# Quantile Regression: Theory and Applications

Cristina Davino, Marilena Furno, Domenico Vistocco (Wiley Series in Probability and Statistics, Wiley, 2013). This book explores the theory and numerous applications of quantile regression, offering empirical data analysis as well as the software tools to implement the methods.

The main focus of this book is to provide the reader with a comprehensive description of the main issues concerning quantile regression; these include basic modeling, geometrical interpretation, estimation and inference for quantile regression, as well as issues on validity of the model, diagnostic tools. Each methodological aspect is explored and followed by applications using real data.

• Presents a complete treatment of quantile regression methods, including, estimation, inference issues and application of methods
• Delivers a balance between methodolgy and application
• Offers an overview of the recent developments in the quantile regression framework and why to use quantile regression in a variety of areas such as economics, finance and computing
• Features a supporting website hosting datasets along with R, Stata and SAS software code

• Front Matter

• Chapter 1: A visual introduction to quantile regression
Quantile regression is a statistical analysis that does not restrict attention to the conditional mean and therefore permits to approximate the whole conditional distribution of a response variable. This chapter offers a visual introduction to quantile regression (QR) starting from the simplest model with a dummy predictor, moving then to the simple regression model with a quantitative predictor, through the case of a model with a nominal regressor. It discusses the basic idea behind quantile regression and the essential notation. Most of the examples presented refer to the Cars93 dataset, which contains information on the sales of cars in the USA in 1993, and it is part of the MASS R package. The chapter restricts itself to simple linear regression models in order to introduce the QR logic. However, it is worth noticing that the extension to multiple regression follows the same line of reasoning of classical regression.

• Chapter 2: Quantile regression: Understanding how and why
The development and dissemination of quantile regression (QR) started with the formulation of the QR problem as a linear programming problem. Such formulation allows to exploit efficient methods and algorithms to solve a complex optimization problem offering the way to explore the whole conditional distribution of a variable and not only its center. After an introduction to the linear programming approach for solving the QR problem, this chapter focuses on the added value of QR exploring its features in the case of regression models with homogeneous, heterogeneous and dependent error models. Subsequently, a set of artificial data is used to show several QR features. A section focuses on the interpretation of the QR estimated coefficients by drawing a parallel between homogeneous and heterogeneous regression models.

• Chapter 3: Estimated coefficients and inference
This chapter shows the behavior of quantile regressions in datasets with different characteristics. Using simulated data, the chapter also shows the empirical distribution of the quantile regression estimator in the case of independent and identically distributed (i.i.d.) errors, non-identically distributed (i.ni.d.) errors and dependent (ni.i.d.) errors. The chapter analyzes only the case of i.i.d. errors. It considers a small size real dataset and a very simple linear regression model where wages depend on education, to compare ordinary least squares and quantile regression estimates when the errors are i.i.d. Then the simple linear regression model is extended to comprise more than one explanatory variable, and elements such as age, gender and type of work, dependent or independent, full time or part time, are included. The tests considered here allow to verify hypotheses on more than one coefficient at a time, in order to evaluate the validity of the selected explanatory variables.

• Chapter 4: Additional tools for the interpretation and evaluation of the quantile regression model
To appreciate the meaningful potentialities of quantile regression (QR), it is necessary to have a greater understanding of the interpretations and the evaluation tools. This chapter deals with some typical issues arising from a real data analysis, highlighting the capability of QR and its differences compared with other methods. It discusses the effect of variable centring and scaling on the interpretation of the results, both from a descriptive and from an inferential point of view. The interpretation of QR results can be refined and enhanced by the estimation of the conditional density of the response variable. The chapter also describes the capabilities of bootstrap methods to estimate standard errors in QR. Homogeneous and heterogeneous variance regression models are used to explain the differences and peculiarities of bootstrap methods.

• Chapter 5: Models with dependent and with non-identically distributed data
This chapter focuses on the quantile regression estimators for models characterized by heteroskedastic and by dependent errors. It considers the precision of the quantile regression model in the case of independent and identically distributed (i.i.d.) errors, taking a closer look at the computation of confidence intervals and hypothesis testing on each estimated coefficient. The chapter extends the analysis to the case of non-identically distributed errors, discussing different ways to verify the presence of heteroskedasticity in the data. It takes into account the case of dependent observations and discusses the estimation of an exchange rate equation characterized by serially correlated errors.

Given its properties, quantile regression (QR) can be considered a versatile method for use in several frameworks, unlike the classical regression model. This chapter deals with the main extensions of QR: its application in nonparametric models and nonlinear relationships among the variables, in the presence of censored and longitudinal data, when data are derived from different groups and when the dependent variable is dichotomous. QR is a helpful alternative to classical regression when dealing with censored data, which are quite widespread in many fields from biostatistics to econometrics and social sciences. Models for binary data are quite widespread in many fields of application. In the regression framework, the most widespread approaches to dealing with binary response data are the linear probability model, the logit model and the probit one.

• Appendix A: Quantile regression and surroundings using R
This chapter considers estimation and inference in case of stationary and non-stationary autoregressive processes as estimated by quantile regressions. It reports the

• Appendix B: Quantile regression and surroundings using SAS
This chapter considers estimation and inference in case of stationary and non-stationary autoregressive processes as estimated by quantile regressions. It reports the

• Appendix C: Quantile regression and surroundings using Stata
This chapter considers estimation and inference in case of stationary and non-stationary autoregressive processes as estimated by quantile regressions. It reports the

## Companion website (hosted by Wiley)

View here the companion website containing the datasets, along with the R, Stata and SAS software code for analysing data through quantile regression, from data pre-processing to graphical tools for result interpretation