Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T07:51:08.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Rules for Seminonparametric Estimators That Achieve Asymptotic Normality

Published online by Cambridge University Press:  11 February 2009

Brian J. Eastwood  and
A. Ronald Gallant
Get access Rights & Permissions [Opens in a new window]

Abstract

In econometrics, seminonparametric (SNP) estimators originated in the consumer demand literature. The Fourier flexible form is a well-known example. The idea is to replace the consumer's indirect utility function with a truncated series expansion and then use a parametric procedure, such as nonlinear multivariate regression, to set a confidence interval on an elasticity. More recently, SNP estimators have been used in nonlinear time series analysis. A truncated Hermite expansion with an ARCH leading term is used as the conditional density of the process. The method of maximum likelihood is used to fit it to data.

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 3 , September 1991 , pp. 307 - 340
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Aldous, D.J. & Eagleson, G.K.. On mixing and stability of limit theorems. The Annals of Probability 6 (1978): 325–331.CrossRefGoogle Scholar
2. Anderson, T.W. The choice of the degree of a polynomial regression as a multiple decision problem. The Annals of Mathematical Statistics 33 (1962): 255–265.CrossRefGoogle Scholar
3. Andrews, Donald W.K. Asymptotic normality of series estimators for nonparametric and semiparametric models. Cowles Foundation Discussion Paper No. 874R, Yale University, New Haven, CT, 1988.Google Scholar
4. Barnett, William A.. New indices of money supply and the flexible Laurent demand system. Journal of Business and Economic Statistics 1 (1983): 7–23.Google Scholar
5. Barnett, William A., Geweke, John & Wolfe, Michael. Seminonparametric Bayesian estimation of the asymptotically ideal production model. Journal of Econometrics, forthcoming.Google Scholar
6. Barnett, William A. & Lee, Yul. The global properties of the minflex Laurent, generalized Leontief, and translog flexible functional forms. Econometrica 53 (1985): 1421–1437.CrossRefGoogle Scholar
7. Bierens, Herman J. Kernel estimators of regression functions. In Bewley, Truman F. (ed.). Advances in Econometrics Fifth World Congress, Volume 1. Cambridge: Cambridge University Press, 1987.Google Scholar
8. Bloomfield, Peter. Fourier Analysis of Time Series: An Introduction. New York: Wiley, 1976.Google Scholar
9. Box, George E.P. & Draper, Norman R.. Empirical Model-Building and Response Surfaces. New York: Wiley, 1987.Google Scholar
10. Breiman, L. & Freedman, D.. How many variables should there be entered in a regression equation. Journal of the American Statistical Association 78 (1983): 131–136.10.1080/01621459.1983.10477941CrossRefGoogle Scholar
11. Cencov, N.N. Evaluation of an unknown distribution density from observations. Soviet Mathematics 3 (1962): 1559–1562.Google Scholar
12. Chalfant, James A. & Ronald Gallant, A.. Estimating substitution elasticities with the Fourier cost function: Some Monte Carlo results. Journal of Econometrics 28 (1985): 205–222.CrossRefGoogle Scholar
13. Christensen, Laurits R., Jorgenson, Dale W. & Lau, Lawrence J.. Transcendental logarithmic utility functions. The American Economic Review 65 (1975): 367–383.Google Scholar
14. Cox, Dennis D. Approximation of linear regression on nested subspaces. The Annals of Statistics 16 (1988): 713–732.10.1214/aos/1176350830CrossRefGoogle Scholar
15. Cox, Dennis D. Approximation of method of regularization estimators. The Annals of Statistics 16 (1988): 694–712.CrossRefGoogle Scholar
