Hostname: page-component-7f64f4797f-pmsz9 Total loading time: 0 Render date: 2025-11-06T14:57:36.177Z Has data issue: false hasContentIssue false
  • English
  • Français

A NONPARAMETRIC HELLINGER METRIC TEST FOR CONDITIONAL INDEPENDENCE

Published online by Cambridge University Press:  04 April 2008

Liangjun Su  and
Halbert White
Liangjun Su
Affiliation:
Singapore Management University
Halbert White*
Affiliation:
University of California, San Diego
*
Address correspondence to Halbert White, Department of Economics, University of California, San Diego, La Jolla, CA 92093-0508, USA; e-mail: hwhite@econ.ucsd.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a nonparametric test of conditional independence based on the weighted Hellinger distance between the two conditional densities, f(y|x,z) and f(y|x), which is identically zero under the null. We use the functional delta method to expand the test statistic around the population value and establish asymptotic normality under β-mixing conditions. We show that the test is consistent and has power against alternatives at distance n−1/2h−d/4. The cases for which not all random variables of interest are continuously valued or observable are also discussed. Monte Carlo simulation results indicate that the test behaves reasonably well in finite samples and significantly outperforms some earlier tests for a variety of data generating processes. We apply our procedure to test for Granger noncausality in exchange rates.

Information

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 4 , August 2008 , pp. 829 - 864
Copyright
Copyright © Cambridge University Press 2008

