The Fourier Flexible Form

Abstract

One perspective on the recent work in flexible functional forms is that the use of such forms represents an attempt to remove the modelinduced augmenting hypothesis that is inevitably linked to parametric statistical inference. The Fourier flexible form is discussed from this perspective. The discussion relies on heuristic and graphical arguments rather than formal mathematics. The generality of parametric statistical inference is limited by the augmenting hypothesis induced by model specification. For instance, to conclude that rejection of the integrability conditions in a translog consumer demand system implies rejection of the theory of consumer demand requires the augmenting hypothesis that all possible consumer demand systems must belong to the translog family (Christensen, Jorgenson, and Lau). This reliance on an assumed parametric model is not only philosophically distasteful but is of practical importance in applications. Estimators can be seriously biased by specification error; a test can reject a null hypothesis with a probability that greatly exceeds its nominal rejection probability (Gallant 1981). One way to interpret the recent work in flexible functional forms is to see it as the use of richer parametric families of models in an attempt to reduce these two sorts of statistical biases--estimator bias and excess rejection probability. Most of the flexible forms that have appeared in the literature are second-order (or Diewert-flexible) forms. Technically, this means that if g(x) is to be approximated by g(xIO), then at any given point xo there is a corresponding choice of parameters 01 such that