Introduction

The last two decades have seen a rapid growth in the application of differential geometry in statistics. Efron (1975) stimulated much of this research with his definition of statistical curvature and he showed that curvature has serious consequences for statistical inference.

Many papers on the application of differential geometry in statistics go straight into defining all the necessary tools, such as the metric and an affine connection on a manifold, and show how they can be used in statistical analysis. The emphasis is predominantly on asymptotic theory and applications are mainly in estimation, information loss and higher-order efficiency. Barndorff-Nielsen, Cox and Reid (1986) give a very accessible account of the relevant ideas in differential geometry and also provide a historical overview; see also Amari (1985) and Okamoto, Amari and Takeuchi (1991) provide a brief recent summary of main achievements.

This chapter is concerned with the effects of curvature on hypothesis testing. However, our approach differs from the differential geometrical approach mentioned above in a number of ways. First, we give a global analysis, i.e. for the whole of the sample and parameter space, not merely in a neighbourhood of a fixed point such as the true parameter value. Secondly, the analysis is valid for all sample sizes, not just asymptotically. Finally, we are concerned with hypothesis testing, which is less common, and use the partitioning of the sample space into critical region and acceptance region to illustrate our arguments graphically, rather than solely investigating the (analytic) properties of test statistic(s).

Introduction

An important objective in econometric modelling is to obtain congruent and encompassing models, which are thus consistent with data evidence and economic theory, and are capable of explaining the behaviour of rival models for the same economic phenomena. Hendry (1995) and Mizon (1995b) inter alia present this view of modelling, as well as recording that a general-to-simple modelling strategy is an efficient way to isolate congruent models. Indeed such a modelling strategy requires the models under consideration to be thoroughly tested and progressively evaluated within a sequence of nested hypotheses, so that each model not rejected in the process is a valid simplification of (i.e. parsimoniously encompasses) all models more general than it in the sequence. Models arrived at as the limit of the reduction process therefore contain all relevant information available in the initial models, thus rendering the latter inferentially redundant. Testing the ability of models to parsimoniously encompass more general ones, thus ensuring that no information is lost in the reduction process, creates a partially ordered sequence of models. Further, Lu and Mizon (1997) point out that the intermediate models considered while testing the parsimonious encompassing ability of models in a nested sequence are mutually encompassing models and hence observationally equivalent to each other. Note, though, that there is an important distinction between observational equivalence in the population and in a sample, which may contain weak evidence.

Introduction

The aim of this chapter is to construct a general geometrical setting, based on Hilbert space, in which one may study various estimation techniques, in particular with respect to efficiency and robustness. Given the sort of data one wishes to study, such as continuous, discrete, etc., the set of data-generation processes (DGPs) capable of generating data of that sort is given the structure of a Hilbert manifold. Statistical models will be treated as submanifolds of this underlying manifold.

Geometrical methods are frequently used in the study of statistical inference. One important strand of the literature is presented in Amari (1990), whose numerous earlier papers were inspired by some very abstract work of Chentsov (1972) and led to the concept of a statistical manifold. Other review papers and books in this tradition include Barndorff-Nielsen, Cox and Reid (1986), Kass (1989), Murray and Rice (1993), and Barndorff-Nielsen and Cox (1994). Most of this work makes use of finite-dimensional differential manifolds, which are usually representations of models in the exponential family.

Infinite-dimensional Hilbert space methods are extensively used in another strand of literature, for which the most suitable recent reference is Small and McLeish (1994). This book contains numerous references to the original papers on which it builds.

Introduction

Since Hansen's (1982) seminal paper, the generalised method of moments (GMM) has become an increasingly important method of estimation in econometrics. Given assumed population moment conditions, the GMM estimation method minimises a quadratic form in the sample counter-parts of these moment conditions. The quadratic form is constructed using a positive definite metric. If this metric is chosen as the inverse of a positive semi-definite consistent estimator for the asymptotic variance matrix of the sample moments, then the resultant GMM estimator is asymptotically efficient. Estimation using GMM is semi-parametric and, therefore, a particular advantage of GMM is that it imposes less stringent assumptions than, for example, the method of maximum like-lihood (ML). Although consequently more robust, GMM is of course generally less efficient than ML. There is some Monte Carlo evidence indicating that GMM estimation may be biased in small samples where the bias seems to arise mainly from the metric used. For example, Altonji and Segal (1996) suggest that GMM estimators using an identity matrix metric may perform better in finite samples than an efficient GMM estimator obtained using the inverse of the estimated asymptotic variance matrix of the sample moments.

