Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T18:50:19.118Z Has data issue: false hasContentIssue false
  • English
  • Français

A NONPARAMETRIC BOOTSTRAP TEST OF CONDITIONAL DISTRIBUTIONS

Published online by Cambridge University Press:  23 May 2006

Yanqin Fan ,
Qi Li  and
Insik Min
Yanqin Fan
Affiliation:
Vanderbilt University
Qi Li
Affiliation:
Texas A&M University
Insik Min
Affiliation:
Kyung Hee University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes a bootstrap test for the correct specification of parametric conditional distributions. It extends Zheng's test (Zheng, 2000, Econometric Theory 16, 667–691) to allow for discrete dependent variables and for mixed discrete and continuous conditional variables. We establish the asymptotic null distribution of the test statistic with data-driven stochastic smoothing parameters. By smoothing both the discrete and continuous variables via the method of cross-validation, our test has the advantage of automatically removing irrelevant variables from the estimate of the conditional density function and, as a consequence, enjoys substantial power gains in finite samples, as confirmed by our simulation results. The simulation results also reveal that the bootstrap test successfully overcomes the size distortion problem associated with Zheng's test.We are grateful for the insightful comments from three referees and a co-editor that greatly improved the paper. Li's research is partially supported by the Private Enterprise Research Center, Texas A&M University. Fan is grateful to the National Science Foundation for research support.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 4 , August 2006 , pp. 587 - 613
Copyright
© 2006 Cambridge University Press

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

Aitchison, J. & C.G.G. Aitken (1976) Multivariate binary discrimination by the kernel method. Biometrika 63, 413–420.Google Scholar
Andrews, D.W.K. (1997) A conditional Kolmogorov test. Econometrica 65, 1097–1128.Google Scholar
Bhattacharya, R.N. & R.R. Rao (1986) Normal Approximations and Asymptotic Expansions. Krieger.
Bierens, H.J. & W. Ploberger (1997) Asymptotic theory of integrated conditional moment tests. Econometrica 65, 1129–1151.Google Scholar
Billingsley, P. (1968) Probability and Measure. Wiley.
Corradi, V. & N. Swanson (2004) Bootstrap conditional distribution tests under dynamic misspecification. Journal of Econometrics, forthcoming.Google Scholar
de Jong, R.M. (1996) The Bierens test under data dependence. Journal of Econometrics 72, 1–32.Google Scholar
Delgado, M.A. & W.G. Manteiga (2001) Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics 29, 1469–1507.Google Scholar
Diebold, F.X., T. Gunther, & A.S. Tay (1998) Evaluating density forecasts with applications to financial risk management. International Economic Review 39, 863–883.Google Scholar
Fan, Y. (1994) Testing the goodness-of-fit of a parametric density function by kernel method. Econometric Theory 10, 316–356.Google Scholar
Fan, Y. (1997) Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function. Journal of Multivariate Analysis 62, 36–63.Google Scholar
Fan, Y. (1998) Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters. Econometric Theory 14, 604–621.Google Scholar
Fan, Y. & Q. Li (1996) Consistent model specification tests: Omitted variables and semi-parametric functional forms. Econometrica 64, 865–890.Google Scholar
Fan, Y. & Q. Li (2000) Consistent model specification tests: Nonparametric versus Bierens' ICM tests. Econometric Theory 16, 1016–1041.Google Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.Google Scholar
Hall, P., J. Racine, & Q. Li (2004) Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association 99, 1015–1026.Google Scholar
Hong, Y. & H. White (1996) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.Google Scholar
Ichimura, H. (2000) Asymptotic Distribution of Non-parametric and Semiparametric Estimators with Data Dependent Smoothing Parameters. Manuscript, University College, London.
Kullback, S. & R.A. Leibler (1951) On information and sufficiency. Annals of Mathematical Statistics 22, 79–86.Google Scholar
Li, F. & G. Tkacz (2004) A consistent bootstrap test for conditional density functions with time-dependent data. Journal of Econometrics, forthcoming.Google Scholar
Mammen, E. (1992) When does bootstrap work? Asymptotic results and simulations. Springer-Verlag.
Ossiander, M. (1987) A central limit theorem under metric entropy with L2 bracketing. Annals of Probability 15, 897–919.Google Scholar
Powell, J.L., J.H. Stock, & T.M. Stoker (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.Google Scholar
Wooldridge, J. (1992) A test for functional form against nonparametric alternatives. Econometric Theory 8, 452–475.Google Scholar
Zheng, J.X. (2000) A consistent test of conditional parametric distributions. Econometric Theory 16, 667–691.Google Scholar