Nonparametric tests for conditional independence using conditional distributions

The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aims to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L2 metric. We use Nadaraya–Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. We ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test which is based on a linear mean-regression, we find that VIX index predicts excess returns both at short and long horizons.

[1]  Thomas M. Stoker,et al.  Goodness-of-fit tests for kernel regression with an application to option implied volatilities , 2001 .

[2]  Jean-Marie Dufour,et al.  Short run and long run causality in time series , 2003 .

[3]  Ľuboš Pástor,et al.  Liquidity Risk and Expected Stock Returns , 2003, Journal of Political Economy.

[4]  L. Summers.  Does the Stock Market Rationally Reflect Fundamental Values , 1986 .

[5]  Qi Li,et al.  Central limit theorem for degenerate U-Statistics of Absolutely Regular Processes with Applications to Model Specification Testing , 1999 .

[6]  Pentti Saikkonen,et al.  ERGODICITY, MIXING, AND EXISTENCE OF MOMENTS OF A CLASS OF MARKOV MODELS WITH APPLICATIONS TO GARCH AND ACD MODELS , 2008, Econometric Theory.

[7]  Tzee-Ming Huang.  Testing conditional independence using maximal nonlinear conditional correlation , 2010, 1010.3843.

[8]  John Y. Campbell,et al.  Asset Prices, Consumption, and the Business Cycle , 1998 .

[9]  Stephen G. Cecchetti,et al.  Asset Pricing with Distorted Beliefs: Are Equity Returns Too Good to Be True? , 1998 .

[10]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[11]  Marcelo Fernandes,et al.  Testing the Markov property with high frequency data , 2007 .

[12]  Jean-Marie Dufour,et al.  Testing Causality Between Two Vectors in Multivariate Arma Models , 1991 .

[13]  E. Nadaraya.  On Estimating Regression , 1964 .

[14]  Christian Gourieroux,et al.  Time Series And Dynamic Models , 1996 .

[15]  C. Granger.  Investigating causal relations by econometric models and cross-spectral methods , 1969 .

[16]  Allan Timmermann,et al.  Is the Distribution of Stock Returns Predictable? , 2008 .

[17]  Robert J. Shiller.  Theories of aggregate stock price movements , 1984 .

[18]  S. Kothari,et al.  Book-to-Market, Dividend Yield, and Expected Market Returns: A Time-Series Analysis , 1997 .

[19]  A. Lo,et al.  Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test , 1987 .

[20]  P. Robinson.  Consistent Nonparametric Entropy-Based Testing , 1991 .

[21]  H. White,et al.  Conditional Independence Specification Testing for Dependent Processes with Local Polynomial Quantile Regression , 2012 .

[22]  H. White,et al.  A Consistent Characteristic-Function-Based Test for Conditional Independence , 2003 .

[23]  Brian Peacock,et al.  Empirical Distribution Function , 2010 .

[24]  Zongwu Cai,et al.  REGRESSION QUANTILES FOR TIME SERIES , 2002, Econometric Theory.

[25]  Liangjun Su,et al.  Nonparametric Testing for Asymmetric Information , 2013 .

[26]  T. Bollerslev,et al.  Expected Stock Returns and Variance Risk Premia , 2007 .

[27]  Helmut Lütkepohl,et al.  Introduction to multiple time series analysis , 1991 .

[28]  J. Lewellen,et al.  Predicting Returns with Financial Ratios , 2002 .

[29]  Kyungchul Song.  Testing Conditional Independence via Rosenblatt Transforms , 2007, 0911.3787.

[30]  Qi Li,et al.  Nonparametric Econometrics: Theory and Practice , 2006 .

[31]  P. Hall.  Central limit theorem for integrated square error of multivariate nonparametric density estimators , 1984 .

[32]  Jean-Marie Dufour,et al.  Short and long run causality measures: Theory and inference , 2008 .

[33]  Jean-Marie Dufour,et al.  Simplified Conditions for Non-Causality Between Vectors in Multivariate Arma Models , 1994 .

[34]  Tae-Hwy Lee,et al.  Money-Income Granger-Causality in Quantiles , 2012 .

[35]  John Geweke,et al.  Inference and causality in economic time series models , 1984 .

[36]  C. J. Stone,et al.  Nonparametric function estimation involving time series , 1992 .

[37]  C. Granger,et al.  Forecasting Economic Time Series. , 1988 .

[38]  Xiaohong Chen,et al.  MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS , 2002, Econometric Theory.

[39]  H. White,et al.  Testing Conditional Independence Via Empirical Likelihood , 2014 .

[40]  J. Cochrane,et al.  New Facts in Finance , 1999 .

[41]  R. Stambaugh,et al.  Predictive Regressions , 1999 .

[42]  Wicher P. Bergsma,et al.  Nonparametric Testing of Conditional Independence by Means of the Partial Copula , 2010, 1101.4607.

[43]  W. Torous,et al.  On Predicting Stock Returns with Nearly Integrated Explanatory Variables , 2004 .

[44]  Michael Wolf,et al.  Stock Returns and Dividend Yields Revisited: A New Way to Look at an Old Problem , 2000 .

[45]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[46]  Herold Dehling,et al.  Central limit theorem and the bootstrap for U-statistics of strongly mixing data , 2008, J. Multivar. Anal..

[47]  Sokbae Lee,et al.  Nonparametric Tests of Conditional Treatment Effects , 2009 .

[48]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[49]  E. Fama,et al.  Dividend yields and expected stock returns , 1988 .

[50]  C. Tenreiro.  Théorèmes limites pour les erreurs quadratiques intégrées des estimateurs à noyau de la densité et de la régression sous des conditions de dépendance , 1995 .