When the variables of interest in the data-generation process (DGP) are independent and identically distributed, an important recent paper (Qin and Lawless (1994)) shows that it is possible to embed the components of the sample moments used in GMM estimation in a non-parametric likelihood function; namely, an empirical likelihood (EL) framework. The resultant maximum EL estimator (MELE) shares the same first-order properties as Hansen's (1982) efficient GMM estimator.

Geometry has always aided intuition in econometrics but has perhaps only very occasionally aided analytic development. However, advances in statistical theory have recently been greatly enhanced through a deeper understanding of differential geometry and the geometrical basis of statistical inference; see, for instance, Kass and Vos (1997), Barndorff-Nielsen and Cox (1994) and McCullagh (1987), as well as a rapidly expanding array of journal articles.

What value do these advances hold for econometrics? How can practical econometric methods be improved by a deeper understanding of the geometrical basis of the problems faced by econometricians?

This volume attempts to stimulate the appetite of econometricians to delve further into the sometimes complex but always enlightening world of differential geometry and its application to econometrics. The reward we have found is not only a deeper appreciation of existing methodologies but the recognition that further advance in a host of different directions can readily be made. It is this potential gained through studying what may be standard problems from an alternative perspective that makes the application of differential geometry to econometrics so exciting. Geometry is mathematically fundamental, and frequently a geometric understanding of a problem highlights the essential issues which can be hidden in an algebraic development of the same issue. Take for instance the basic notion of orthogonal projection that underlies much of econometric and statistical method and its extension to curved statistical spaces and the question of invariance to different parameterisations, the interpretation of higher-order asymptotic expansions and statistical curvature and the role of ancillarity and exogeneity.

Differential geometry has found fruitful application in statistical inference. In particular, Amari's (1990) expected geometry is used in higherorder asymptotic analysis and in the study of sufficiency and ancillarity. However, we can see three drawbacks to the use of a differential geometric approach in econometrics and statistics more generally. First, the mathematics is unfamiliar and the terms involved can be difficult for the econometrician to appreciate fully. Secondly, their statistical meaning can be less than completely clear. Finally, the fact that, at its core, geometry is a visual subject can be obscured by the mathematical formalism required for a rigorous analysis, thereby hindering intuition. All three drawbacks apply particularly to the differential geometric concept of a non-metric affine connection.

The primary objective of this chapter is to attempt to mitigate these drawbacks in the case of Amari's expected geometric structure on a full exponential family. We aim to do this by providing an elementary account of this structure that is clearly based statistically, accessible geometrically and visually presented.

Statistically, we use three natural tools: the score function and its first two moments with respect to the true distribution. Geometrically, we are largely able to restrict attention to tensors; in particular, we are able to avoid the need formally to define an affine connection. To emphasise the visual foundation of geometric analysis we parallel the mathematical development with graphical illustrations using important examples of full exponential families.

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.

Introduction

In a seminal paper Rao (1945) introduced the concept of ‘Geodesic Distance’ (or ‘Rao Distance’) into statistics. This concept has important theoretical properties and is based on the demanding differential-geometrical approach to statistics. These mathematical requirements and difficulties in its application are responsible for the low level of familiarity by econometricians with this generalisation of the well-known Mahalanobis distance. Econometricians require some detailed knowledge of Riemannian geometry to gain a complete understanding of Rao distances. Section 2 provides a short introduction to the necessary theory on Rao distances, hyperbolic curvature, isocircles and Rao distance tests.

Since their original development in the paper by Aigner and Chu (1968), frontier functions have served almost exclusively for estimating production and cost functions consistently with an economic theory of optimising behaviour. In section 3 of this chapter, an extended human capital model is estimated as a stochastic earnings frontier with data from the German socio-economic panel. This provides a deeper interpretation of the deviations of observed income from estimated income. Distinguishing between ‘potential human capital’ and ‘active human capital’ accounts for the partial failure of human capital models when estimated as average functions.

There has been a tendency in Germany for the number of years of education attained to increase, in part owing to the combination of an apprenticeship with university studies. Büchel and Helberger (1995) call this combination an inefficient ‘insurance strategy’ pursued mainly by children from low-income and low-education families because of low efficiency or high risk-aversion.