M.Sc. (Mathematical Statistics) / While a typical regression model describes how the mean value of a response variable varies with a set of explanatory variables, quantile regression describes the variation in the quantiles of the response. When the response distribution di ers substantially from normality the quantiles provide a substantially richer description of the distribution than can be obtained by standard regression, and is obtainable without making any assumptions on the form of the underlying distribution. In this dissertation we study the theory of quantile regression models, with particular focus on the application of quantile regression methods to censored survival data. While the statistical literature on censored quantile regression methods is extensive, the computational di culties and complicated inferential and asymptotic arguments associated with many of these approaches present a considerable stumbling block in the routine application of the methodology. We discuss in detail a more recent approach which is based on counting processes and martingale properties associated with counting processes. The inferential and asymptotic properties of this method provides some notable advantages over comparable methods. The performance of the method is examined using Monte Carlo Simulation, as well as an application to a large loan portfolio of a nancial institution.

The goal of this paper is to do some basic proofs for lasso and have a deep understanding of linear regression. In this paper, firstly I give a review of methods in linear regression, and most concerns with the method of lasso. Lasso for ¡®least absolute shrinkage and selection operator¡¯ is a regularized version of method adds a constraint which uses norm less or equal to a given value t. By doing so, some predictor coefficients would be shrank and some others might be set to 0. We can attain good interpretation and prediction accuracy by using lasso method. Secondly, I provide some basic proofs for lasso, which would be very helpful in understanding lasso. Additionally, some geometric graphs are also given and one example is illustrated.

Thesis advisor: Zhijie Xiao / This dissertation contains three essays. It provides an application of quantile regression in Financial Economics. The first essay investigates whether tail dependence makes a difference in the estimation of systemic risk. This chapter develops a common framework based on a copula model to estimate several popular return-based systemic risk measures: Delta Conditional Value at Risk (ΔCoVaR) and its modification; and Marginal Expected Shortfall (MES) and its extension, systemic risk measure (SRISK). By eliminating the discrepancy of the marginal distribution, copula models provide the flexibility to concentrate only on the effects of dependence structure on the systemic risk measure. We estimate the systemic risk contributions of four financial industries consisting of a large number of institutions for the sample period from January 2000 to December 2010. First, we found that the linear quantile regression estimation of ΔCoVaR, proposed by Adrian and Brunnermeier (AB hereafter) (2011), is inadequate to completely capture the non-linear contagion tail effect, which tends to underestimate systemic risk in the presence of lower tail dependence. Second, ΔCoVaR originally proposed by AB (2011) is in conflict with dependence measures. By comparison, the modified version of ΔCoVaR put forward by Girardi et al. (2011) and MES, proposed by Acharya et al. (2010), are more consistent with dependence measures, which conforms with the widely held notion that stronger dependence strength results in higher systemic risk. Third, the modified ΔCoVaR is observed to have a strong correlation with tail dependence. In contrast, MES is found to have a strong empirical relationship with firms' conditional CAPM beta. SRISK, however, provides further connection with firms' level characteristics by accounting for information on market capitalization and liability. This stylized fact seems to imply that ΔCoVaR is more in line with the ``too interconnected to fail" paradigm, while SRISK is more related to the ``too big to fail" paradigm. In contrast, MES offers a compromise between these two paradigms. The second essay proposes a quantile regression approach to stock return prediction. I show that incorporating distributional information together with combining model information can produce a superior forecast for the conditional mean as well as the entire distribution of future equity premium, which significantly outperforms the forecast that utilizes either source of information alone. Meanwhile, the order of combination strategies appears to make a difference in the efficiency of pooling both distributional information and model information. It turns out that aggregating distributional information in the first step, followed by combining model information in the second step is more advantageous in return forecast than the alternative combination strategies which reverse the order of combination strategy. Furthermore, the forecast based on LASSO model selection can be significantly improved as well if the distributional information is further incorporated. In other word, aggregating distributional information via combining multiple quantiles estimators contributes to the improvement of forecasts obtained either from model combination or model selection. This paper not only investigates the forecast of conditional mean, but also studies the forecast of the whole distribution of future stock returns. The approaches of quantile combination together with either model combination or model selection turn out to deliver statistically and economically significant out-of-sample forecasts relative to a historical average benchmark. The third essay proposes a quantile-based approach to efficiently estimate the conditional beta coefficient without assuming a parametric structure on the distribution of data generating process. Multiple quantiles estimates are combined in a weighting scheme to utilize distributional information across different quantile of the distribution. Monte Carlo simulation demonstrated that combining multiple quantile estimates can substantially improve the estimation efficiency for beta risk estimates in the absence of Gaussian distribution. The robustness of quantile-based beta estimates are pronounced during financial crisis when the distribution of stock returns deviates most from normality. I also explored the performance of different beta estimators in an application of portfolio management analysis and found that beta estimates from the proposed quantile combination approaches are superior to the OLS estimates in constructing Global Minimum Variance Portfolio, which generates lower variance of portfolio but does not come at the expense of persistent lower returns. / Thesis (PhD) — Boston College, 2013. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.