16. Cox, Dennis D. & O'Sullivan, Finbarr. Analysis of penalized likelihood-type estimators with applications to generalized smoothing in Sobolev spaces. Department of Statistics Technical Report No. 51, University of California, Berkeley, CA, 1985.Google Scholar
17. Devroye, Luc & Gyorfi, Laszlo. Nonparametric Density Estimation, the L 1 View. New York: Wiley, 1985.Google Scholar
18. Donoho, David & Johnstone, Iain M.. Projection-based approximation and a duality with kernel methods. The Annals of Statistics 17 (1989): 58–106.CrossRefGoogle Scholar
19. Eastwood, Brian J. Asymptotic normality and consistency of semi-nonparametric regression estimators using an upwards Ftest truncation rule. Journal of Econometrics, forthcoming.Google Scholar
20. Eastwood, Brian J. Confidence interval construction in semi-nonparametric regression estimation. Ph.D. Dissertation, North Carolina State University, Department of Statistics, Raleigh, NC, 1987.Google Scholar
21. Edmunds, D.E. & Moscatelli, V.B.. Fourier approximation and embeddings of Sobolev spaces. Dissertationes Mathematicae CXLV, 1977.Google Scholar
22. Elbadawi, Ibrahim A., Gallant, Ronald & Souza, Geraldo. An elasticity can be estimated consistently without a priori knowledge of functional form. Econometrica 51 (1983): 1731–1751.CrossRefGoogle Scholar
23. Eubank, Randall L. Spline Smoothing and Nonparametric Regression. New York: Marcel Dekker, 1988.Google Scholar
24. Ewis, Nabil A. & Fisher, Douglas. Towards a consistent estimate of the substitutability between money and near money: An application of the Fourier flexible functional form. Journal of Macroeconomics 7 (1985): 151–174.CrossRefGoogle Scholar
25. Fabian, Vaclav Polynomial estimation of regression functions with the supremum norm error. The Annals of Statistics 16 (1988): 1345–1368.CrossRefGoogle Scholar
26. Fuller, Wayne A.. Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
27. Gallant, A. Ronald. On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics 15 (1981): 211–245.10.1016/0304-4076(81)90115-9CrossRefGoogle Scholar
28.Gallant, A. Ronald. Unbiased determination of production technologies. Journal of Econometrics 20 (1982): 285–323.10.1016/0304-4076(82)90022-7CrossRefGoogle Scholar
29. Gallant, A. Ronald, Hansen, Lars Peter & Tauchen, George E.. Using conditional moments of asset payoffs to infer the volatility of intertemporal marginal rates of substitution. Journal of Econometrics, forthcoming.Google Scholar
30. Gallant, A. Ronald, Hsieh, David A. & Tauchen, George E.. On fitting a recalcitrant series: The pound/dollar exchange rate, 1974–83. In Barnett, William A., Powell, James & Tauchen, George E. (eds.). Nonparametric and Semiparametric Methods in Econometrics and Statistics, Proceedings of the Fifth International Symposium in Economic Theory and Econometrics. Cambridge: Cambridge University Press, forthcoming.Google Scholar
31. Gallant, A. Ronald & Nychka, Douglas W.. Semi-nonparametric maximum likelihood estimation. Econometrica 55 (1987): 363–390.CrossRefGoogle Scholar
32. Gallant, A. Ronald, Rossi, Peter E. & Tauchen, George E.. Stock prices and volume. Working paper. Graduate School of Business, University of Chicago, Chicago, IL, 1990.Google Scholar
33. Gallant, A. Ronald & Souza, Geraldo. On the asymptotic normality of Fourier flexible form estimates. Institute of Statistics Mimeograph Series No. 1957, North Carolina State University, Raleigh, NC, 1989.Google Scholar
34. Gallant, A. Ronald & Tauchen, George. Seminonparametric estimation of conditionally constrained heterogeneous processes: Asset pricing applications. Econometrica 57 (1989): 1091–1120.CrossRefGoogle Scholar
35. Gallant, A. Ronald & Tauchen, George. A nonparametric approach to nonlinear time series analysis: Estimation and simulation. Institute for Mathematics and Its Applications, Preprint Series #705, University of Minnesota, Minneapolis, MN, 1990.Google Scholar
