Introduction

In this introductory chapter we seek to cover sufficient differential geometry in order to understand its application to econometrics. It is not intended to be a comprehensive review either of differential geometric theory, or of all the applications that geometry has found in statistics. Rather it is aimed as a rapid tutorial covering the material needed in the rest of this volume and the general literature. The full abstract power of a modern geometric treatment is not always necessary and such a development can often hide in its abstract constructions as much as it illuminates.

In section 2 we show how econometric models can take the form of geometrical objects known as manifolds, in particular concentrating on classes of models that are full or curved exponential families.

This development of the underlying mathematical structure leads into section 3, where the tangent space is introduced. It is very helpful to be able to view the tangent space in a number of different but mathematically equivalent ways, and we exploit this throughout the chapter.

Section 4 introduces the idea of a metric and more general tensors illustrated with statistically based examples. Section 5 considers the most important tool that a differential geometric approach offers: the affine connection. We look at applications of this idea to asymptotic analysis, the relationship between geometry and information theory and the problem of the choice of parameterisation. Section 6 introduces key mathematical theorems involving statistical manifolds, duality, projection and finally the statistical application of the classic geometric theorem of Pythagoras.