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ROBUST OPTIMAL TESTS FOR CAUSALITY IN MULTIVARIATE TIME SERIES

Published online by Cambridge University Press:  04 April 2008

Abdessamad Saidi  and
Roch Roy
Abdessamad Saidi
Affiliation:
Université de Montréal
Roch Roy*
Affiliation:
Université de Montréal
*
Address correspondence to Roch Roy, Département de mathématiques et de statistique, Université de Montréal, CP 6128, succursale Centre-ville, Montréal, Québec, H3C 3J7, Canada; e-mail: royr@dms.umontreal.ca.
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Abstract

Here, we derive optimal rank-based tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one direction causality hypotheses are tested. Using the characterization of noncausality in the VAR context, the local asymptotic normality (LAN) theory described in Le Cam (1986, Asymptotic Methods in Statistical Decision Theory) allows for constructing locally and asymptotically optimal tests for the null hypothesis of noncausality in one or both directions. These tests are based on multivariate residual ranks and signs (Hallin and Paindaveine, 2004a, Annals of Statistics 32, 2642–2678) and are shown to be asymptotically distribution free under elliptically symmetric innovation densities and invariant with respect to some affine transformations. Local powers and asymptotic relative efficiencies are also derived. The level, power, and robustness (to outliers) of the resulting tests are studied by simulation and are compared to those of the Wald test. Finally, the new tests are applied to Canadian money and income data.