University of Macau / Faculty of Science and Technology. / Department of Mathematics

by Wai-leung Li. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-77). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter § 1.1 --- Why transformation of variables in regression analysis is needed? --- p.1 / Chapter § 1.2 --- Suggested functional transformation -- Box-Cox Transformation --- p.3 / Chapter § 1.3 --- Methodology --- p.5 / Chapter § 1.4 --- General theory of constructing asymptotic confidence intervals and confidence regions --- p.9 / Chapter § 1.4.1 --- Method based on the log-likelihood ratio statistic --- p.9 / Chapter § 1.4.2 --- Method based on the asymptotic normality of the maximum likelihood estimates --- p.13 / Chapter § 1.4.3 --- Method based on the score statistic --- p.15 / Chapter § 1.5 --- General theory of constructing exact confidence intervals and confidence regions --- p.17 / Chapter § 1.6 --- Summary --- p.23 / Chapter Chapter 2 --- Confidence Intervals for the non-linear parameter λ in the Box-Cox transformation models --- p.24 / Chapter § 2.1 --- Confidence intervals based on the log-likelihood ratio statistics --- p.26 / Chapter § 2.1.1 --- Asymptotically equivalent forms for constructing confidence intervals based on the log-likelihood ratio statistics --- p.30 / Chapter § 2.2 --- Confidence intervals based on the asymptotic normality of the maximum likelihood estimates --- p.31 / Chapter § 2.3 --- Confidence intervals based on the score statistics --- p.35 / Chapter § 2.4 --- Confidence intervals based on the exact test --- p.36 / Chapter § 2.5 --- Small simulation studies of constructing confidence intervals for A based on the four different methods --- p.37 / Chapter § 2.5.1 --- Design of the simulation studies --- p.40 / Chapter § 2.5.2 --- Simulation results --- p.41 / Chapter § 2.6 --- Summary --- p.44 / Chapter Chapter 3 --- Confidence Regions for the parameters in the Box-Cox transformation models --- p.45 / Chapter § 3.1 --- Confidence regions based on the log-likelihood ratio statistics --- p.45 / Chapter § 3.1.1 --- "Confidence region for (λ,ζ1）based on the log-likelihood ratio statistics" --- p.46 / Chapter § 3.1.2 --- Confidence region for (ζ1）based on the log-likelihood ratio statistics --- p.51 / Chapter § 3.2 --- Confidence regions based on the asymptotic normality of the maximum likelihood estimates --- p.53 / Chapter § 3.2.1 --- "Confidence region for (λ,ζ1）based on the asymptotic normality of the maximum likelihood estimates" --- p.53 / Chapter § 3.2.2 --- Confidence region for (ζ1）based on the asymptotic normality of the maximum likelihood estimates --- p.57 / Chapter § 3.3 --- Confidence regions based on the score statistics --- p.58 / Chapter § 3.3.1 --- "Confidence region for (λ,ζ1） based on the score statistic" --- p.59 / Chapter § 3.3.2 --- Confidence region for (ζ1 ) based on the score statistic --- p.60 / Chapter § 3.4 --- Confidence region based on the exact test --- p.61 / Chapter § 3.5 --- Small simulation studies of constructing confidence regions for the parameters of interest based on the four different methods --- p.62 / Chapter Chapter 4 --- Robustness and Discussion --- p.67 / Chapter §4.1 --- Contamination normal distribution --- p.67 / Chapter § 4.1.1 --- Confidence intervals for the non- linear parameter λ based on the contamination normal distribution of error terms --- p.68 / Chapter § 4.1.2 --- Confidence regions for the parameters of interest based on the contamination normal distribution of the error terms --- p.70 / Chapter § 4.2 --- Summary --- p.72 / References --- p.74 / Figures / Appendix A / Appendix B / Appendix C / Appendix D

The goal of this thesis is to propose a new quantile regression approach to identify and estimate the quantiles of the private value conditional distribution in ascending and rst price auctions under the Independent Private Value (IPV) paradigm. The quantile regression framework provides a exible and convenient parametrization of the private value distribution, which is not a ected by the curse of dimensionality. The rst Chapter of the thesis introduces a quantile regression methodology for ascending auctions. The Chapter focuses on revenue analysis, optimal reservation price and its associated screening level. An empirical application for the USFS timber auctions suggests an optimal reservation price policy with a probability of selling the good as low as 58% for some auctions with two bidders. The second Chapter tries to address this issue by considering a risk averse seller with a CRRA utility function. A numerical exercise based on the USFS timber auctions shows that increasing the CRRA of the sellers is su cient to give more reasonable policy recommendations and a higher probability of selling the auctioned timber lot. The third Chapter develops a quantile regression methodology for rst-price auction. The estimation method combines local polynomial, quantile regression and additive sieve methods. It is shown in addition that the new quantile regression methodology is not subject to boundary issues. The choice of smoothing parameters is also discussed.

by Maggie Wai Ling Poon. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 70-78). / Abstract and questionnaire also in Chinese. / Chapter 1. --- ABSTRACT --- p.3 / Chapter 2. --- ACKNOWLEDGEMENTS --- p.4 / Chapter 3. --- INTRODUCTION --- p.5 / Cognitive factors / The content specificity hypothesis / Empirical support of the cognitive model / Purpose of the present study / Chapter 4. --- METHOD --- p.21 / Subjects / Measures / Chapter 5. --- RESULTS --- p.28 / Psychometric properties / Correlational analysis / Hierarchical regression analyses / Chapter 6. --- DISCUSSION --- p.53 / Psychometric properties of instruments / Empirical support to the cognitive model / Implications of the study / Summary and conclusion / Comments and future direction / Chapter 7. --- REFERENCES --- p.70 / Chapter 8. --- APPENDICES --- p.79

Digitized by Kansas Correctional Industries

Typescript (photocopy). / Digitized by Kansas Correctional Industries

Wenyang Zhang. / "October 1998." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (p. 190-196). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.