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Aït-Sahalia, Y., Bickel, P.J., ’ Stoker, T.M. (2001) Goodness-of-fit for kernel regression with an application to option implied volatilities. Journal of Econometrics 105, 363–412.CrossRefGoogle Scholar
Andrews, D.W.K. (1995) Nonparametric kernel estimation for semiparametric models. Econometric Theory 11, 560–596.CrossRefGoogle Scholar
Angrist, J.D. (1997) Conditional independence in sample selection models. Economics Letters 54, 103–112.CrossRefGoogle Scholar
Baek, E. ’ Brock, W. (1992) A General Test for Nonlinear Granger Causality: Bivariate Model. Working paper, Iowa State University and University of Wisconsin, Madison.Google Scholar
Bell, D., Kay, J., ’ Malley, J. (1996) A non-parametric approach to non-linear causality testing. Economics Letters 51, 7–18.CrossRefGoogle Scholar
Bickel, P.J. ’ Rosenblatt, M. (1973) On some global measures of the deviations of density function estimates. Annals of Statistics 1, 1071–1095.CrossRefGoogle Scholar
Blum, J.R., Kiefer, J., ’ Rosenblatt, M. (1961) Distribution free tests of independence based on the sample distribution function. Annals of Mathematical Statistics 32, 485–498.CrossRefGoogle Scholar
Brock, W., Dechert, W., Scheinkman, J., ’ LeBaron, B. (1996) A test for independence based on the correlation dimension. Econometric Reviews 15, 197–235.CrossRefGoogle Scholar
Chen, X., Linton, O., ’ Robinson, P.M. (2001) The Estimation of Conditional Densities. STICERD Econometrics Discussion paper, London School of Economics and Political Science.CrossRefGoogle Scholar
Dawid, A.D. (1979) Conditional independence in statistical theory. Journal of the Royal Statistical Society, Series B 41, 1–31.Google Scholar
Delgado, M.A. ’ González-Manteiga, W. (2001) Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics 29, 1469–1507.CrossRefGoogle Scholar
Fan, Y. ’ Li, Q. (1996) Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64, 865–890.CrossRefGoogle Scholar
Fan, Y. ’ Li, Q. (1999a) Central limit theorem for degenerate U-statistics of absolutely regular processes with applications to model specification testing. Journal of Nonparametric Statistics 10, 245–271.CrossRefGoogle Scholar
Fan, Y. ’ Li, Q. (1999b) Root-n-consistent estimation of partially linear time series models. Journal of Nonparametric Statistics 11, 251–269.CrossRefGoogle Scholar
Fernandes, M. (2000) Nonparametric Entropy-Based Tests of Independence between Stochastic Processes. Discussion paper, European University Institute.Google Scholar
Fernandes, M. ’ Flores, R.G. (2000) Tests for Conditional Independence, Markovian Dynamics, and Noncausality. Discussion paper, European University Institute.Google Scholar
Florens, J.P. ’ Fougere, D. (1996) Noncausality in continuous time. Econometrica 64, 1195–1212.CrossRefGoogle Scholar
Florens, J.P. ’ Mouchart, M. (1982) A note on causality. Econometrica 50, 583–591.CrossRefGoogle Scholar
Gasser, T., Müller, H.-G., ’ Mammitzsch, V. (1985) Kernels for nonparametric curve estimation. Journal of the Royal Statistical Society Series B 47, 238–252.Google Scholar
Gouriéroux, C. ’ Tenreiro, C. (2001) Local power properties of kernel based goodness of fit tests. Journal of Multivariate Analysis 78, 161–190.CrossRefGoogle Scholar
Granger, C.W.J. (1980) Testing for causality: A personal viewpoint. Journal of Economic Dynamics and Control 2, 329–352.CrossRefGoogle Scholar
Gronau, R. (1974) Wage comparisons: A selectivity bias. Journal of Political Economy 82, 1119–1144.CrossRefGoogle Scholar
Hall, P. (1984) Central limit theorem for integrated square error of multivariate nonparametric density estimators. Journal of Multivariate Analysis 14, 1–16.CrossRefGoogle Scholar
Härdle, W. ’ Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Heckman, J.J. (1974) Shadow prices, market wages, and labor supply. Econometrica 42, 679–694.CrossRefGoogle Scholar
Hiemstra, C. ’ Jones, J.D. (1994) Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance 49, 1639–1664.Google Scholar
Hoeffding, W. (1948) A nonparametric test of independence. Annals of Mathematical Statistics 58, 546–557.CrossRefGoogle Scholar
Hong, Y. (2001) A test for volatility spillover with application to exchange rates. Journal of Econometrics 103, 183–224.CrossRefGoogle Scholar
Hong, Y. ’ White, H. (2005) Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73, 837–901.CrossRefGoogle Scholar
Horowitz, J.L. (1992) A smoothed maximum score estimator for the binary response model. Econometrica 60, 505–531.CrossRefGoogle Scholar
Horowitz, J.L. ’ Spokoiny, V.G. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–631.CrossRefGoogle Scholar
Klein, R.W. ’ Spady, R.H. (1993) An efficient semiparametric estimator for discrete choice models. Econometrica 61, 387–421.CrossRefGoogle Scholar
Li, Q. (1999) Consistent model specification tests for time series econometric models. Journal of Econometrics 92, 101–147.CrossRefGoogle Scholar
Liebscher, E. (1996) Strong convergence of sums of α-mixing random variables with applications to density estimation. Stochastic Process and Their Applications 65, 69–80.CrossRefGoogle Scholar
Linton, O. (2002) Edgeworth approximations for semiparametric instrumental variable estimators and test statistics. Journal of Econometrics 106, 325–368.CrossRefGoogle Scholar
Linton, O. ’ Gozalo, P. (1997) Conditional Independence Restrictions: Testing and Estimation. Discussion paper, Cowles Foundation for Research in Economics, Yale University.Google Scholar
Manski, C.F. (1975) The maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3, 205–228.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Neumann, M.H. ’ Paparoditis, E. (2000) On bootstrapping L 2-type statistics in density testing. Statistics and Probability Letters 50, 137–147.CrossRefGoogle Scholar
Newey, W.K. (1994) Kernel estimation of partial means and a general variance estimator. Econometric Theory 10, 233–253.CrossRefGoogle Scholar
Nishiyama, Y. ’ Robinson, P.M. (2000) Edgeworth expansions for semiparametric averaged derivatives. Econometrica 68, 931–980.CrossRefGoogle Scholar
Paparoditis, E. ’ Politis, D.N. (2000) The local bootstrap for kernel estimators under general dependence conditions. Annals of the Institute of Statistical Mathematics 52, 139–159.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Conditional and unconditional statistical independence. Journal of Econometrics 38, 341–348.CrossRefGoogle Scholar
Pitman, E.J.G. (1979) Some Basic Theory for Statistical Inference. Wiley.Google Scholar
Robinson, P.M. (1988) Root-n-consistent semiparametric regression. Econometrica 56, 931–954.CrossRefGoogle Scholar
Robinson, P.M. (1991) Consistent nonparametric entropy-based testing. Review of Economic Studies 58, 437–453.CrossRefGoogle Scholar
Rosenbaum, P.R. (1984) Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika 49, 425–435.CrossRefGoogle Scholar
Rosenblatt, M. (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Annals of Statistics 3, 1–14.CrossRefGoogle Scholar
Singh, R.S. (1987) MISE of kernel estimates of a density and its derivatives. Statistics and Probability Letters 5, 153–159.CrossRefGoogle Scholar
Skaug, H.J. ’ Tjøstheim, D. (1993) A nonparametric test of serial independence based on the empirical distribution function. Biometrika 80, 591–602.CrossRefGoogle Scholar
Tenreiro, C. (1997) Loi asymptotique des erreurs quadratiques intégrées des estimateurs a noyau de la densité et de la régression sous des conditions de dependance. Portugaliae Mathematica 54, 197–213.Google Scholar
White, H. ’ Hong, Y. (1999) M-testing using finite and infinite dimensional parameter estimators. In Engle, R. ’ White, H. (eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W.J. Granger, pp. 326–345. Oxford University Press.CrossRefGoogle Scholar
Yao, Q. ’ Tritchler, D. (1993) An exact analysis of conditional independence in several 2 × 2 contingency tables. Biometrics 49, 233–236.CrossRefGoogle Scholar
Yoshihara, K. (1976) Limiting behavior of U-statistics for stationary, absolutely regular processes. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 35, 237–252.CrossRefGoogle Scholar
Yoshihara, K. (1989) Limiting behavior of generalized quadratic forms generated by absolutely regular processes. In Mandl, P. ’ Hušková, M. (eds.), Proceedings of the Fourth Prague Symposium on Asymptotic Statistics, pp. 539–547. Charles University Press.Google Scholar