Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T07:20:05.141Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERISTIC FUNCTION BASED TESTING FOR CONDITIONAL INDEPENDENCE: A NONPARAMETRIC REGRESSION APPROACH

Published online by Cambridge University Press:  11 April 2017

Xia Wang  and
Yongmiao Hong
Xia Wang*
Affiliation:
Sun Yat-Sen University
Yongmiao Hong*
Affiliation:
Cornell University Xiamen University
*
†Xia Wang, Lingnan (University) College, Sun Yat-Sen University, Guangzhou 510275, China.
*Address correspondence to Yongmiao Hong, Department of Economics and Department of Statistical Sciences, Cornell University, 424 Uris Hall, Ithaca, NY 14850, USA, and Wang Yanan Institute for Studies in Economics, Economic Building, Xiamen University, Xiamen 361005, China; e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a characteristic function based test for conditional independence, applicable to both cross-sectional and time series data. We also derive a class of derivative tests, which deliver model-free tests for such important hypotheses as omitted variables, Granger causality in various moments and conditional uncorrelatedness. The proposed tests have a convenient asymptotic null N (0, 1) distribution, and are asymptotically locally more powerful than a variety of related smoothed nonparametric tests in the literature. Unlike other smoothed nonparametric tests for conditional independence, we allow nonparametric estimators for both conditional joint and marginal characteristic functions to jointly determine the asymptotic distributions of the test statistics. Monte Carlo studies demonstrate excellent power of the tests against various alternatives. In an application to testing Granger causality, we document the existence of nonlinear relationships between money and output, which are missed by some existing tests.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 4 , August 2018 , pp. 815 - 849
Copyright
Copyright © Cambridge University Press 2017 

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.)

Footnotes

Wang acknowledges financial supports from the National Science Foundation of China (No. 71401160), the Ministry of Education of Humanities and Social Sciences Project of China (No. 14YJC790120), Fujian Provincial Key Laboratory of Statistics (Xiamen University, No. 2016003). We thank the editor Peter C. B. Phillips, the co-editor Liangjun Su, two referees, and seminar participants at Chinese Academy of Sciences, Kansas Econometrics Workshop, New York Econometrics Camp, Singapore Management University, and UC Riverside for their comments and suggestions. We are most grateful to Peter C. B. Phillips, who has gone beyond the call of duty for an editor in carefully correcting our English. We also thank Liangjun Su for providing us the computer codes to implement his tests. All remaining errors are solely our own.

