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NONPARAMETRIC STUDY OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

Published online by Cambridge University Press:  12 December 2005

Anne Vanhems
Anne Vanhems
Affiliation:
Toulouse Business School
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Abstract

The solution of differential equations lies at the heart of many problems in structural economics. In econometrics the general nonparametric analysis of consumer welfare is historically the most obvious application, but there are also many applications in finance and other fields. This work considers the general nonparametric form for these problems and identification conditions. It derives a kernel-based estimator and shows consistency and asymptotic normality.

In particular, the link with inverse problems allows us to define it in terms of a well-posed inverse problem and to stress the regularity properties of the estimated solution.I thank Jean-Pierre Florens, Richard Blundell, Eric Renault, and Christine Thomas-Agnan for stimulating conversations, suggestions, and advice. I am very grateful to Oliver Linton and two anonymous referees for most helpful comments.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 1 , February 2006 , pp. 127 - 157
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Aït-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.CrossRefGoogle Scholar
Blundell, R. & J.L. Powell (2000) Endogeneity in nonparametric and semiparametric regression models. Invited lecture at the 8th World Congress of the Econometric Society.
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes. 2nd ed. Springer-Verlag.
Choquet, G. (1964) Cours d'analyse tome II Topologie. Masson et Cie.
Darolles, S., J.P. Florens, & E. Renault (2000) Nonparametric Instrumental Regression. Preprint.
Fan, J. & I. Gijbels (1996) Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. Chapman and Hall.
Florens, J.P. (2003) Inverse problems and structural econometrics: The example of instrumental variables. In M. Dewatripont, L.P. Hansen, & S.J. Turnovsky (eds.), Advances in Economics and Econometrics, vol. 2, pp. 284–311. Cambridge University Press.
Florens, J.P. & A. Vanhems (2000) Signal Recovering from Observations of Its Derivatives under Disconnected Intervals. Preprint.
Garza, J., P. Hall, & F.H. Ruymgaart (1999) A New Method of Solving Abel Type Equations. Preprint.
Green, P.J. & B.W. Silverman (1994) Nonparametric Regression and Generalized Linear Models. Monographs on Statistics and Applied Probability 58. Chapman and Hall.
Groetsch, C. (1984) The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman.
Hastie, T.J. & R.J. Tibshirani (1990) Generalized Additive Models. Monographs on Statistics and Applied Probability 43. Chapman and Hall.
Hausman, J. (1981) Exact consumer's surplus and deadweight loss. American Economic Review 71, 662–676.Google Scholar
Hausman, J. (1985) The econometrics of nonlinear budget sets. Econometrica 53, 1255–1282.CrossRefGoogle Scholar
Hausman, J. & W.K. Newey (1995) Nonparametric estimation of exact consumers surplus and deadweight loss. Econometrica 63, 1445–1476.CrossRefGoogle Scholar
Horowitz, J. (1998) Semiparametric Methods in Econometrics. Lecture Notes in Statistics 131. Springer-Verlag.
Horowitz, J. (2001a) Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69, 499–513.Google Scholar
Horowitz, J. (2001b) The bootstrap. In J.J. Heckman & E. Leamer (eds.), Handbook of Econometrics, vol. 5, pp. 3160–3228. Elsevier Science.
Ichimura, H. (1987) Estimation of single index models. Ph.D. Dissertation, Massachusetts Institute of Technology.
Kunze, H.E. & E.R. Vrscay (1999) Solving inverse problems for ordinary differential equations using the Picard contraction mapping. Inverse Problems 15, 745–770.CrossRefGoogle Scholar
Linton, O. & J.P. Nielsen (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93–100.CrossRefGoogle Scholar
Loubes, J.M. & A. Vanhems (2002) Differential Equations with Endogeneous Effects. Preprint.
Newey, W.K. (1994) Kernel estimation of partial means and a general variance estimator. Econometric Theory 10, 233–253.Google Scholar
Newey, W., J. Powell, & F. Vella (1999) Nonparametric estimation of triangular simultaneous equations models. Econometrica 67, 565–604.CrossRefGoogle Scholar
O'Sullivan, F. (1986) A statistical perspective on ill-posed inverse problems. Statistical Science 4, 502–527.CrossRefGoogle Scholar
Pagan, A. & A. Ullah (1999) Nonparametric Econometrics. Themes in Modern Econometrics. Cambridge University Press.
Robinson, P.M. (1988) Root-N-consistent semiparametric regression. Econometrica 56, 931–954.CrossRefGoogle Scholar
Schwartz, L. (1970) Analyse, Topologie générale et analyse fonctionnelle. Collection Enseignement des Sciences. Hermann.
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley.
Sibony, M. & J.-Cl. Mardon (1984) Approximations et equations differentielles. Hermann.
Tikhonov, A.N. & V.Y. Arsenin (1977) Solutions of Ill-Posed Problems. Winston.
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
van der Vaart, A.W. & J.A. Wellner (1996) Weak Convergence and Empirical Processes. Springer-Verlag.
Vanhems, A. (2001) Random Differential Equations and Heterogeneity between Consumers. Preprint.
Van Rooij, A.O. & F.H. Ruymgaart (1999) On inverse estimation. In S. Ghosh (ed.), Asymptotics, Nonparametrics, and Time Series, pp. 579–613. Dekker.
Vartia, Y. (1984) Efficient methods of measuring welfare change and compensated income in terms of ordinary demand functions. Econometrica 51, 79–98.Google Scholar
Wand, M.P. & M.C. Jones (1995) Kernel Smoothing. Monographs on Statistics and Applied Probability 60. Chapman and Hall.