Estimating Functions and Equations: An Essay on Historical Developments with Applications to Econometrics

Abstract

1We are most grateful to Kerry Patterson for his constant encouragement and very help-ful comments on an earlier draft. Without his many kind and gentle proddings this chapter would not have been completed. We are also most thankful to Peter Phillips and Jun Yan for many pertinent comments and suggestions. However, we retain the responsibility for any remaining errors. A part of the work was done during a visit by the first author to the Department of Economics, University of Cyprus, whose financial support is gratefully acknowledged. The idea of using estimating functions goes a long way back, at least to Karl Pearson’s introduction to the method of moments in 1894. It is now a very active area of research in the statistics literature. One aim of this chapter is to provide an account of the developments relating to the theory of estimating functions. Starting from the simple case of a single parameter under independence, we cover the multi-parameter, presence of nuisance parameters and dependent data cases. Application of the estimating functions technique to econometrics is still at its infancy. However, we illustrate how this estimation approach could be used in a number of time series models, such as random coefficient, threshold, bilinear, autoregressive conditional heteroscedasticity models, in models of spatial and longitudinal data, and median regression analysis. The chapter is concluded with some remarks on the place of estimating functions in the history of estimation.