References

REFERENCES

Aït-Sahalia, Y., Bickel, P.J., & Stoker, T.M. (2001) Goodness-of-fit tests for kernel regression with an application to option implied volatilities. Journal of Econometrics 105, 363–412.CrossRefGoogle Scholar
Bae, Y. & de Jone, R. (2007) Money demand function estimation by nonlinear cointegration. Journal of Applied Econometrics 22, 767–793.CrossRefGoogle Scholar
Blundell, R. & Horowitz, J.L. (2007) A non-parametric test of exogeneity. Review of Economic Studies 74, 1035–1058.CrossRefGoogle Scholar
Bouezmarni, T., Rombouts, J.V.K., & Taamouti, A. (2012) Nonparametric copula-based test for conditional independence with applications to granger causality. Journal of Business and Economic Statistics 30, 275–287.CrossRefGoogle Scholar
Bouezmarni, T. & Taamouti, A. (2014) Nonparametric tests for conditional independence using conditional distributions. Journal of Nonparametric Statistics 26, 697–719.CrossRefGoogle Scholar
Campbell, J.Y. (1992) No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics 31, 281–318.CrossRefGoogle Scholar
Casillas-Olvera, G. & Bessler, D.A. (2006) Probability forecasting and central bank accountability. Journal of Policy Modeling 28, 223–234.CrossRefGoogle Scholar
Chen, B. & Hong, Y. (2010) Characteristic function-based testing for multifactor continuous-time markov models via nonparametric regression. Econometric Theory 26, 1115–1179.CrossRefGoogle Scholar
Chen, R. & Tsay, R.S. (1993) Functional-coefficient autoregressive models. Journal of American Statistical Association 88, 298–308.Google Scholar
Christiano, L.J. & Ljungqvist, L. (1988) Money does Granger-cause output in the bivariate money-output relation. Journal of Monetary Economics 22, 217–235.CrossRefGoogle Scholar
Clements, M.P. (2004) Evaluating the Bank of England density forecasts of inflation. Economic Journal 114, 844–866.CrossRefGoogle Scholar
Dawid, A.P. (1979) Conditional independence in statistical theory. Journal of Royal Statistical Society: Series B 41, 1–31.Google Scholar
Dawid, A.P. (1980) Conditional independence for statistical operations. Annals of Statistics 8, 598–617.CrossRefGoogle Scholar
Diebold, F.X., Hahn, J., & Tay, A.S. (1999) Multivariate density forecast evaluation and calibration in financial risk management: High-frequency returns of foreign exchange. Review of Economics and Statistics 81, 661–673.CrossRefGoogle Scholar
Easley, D. & O’Hara, M. (1987) Price, trade size, and information in securities markets. Journal of Financial Economics 19, 69–90.CrossRefGoogle Scholar
Engle, R.F., Lilien, D.M., & Robins, R.P. (1987) Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica 55, 391–407.CrossRefGoogle Scholar
Fan, Y. & Gijbels, I. (1996) Local Polynomial Modelling and its Applications. Chapman and Hall.Google Scholar
Fan, Y. & Li, Q. (1996) Consistent model specification tests: Omitted varibales and semiparametric function forms. Econometrica 64, 865–890.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (1999) Root –n– consistent estimation of partially linear time series models. Journal of Nonparametric Statistics 10, 245–271.CrossRefGoogle Scholar
Friedman, B.M. & Kutter, K.N. (1993) Another look at the evidence on money-income causality. Journal of Econometrics 57, 189–203.CrossRefGoogle Scholar
Gao, J. & Gijbels, I. (2008) Bandwidth selection in nonparametric kernel testing. Journal of American Statistical Association 103, 1584–1594.CrossRefGoogle Scholar
Granger, C.W.J. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.CrossRefGoogle Scholar
Granger, C.W.J. (1980) Testing for causality: A personal viewpoint. Journal of Economic Dynamics and Control 2, 329–352.CrossRefGoogle Scholar
Hahn, J., Todd, P., & Klaauw, W.V. (2001) Identification and estimation of treatment effects with a regression-discontinuity design. Econometrica 69, 201–209.CrossRefGoogle Scholar
Heckman, J.D. (1976) The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurement 5, 475–492.Google Scholar
Hjellvik, V., Yao, Q., & Tjøstheim, D. (1998) Linearity testing using local polynomial approximation. Journal of Statistical Planning and Inference 68, 295–321.CrossRefGoogle Scholar
Hong, Y. (1999) Hypothesis testing in time series via the empirical characteristic function: A generalized spectral density approach. Journal of American Statistical Association 94, 1201–1220.CrossRefGoogle Scholar
Horowitz, J.L. & Manski, C.F. (2000) Nonparametric analysis of randomized experiments with missing covariate and outcome data. Journal of American Statistical Association 95, 77–84.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
Huang, T.M. (2010) Testing conditional independence using maximal nonlinear conditional correlation. Annals of Statistics 38, 2047–2091.CrossRefGoogle Scholar
Huang, M., Sun, Y., & White, H. (2016) A flexible nonparametric test for conditional independence, Econometric Theory 32, 1–49.CrossRefGoogle Scholar
Kandil, M. (1995) Asymmetric norminal flexibility and economic fluctuations. Southern Economic Journal 61, 674–695.CrossRefGoogle Scholar
Kim, C.J. & Nelson, C.R. (2006) Estimation of a forward-looking monetary policy rule: A time-varying parameter model using ex post data. Journal of Monetary Economics 53, 1949–1966.CrossRefGoogle Scholar
Lavergne, P. & Vuong, Q. (2000) Nonparametric siginificance testing. Econometric Theory 16, 576–601.CrossRefGoogle Scholar
