Technical Working Paper 03-110

Imposing Regional Regularity on Flexible Functional Forms, H. Wolff, T. Heckelei, and R.C. Mittelhammer July 2003

Abstract

In many areas of economic analysis, theory restricts the shape of functions. Obvious examples are monotonicity and curvature conditions which apply to indirect utility, profit, and cost functions. The imposition of such regularity conditions requires an econometric method capable of estimating parameters subject to many nonlinear inequality constraints--a task defined by Diewert and Wales (1987) as 'one of the most vexing problems applied economists have encountered'. Commonly regularity conditions are either imposed locally or globally.

We extend methods by Gallant and Golub (1984) and Terrell (1996) and define the theoretical restrictions at a connected subset of the regressor space, labeled as the regional approach. It has important advantages by ensuring theoretical consistency not only locally, but in the whole empirically relevant region of the regressor space and by providing higher functional flexibility compared to the global imposition of regularity. We implement the method using Metropolis-Hasting Accept-Reject Algorithm. As an extension to Terrell's 'approximation' of the regular region by a grid of singular local points, we demonstrate under what conditions regularity can be imposed for a connected set. Related theorems also contribute to enhancing computational speed by limiting the number of regularity checks. Furthermore, we show that the common practice of defining point estimates as the mean of posterior distributions can lead to inconsistent specifications. We propose two alternative regularity-retaining estimators. We illustrate the technique by estimating a dual cost function using 'second order' flexible and 'globally' flexible functional forms and numerical comparisons demonstrate the relevance of our methological contributions.