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On the asymptotic normality of Fourier flexible form estimates. (English) Zbl 0792.62102
The paper deals with the determination of a class of rules for increasing the number of parameters of the Fourier factor demand system that imply asymptotically normal elasticity estimates. After assuming a deterministic version of factor demand theory satisfying that observed factor cost shares follow some distribution with certain location parameter, a theorem is proved related to the asymptotic normality of the relative error.
It is interesting to stress the originality of the proof strategy followed by the authors, consisting in relating the minimum eigenvalue of the sample sum of squares and cross-product matrix to the minimum eigenvalue of the population matrix via a uniform strong law.
If the rate at which parameters may increase is slower than any fractional power of the sample size, the rate is coincident with the one given by D. W. K. Andrews [see ibid. 47, No. 2/3, 359-377 (1991; Zbl 0734.62069); Econometrika 59, No. 3, 817-858 (1991; Zbl 0732.62052) and ibid., No. 2, 307-345 (1991; Zbl 0727.62047)]. This is the case when the results are applied to a multivariate regression with minimum eigenvalue that is bounded or declines at a polynomial rate. In the referred conditions, the method proposed in this paper gives faster rates than Andrews’, and it seems therefore to be “preferable” although it does not cover the heteroscedastic case.
Reviewer: R.Pérez (Oviedo)

62P20 Applications of statistics to economics
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
91B42 Consumer behavior, demand theory
Full Text: DOI
[1] Adams, Robert A., Sobolev spaces, (1975), Academic Press New York, NY · Zbl 0314.46030
[2] Andrews, Donald W.K., Asymptotic normality of series estimators for nonparametric and semiparametric models, Econometrica, (1991), forthcoming · Zbl 0727.62047
[3] Billingsley, Patrick, Probability and measure, (1979), Wiley New York, NY · Zbl 0411.60001
[4] Christensen, Laurits R.; Jorgenson, Dale W.; Lau, Lawrence J., Transcendental logarithmic utility functions, American economic review, 65, 367-383, (1975)
[5] Deaton; Angus; Muelbauer, John, Economics and consumer behavior, (1980), Cambridge University Press Cambridge · Zbl 0444.90002
[6] Eastwood, Brian J., Asymptotic normality and consistency of seminonparametric regression estimators using an upwards F-test truncation rule, Journal of econometrics, 48, 151-181, (1991) · Zbl 0735.62038
[7] Eastwood, Brian J; Ronald Gallant, A., Adaptive truncation rules for seminonparametric estimators that achieve asymptotic normality, Economic theory, (1987), forthcoming
[8] Edmunds, D.E.; Moscatelli, V.B., Fourier approximation and embeddings of Sobolev spaces, Dissertationes mathematicae, CXLV, 1-46, (1977) · Zbl 0359.42011
[9] Eicker, Friedhelm, Limit theorems for regressions with unequal and dependent errors, Proceedings of the fifth Berkeley symposium on probability and mathematical statistics, 1, 59-82, (1967)
[10] Elbadawi, Ibrahim; Ronald Gallant, A.; Souza, Geraldo, An elasticity can be estimated consistently without a priori knowledge of functional form, Econometrica, 51, 1731-1752, (1983) · Zbl 0567.62100
[11] Gallant, A.Ronald, On the bias in flexible functional forms and an essentially unbiased form: the Fourier flexible form, Journal of econometrics, 15, 211-245, (1981) · Zbl 0454.62096
[12] Gallant, A.Ronald, Unbiased determination of production technologies, Journal of econometrics, 20, 285-323, (1982) · Zbl 0521.62096
[13] Gallant, A.Ronald; Monahan, John F., Explicitly infinite dimensional Bayesian analysis of production technologies, Journal of econometrics, 30, 171-201, (1985) · Zbl 0633.62114
[14] Huber, Peter J., Robust regression: asymptotics, conjectures and Monte Carlo, Annals of statistics, 1, 799-821, (1973) · Zbl 0289.62033
[15] Kolmogorov, A.N.; Tihomirov, V.M., Ε-entropy and ε-capacity of function spaces, Uspehi mat nauk, 14, 3-86, (1959) · Zbl 0090.33503
[16] Kolmogorov, A.N.; Tihomirov, V.M., English translation, Mathematical society translations, 17, 277-364, (1961)
[17] Pollard, David, Convergence of stochastic processes, (1984), Springer-Verlag New York, NY · Zbl 0544.60045
[18] Portnoy, Stephen, Asymptotic behavior of M-estimators of p regression parameters when p2/n is large: II. normal approximation, Annals of statistics, 13, 1403-1417, (1985) · Zbl 0601.62026
[19] Smolyak, S.A., The ε-entropy of classes \(E\)^α, k(B) and \(W\)αs(B) in the \(L\)2-metric, Soviet mat dokl, 1, 192-195, (1960)
[20] Tolstov, Georgi P., Fourier series, (1962), Dover Publications New York, NY · Zbl 0358.42001
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