36. Guiasu, Silvia. On the asymptotic distribution of the sequences of random variables with random indices. The Annals of Mathematical Statistics 6 (1971): 2018–2028.CrossRefGoogle Scholar
37. Hart, Jeffrey D. On the choice of a truncation point in Fourier series density estimation. Journal of Statistical Computer Simulation 21 (1985): 95–116.CrossRefGoogle Scholar
38. Huber, Peter J. Robust regression: Asymptotics, conjectures and Monte Carlo. The Annals of Statistics 1 (1973): 799–821.CrossRefGoogle Scholar
39. Hussey, Robert M. Nonparametric evidence on asymmetry in business cycles using aggregate employment time series. Working paper. The Jesuits at Wernersville, PA, 1989.Google Scholar
40.International Mathematical and Statistical Libraries, Inc. IMSL Library, Version 9.2. International Mathematical and Statistical Libraries, Inc., Houston, 1981.Google Scholar
41. Krommal, R. & Tarter, M.. The estimation of probability densities and cumulatives by Fourier series methods. Journal of the American Statistical Association 63 (1968): 925–952.Google Scholar
42. Kumm, Mark. Comparison of the generalized Box-Cox and Fourier functional forms of analysis of time-of-use electricity pricing experiments. Ph.D. dissertation, Duke University, Department of Economics, Durham, NC, 1981.Google Scholar
43. Mallows, C.L. Some comments on Cp. Technometrics 15 (1973): 661–675.Google Scholar
44. Nychka, Douglas. Bayesian confidence intervals of smoothing splines. Journal of the American Statistical Association 83 (1988): 1134–1143.10.1080/01621459.1988.10478711CrossRefGoogle Scholar
45. Portnoy, Stephen. Asymptotic behavior of M-estimators of p regression parameters when p 2/n is large. I. Consistency. The Annals of Statistics 12 (1984): 1298–1309.CrossRefGoogle Scholar
46. Portnoy, Stephen. Asymptotic behavior of W-estimators of p regression parameters when p 2/n is large. II. Normal approximation. The Annals of Statistics 13 (1985): 1403–1417.CrossRefGoogle Scholar
47. Rao, M.M. Probability Theory with Applications. Orlando, FL: Academic Press, 1984.Google Scholar
48. Rossi, Peter E. Specification and analysis of econometric production models. Ph.D. dissertation, University of Chicago, Chicago, IL, 1983.Google Scholar
49. Serfling, Robert J. Approximation Theorems of Mathematical Statistics. New York: Wiley, 1980.CrossRefGoogle Scholar
50. Sims, Christopher A. Macroeconomics and reality. Econometrica 48 (1980): 1–48.10.2307/1912017CrossRefGoogle Scholar
51. Thompson, Mary L. Selection of variables in multiple regression: Part I. A review and evaluation. International Statistical Review 46 (1978): 1–19.CrossRefGoogle Scholar
52. Thompson, Mary L. Selection of variables in multiple regression: Part II. Chosen procedures, computations and examples. International Statistical Review 46 (1978): 129–146.10.2307/1402809CrossRefGoogle Scholar
53. Tolstov, Georgi P.. Fourier Series. New York: Dover Publications, 1962.Google Scholar
54. Wallace, T. Dudley. Pretest estimation in regression: A survey. American Journal of Agricultural Economics 59 (1977): 431–443.10.2307/1239645CrossRefGoogle Scholar
55. White, Halbert. Using least squares to approximate unknown regression functions. International Economic Review 21 (1980): 149–170.10.2307/2526245CrossRefGoogle Scholar
56. Wolfinger, Russell D. & Gallant, A. Ronald. Varying degree polynomial regression, I. Rates of convergence. Institute of Statistics Mimeograph Series No. 1974, North Carolina State University, Raleigh, NC, 1989.Google Scholar
57. Wolfinger, Russell D. & Gallant, A. Ronald. Varying degree polynomial regression, II. Asymptotic normality. Institute of Statistics Mimeograph Series No. 1975, North Carolina State University, Raleigh, NC, 1989.Google Scholar