sample was drawn from either Zwicky or UGC catalogues based on a combination of magnitude, type and ellipticity. These provided spiral duster member candidates over the entire duster region to a magnitude limit of 16 in the b-band. The second sample was selected from published photographic plate scans of the central areas of each duster. This sample had a fainter magnitude limit of 18 b-band mags but covered a much smaller area (approx. 2ºx2º). The galaxies were observed over two observing runs in May and August of 1993 on the JKT and INT Isaac Newton Group Telescopes simultaneously. I-band CCD images and optical long-slit spectra were taken of 65% of the sdected objects. Isophotal ellipse fitting of the images was used to produce surface brightness profiles. From these, isophotal magnitudes and diameters woe extracted. From, consideration of the surface brightness, ellipticity and position angle a "disk region" of each profile was selected and used to calculate extrapolated total magnitudes. Gaussian fitting of Ha emission lines of the long-split spectra produced optical rotation curves for each galaxy. Maximum rotation velocities were calculated from these curves. Corrections found in the literature were applied to the total magnitude and rotation velocity of each galaxy. These compensated for internal dust extinction and the inclination of the disk to the line-of sight. Numerical simulations of the fitting procedures demonstrated that an inverse regression of log-rotation velocity on magnitude provided a fit tothe relationship free from selection bias. Residuals around this fit woe used to choose forms of the corrections that produced the minimum scatter. A fullerror budget was compiled and an error weighted fit to the data yielded relationships with a mean scatter of 0.35 mags rms. A combination of all sources of measurement error, considering inter-correlation, produced a value of 027 mags rms, as an estimate of the contribution to the scatter. It was shown that uncertain cluster membership was not a significant source of scatter. In addition, the "expanding duster" model correction suggested in the literature did not significantly reduce the scatter. The most important source of scatter in the relationship was found to be the symmetry and extent of rotation curves used. A significant correlation was shown to exist between rotation curve extent in terms of disk scale lengths and the TF fit residuals. When only the highest quality data were used, the typical scatter was reduced to 020 mags rms. Consideration of the remaining measurement errors produced an upper limit of 0.12 mags rms for the intrinsic scatter within the TF relation. Monte-Carlo modelling indicated that the observed difference in TF slope between the Coma and Abell 2634 samples was significant, The possibility that this difference is the result of systematic errors in the dataset was ruled out It is concluded that the change in gradients is due to real variations in the underlying slope influenced by differences in duster environment.

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This thesis investigates higher order asymptotic properties of a semiparametric averaged derivative estimator. Classical parametric models assume that we know the distribution function of random variables of interest up to finite dimensional parameters, while nonparametric models do not assume this knowledge. Parametric estimators typically enjoy - consistency and asymptotic normality under certain conditions, while nonparametric estimators converge to the true functionals of interest slower than parametric ones. Semiparametric estimators, a compromise between the two, have been intensively studied since the 1970s. Some of them have been shown to have the same convergence rate as parametric estimators despite involving nonparametric functional estimates. Semiparametric methods often suit econometrics because economic theory typically does not provide the whole information on economic variables which parametric methods require, and a sample of very large size is rarely available in econometrics. This thesis treats a semiparametric averaged derivative estimator of single index models. Its first order asymptotic theory has been studied since late 1980s. It has been shown to be n-consistent and asymptotically normally distributed under certain regularity conditions despite involving a nonparametric density estimate. However its higher order properties could be affected by the property of nonparametric estimates. We obtain valid Edgeworth expansions for both normalized and studentized estimators, and moreover show the bootstrap distribution approximates the exact distribution of the estimator asymptotically as well as the Edgeworth expansion for the normalized statistics. We propose optimal bandwidth choices which minimize the normal approximation error using the expansion. We also examine the finite sample performance of the Edgeworth expansions by a Monte Carlo study.

