Book Reviews

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Analyzing Spatial Models of Choice and Judgment with R

David A. ARMSTRONG II, Ryan BAKKER, Royce CARROLL, Christopher HARE, Keith T. POOLE, and Howard ROSENTHAL. Boca Raton, FL: CRC Press 2014, xx+336 pp., $69.95(H), ISBN: 978-1-4665-1715-8.

The purpose of this book is to give a comprehensive introduction to estimating spatial models from political choice data using R. The authors use the term “spatial” to describe distance-based methods which assume that it is possible to compute a distance (or similarity) between each pair of objects in the domain. Such data are also called relational data and spatial models are used to produce a spatial map which contains the same information as the list of distances, but is more interpretable. Spatial models are also used to postulate latent quantities using observed variables. The dimensionality of such a latent space can be interpreted as the number of separate substantial sources of variation among the subjects or objects. In this case, researchers use the relative positions of points in an abstract space to discover and present patterns in the data.

The methodology discussed in the book was developed in fields such as psychology, economics, and political science. In political science, for example, spatial models can be used to measure a voter’s level of conservatism via her position on an ideological scale using a series of survey questions. The spatial theory of voting is surveyed in Chapter 1.

The book considers various data types: data from issue scales, similarity and dissimilarity data, rating scale data, and binary choice data. Each chapter is dedicated to a specific data type and the corresponding spatial models. The authors discuss the assumptions the models make about the data-generating process and about the utility functions used to formalize preferences. Since this is a survey book, the material reflects the state of the art in spatial modeling of choice and judgment. Some models and the subsequent inference enjoy quite a sophisticated treatment and some are somewhat heuristic. In fact, some methods lack formal statistical inference including uncertainty quantification and hypothesis testing. The authors do, however, discuss the use of nonparametric (Chapter 3) and parametric (Chapter 5) bootstrap to estimate uncertainty in a few models. In models for which a Bayesian formulation is given, the uncertainty is estimated from the posterior distribution. Chapters 3, 4 and 6 address Bayesian treatments of issue scales, perceptual, and preferential data models, respectively.

Chapter 7, the last chapter, discusses Bayesian extensions of spatial voting models. The Bayesian analyses reviewed rely heavily on Markov chain Monte Carlo (MCMC) to explore high-dimensional posterior distributions. The use of Bayes Theorem and the incorporation of prior beliefs are not emphasized. While MCMC is a useful tool, users may face difficulties with complex posterior distributions or large datasets. In many cases, the careful use of convergence diagnostics and burn-in that the authors advocate will be sufficient to ensure proper inference. Assessing convergence, however, can be difficult in problems with many parameters and users face additional challenges when the Markov chain exhibits high autocorrelation, even after convergence. Neither of these limitations nor recently developed tools such as variational approximations, which can be used to facilitate the application of Bayesian models, are discussed. The authors do discuss the sensitivity of Bayesian methods to the choice of the prior distribution and of the more general specification of the model, and offer advice on Monte Carlo methods to study the consequences of model misspecification. (Although the book does not discuss robust Bayesian methods it does describe the use of frequentist methods such as nonparametric procedures which are robust against misspecification of the error distribution in classification problems in Chapter 6.)

The book is well organized. Each chapter starts with a description of a particular data type and motivates it using examples. Then, the authors explain the basic theory behind the relevant methods along with their historical developments. Estimation is discussed and demonstrated with the use of examples. The implementation in R is presented along with explanation of the computer code. The authors do not assume the reader is familiar with R; they provide a description of R programming environment in Chapter 2. The R code in the book is well documented and the R outputs are clearly interpreted. The authors finish each chapter with a review where they give a critical analysis of the methodology discussed. Finally, each chapter contains a number of exercises, so that the book can be used as a textbook for a graduate level class in Political Science.

The book is accessible to applied researchers who are more interested in applying the methods than in delving into their underlying theory. The step-by-step instructions given allow the reader to directly apply the methods. The understanding of the theoretical arguments, however, only requires college level algebra.

Tatiyana V. APANASOVICH

George Washington University

Discrete Models of Financial Markets

Marek CAPINSKI and Ekkehard KOPP. Cambridge: Cambridge University Press 2012, ix+181 pp., $44.99(P), ISBN: 978-0-52–117572-2.

