Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study
-
Efthymios G. Pavlidis
,
Ivan Paya
Abstract
This paper assesses the performance of linear and nonlinear causality tests in the presence of multivariate conditional heteroskedasticity, exogenous volatility regressors, and additive volatility outliers. Monte Carlo simulations show that tests based on the least squares covariance matrix estimator can frequently lead to finding spurious Granger causality. The degree of oversizing tends to increase with the sample size and is substantially larger for the nonlinear test. On the other hand, heteroskedasticity-robust tests which are based on the fixed design wild bootstrap perform adequately in terms of size and power. Consequently, reliable causality in mean tests can be conducted without the need to specify a conditional variance function. As an empirical application, we re-examine the return-volume relationship.
- 1
It also holds that the performance of tests for causality in variance depends crucially on the conditional mean equation (see, e.g., Pantelidis and Pittis 2004).
- 2
It is assumed that the function under consideration has convergent Taylor expansions at any point in the sample space for every parameter vector in the parameter space (see assumption A4 in Péguin-Feissolle, Strikholm, and Teräsvirta etal. 2012, p. 4).
- 3
Péguin-Feissolle, Strikholm, and Terésvirta (2012) employ a Lagrange multiplier test rather than a Wald test. As proven by Engle (1984), Wald, Lagrange multiplier and likelihood ratio tests are asymptotically equivalent. We employ the Wald test to simplify the exposition when robustifying for heteroskedasticity.
- 4
Chesher and Jewitt (1987) show that as the maximum htt approaches unity HC1 becomes extremely biased. HC2, HC3 and HC4 circumvent this problem by attaching less weight to influential observations.
- 5
It is also worth noting that because the effect of outliers is amplified by including high powers of y and x in the test Equation (2), nonlinear tests based on HC1 are expected to perform worse than their linear counterparts. We show that this is indeed the case in Section 3.1.
- 6
The results are available from the authors upon request.
- 7
All experiments are conducted on the High End Computing Cluster of Lancaster University in R 2.11.1. We are grateful to the administrator, Mike Pacey, for his valuable assistance.
- 8
Some of the contemporaneous results should be interpreted with caution due to the overlap in trading. More specifically, there is a 2.5 h overlap between London and New York. As Lee and Rui (2002) argue, this overlap may create problems when examining contemporaneous (but not dynamic) relations.
- 9
The results reported in Table 3 are similar but not identical to those of Lee and Rui (2002). The small differences can be attributed to data revisions. From February 20th, 2003 the history of the Datastream S&P500 indices was recalculated back to the base date of 2nd January 1973 using the current constituents.
- 10
On the contrary, the LS method indicates bidirectional causality. Nonlinear LS causality tests, in particular, reject the null at the 1% significance level in all cases but one.
References
Andersen, T. G. (1996), “Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility.” Journal of Finance 51: 169–204.10.1111/j.1540-6261.1996.tb05206.xSuche in Google Scholar
Andrews, D. W. K. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica 59: 817–858.10.2307/2938229Suche in Google Scholar
Beran, R. 1988. “Prepivoting Test Statistics: A Bootstrap View of Asymptotic Refinements.” Journal of the American Statistical Association 83: 687–697.10.1080/01621459.1988.10478649Suche in Google Scholar
Blume, L., D. Easley, and M. O’Hara. 1994. “Market Statistics and Technical Analysis: The Role of Volume.” Journal of Finance 49: 153–181.10.1111/j.1540-6261.1994.tb04424.xSuche in Google Scholar
Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Suche in Google Scholar
Bollerslev, T. 2001. “Financial Econometrics: Past Developments and Future Challenges.” Journal of Econometrics 100: 41–51.10.1016/S0304-4076(00)00052-XSuche in Google Scholar
Carnero, M.A., D. Peña, and E. Ruiz. 2007. “Effects of Outliers on the Identification an Destimation of GARCH Models.” Journal of Time Series Analysis 28: 471–497.10.1111/j.1467-9892.2006.00519.xSuche in Google Scholar
Chesher, A. and I. Jewitt. 1987. “The Bias of a Heteroskedasticity Consistent Covariance Matrix Estimator.” Econometrica 55: 1217–1222.10.2307/1911269Suche in Google Scholar
Cheung, Y.-W. and L. K. Ng. 1996. “A Causality-in-variance Test and its Application to Financial Market Prices.” Journal of Econometrics 72: 33–48.10.1016/0304-4076(94)01714-XSuche in Google Scholar
