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An introduction to differential geometry in econometrics
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Marriott, Paul and Salmon, Mark H. (2000) An introduction to differential geometry in econometrics. Working Paper. University of Warwick: Warwick Business School Financial Econometrics Research Centre. Working Papers Series, Vol.1999 (No.10).

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Abstract
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 of
either differential 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 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. 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).
Item Type:  Working or Discussion Paper (Working Paper) 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Social Sciences > Warwick Business School > Financial Econometrics Research Centre Faculty of Social Sciences > Warwick Business School 
Library of Congress Subject Headings (LCSH):  Geometry, Differential, Geometrical models in statistics 
Series Name:  Working Papers Series 
Publisher:  Warwick Business School Financial Econometrics Research Centre 
Place of Publication:  University of Warwick 
Official Date:  22 January 2000 
Volume:  Vol.1999 
Number:  No.10 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
References:  Amari SI, (1987). Differential geometric theory of statistics. In Differential Geometry in 
URI:  http://wrap.warwick.ac.uk/id/eprint/1839 
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