[51]  H. White,et al.  A NONPARAMETRIC HELLINGER METRIC TEST FOR CONDITIONAL INDEPENDENCE , 2008, Econometric Theory.

[52]  Jeffrey S. Racine,et al.  Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions , 2013 .

[53]  H Lawrence,et al.  SUMMERS, . Does the Stock Market Rationally Reflect Fundamental Values?, The Journal of Finance, , . , 1986 .

[54]  M. Shapiro,et al.  Do We Reject Too Often? Small Sample Properties of Tests of Rational Expectations Models , 1985 .

[55]  Jean-Pierre Florens,et al.  Noncausality in Continuous Time , 1996 .

[56]  Jean-Marie Dufour,et al.  Short-Run and Long-Rub Causality in Time Series: Theory. , 1998 .

[57]  J. Rombouts,et al.  Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality , 2012 .

[58]  J. Florens,et al.  A Note on Noncausality , 1982 .

[59]  Jeffrey S. Racine,et al.  Nonparametric Estimation of Conditional CDF and Quantile Functions With Mixed Categorical and Continuous Data , 2008 .

[60]  R. Shiller,et al.  The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors , 1986 .

[61]  Qi Li,et al.  Consistent Model Specification Tests : Kernel-Based Tests versus Bierens ' ICM Tests , 2008 .

[62]  W. Hoeffding.  A Non-Parametric Test of Independence , 1948 .

[63]  Rodney C. Wolff,et al.  Methods for estimating a conditional distribution function , 1999 .

[64]  Qi Li,et al.  A nonparametric test for equality of distributions with mixed categorical and continuous data , 2009 .

[65]  J. Kiefer,et al.  DISTRIBUTION FREE TESTS OF INDEPENDENCE BASED ON THE SAMPLE DISTRIBUTION FUNCTION , 1961 .

[66]  Joel L. Horowitz,et al.  An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model Against a Nonparametric Alternative , 2001 .

[67]  Jean-Pierre Florens,et al.  Non Causality in Continuous Time , 1996 .

[68]  M. Denker,et al.  On U-statistics and v. mise’ statistics for weakly dependent processes , 1983 .

[69]  Herman J. Bierens,et al.  Asymptotic Theory of Integrated Conditional Moment Tests , 1997 .

[70]  H. White,et al.  ASYMPTOTIC DISTRIBUTION THEORY FOR NONPARAMETRIC ENTROPY MEASURES OF SERIAL DEPENDENCE , 2005 .

[71]  K. French,et al.  Stock return variances: The arrival of information and the reaction of traders , 1986 .

[72]  Asymptotic Distribution of Smoothers Based on Local Means and Local Medians under Dependence , 1995 .

[73]  M. Rosenblatt.  A Quadratic Measure of Deviation of Two-Dimensional Density Estimates and A Test of Independence , 1975 .

[74]  Larry D. Haugh,et al.  Causality in temporal systems: Characterization and a survey , 1977 .

[75]  J. Campbell,et al.  By Force of Habit: A Consumption‐Based Explanation of Aggregate Stock Market Behavior , 1995, Journal of Political Economy.

[76]  O. Linton,et al.  Testing Conditional Independence Restrictions , 2014 .

[77]  C. Granger.  Testing for causality: a personal viewpoint , 1980 .

[78]  Yanqin Fan,et al.  Consistent model specification tests : Omitted variables and semiparametric functional forms , 1996 .

[79]  Roch Roy,et al.  ROBUST OPTIMAL TESTS FOR CAUSALITY IN MULTIVARIATE TIME SERIES , 2008, Econometric Theory.

[80]  Andrew Ang,et al.  Stock Return Predictability: Is it There? , 2001 .

[81]  Ravi Bansal,et al.  Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles , 2000 .

[82]  Efstathios Paparoditis,et al.  The Local Bootstrap for Kernel Estimators under General Dependence Conditions , 2000 .

[83]  Markku Lanne,et al.  Testing the Predictability of Stock Returns , 2002, Review of Economics and Statistics.

[84]  H. White,et al.  A FLEXIBLE NONPARAMETRIC TEST FOR CONDITIONAL INDEPENDENCE , 2013, Econometric Theory.

[85]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[86]  Wenceslao González Manteiga,et al.  Significance testing in nonparametric regression based on the bootstrap , 2001 .

[87]  R. Hodrick.  Dividend Yields and Expected Stock Returns: Alternative Procedures for Interference and Measurement , 1991 .

[88]  Hans J. Skaug,et al.  A nonparametric test of serial independence based on the empirical distribution function , 1993 .

[89]  J. Simonoff.  Smoothing Methods in Statistics , 1998 .

[90]  K. Yoshihara.  Limiting behavior of U-statistics for stationary, absolutely regular processes , 1976 .

[91]  C. Nelson,et al.  Predictable Stock Returns: The Role of Small Sample Bias , 1993 .

[92]  P. Doukhan.  Mixing: Properties and Examples , 1994 .

[93]  C. Tenreiro.  LOI ASYMPTOTIQUE DES ERREURS QUADRATIQUES INTÉGRÉES DES ESTIMATEURS À NOYAU DE LA DENSITÉ ET DE LA RÉGRESSION SOUS DES CONDITIONS DE DÉPENDANCE* , 2022 .

[94]  Rossen Valkanov.  Long-horizon regressions: theoretical results and applications , 2003 .

[95]  Jean-Marie Dufour,et al.  Testing Causality between Two Vectors in Multivariate Autoregressive Moving Average Models , 1992 .

[96]  I. Gijbels,et al.  Bandwidth Selection in Nonparametric Kernel Testing , 2008 .