Cribari-Neto, F. 2004. “Asymptotic Inference Under Heteroskedasticity of Unknown Form.” Computational Statistics & Data Analysis 45: 215–233.10.1016/S0167-9473(02)00366-3Suche in Google Scholar
Davidson, R. and E. Flachaire. 2008. “The Wild Bootstrap, Tamed at Last.” Journal of Econometrics 146: 162–169.10.1016/j.jeconom.2008.08.003Suche in Google Scholar
Davidson, R. and J. G. MacKinnon. 1999. “The Size Distortion Of Bootstrap Tests.” Econometric Theory 15: 361–376.10.1017/S0266466699153040Suche in Google Scholar
Dominguez, K. M. 1993. “Does Central Bank Intervention Increase the Volatility of Foreign Exchange Rates?” Nber working papers, National Bureau of Economic Research, Inc.10.3386/w4532Suche in Google Scholar
Doornik, J. A. and M. Ooms. 2005. “Outlier Detection in GARCH Models.” Tinbergen institute discussion papers, Tinbergen Institute.Suche in Google Scholar
Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity With Estimates of the Variance of United Kingdom Inflation.” Econometrica 50: 987–1007.10.2307/1912773Suche in Google Scholar
Engle, R. F. 1984. “Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics.” In Handbook of Econometrics, edited by Z. Grilichesâ, and M. D. Intriligator, 775–826. Vol. 2, Chap. 13. Amsterdam: Elsevier.10.1016/S1573-4412(84)02005-5Suche in Google Scholar
Engle, R. 2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20: 339–350.10.1198/073500102288618487Suche in Google Scholar
Engle, R. F. and A. J. Patton. 2001. “What Good is a Volatility Model?” Quantitative Finance 1: 237–245.10.1088/1469-7688/1/2/305Suche in Google Scholar
Flachaire, E. 2005. “Bootstrapping Heteroskedastic Regression Models: Wild Bootstrap vs. Pairs Bootstrap.” Computational Statistics and Data Analysis 49: 361–376.10.1016/j.csda.2004.05.018Suche in Google Scholar
Godfrey, L. G. and A. R. Tremayne. 2005. “The Wild Bootstrap and Heteroskedasticity-Robust Tests for Serial Correlation in Dynamic Regression Models.” Computational Statistics & Data Analysis 49: 377–395.10.1016/j.csda.2004.05.020Suche in Google Scholar
Gonçalves, S. and L. Kilian. 2004. “Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form.” Journal of Econometrics 123: 89–120.10.1016/j.jeconom.2003.10.030Suche in Google Scholar
Granger, C. W. J. and T. Teräsvirta. 1993. Modelling Nonlinear Economic Relationships. Oxford University Press.Suche in Google Scholar
Granger, C. W. J. 1969. “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods.” Econometrica 37: 424–438.10.2307/1912791Suche in Google Scholar
He, C. and T. Teräsvirta. 2004. “An Extended Constant Conditional Correlation Garch Model And Its Fourth-Moment Structure.” Econometric Theory 20: 904–926.10.1017/S0266466604205059Suche in Google Scholar
Hiemstra, C. and Jones, J. D. 1994. “Testing for Linear and Nonlinear Granger Causality in the Stock Price-Volume Relation.” Journal of Finance 49: 1639–1664.Suche in Google Scholar
Hurn, S. and R. Becker. 2007. “Testing for Nonlinearity in Mean in the Presence of Heteroskedasticity.” Working Paper Series 8, NCER.Suche in Google Scholar
Jain, P. C. and G. Joh. 1988. “The Dependence between Hourly Prices and Trading Volume.” Journal of Financial and Quantitative Analysis 23: 269–283.10.2307/2331067Suche in Google Scholar
Jeantheau, T. 1998. “Strong Consistency of Estimators for Multivariate ARCH Models.” Econometric Theory 14: 70–86.10.1017/S0266466698141038Suche in Google Scholar
Jennings, R. H., L. T. Starks, and J. C. Fellingham. 1981. “An Equilibrium Model of Asset Trading with Sequential Information Arrival.” Journal of Finance 36: 143–161.10.1111/j.1540-6261.1981.tb03540.xSuche in Google Scholar
Kreiss, J.-P. 1997. “Asymptotic Properties of Residual Bootstrap for Autoregressions.” Manuscript, Institute for Mathematical Stochastics, Technical University of Braunschweig, Germany.Suche in Google Scholar
Lamoureux, C. G. and W. D. Lastrapes. 1990. “Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects.” Journal of Finance 45: 221–229.10.1111/j.1540-6261.1990.tb05088.xSuche in Google Scholar
Lamoureux, C. G. and W. D. Lastrapes. 1994. “Endogenous Trading Volume and Momentum in Stock-Return Volatility.” Journal of Business & Economic Statistics 12: 253–260.Suche in Google Scholar
Laurent, S., L. Bauwens, and J. V. K. Rombouts. 2006. “Multivariate GARCH Models: A Survey.” Journal of Applied Econometrics 21: 79–109.10.1002/jae.842Suche in Google Scholar
Lee, B.-S. and B.-S. Rui. 2002. “The Dynamic Relationship between Stock Returns and Trading Volume: Domestic and Cross-country Evidence.” Journal of Banking & Finance 26: 51–78.10.1016/S0378-4266(00)00173-4Suche in Google Scholar
