## Journal of Econometrics

**Volume 50, Issue 3, 01-December-1991**

Journal Of Econometrics Vol. 50 (3) pp. 329-353

## On the asymptotic normality of Fourier flexible form estimates

A.R. Gallant

*North Carolina State University, Raleigh, NC 27695, USA*

G. Souza

*EMBRAPA, 70770 Brazilia, DF, Brazil*

Received 1 September 1989; Revised 1 August 1990

**Abstract**

Rates of increase in the number of parameters of a Fourier factor
demand system that imply asymptotically normal elasticity estimates
are characterized. This is the multivariate analog of work by Andrews
(1991). Our proof strategy is new and consists of relating the minimum
eigenvalue of the sample sum of squares and cross-products matrix to
the minimum eigenvalue of the population matrix via a uniform strong
law with rate that is established using results from the empirical
processes literature. In its customary form, the minimum eigenvalue of
the Fourier sum of squares and cross-products matrix, considered as a
function of the number of parameters, decreases faster than any
polynomial. The consequence is that the rate at which parameters may
increase is slower than any fractional power of the sample size. In
this case, we get the same rate as Andrews. When our results are
applied to multivariate regressions with a minimum eigenvalue that is
bounded or declines at a polynomial rate, the rate on the parameters
is a fractional power of the sample size. In this case, our method of
proof gives faster rates than Andrews. Andrews' results cover the
heteroskedastic case, ours do not.