On the asymptotic normality of Fourier flexible form estimates

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.