ON THE TESTING MULTI-VALUED MARTINGALE DIFFERENCE HYPOTHESIS
Keywords:Martingale Difference Hypothesis, Multi-Values Martingale Difference, Generalized Spectral Analysis, Exchange Rates
AbstractThis paper presents a definition of Multi-Valued Martingale Difference (MVMD) based on Castaing representation of a multi-valued martingale that consists of martingale difference selections. Testing the Multi-Valued Martingale Difference Hypothesis (MVMDH) then examined. Testing the Martingale Difference Hypothesis (MDH) earlier was based on linear measures then later developed two directions in order to account for the existing nonlinearity in economic and financial data. First, the classical approaches have been modified by take into account the possible nonlinear dependence. Second, the use of more sophisticated statistical tools such as those based generalized spectral analysis. According to this article, both these developments in MDH are modified for MVMDH and applies them to exchange rate data and returns of stock market data.
A. Aswani, “Statistics with set-valued functions: applications to inverse approximate optimization,” Mathematical Programming, pp. 1–27, 2017.
M. R. Borges, “Eﬃcient market hypothesis in european stock markets,” The European Journal of Finance, vol. 16, no. 7, pp. 711–726, 2010.
G. E. P. Box and D. A. Pierce, “Distribution of residual autocorrelations in autoregressive-integrated moving average time series models,” Journal of the American Statistical Association, vol. 65, no. 332, pp. 1509–1526, 1970.
P. J. Brockwell and R. A. Davis, “Time series: Theory and methods,” Springer Series in Statistics, Berlin, New York: Springer,— c1991, 2nd ed., 1991.
C. Castaing, N. Van Quang, and N. T. Thuan, “A new family of convex weakly compact valued random variables in banach space and applications to laws of large numbers,” Statistics & Probability Letters, vol. 82, no. 1, pp. 84–95, 2012.
W. W. Chen and R. S. Deo, “The variance ratio statistic at large horizons,” Econometric Theory, vol. 22, no. 2, pp. 206–234, 2006.
I. Choi, “Testing the random walk hypothesis for real exchange rates,” Journal of Applied Econometrics, vol. 14, no. 3, pp. 293–308, 1999.
K. V. Chow and K. C. Denning, “A simple multiple variance ratio test,” Journal of Econometrics, vol. 58, no. 3, pp. 385–401, 1993.
R. S. Deo, “Spectral tests of the martingale hypothesis under conditional heteroscedasticity,” Journal of Econometrics, vol. 99, no. 2, pp. 291–315, 2000.
M. A. Dom´ınguez and I. N. Lobato, “Testing the martingale diﬀerence hypothesis,” Econometric Reviews, vol. 22, no. 4, pp. 351–377, 2003.
S. N. Durlauf, “Spectral based testing of the martingale hypothesis,” Journal of Econometrics, vol. 50, no. 3, pp. 355–376, 1991.
J. C. Escanciano and I. N. