This dissertation contains four topics in time series data analysis. First, we propose the oracle model selection for autoregressive time series when the observations are contaminated with trend. An adaptive least absolute shrinkage and selection operator (LASSO) type model selection method is used after the trend is estimated by non-parametric B-splines method. The first step is to estimate the trend by B-splines method and then calculate the detrended residuals. The second step is to use the residuals as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an Adaptive Lasso model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The extensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements. Second, we generalize our first topic to a more general class of time series using the autoregressive and moving-average (ARMA) model. The ARMA model class is the building block for stationary time series analysis. We adopt the two-step method non-parametric trend estimation with B-spline and model selection and model estimation with the adaptive LASSO. We prove that such model selection and model estimation procedure possesses the oracle properties. Another important objective of this topic is forecasting time series with trend. We approach the forecasting problem by two methods: the empirical method by using the one-step ahead prediction in time series and the bagging method. Our simulation studies show that both methods are efficient with the decreased mean square error when the sample size increases. Simulation studies are adopted to illustrate the asymptotic result of our proposed method for model selection and model estimation with twelve ARMA(p, q) models, in which p an q are in the range from 1 to 15. The method is also illustrated by two time series data sets from the New York State Energy Research and Development Authority (NYSERDA), a public benefit corporation which offers data and analysis to help New Yorkers increase energy efficiency. Third, we propose a new model class, which is motivated by lag effects of covariates on the dependent variable. Our paper aims at providing more accurate statistical analysis for the relationship, for example, between the outcome of an event that occurs once every several years and the covariates that have observations every year. Lag effects have received a great deal of attention since Almon (1965) proposed linear distributed lag models to model the dependence of time series on several regressors from a correlated sequence. Motivated by the linear distributed lag model, we propose distributed generalized linear models as well as the estimation procedure for the model coefficients. The estimators from our proposed procedure are shown to be oracle or asymptotically efficient. Simulation studies confirm the asymptotic properties of the estimators and present consequences of model misspecification as well as better model prediction accuracy. The application is illustrated by analysis of the presidential election data in 2016. Fourth, we aim to provide an easy-to-use data analysis procedure for linear regression with non-independent errors. In practice, errors in regression model may be non-independent. In such situation, it is usually suitable to assume that the error terms for the model follow a time series structure. In fact, this type of model structure (reffered as RegARMA) has received great interests from researchers. Pierce (1971) discussed a nonlinear least squares estimation of RegARMA; Greenhouse et al. (1987) studied biological rhythm data by using the RegARMA model. Recently, Wu and Wang (2012) used the shrinkage estimation procedure to analyze data using RegARMA. However, in the literature the trend factor of the time series has not been considered. We will use the same idea of the two step-procedure as in the first project and the second project for our approach. We first estimate the trend of the time series by using a non-parametric method such as B-spline or linear Kernel. We then use the adaptive LASSO method for model selection and model estimation of the linear part and the time series error part. Simulation results show that our approach works quite well. However, it would be very interesting and challenging to improve the estimations and extend the estimation method to more complicated models, which will be the focus of the future research.

Shrinkage and selection. In this article, we extend its application to the REGression model with AutoRegressive errors (REGAR). Two types of lasso estimators are carefully studied. The first is similar to the traditional lasso estimator with only two tuning parameters (one for regression coefficients and the other for autoregression coefficients). These tuning parameters can be easily calculated via a data driven method, but the resulting lasso estimator may not be fully efficient (Fan and Li, 2001). In order to overcome this limitation, we propose a second lasso estimator which uses different tuning parameters for each coefficient. We show that this modified lasso is able to produce the estimator as efficiently as the oracle. Moreover, we propose an algorithm for tuning parameter estimates to obtain the modified lasso estimator. Simulation studies demonstrate that the modified estimator is superior to the traditional one. One empirical example is also presented to illustrate the usefulness of lasso estimators. The extension of lasso to the autoregression with exogenous variables (ARX) model is briefly discussed.