This first volume in the Mastering Mathematical Finance series provides an excellent and gentle introduction to the most important concepts in asset pricing theory. The book should be readily accessible and useful to advanced undergraduate students and will complement nicely the standard graduate level texts on the subject. The coverage starts from first principles and builds on quickly toward the first and second fundamental theorem of asset pricing linking the concepts of absence of arbitrage, risk-neutral probabilities, state prices, and replicating portfolios as well as their practical application. Each chapter is peppered with a number of useful exercises with solutions available on the Website that would keep the interested reader busy for some time and curious about the upcoming volumes in the series.

Chapter 1 presents a brief introduction to the concepts and topics that will be covered in the book.

Chapter 2 introduces single-step binomial and trinomial option pricing trees. The authors introduce early on the concept of arbitrage and discuss the restrictions on the binomial model parameters that are needed to avoid arbitrage. The concept of a replicating portfolio coupled with the law of one price quickly lead to the no-arbitrage value of a European call option. The authors also discuss the issues arising in derivative pricing in incomplete markets when there are three states of the world and only two underlying securities available to construct a replicating portfolio. They discuss the sub-replicating and super-replicating strategies and their prices. The chapter also has a brilliant discussion surrounding the limiting case of market completion when the sub-replicating price converges the super-replicating price leading to the case of complete markets. There are also a number of exercises to convince the reader as to the properties of option prices, namely, convexity in the underlying price as well as the strike price and the important concept of put-call parity linking the values of European call and put option to the underlying stock price and the strike price.

Chapter 3 extends the idea of binomial option pricing to two periods and introduces the important notion of filtration as a way of modeling the unfolding of uncertainty over time in a discrete setting. The chapter introduces the workhorse model of discrete time option pricing following Cox, Ross, and Rubinstein (1979) as well as the important concept of hedging the risk of a derivative security using the option’s delta defined in a discrete setting. The authors also present and discuss the concept of martingale as an important property of pricing financial securities under the risk-neutral probability measure.

Chapter 4 extends the ideas of martingales and conditional expectations developed in previous chapters to discuss the pricing of derivatives with multiple-step binomial models. They also introduce the most important theorem in asset pricing, namely, the fundamental theorem of asset pricing linking the absence of arbitrage to a unique risk-neutral probability measure as well as a set of conforming Arrow–Debreu state prices. Several applications are included on calibrating models to match the observed second moments of securities’ returns as well as the risk-free rate of return. The authors also discuss the pricing of forward and futures contracts as well as more complex derivatives like knock-ins, knock-outs, and lookback options in the binomial pricing framework. There are also a few examples of a protective put strategy, a covered call and a butterfly spread.

Chapter 5 discusses the pricing of American-type option where execution can take place at any point prior to the maturity of the option. It introduces the concept of Snell envelope as the discrete-time version of the free boundary of early exercise and discusses its supermartingale properties. The chapter then focuses on optimal stopping times and their properties in discrete time models, especially, the case of an American call option on a dividend-paying stock. The incorporation of the Doob decomposition and its importance to the optimal exercise times for American-type options is of special interest. Finally, the chapter concludes with the presentation of a few bounds on option prices when early exercise is permitted. This includes the practically important generalization to put-call parity to the case of American options where instead of an equality only two inequalities obtain.

Chapter 6 applies the ideas of a binomial process to the pricing of bonds in a multi-period discrete-time setting. It introduces the important concepts of spot and forward interest rates as well as the spot interest rate. The authors apply the binomial model to the pricing of fixed-coupon bonds, floating-rate bonds as well as interest rate swaps. The chapter also contains a detailed discussion of the Ho and Lee (1986) model of the term structure of interest rates as well as a lot of detail on its calibration and properties, notably, the constant variance of the short rate. It would have been nice to also see a brief mention or discussion of other popular fixed income models like Black, Derman, and Toy (1990), Black and Karasinski (1991), and Heath, Jarrow, and Morton (1990, 1992) but perhaps space limitations constrained the choice of the leading fixed income model for the authors to focus on.