Liu, R. 1988. “Bootstrap Procedure Under Some Non i.i.d. Models.” Annals of Statistics 16: 1696–1708.10.1214/aos/1176351062Suche in Google Scholar
Long, J. S. and L. H. Ervin. 2000. “Using Heteroscedasticity Consistent Standard Error sin the Linear Regression Model.” The American Statistician 54: 217–224.Suche in Google Scholar
MacKinnon, J. G. and H. White. 1985. “Some Heteroskedasticity Consistent Covariance Matrix Estimators With Improved Finite Sample Properties.” Journal of Econometrics 29: 305–325.10.1016/0304-4076(85)90158-7Suche in Google Scholar
Mammen, E. 1993. “Bootstrap and Wild Bootstrap for High Dimensional Linear Models.” Annals of Statistics 21: 255–285.10.1214/aos/1176349025Suche in Google Scholar
Mandelbrot, B. 1963. “The Variation of Certain Speculative Prices.” Journal of Business 36: 394–419.10.1086/294632Suche in Google Scholar
McAleer, M., S. Hoti, and F. Chan. 2009. “Structure and Asymptotic Theory for Multivariate Asymmetric Conditional Volatility.” Econometric Reviews 28: 422–440.10.1080/07474930802467217Suche in Google Scholar
Newey, W. K. and W. K. West. 1987. “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica 55: 703–708.10.2307/1913610Suche in Google Scholar
Pantelidis, T. and N. Pittis. 2004. “Testing for Granger Causality Invariance in the Presence of Causality in Mean.” Economics Letters 85: 201–207.10.1016/j.econlet.2004.04.006Suche in Google Scholar
Pavlidis, E. G., I. Paya, and D. A. Peel. 2010. “Specifying Smooth Transition Regression Models in the Presence of Conditional Heteroskedasticity of Unknown Form.” Studies in Nonlinear Dynamics & Econometrics 14: 3.10.2202/1558-3708.1702Suche in Google Scholar
Péguin-Feissolle, A., B. Strikholm, and T. Teräsvirta. 2012. “Testing the Granger Noncausality Hypothesis in Stationary Nonlinear Models of Unknown Functional Form.” Forthcoming in Communications in Statistics - Simulation and Computation.10.1080/03610918.2012.661500Suche in Google Scholar
Räıssi, H. 2011. “Testing Linear Causality in Mean when the Number of Estimated Parametersis High. Electronic Journal of Statistics 5: 507–533.10.1214/11-EJS617Suche in Google Scholar
Rogalski, R. J. 1978. “The Dependence of Prices and Volume.” The Review of Economics and Statistics 60: 268–274.10.2307/1924980Suche in Google Scholar
Sakata, S. and H. White. 1998. “High Breakdown Point Conditional Dispersion Estimation with Application to S&P 500 Daily Returns Volatility.” Econometrica 66: 529–568.10.2307/2998574Suche in Google Scholar
Teräsvirta, T., C.-F. Lin, and C. W. J. Granger. 1993. “Power of the Neural Network Linearity Test.” Journal of Time Series Analysis 14: 209–220.10.1111/j.1467-9892.1993.tb00139.xSuche in Google Scholar
Vilasuso, J. R. 2001. “Causality Tests and Conditional Heteroskedasticity: Monte Carlo evidence.” Journal of Econometrics 101: 25–35.10.1016/S0304-4076(00)00072-5Suche in Google Scholar
Wang, J. 1994. “A Model of Competitive Stock Trading Volume.” Journal of Political Economy 102: 127–168.10.1086/261924Suche in Google Scholar
Weiner, N. 1956. “The Theory of Prediction.” In Modern Mathematics for the Engineer, edited by Edwin F. Beckenback, 165–190, New York: McGraw-Hill.Suche in Google Scholar
White, H. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica 48: 817–838.10.2307/1912934Suche in Google Scholar
Wu, C. F. J. 1986. “Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis (with discussion).” Annals of Statististics 14: 1261–1350.Suche in Google Scholar
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Artikel in diesem Heft
- Masthead
- Masthead
- The Danish krone-euro exchange rate and Danmark Nationalbank intervention operations
- Common large innovations across nonlinear time series
- The forward rate premium puzzle: a case of misspecification?1)
- A smooth transition long-memory model
- Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study
- Threshold linkages between volatility and trading volume: evidence from developed and emerging markets
- Inventory investment and the business cycle: the usual suspect
Artikel in diesem Heft
- Masthead
- Masthead
- The Danish krone-euro exchange rate and Danmark Nationalbank intervention operations
- Common large innovations across nonlinear time series
- The forward rate premium puzzle: a case of misspecification?1)
- A smooth transition long-memory model
- Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study
- Threshold linkages between volatility and trading volume: evidence from developed and emerging markets
- Inventory investment and the business cycle: the usual suspect