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Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 4 , August 2008 , pp. 948 - 987
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Boudjellaba, H., Dufour, J.-M., ’ Roy, R. (1992) Testing causality between two vectors in multivariate autoregressive moving average models. Journal of the American Statistical Association 87, 1082–1090.CrossRefGoogle Scholar
Boudjellaba, H., Dufour, J.-M., ’ Roy, R. (1994) Simplified conditions for noncausality between vectors in multivariate ARMA models. Journal of Econometrics 63, 271–287.CrossRefGoogle Scholar
Dolado, J.J. ’ Lütkepohl, H. (1996) Making Wald tests work for cointegrated VAR systems. Econometric Reviews 15, 369–386.Google Scholar
Drost, F.C., Klaassen, C.A.J., ’ Werker, B.J.M. (1997) Adaptive estimation in time-series models. Annals of Statistics 25, 786–818.Google Scholar
Dufour, J.-M., Pelletier, D., ’ Renault, E. (2006) Short run and long run causality in time series: Inference. Journal of Econometrics 133, 337–362.Google Scholar
Dufour, J.-M. ’ Renault, E. (1998) Short run and long run causality in time series: Theory. Econometrica 66, 1099–1125.CrossRefGoogle Scholar
Fang, K.T., Kotz, S., ’ Ng, K.W. (1990) Symmetric Multivariate and Related Distributions. Chapman and Hall.Google Scholar
Garel, B. ’ Hallin, M. (1995) Local asymptotic normality of multivariate ARMA processes with a linear trend. Annals of the Institute of Statistical Mathematics 47, 551–579.CrossRefGoogle Scholar
Geweke, J. (1984) Inference and causality in economic time series. In Griliches, Z. ’ Intrilligator, M.D. (eds.), Handbook of Econometrics, pp. 1102–1144. North-Holland.Google Scholar
Gouriéroux, C. ’ Monfort, A. (1990) Séries temporelles et modèles dynamiques. Economica.Google Scholar
Granger, C.W.J. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–459.Google Scholar
Hájek, J. ’ S̆idák, Z. (1967) Theory of Rank Tests. Academic Press.Google Scholar
Hallin, M. ’ Paindaveine, D. (2002a) Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Annals of Statistics 30, 1103–1133.Google Scholar
Hallin, M. ’ Paindaveine, D. (2002b) Multivariate signed ranks: Randles' interdirections or Tyler's angles? In Dodge, Y. (ed.), Statistical Data Analysis Based on the L1-norm and Related Methods, pp. 271–282. Basel.Google Scholar
Hallin, M. ’ Paindaveine, D. (2002c) Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence. Bernoulli 8, 787–815.Google Scholar
Hallin, M. ’ Paindaveine, D. (2003) Affine invariant linear hypotheses for the multivariate general linear model with ARMA error terms. In Moore, M., Froda, S., ’ Léger, C. (eds.), Mathematical Statistics and Applications: Festschrift for Constance van Eeden, I.M.S. Lecture Notes—Monograph Series, pp. 417–434. Hayward.Google Scholar
Hallin, M. ’ Paindaveine, D. (2004a) Rank-based optimal tests of the adequacy of an elliptic VARMA model. Annals of Statistics 32, 2642–2678.CrossRefGoogle Scholar
Hallin, M. ’ Paindaveine, D. (2004b) Multivariate signed rank tests in vector autoregressive order identification. Statistical Science 19, 697–711.CrossRefGoogle Scholar
Hallin, M. ’ Paindaveine, D. (2005) Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors. Journal of Multivariate Analysis 93, 122–163.Google Scholar
Hallin, M. ’ Puri, M.L. (1992) Rank tests for time-series analysis: A survey. In Brillinger, D., Parzen, E., ’ Rosenblatt, M. (eds.), New Directions in Time Series Analysis, vol. 45, pp. 111–153. Springer-Verlag.Google Scholar
Hallin, M. ’ Puri, M.L. (1994) Aligned rank tests for linear models with autocorrelated errors. Journal of Multivariate Analysis 50, 175–237.Google Scholar
Hallin, M. ’ Werker, B.J.M. (1999) Optimal testing for semi-parametric autoregressive models: From Gaussian Lagrange multipliers to regression rank scores and adaptive tests. In Ghosh, S. (ed.), Asymptotics, Nonparametrics, and Time Series, pp. 295–358. M. Dekker.Google Scholar
Hallin, M. ’ Werker, B.J.M. (2003) Semiparametric efficiency, distribution-freeness and invariance. Bernoulli 9, 137–165.Google Scholar
Hsiao, C. (1979) Autoregressive modeling of Canadian money and income data. Journal of the American Statistical Association 74, 553–560.CrossRefGoogle Scholar
Hwang, S.Y. ’ Basawa, I.V. (1993) Asymptotic optimal inference for a class of nonlinear time series. Stochastic Processes and Their Applications 46, 91–114.CrossRefGoogle Scholar
Koul, H.L. ’ Schick, A. (1996) Adaptive estimation in a random coefficient autoregressive model. Annals of Statistics 24, 1025–1052.Google Scholar
Koul, H.L. ’ Schick, A. (1997) Efficient estimation in nonlinear autoregressive time series models. Bernoulli 3, 247–277.Google Scholar
Kreiss, J.-P. (1987) On adaptative estimation in stationary ARMA processes. Annals of Statistics 15, 112–133.Google Scholar
Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. Springer-Verlag.CrossRefGoogle Scholar
Lütkepohl, H. (1991) Introduction to Multiple Time Series Analysis. Springer-Verlag.CrossRefGoogle Scholar
Lütkepohl, H. (1993) Testing for causation between two variables in higher dimensional VAR models. In Schneeweis, H. ’ Zimmerman, K. (eds.), Studies in Applied Econometrics, pp. 75–91. Springer-Verlag.Google Scholar
Martin, R.D. ’ Yohai, V.J. (1985) Robustness in time series and estimating ARMA models. In Hannan, E.J., Krishnaiah, P.R., ’ Rao, M.M. (eds.), Handbook of Statistics, vol. 5, Time Series in the Time Domain, pp. 119–155. North-Holland.CrossRefGoogle Scholar
Newbold, P. (1982) Causality testing in economics. In Anderson, O.D. (ed.), Time Series Analysis: Theory and Practice 1, pp. 701–716. North-Holland.Google Scholar
Oja, H. (1999) Affine invariant multivariate sign and rank tests and corresponding estimates. Scandinavian Journal of Statistics 26, 319–343.Google Scholar
Oja, H. ’ Paindaveine, D. (2005) Optimal signed-rank tests based on hyperplanes. Journal of Statistical Planning and Inference 135, 300–323.CrossRefGoogle Scholar
Peters, D. ’ Randles, R.H. (1990) A multivariate signed-rank test for the one-sample location problem. Journal of the American Statistical Association 85, 552–557.Google Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283–306.CrossRefGoogle Scholar
Pierce, D.A. ’ Haugh, L.D. (1977) Causality in temporal systems: Characterization and survey. Journal of Econometrics 5, 265–293.Google Scholar
Puri, M.L. ’ Sen, P.K. (1971) Nonparametric Methods in Multivariate Analysis. Wiley.Google Scholar
Randles, R.H. (1989) A distribution-free multivariate sign test based on interdirections. Journal of the American Statistical Association 84, 1045–1050.Google Scholar
Rousseeuw, P.J. ’ Leroy, A.M. (1987) Robust Regression and Outlier Detection. Wiley.Google Scholar
Saidi, A. ’ Roy, R. (2006) Robust Optimal Tests for Causality in Multivariate Time Series. Technical report CRM-3228, Centre de recherches mathématiques, Université de Montréal.Google Scholar
Sims, C.A., Stock, J.H., ’ Watson, M.W. (1990) Interference in linear time series models with some unit roots. Econometrica 58, 113–144.Google Scholar
Swensen, A.R. (1985) The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis 16, 54–70.Google Scholar
Taniguchi, M. ’ Kakizawa, Y. (2000) Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag.Google Scholar
Taylor, S.A. (1989) A comparison of classical tests of criterion when determining Granger causality with a bivariate ARMA models. Empirical Economics 14, 257–271.Google Scholar
Tiao, G.C. ’ Box, G.E.P. (1981) Modeling multiple time series with applications. Journal of the American Statistical Association 76, 802–816.Google Scholar
Tjøstheim, D. (1981) Granger causality in multiple time series. Journal of Econometrics 17, 157–176.Google Scholar
Toda, H.Y. ’ Phillips, P.C.B. (1993) Vector autoregressions and causality. Econometrica 61, 1367–1393.CrossRefGoogle Scholar
Toda, H.Y. ’ Phillips, P.C.B. (1994) Vector autoregressions and causality: A theoretical overview and simulation study. Econometric Reviews 13, 259–285.CrossRefGoogle Scholar
Tsay, R.S., Peña, D., ’ Pankratz, A.E. (2000) Outliers in multivariate time series. Biometrika 87, 789–804.Google Scholar
Tyler, D.E. (1987) A distribution-free M-estimate of multivariate scatter. Annals of Statistics 15, 234–251.CrossRefGoogle Scholar
Wiener, N. (1956) The theory of prediction. In Beckenback, E.F. (ed.), Modern Mathematics for Engineers, pp. 165–190. McGraw-Hill.Google Scholar