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Testing the Martingale Hypothesis

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Abstract

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov-Smirnov and Cramer-von Mises tests. The tests are distribution free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The paper develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, non-martingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroskedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research.

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  • Peter C.B. Phillips & Sainan Jin, 2013. "Testing the Martingale Hypothesis," Cowles Foundation Discussion Papers 1912, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1912
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    1. Hall, Robert E, 1978. "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence," Journal of Political Economy, University of Chicago Press, vol. 86(6), pages 971-987, December.
    2. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2014. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behaviour," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(3), pages 315-333, June.
    3. Herman J. Bierens & Werner Ploberger, 1997. "Asymptotic Theory of Integrated Conditional Moment Tests," Econometrica, Econometric Society, vol. 65(5), pages 1129-1152, September.
    4. Magdalinos, Tassos & Phillips, Peter C.B., 2009. "Limit Theory For Cointegrated Systems With Moderately Integrated And Moderately Explosive Regressors," Econometric Theory, Cambridge University Press, vol. 25(2), pages 482-526, April.
    5. repec:cte:wsrepe:ws035312 is not listed on IDEAS
    6. Escanciano, J. Carlos & Lobato, Ignacio N., 2009. "An automatic Portmanteau test for serial correlation," Journal of Econometrics, Elsevier, vol. 151(2), pages 140-149, August.
    7. Yongmiao Hong & Yoon-Jin Lee, 2005. "Generalized Spectral Tests for Conditional Mean Models in Time Series with Conditional Heteroscedasticity of Unknown Form," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(2), pages 499-541.
    8. Phillips, Peter C.B., 2009. "Local Limit Theory And Spurious Nonparametric Regression," Econometric Theory, Cambridge University Press, vol. 25(6), pages 1466-1497, December.
    9. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    10. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-161, January.
    11. Lobato, Ignacio & Nankervis, John C & Savin, N E, 2001. "Testing for Autocorrelation Using a Modified Box-Pierce Q Test," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(1), pages 187-205, February.
    12. Charles, Amélie & Darné, Olivier & Kim, Jae H., 2011. "Small sample properties of alternative tests for martingale difference hypothesis," Economics Letters, Elsevier, vol. 110(2), pages 151-154, February.
    13. Lobato, I.N. & Nankervis, John C. & Savin, N.E., 2002. "Testing For Zero Autocorrelation In The Presence Of Statistical Dependence," Econometric Theory, Cambridge University Press, vol. 18(3), pages 730-743, June.
    14. Durlauf, Steven N., 1991. "Spectral based testing of the martingale hypothesis," Journal of Econometrics, Elsevier, vol. 50(3), pages 355-376, December.
    15. Deo, Rohit S., 2000. "Spectral tests of the martingale hypothesis under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 99(2), pages 291-315, December.
    16. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(3), pages 269-298, June.
    17. Liu, Christina Y & He, Jia, 1991. "A Variance-Ratio Test of Random Walks in Foreign Exchange Rates," Journal of Finance, American Finance Association, vol. 46(2), pages 773-785, June.
    18. Kuan Chung-Ming & Lee Wei-Ming, 2004. "A New Test of the Martingale Difference Hypothesis," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 8(4), pages 1-26, December.
    19. Yongmiao Hong & Tae-Hwy Lee, 2003. "Inference on Predictability of Foreign Exchange Rates via Generalized Spectrum and Nonlinear Time Series Models," The Review of Economics and Statistics, MIT Press, vol. 85(4), pages 1048-1062, November.
    20. Fama, Eugene F, 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, American Finance Association, vol. 25(2), pages 383-417, May.
    21. Shi, Xiaoxia & Phillips, Peter C.B., 2012. "Nonlinear Cointegrating Regression Under Weak Identification," Econometric Theory, Cambridge University Press, vol. 28(3), pages 509-547, June.
    22. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    23. Escanciano, J. Carlos, 2007. "Weak convergence of non-stationary multivariate marked processes with applications to martingale testing," Journal of Multivariate Analysis, Elsevier, vol. 98(7), pages 1321-1336, August.
    24. Fong, Wai Mun & Koh, Seng Kee & Ouliaris, Sam, 1997. "Joint Variance-Ratio Tests of the Martingale Hypothesis for Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 51-59, January.
    25. Nankervis, John C. & Savin, N. E., 2010. "Testing for Serial Correlation: Generalized Andrews–Ploberger Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(2), pages 246-255.
