Regression Modeling with Actuarial and Financial Applications

Edward (Jed) Frees, University of Wisconsin - Madison

Preface

This is the online version of the book published by Cambridge University Press in 2009. You can purchase a copy of this book directly from Cambridge or your preferred bookstore.

Forward

Actuaries and other financial analysts quantify situations using data – we are ‘numbers’ people. Many of our approaches and models are stylized, based on years of experience and investigations performed by legions of analysts. However, the financial and risk management world evolves rapidly. Many analysts are confronted with new situations in which tried-and-true methods simply do not work. This is where a toolkit like regression analysis comes in.

Regression is the study of relationships among variables. It is a generic statistics discipline that is not restricted to the financial world – it has applications in fields of social, biological and physical sciences. You can use regression techniques to investigate large and complex data sets. To familiarize you with regression, this book explores many examples and data sets based on actuarial and financial applications. This is not to say that you will not encounter applications outside of the financial world (for example, an actuary may need to understand the latest scientific evidence on genetic testing for underwriting purposes). However, as you become acquainted with this toolkit, you will see how regression can be applied in many (and sometimes new) situations.

Who Is This Book For?

This book is written for financial analysts who face uncertain events and wish to quantify these events using empirical information. No industry knowledge is assumed although readers will find the reading much easier if they have an interest in the applications discussed here! This book is designed for students who are just being introduced to the field as well as industry analysts who would like to brush up on old techniques and (for the later chapters) get an introduction to new developments.

To read this book, I assume knowledge comparable to a one semester introduction to probability and statistics - Appendix A1 provides a brief review to brush up if you are rusty. Actuarial students in North America will have a one-year introduction to probability and statistics - this type of introduction will help readers grasp concepts more quickly than a one semester background. Finally, readers will find matrix, or linear, algebra helpful, although not a prerequisite for reading this text.

Different readers are interested in understanding statistics at different levels. This book is written to accommodate the ‘armchair reader,’ that is, one who passively reads and does not get involved by attempting the exercises in the text. Consider an analogy to football, or any other game. Just like the armchair quarterback of football, there is a great deal that you can learn about the game just by watching. However, if you want to sharpen your skills, you have to go out and play the game. If you do the exercises or reproduce the statistical analyses in the text, you will become a better player. Still, this text is written by interweaving examples with the basic principles. Thus, even the armchair reader can obtain a solid understanding of regression techniques through this text.

What Is This Book About?

The Table of Contents provides an overview of the topics covered, organized into four parts. The first part introduces linear regression. This is the core material of the book, with refreshers on mathematical statistics, distributions and matrix algebra woven in as needed.

The second part is devoted to topics in times series. Why integrate time series topics into a regression book? The reasons are simple, yet compelling; most accounting, financial and economic data become available over time. Although cross-sectional inferences are useful, business decisions need to be made in real time with currently available data. Chapters 7-10 introduce time series techniques that can be readily accomplished using regression tools (and there are many).

Nonlinear regression is the subject of the third part. Many modern day ‘predictive modeling’ tools are based on nonlinear regression - these are the workhorses of statistical shops in the financial and risk management industry.

The fourth part concerns ‘actuarial applications,’ topics that I have found relevant in my research and consulting work in financial risk management. The first four chapters of this part consists of variations of regression models that are particularly useful in risk management. The last two chapters focus on communications, specifically, report writing and designing graphs. Communicating information is an important aspect of every technical discipline and statistics is certainly no exception.

How Does This Book Deliver Its Message?

Chapter Development. Each chapter has several examples interwoven with theory. In chapters where a model is introduced, I begin with an example and discuss the data analysis without regard to the theory. This analysis is presented at an intuitive level, without reference to a specific model. This is straightforward, because it amounts to little more than curve fitting. The theme is to have students summarize data sensibly without having the notion of a model obscure good data analysis. Then, an introduction to the theory is provided in the context of the introductory example. One or more additional examples follow that reinforce the theory already introduced and provide a context for explaining additional theory. In Chapters 5 and 6, that do not introduce models but rather techniques for analysis, I begin with an introduction of the technique. This introduction is then followed by an example that reinforces the explanation. In this way, the data analysis can be easily omitted without loss of continuity, if time is a concern.

Real Data. Many of the exercises ask the reader to work with real data. The need for working with real data is well documented; for example, see Hogg (1972) or Singer and Willett (1990). Some criteria of Singer and Willett for judging a good data set include: (1) authenticity, (2) availability of background information, (3) interest and relevance to substantive learning, and (4) availability of elements with which readers can identify. Of course, there are some important disadvantages to working with real data. Data sets can quickly become outdated. Further, the ideal data set to illustrate a specific statistical issue is difficult to find. This is because with real data, almost by definition, several issues occur simultaneously. This makes it difficult to isolate a specific aspect. I particularly enjoy working with large data sets. The larger the data set, the greater is the need for statistics to summarize the information content.

