On the asymptotic normality of Fourier flexible form estimates

Abstract

Rates of increase in the number of parameters of a Fourier factor demand system that imply asymptotically normal elasticity estimates are characterized. The main technical problem in achieving this characterization is caused by the fact that the eigenvalues of the sample sum of squares and cross products matrix of the generalized least squares estimator are not bounded from below. This problem is addressed by establishing a uniform strong law with rate for the eigenvalues of this matrix so as to relate them to the eigenvalues of the expected sum of squares and cross products matrix. Because the minimum eigenvalue of the latter matrix considered as a function of the number of parameters decreases faster than any polynomial, the rate at which parameters may increase is slower than any fractional power of the sample size.

Environmental degradation in France: The effects of FDI, financial development, and energy innovations

“…A number of remarkable studies, for instance, Gallant (1981), Davies (1987), Gallant andSouza (1991), and Bierens (1997), empirically show that with a small number of frequency components, we can often capture the essential characteristics of an unknown functional form smooth break while employing the Fourier approximation. Furthermore, as it is vital to accommodate the evolution of the nonlinear trend to be gradual, hence, n should be small.…”

Chapter 37 Empirical process methods in econometrics

“…Gallant (1989) and Gallant and Souza (1991) use these methods to establish the asymptotic normality of certain seminonparametric (i.e. nonparametric series) estimators.…”

Reducing Approximation Error in the Fourier Flexible Functional Form

“…While the Fourier series component is sufficient for close approximation in the Sobolev norm, the addition of the polynomial serves to (1) limit the number of required Fourier series terms, (2) allow for nested testing of the translog and Cobb-Douglas functional forms and (3) decrease approximation error at the boundaries of the domain (Gallant 1981(Gallant , 1982Gallant and Souza 1991). In the next section, we argue that a generalization of this polynomial allows for further reductions in approximation error and allows for a wider range of nested tests of functional form.…”