## An Elasticity can be Estimated Consistently without a Priori
Knowledge of Functional Form

Ibrahim Elbadawi, A. Ronald Gallant,
Geraldo Souza

*Econometrica*, Vol. 51, No. 6. (Nov., 1983),
pp. 1731-1752.

### Abstract

We consider an open question in applied price theory: Without a priori
knowledge of a firm's cost function or a consumer's indirect utility
function, it is possible to estimate price and substitution
elasticities consistently by observing a demand system? As the work of
White [30], Guilkey, Lovell, and Sickles [19], and others has shown,
ordinary flexible functional forms such as the translog cannot achieve
this objective. We find that if one is prepared to assume that
elasticities of substitution cannot oscillate wildly over the region
of interest then consistent estimation is possible using the Fourier
flexible form provided the number of fitted parameters increases as
the number of observations increases. This result obtains with any of
the commonly used statistical methods as, for example, multivariate
least squares, maximum likelihood, and three-stage least squares. It
obtains if the number of fitted parameters is chosen adaptively by
observing the data or chosen deterministically according to some fixed
rule. We approach the problem along the classical lines of
estimability considerations as used in the study of less than full
rank linear statistical models and thereby discover that the problem
has a fascinating structure which we explore in detail.