Statistical Software and Data. My goal in writing this text is to reach a broad group of students and industry analysts. Thus, to avoid excluding large segments, I chose not to integrate any specific statistical software package into the text. Nonetheless, because of the applications orientation, it is critical that the methodology presented be easily accomplished using readily available packages. For the course taught at the University of Wisconsin, I use the statistical packages SAS and R. On the

Book Web Site

users will find scripts written in SAS and R for the analysis presented in the text. The data are available in text format, allowing readers to employ any statistical packages that they wish. When you see a display such as this in the margin, you will also be able to find this data set (TermLife) on the book web site.

Code in the Online Version. Most of the illustrative calculations for the book, written about 2008, were done in SAS. In keeping with modern approach to analytics, the illustrative code provided in the online version are in the statistical software R. Thus, users should expect to see some some discrepancies between the results in the book and those that are calculated using the R software.

Technical Supplements. The technical supplements reinforce and extend the results in the main body of the text by giving a more formal, mathematical treatment of the material. This treatment is in fact a supplement because the applications and examples are described in the main body of the text. For readers with sufficient mathematical background, the supplements provide additional material that is useful in communicating to technical audiences. The technical supplements provide a deeper, and broader, coverage of applied regression analysis.

I believe that analysts should have an idea of ‘what is going on under the hood,’ or ‘how the engine works.’ Most of these topics will be omitted from the first reading of the material. However, as you work with regression, you will be confronted with questions on ‘Why?’ and you will need to get into the details to see exactly how a certain technique works. Further, the technical supplements provide a menu of optional items that an instructor may wish to cover.

Suggested Courses. There is a wide variety of topics that can go into a regression course. Here are some suggested courses. The course that I teach at the University of Wisconsin is the first on the list in the following table.

\[ {\small \begin{array}{ll} \begin{array}{lll} \hline \textbf{Audience} & \textbf{Nature of Course}& \textbf{Suggested Chapters} \\ \hline \text{One-year background in} & \text{Survey of regression and} & \text{Chapters } 1-8 ,11-13, 20-21,\\ ~~~\text{probability and statistics} & ~~~\text{time series models} & ~~~\text{main body of text only} \\ \text{One-year background in} & \text{Regression and time} & \text{ Chapters } 1-8, 20-21, \text{selected} \\ ~~~\text{probability and statistics} & ~~~\text{series models} & ~~~\text{portions of technical supplements} \\ \text{One-year background in} & \text{Regression modeling} & \text{Chapters } 1-6, 11-13, 20-21, \text{selected} \\ ~~~\text{probability and statistics} & & ~~~ \text{portions of technical supplements} \\ \text{Background in statistics}& \text{Actuarial regression}& \text{ Chapters } 10-21, \text{selected} \\ ~~~\text{and linear regression}& ~~~\text{models} & ~~~ \text{portions of technical supplements} \\ \hline \end{array} \end{array} } \]

In addition to these suggested courses, this book is designed for supplemental reading for a time series course as well as a reference book for industry analysts. My hope is that college students who use the beginning parts of the book in their university course will find the later chapters helpful in their industry positions. In this way I hope to promote life-long learning!

Acknowledgements

It is appropriate to begin the acknowledgement section by thanking the students in the actuarial program here at the University of Wisconsin; students are important partners in the knowledge creation and dissemination business at universities. Through their questions and feedback, I have learned a tremendous amount over the years. I have also benefited from excellent assistance from those who have helped me pull together all the pieces for this book, specifically, Missy Pinney, Peng Shi, Yunjie (Winnie) Sun and Ziyan Xie.

I have enjoyed working with several former students and colleagues on regression problems over the recent years, including Katrien Antonio, Jie Gao, Paul Johnson, Margie Rosenberg, Jiafeng Sun, Emil Valdez and Ping Wang. Their contributions are reflected indirectly throughout the text. Because of my long association with the University of Wisconsin-Madison, I am reluctant to go back further in time and provide a longer list for fear of missing important individuals. I have also been fortunate to have a more recent association with the Insurance Services Office (ISO). Colleagues at ISO have provided me with important insights into applications. Through this text that features applications of regression into actuarial and financial industry problems, I hope to encourage the fostering of additional partnerships between academia and industry.

I am pleased to acknowledge detailed reviews that I have received from colleagues Tim Welnetz and Margie Rosenberg. I also wish to thank Bob Miller for permission to include our joint work on designing effective graphs in Chapter 21. Bob has taught me a lot about regression over the years.

Moreover, I am happy to acknowledge financial support through the Assurant Health Professorship in Actuarial Science at the University of Wisconsin-Madison.

Saving the most important for last, I thank my family for their support. Ten thousand thanks to my mother Mary, brothers Randy, Guy and Joe, my wife Deirdre and our sons Nathan and Adam.

Dedication

There is an old saying, attributed to Sir Issac Newton and that can be found on web at Google Scholar,

If I have seen far, it is by standing on the shoulders of giants.

I dedicate this book to the memory of two giants who helped me, and everyone who knew them, see farther and live better lives:

James C. Hickman

and

Joseph P. Sullivan.