Skip to main content
Article thumbnail
Location of Repository

An introduction to differential geometry in econometrics

By Paul Marriott and Mark H. Salmon


In this introductory chapter we seek to cover sufficient differential geometry in order to understand\ud its application to Econometrics. It is not intended to be a comprehensive review of\ud either differential geometric theory, nor of all the applications which geometry has found in\ud statistics. Rather it is aimed as a rapid tutorial covering the material needed in the rest of this\ud volume and the general literature. The full abstract power of a modern geometric treatment\ud is not always necessary and such a development can often hide in its abstract constructions\ud as much as it illuminates.\ud In Section 2 we show how econometric models can take the form of geometrical objects\ud known as manifolds, in particular concentrating on classes of models which are full or curved\ud exponential families.\ud This development of the underlying mathematical structure leads into Section 3 where the\ud tangent space is introduced. It is very helpful, to be able view the tangent space in a number\ud of different, but mathematically equivalent ways and we exploit this throughout the chapter.\ud Section 4 introduces the idea of a metric and more general tensors illustrated with statistically\ud based examples. Section 5 considers the most important tool that a differential\ud geometric approach offers, the affine connection. We look at applications of this idea to\ud asymptotic analysis, the relationship between geometry and information theory and the problem\ud of the choice of parameterisation. The last two sections look at direct applications of this\ud geometric framework. In particular at the problem of inference in curved families and at the\ud issue of information loss and recovery.\ud Note that while this chapter aims to give a reasonably precise mathematical development\ud of the required theory an alternative and perhaps more intuitive approach can be found in\ud the chapter by Critchley, Marriott and Salmon later in this volume. For a more exhaustive\ud and detailed review of current geometrical statistical theory see Kass and Vos (1997) or from\ud a more purely mathematical background, see Murray and Rice (1993)

Topics: QA
Publisher: Warwick Business School Financial Econometrics Research Centre
Year: 2000
OAI identifier:

Suggested articles


  1. A Panel-Based Investigation into the Relationship Between Stock Prices and Dividends,
  2. An Analysis of the Performance of European Foreign Exchange Forecasters, doi
  3. An Elementary Account of Amari's Expected Geometry, doi
  4. An Introduction to Differential Geometry in Econometrics, doi
  5. (1989). Applications of di erential geometry to statistics Ph.D. dissertation.
  6. Currency Spillovers and Tri-Polarity: a Simultaneous Model doi
  7. (1987). Di erential geometric theory of statistics.
  8. Finite Sample Inference for Extreme Value Distributions,
  9. Forecasting Inflation with a Non-linear Output Gap Model,
  10. Forecasting T-Bill Yields: Accuracy versus Profitability, WP98-03 4. Adam Kurpiel and Thierry Roncalli , Option Hedging with Stochastic Volatility, doi
  11. Forecasting Volatility using LINEX Loss Functions, doi
  12. From Market Micro-structure to Macro Fundamentals: is there Predictability in the Dollar-Deutsche Mark Exchange Rate?,
  13. Hopscotch Methods for Two State Financial Models, doi
  14. How do UK-Based Foreign Exchange Dealers Think Their Market Operates?, doi
  15. Implied Volatility Forecasting: A Compaison of Different Procedures Including Fractionally Integrated Models with Applications to UK Equity Options,
  16. Improved Testing for the Efficiency of Asset Pricing Theories in Linear Factor Models,
  17. Market Risk and the Concept of Fundamental Volatility: Measuring Volatility Across Asset and Derivative Markets and Testing for the Impact of Derivatives Markets on Financial Markets, doi
  18. Modelling Emerging Market Risk Premia Using Higher Moments, doi
  19. On the Evolution of Credibility and Flexible Exchange Rate Target Zones,
  20. Rationality Testing under Asymmetric Loss, doi
  21. Soosung Hwang and Stephen Satchell, Using Bayesian Variable Selection Methods to Choose Style Factors in Global Stock Return Models, doi
  22. Technical Analysis and Central Bank Intervention, doi
  23. The Effects of Systematic Sampling and Temporal Aggregation on Discrete Time Long Memory Processes and their Finite Sample Properties, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.