An Introduction to Di¤erential Geometry in Econometrics

In this introductory chapter we seek to cover su¢cient di¤erential geometry in order to understand its application to Econometrics. It is not intended to be a comprehensive review of either di¤erential geometric theory, nor of all the applications which 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 which 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 view the tangent space in a number of di¤erent, 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 di¤erential geometric approach o¤ers, the a¢ne 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. The last two sections look at direct applications of this geometric framework. In particular at the problem of inference in curved families and at the issue of information loss and recovery. Note that while this chapter aims to give a reasonably precise mathematical development of the required theory an alternative and perhaps more intuitive approach can be found in the chapter by Critchley, Marriott and Salmon later in this volume. For a more exhaustive and detailed review of current geometrical statistical theory see Kass and Vos (1997) or from a more purely mathematical background, see Murray and Rice (1993).