By Paul Marriott, Mark Salmon

Differential geometry has turn into a customary device within the research of statistical types, delivering a deeper appreciation of present methodologies and highlighting the problems that may be hidden in an algebraic improvement of an issue. This quantity is the 1st to use those thoughts to econometrics. An introductory bankruptcy offers a quick educational for these surprising with the instruments of differential geometry. the subsequent chapters supply purposes of geometric the way to useful strategies and provide perception into difficulties of econometric inference.

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**Example text**

All geodesics are the one-dimensional af®ne subspaces. This is the simplest possible geometry. When the Riemann curvature is non-zero, the fundamental geometry of the manifold creates an obstruction to the existence of 52 Paul Marriott and Mark Salmon such an af®ne parameterisation. In this case the geodesic normal parameterisation is, in a sense, closest to being truly af®ne. In the case where the connections come from the expected geometric statistical manifold, then af®ne parameterisations, when they exist, and geodesic normal parameters, when they do not exist, have useful statistical properties.

In the case of a full exponential family, the ®rst-order bias of the maximum likelihood estimate is given by bi À1 ij kl À1 g g Àjkl : 2n 27 Consider this formula in the context of linear models. Example 1. The linear model (continued) There is a natural geometric interpretation for ordinary least squares (OLS) estimation, which connects Euclidean geometry and regression (see, for example, Bates and Watts (1988)). For presentational simplicity we consider the case where the variance parameter is ®xed and known.

Thus we have the expression h ; i gab da db ; where gab h@a ; @b i: Just as we write any tangent vector in terms of its components with respect to the basis f@1 ; F F F ; @r g, the metric is often simply denoted by Introduction to differential geometry 31 its components gab gab relative to the basis fd1 ; F F F ; dr g. It is important then to understand how this `component form' representation of the metric changes with a change of parameters. Suppose that is another parameterisation, then we have h ; i gab @i a @j b di dj g~ ij di dj ; where g~ij is the coordinate form for the -parameterisation.