    26. Chow, K. Victor & Denning, Karen C., 1993. "A simple multiple variance ratio test," Journal of Econometrics, Elsevier, vol. 58(3), pages 385-401, August.
    27. Chen, Willa W. & Deo, Rohit S., 2006. "The Variance Ratio Statistic At Large Horizons," Econometric Theory, Cambridge University Press, vol. 22(2), pages 206-234, April.
    28. Donald W. K. Andrews, 1997. "A Conditional Kolmogorov Test," Econometrica, Econometric Society, vol. 65(5), pages 1097-1128, September.
    29. Wright, Jonathan H, 2000. "Alternative Variance-Ratio Tests Using Ranks and Signs," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(1), pages 1-9, January.
    30. Kim, Jae H., 2006. "Wild bootstrapping variance ratio tests," Economics Letters, Elsevier, vol. 92(1), pages 38-43, July.
    31. Belaire-Franch, Jorge & Opong, Kwaku K., 2005. "Some evidence of random walk behavior of Euro exchange rates using ranks and signs," Journal of Banking & Finance, Elsevier, vol. 29(7), pages 1631-1643, July.
    32. Bierens, Herman J, 1990. "A Consistent Conditional Moment Test of Functional Form," Econometrica, Econometric Society, vol. 58(6), pages 1443-1458, November.
    33. Meese, Richard A. & Rogoff, Kenneth, 1983. "Empirical exchange rate models of the seventies : Do they fit out of sample?," Journal of International Economics, Elsevier, vol. 14(1-2), pages 3-24, February.
    34. Peter C. B. Phillips & Shu-Ping Shi & Jun Yu, 2011. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior," Working Papers 15-2011, Singapore Management University, School of Economics.
    35. Joon Y. Park & Peter C. B. Phillips, 2000. "Nonstationary Binary Choice," Econometrica, Econometric Society, vol. 68(5), pages 1249-1280, September.
    36. Choi, In, 1999. "Testing the Random Walk Hypothesis for Real Exchange Rates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(3), pages 293-308, May-June.
    37. Escanciano, J. Carlos & Velasco, Carlos, 2006. "Generalized spectral tests for the martingale difference hypothesis," Journal of Econometrics, Elsevier, vol. 134(1), pages 151-185, September.
    38. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    39. Hsieh, David A., 1988. "The statistical properties of daily foreign exchange rates: 1974-1983," Journal of International Economics, Elsevier, vol. 24(1-2), pages 129-145, February.
    40. Yilmaz, Kamil, 2003. "Martingale Property of Exchange Rates and Central Bank Interventions," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(3), pages 383-395, July.
    41. Phillips, Peter C.B. & Magdalinos, Tassos, 2008. "Limit Theory For Explosively Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 24(4), pages 865-887, August.
    42. Horowitz, Joel L. & Lobato, I.N. & Nankervis, John C. & Savin, N.E., 2006. "Bootstrapping the Box-Pierce Q test: A robust test of uncorrelatedness," Journal of Econometrics, Elsevier, vol. 133(2), pages 841-862, August.
    43. Fong, Wai Mun & Ouliaris, Sam, 1995. "Spectral Tests of the Martingale Hypothesis for Exchange Rates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 10(3), pages 255-271, July-Sept.
    44. Whang, Yoon-Jae, 2000. "Consistent bootstrap tests of parametric regression functions," Journal of Econometrics, Elsevier, vol. 98(1), pages 27-46, September.
    45. Bierens, Herman J., 1984. "Model specification testing of time series regressions," Journal of Econometrics, Elsevier, vol. 26(3), pages 323-353, December.
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    4. Adeyeye Patrick Olufemi & Aluko Olufemi Adewale & Migiro Stephen Oseko, 2017. "Efficiency of Foreign Exchange Markets in Sub-Saharan Africa in the Presence of Structural Break: A Linear and Non-Linear Testing Approach," Journal of Economics and Behavioral Studies, AMH International, vol. 9(4), pages 122-131.
    5. Daniel Borup & Martin Thyrsgaard, 2017. "Statistical tests for equal predictive ability across multiple forecasting methods," CREATES Research Papers 2017-19, Department of Economics and Business Economics, Aarhus University.
    6. Yi He & Sombut Jaidee & Jiti Gao, 2020. "Most Powerful Test against High Dimensional Free Alternatives," Monash Econometrics and Business Statistics Working Papers 13/20, Monash University, Department of Econometrics and Business Statistics.
    7. Guo, Xu & Li, Gao Rong & Wong, Wing Keung, 2014. "Specification Testing of Production Frontier Function in Stochastic Frontier Model," MPRA Paper 57999, University Library of Munich, Germany.
    8. Chang, Jinyuan & Jiang, Qing & Shao, Xiaofeng, 2023. "Testing the martingale difference hypothesis in high dimension," Journal of Econometrics, Elsevier, vol. 235(2), pages 972-1000.

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    More about this item

    Keywords

    Brownian functional; Martingale hypothesis; Kolmogorov-Smirnov test; Cramer-von Mises test; Explosive process; Exchange rates;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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