Lee, J. (2013) A Consistent Nonparametric Bootstrap Test of Exogeneity. Working paper, University of St Andrews.Google Scholar
Lee, T. & Yang, W. (2012) Money-income granger-causality in quantiles. In Millimet, D. & Terrell, D. (eds.), Advances in Econometrics, vol. 30, pp. 383–407. Emerald Publishers.Google Scholar
Linton, O. & Gozalo, P. (2014) Testing conditional independence restrictions. Econometric Reviews 33, 523–552.CrossRefGoogle Scholar
Little, R.J.A. (1985) A note about models for selectivity bias. Econometrica 53, 1469–1474.CrossRefGoogle Scholar
Manski, C.F. (2000) Identification problems and decisions under ambiguity: Empirical analysis of treatment response and normative analysis of treatment choice. Journal of Econometrics 95, 415–442.CrossRefGoogle Scholar
Manski, C.F. (2003) Partial Identification of Probability Distribution. Springer-Verlag.Google Scholar
Manski, C.F. (2007) Identification for Prediction and Decision. Princeton University Press.Google Scholar
Masry, E. (1996a) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Masry, E. (1996b) Multivariate regression estimation: local polynomial fitting for time series. Stochastic Processes and Their Applications 65, 81–101.CrossRefGoogle Scholar
Nishiyama, Y., Hitomi, K., Kawasaki, Y., & Jeong, K. (2011) A consistent nonparametric test for nonlinear causalityłspecification in time series regression. Journal of Econometrics 165, 112–127.CrossRefGoogle Scholar
Nobay, R.A. & Peel, D.A. (2003) Optimal discretionary monetary policy in a model of asymmetric central bank preferences. Economic Journal 113, 657–665.CrossRefGoogle Scholar
Paparodits, E. & Politis, D.N. (2000) The local bootstrap for kernel estimators under general dependence conditions. Annals of the Institute of Statistical Mathmatics 52, 139–159.CrossRefGoogle Scholar
Peiró, A. (1999) Skewness in financial returns. Journal of Banking and Finance 23, 847–862.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Conditional and unconditional statistical independence. Journal of Econometrics 38, 341–348.CrossRefGoogle Scholar
Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis. Academic Press.Google Scholar
Psaradakis, Z., Ravn, M.O., & Sola, M. (2005) Markov switching causality and the money-output relationship. Journal of Applied Econometrics 20, 665–683.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
Rubin, D.B. (1976) Inference and missing data. Biometrika 63, 581–592.CrossRefGoogle Scholar
Rust, J. (1994) Structural estimation of markov decision processes. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 3081–3143. Elsevier.Google Scholar
Sims, C.A. (1972) Money, income, and causality. American Economic Review 62, 540–552.Google Scholar
Sims, C.A. (1980) Macroeconomics and reality. Econometrica 48, 1–48.CrossRefGoogle Scholar
Song, K. (2009) Testing conditional independence via rosenblatt transforms. Annals of Statistics 37, 4011–4045.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1989) Interpretating the evidence on money-income causality. Journal of Econometrics 40, 161–181.CrossRefGoogle Scholar
Su, L. & Ullah, A. (2009) Testing conditional uncorrelatedness. Journal of Business and Economic Statistics 27, 18–29.CrossRefGoogle Scholar
Su, L. & White, H. (2007) A consistent characteristic function-based test for conditional independence. Journal of Econometrics 141, 807–834.CrossRefGoogle Scholar
Su, L. & White, H. (2008) A nonparametric hellinger metric test for conditional independence. Econometric Theory 24, 829–864.CrossRefGoogle Scholar
Su, L. & White, H. (2012) Conditional independence specification testing for dependent process with local polynomial quantile regression. Advances in Econometrics 29, 355–434.CrossRefGoogle Scholar
Su, L. & White, H. (2014) Testing conditional independence via empirical likelihood. Journal of Econometrics 182, 27–44.CrossRefGoogle Scholar
Sun, Y., Phillips, P., & Jin, S. (2008) Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica 76, 175–194.CrossRefGoogle Scholar
Taamouti, A., Bouezmarni, T., & El Ghouch, A. (2014) Nonparametric estimation and inference for conditional density based granger causality measures. Journal of Econometrics 180, 251–264.CrossRefGoogle Scholar
Taylor, J. (1993) Discretion versus policy rules in practice. Carnegie-Rochester conference Series on Public Policy 39, 195–214.CrossRefGoogle Scholar
Tenreiro, C. (1997) Loi asymptotique des erreurs quadratiques intégrées des estimateurs a noyau de la densité et de la régression sou des conditions de dependance. Portugaliae Mathematica 54, 187–213.Google Scholar
Tong, H. & Lim, K.S. (1980) Threshold autoregression, limit cycles and cyclical data. Journal of Royal Statistical Society, Series B 42, 245–292.Google Scholar
Uhlig, H. (2005) What are the effects of monetary policy on output? results from an agnostic identification procedure. Journal of Monetary Economics 52, 381–419.CrossRefGoogle Scholar
Wang, Q., Linton, O., & Hardle, W. (2004) Semiparametric regression analysis with missing response at random. Journal of American Statistical Association 99, 334–345.CrossRefGoogle Scholar
Supplementary material: PDF

Wang and Hong Supplementary Material

Supplementary Material

Download Wang and Hong Supplementary Material(PDF)
PDF 499.9 KB