This book fills an important gap in the literature between standard undergraduate finance texts and graduate level finance texts. The steep gradient between those two sets of texts presents a void ripe for the taking. In the mind of this reviewer, Discrete Models in Financial Markets fills this void rather nicely and leaves the interested reader with a glimpse of what is to come in the follow-up volumes in the series.

Paskalis GLABADANIDIS

University of Adelaide

Linear Algebra and Matrix Analysis for Statistics

Sudipto BANERJEE and Anindya ROY. Boca Raton, FL: CRC Press 2014, xvii+565 pp., $79.95(H), ISBN: 978-1-4200-9538-8.

There are many books on linear algebra intended for statisticians. After all, linear algebra is the foundation of much of our field and a firm understanding of matrix manipulations, vector spaces, orthogonality, projections, and various matrix decompositions is essential for graduate study in statistics, biostatistics, and related disciplines. The most direct connections between linear algebra and statistics occur in the theory of the linear model, which is undoubtedly why so many linear models experts have written texts on linear/matrix algebra. Books in this category include works by Searle (1982), Harville (1997), Graybill (2001) and Seber (2008). Other matrix analysis texts oriented to statistics include books by Schott (2005) and Gentle (2007). Despite this crowded landscape, Bannerjee and Roy have managed to offer a unique and remarkable book, Linear Algebra and Matrix Analysis for Statistics (or LAMAS hereafter), which has much to offer that is not found elsewhere.

In the preface of LAMAS, the authors distinguish their approach from many of their potential competitors. Their goal was to write a self-contained book that does not assume prior knowledge of linear algebra, suitable for beginning statistics or biostatistics students who may come to these fields from disciplines other than mathematics. While accommodating such an audience, they were mindful of not “dumbing down the subject” (p.xv). Indeed, their book could never be accused of that fault; instead it is rigorous in its presentation and comprehensive in its coverage. The authors suggest that LAMAS could be used “as a companion text in the more theoretical courses on linear regression” or as the basis for “a one-semester course devoted to linear algebra for statistics and econometrics.” (p.xvii)

The book comprises 16 chapters, starting with a basic introduction to matrices, vectors and operations on them (Ch. 1); followed by two chapters on systems of linear equations covering Gaussian and Gauss-Jordan elimination, matrix inverses (without determinants), LU, LDU and Cholesky factorizations (Chs. 2 and 3); a chapter on (Euclidean) vector spaces and the four fundamental subspaces (Ch. 4); a chapter on matrix rank (Ch. 5); a chapter on complimentary subspaces (Ch. 6); and two chapters on orthogonality, including orthogonal subspaces, matrices, and projections (Chs. 7 and 8). Generalized inverses are introduced in Chapter 9, while Chapter 10 is entirely devoted to determinants. Eigenvalues and eigenvectors are the focus of Chapter 11, and singular value and Jordan decompositions are discussed in Chapter 12. The next two chapters deal with quadratic forms (Ch. 13), and the Kronecker product and related tools (Ch. 14). Chapter 15 discusses vector and matrix norms, linear iterative systems of equations, and matrix convergence, and offers a rare (in this book) application of these ideas involving internet search algorithms. The final chapter (Ch. 16) discusses abstract linear algebra over fields, ending with a brief introduction to Hilbert spaces.

The topical outline of the book should make it clear that it is ambitious in scope. While the reader is allowed to start with the basics, s/he is drawn rapidly along into the more advanced topics of linear algebra. In addition, the presentation is no thumbnail sketch. Topics are developed systematically and thoroughly, with a great deal of careful and well-written explanation. Often, important results or concepts are revisited repeatedly after new methods and ideas have been introduced that allow fresh perspectives or insights. Nearly all results are proved in the text, and in many cases the authors offer several proofs of the same theorem to highlight connections between newly introduced ideas and material from earlier in the book.

The book is about matrix algebra per se. While the topics within that field were chosen and given emphasis according to their importance to statistics, the authors do not often explore the statistical problems to which these topics apply. Certainly one reason for this must have been to limit the size and scope of the text; as is, the book is a substantial volume. Nevertheless, I would have liked to learn where some of the more advanced topics connect with statistical methodology. Identifying such links more frequently, even if not pursuing them to any great degree, would be a welcome addition if a second edition is ever pursued.

