Although econometrics is a staple course in the business curriculum, differential geometry has only recently taken its place as a legitimate tool in statistical analysis. This is one of the first texts to encourage the use of differential geometry in the study and use of econometrics in a variety of quantitative models.

The introductory chapter is basically a tutorial in differential geometry and its application. The treatment is not exhaustive, but it is sufficient to provide an understanding of the remainder of the text.

Chapter 2 deals with nested models, orthogonal projection, and encompassing. The main focus of the chapter is on congruent and encompassing models. After a brief introduction, nesting and restricted models are covered. Subsequently, the concept of parsimonious encompassing models is detailed. Section 4 deals with Hilbert space.

Chapter 3 is an exhaustive treatment of the exact properties of the maximum likelihood estimator in exponential regression models. Chapter 4 continues this material by covering empirical likelihood estimation and inference.

The aim of chapter 5 is the construction of a general geometrical setting, based on Hilbert space, in which the researcher may study various estimation techniques, with an emphasis on efficiency and robustness. Major topics in chapter 6 include Roa distances, the human capital model as a stochastic earnings frontier, and differentials in earnings and inefficiency. Chapter 7 deals with first-order optimal predictive densities, with an emphasis on the form of optimal solutions. Chapter 8 is a bit of a departure: it covers the application of differential geometry to statistics, and especially the effects of curvature on hypothesis testing.

Chapter 9, “Testing for Units in AR and MA Models,” looks at three examples that arise in nonstationary time series analysis: testing for a unit root in a first-order autoregressive model and in two variants of a first-order moving-average model. The text concludes with chapter 10, whose primary objective is to mitigate certain drawbacks in the use of differential geometry in the context of Amari’s expected geometric structure on a full exponential family.

The book is sophisticated. However, if differential geometry is your area of expertise or interest, the book will extend that interest. The bibliographies at the end of each chapter are extensive.