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Please read the abstract in the dissertation. / Dissertation (MSc)--University of Pretoria, 2011. / Statistics / unrestricted

Thesis advisor: Arthur Lewbel / In my doctoral research, I developed econometric estimators with strong applications in analysis of heterogeneous consumer demand. The first chapter develops an estimator for grouped patterns of heterogeneity in an approximately sparse setting. This setting is used to estimate demand shocks, competition sets and own-price elasticities for different groups of consumers. The second chapter, which is joint work with Stefan Hoderlein and Alexander Meister, develops a nonparametric estimator of the marginal effects in a panel data even if there are only a small number of time periods. This is used to estimate the heterogeneous marginal effects of increasing income on consumption of junk food. The third chapter, which is joint work with Stefan Hoderlein and Solvejg Wewal, is the first difference-in-differences model for binary choice outcome variables when treatment effects are heterogeneous. We apply this estimator to examine the heterogeneous effects of a soda tax. Chapter 1: ``Approximately Sparse Models and Methods with Grouped Patterns of Heterogeneity with an Application to Consumer Demand" introduces post-Lasso methods to time-varying grouped patterns of heterogeneity in linear panel data models with heterogeneous coefficients. Group membership is left unrestricted and the model is approximately sparse, meaning the conditional expectation of the variables given the covariates can be well-approximated by a subset of the variables whose identities may be unknown. I estimate the parameters of the model using a “grouped fixed-effects” estimator that minimizes a post-Lasso least-squares criterion with respect to all possible groupings of the cross-sectional units. I provide conditions under which the estimator is consistent as both dimensions of the panel tend to infinity and provide inference methods. Under reasonable assumptions, applying this estimator to a consumer demand application allows me to partition consumers into groups, deal with price endogeneity without instrumental variables, estimate demand shocks, and identify compliments and substitutes for each group. I then use this estimator to estimate demand for soda by identifying different groups' competition sets as well as demand shocks using Homescan data. Chapter 2: In ``A Panel Data Estimator for the Distribution and Quantiles of Marginal Effects in Nonlinear Structural Models with an Application to the Demand for Junk Food", we propose a framework to estimate the distribution of marginal effects in a general class of structural models that allow for arbitrary smooth nonlinearities, high dimensional heterogeneity, and unrestricted correlation between the persistent components of this heterogeneity and all covariates. The main idea is to form a derivative dependent variable using two periods of the panel, and use differences in outcome variables of nearby subpopulations to obtain the distribution of marginal effects. We establish constructive nonparametric identification for the population of ``stayers" (Chamberlain 1982), and show generic non-identification for the ``movers". We propose natural semiparametric sample counterparts estimators, and establish that they achieve the optimal (minimax) rate. Moreover, we analyze their behavior through a Monte-Carlo study, and showcase the importance of allowing for nonlinearities and correlated heterogeneity through an application to demand for junk food. In this application, we establish profound differences in marginal income effects between poor and wealthy households, which may partially explain health issues faced by the less privileged population. Chapter 3: In ``A Binary Choice Difference-in-Differences Model with Heterogeneous Treatment Effects and an Application on Soda Taxes", we answer how should Differences-in-Differences be implemented when outcomes are binary and we expect heterogeneous effects. The scope for applications is clearly vast, including labor force participation, product purchase decisions, enrollment in health insurance and much more. However, assumptions necessary to measure heterogeneous effects in classic Difference-in-Difference models break down with a binary dependent variable. We propose a model with a nonparametric random coefficient formulation that allows for heterogeneous treatment effects with a binary dependent variable. We provide identification of the average treatment effect on the treated (ATT) along with identification of the joint distribution of the actual and counterfactual latent outcome variable in the treatment group which allows us to show the heterogenous treatment effects. We suggest an estimator for the treatment effects and evaluate its finite sample properties with the help of Monte Carlo simulations. We further provide extensions that allow for more flexible empirical applications, such as including covariates. We apply our estimator to analyze the effect of a soft drink tax on consumer's likelihood to consume soda and find heterogeneous effects. The tax reduced the likelihood of consumption for the most consumers but not for those who were most likely to be consuming previously. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.