The broad scope and detailed presentation of LAMAS is both a great strength and a weakness of the text. Most alternatives to this book offer a limited survey of linear algebra, focusing on areas of the field that are most relevant to linear models and traditional multivariate analysis. Bannerjee and Roy’s book is much more comprehensive and offers the reader a deeper and broader treatment of linear algebra that will provide some of the necessary background for more modern and advanced topics in statistics such as functional data analysis, large-p small-n methods, and dimension reduction. And while many of the books on matrix algebra for statisticians provide useful handbooks for quick reference, Bannerjee and Roy’s book provides the basis for systematic and detailed study of the topic. Indeed this is not the source for concise listings of the properties of the trace/determinant/fill-in-the-blank. Given the ease with which such results can be found on the internet, this is perhaps not much of a drawback.

The downside of Bannerjee and Roy’s thorough approach is that the book will provide information overload for many potential readers. While, in principle, a student could start from a modest mathematical background and learn all of the linear algebra that most statisticians would ever need to know (and more) from this book, this would require dedicated study over considerably longer than a single semester. For most students (and most degree programs), a two-pass approach is more feasible in which students learn the basics as an undergraduate or in a remedial graduate course and then return to the subject later to study more advanced topics as needed. Therefore, I am not enthusiastic about this book as a primary text in a course on linear/matrix algebra for statisticians. Rather I think it is an excellent choice as a supplementary resource in courses on linear model theory and more advanced topics, and as the definitive resource on linear algebra for research-oriented statisticians whose work intersects with this branch of mathematics. Whether as a course text or for self-study, the reader will benefit from numerous exercises at the end of each chapter (averaging 27 per chapter).

In Linear Algebra and Matrix Analysis for Statistics, Sudipto Bannerjee and Anindya Roy have raised the bar for textbooks in this genre. For me, this book will be an invaluable resource for my teaching and research. While I do not think that it is the ideal introduction to linear algebra for young statisticians or the best reference for practitioners, it is an outstanding choice for research-oriented statisticians who want a comprehensive theoretical treatment of the subject that will take them well beyond the prerequisites for the study of linear models.

Daniel B. HALL

University of Georgia

Richly Parameterized Linear Models: Additive, Time Series, and Spatial Models Using Random Effects

James S. HODGES. Boca Raton, FL: CRC Press, 2013, xxxviii+431 pp., $94.95(H), ISBN: 978-1-4398-6683-2.

Richly Parameterized Linear Models is a strong addition to the literature on mixed models, offering a much more unified treatment and broader scope than many existing texts. The book is intended for applied researchers and covers a range of individual topics, including penalized splines, spatial analysis, time series, and even Bayesian analysis. The book also provides an excellent treatment of diagnostics for mixed models. Although Chapter 1 provides a brief survey of mixed linear models, researchers already experienced in generalized linear models and mixed linear models will benefit the most from this book.

A unique aspect of the book is its informal narrative. This is a refreshing break from existing texts but is also counterproductive at times. For example, the book regularly discusses “old-style” and “new-style” random effects. The distinction between the two is necessary to treat several models under a common framework, but this characterization may ultimately impede readers looking for a more clear-cut distinction between different modeling assumptions.

Broadly, the book centers around two objectives. The first is to organize under a common framework a series of models typically examined separately. In particular, Chapters 1–7 serve to reexpress several models as a variation of a standard mixed linear model. With its broad scope, the book necessarily sacrifices some theoretical details; however, for applied researchers most interested in the mechanics of mixed linear models and extensions, the book largely succeeds in this first endeavor.

The second objective is to examine in detail the underlying assumptions of different models and the effects of such assumptions on results in practice. The book’s opening quote characterizes this objective succinctly, “if you believe in things that you don’t understand, then you suffer.” This theme persists in all of the book’s content, and is particularly prevalent in Chapters 8–19, which provide an excellent treatment of open questions in the literature and difficult problems encountered in practice.

The literature on mixed models has expanded in recent decades, often in disparate ways. Richly Parameterized Linear Models provides a step toward unifying this growing area of research and serves as an excellent resource for applied researchers with experience and interest in mixed models.

Ian M. MCCARTHY

Emory University