A comprehensive resource that draws a balance between theory and applications of nonlinear time series analysis Nonlinear Time Series Analysis offers an important guide to both parametric and nonparametric methods, nonlinear state-space models, and Bayesian as well as classical approaches to nonlinear time series analysis. The authors—noted experts in the field—explore the advantages and limitations of the nonlinear models and methods and review the improvements upon linear time series models. The need for this book is based on the recent developments in nonlinear time series analysis, statistical learning, dynamic systems and advanced computational methods. Parametric and nonparametric methods and nonlinear and non-Gaussian state space models provide a much wider range of tools for time series analysis. In addition, advances in computing and data collection have made available large data sets and high-frequency data. These new data make it not only feasible, but also necessary to take into consideration the nonlinearity embedded in most real-world time series. This vital guide: • Offers research developed by leading scholars of time series analysis • Presents R commands making it possible to reproduce all the analyses included in the text • Contains real-world examples throughout the book • Recommends exercises to test understanding of material presented • Includes an instructor solutions manual and companion website Written for students, researchers, and practitioners who are interested in exploring nonlinearity in time series, Nonlinear Time Series Analysis offers a comprehensive text that explores the advantages and limitations of the nonlinear models and methods and demonstrates the improvements upon linear time series models.

Part I. Unit roots and trend breaks -- Part II. Structural change

Introduction to Time Series Using Stata, Revised Edition, by Sean Becketti, is a practical guide to working with time-series data using Stata. In this book, Becketti introduces time-series techniques--from simple to complex--and explains how to implement them using Stata. The many worked examples, concise explanations that focus on intuition, and useful tips based on the author's experience make the book insightful for students, academic researchers, and practitioners in industry and government.Becketti is a financial industry veteran with decades of experience in academics, government, and private industry. He was also a developer of Stata in its infancy and has been a regular Stata user since its inception. He wrote many of the first time-series commands in Stata. With his abundant knowledge of Stata and extensive experience with real-world time-series applications, Becketti provides readers with unique insights and motivation throughout the book.For those new to Stata, the book begins with a mild yet fast-paced introduction to Stata, highlighting all the features you need to know to get started using Stata for time-series analysis. Before diving into analysis of time series, Becketti includes a quick refresher on statistical foundations such as regression and hypothesis testing.The discussion of time-series analysis begins with techniques for smoothing time series. As the moving-average and Holt-Winters techniques are introduced, Becketti explains the concepts of trends, cyclicality, and seasonality and shows how they can be extracted from a series. The book then illustrates how to use these methods for forecasting. Although these techniques are sometimes neglected in other time-series books, they are easy to implement, can be applied quickly, often produce forecasts just as good as more complicated techniques, and, as Becketti emphasizes, have the distinct advantage of being easily explained to colleagues and policy makers without backgrounds in statistics.Next, the book focuses on single-equation time-series models. Becketti discusses regression analysis in the presence of autocorrelated disturbances as well as the ARIMA model and Box-Jenkins methodology. An entire chapter is devoted to applying these techniques to develop an ARIMA-based model of U.S. GDP; this will appeal to practitioners, in particular, because it goes step by step through a real-world example: here is my series, now how do I fit an ARIMA model to it? The discussion of single-equation models concludes with a self-contained summary of ARCH/GARCH modeling.In the final portion of the book, Becketti discusses multiple-equation models. He introduces VAR models and uses a simple model of the U.S. economy to illustrate all key concepts, including model specification, Granger causality, impulse-response analyses, and forecasting. Attention then turns to nonstationary time-series. Becketti masterfully navigates the reader through the often-confusing task of specifying a VEC model, using an example based on construction wages in Washington, DC, and surrounding states.Introduction to Time Series Using Stata, Revised Edition, by Sean Becketti, is a first-rate, example-based guide to time-series analysis and forecasting using Stata. This is a must-have resource for researchers and students learning to analyze time-series data and for anyone wanting to implement time-series methods in Stata. [ed.]

This book teaches multiple regression and time series and how to use these to analyze real data in risk management and finance.

This volume in the Challenges in Machine Learning series gathers papers from the Mini Symposium on Causality in Time Series, which was part of the Neural Information Processing Systems (NIPS) confernce in 2009 in Vancouver, Canada. These papers present state-of-the-art research in time-series causality to the machine learning community, unifying methodological interests in the various communities that require such inference.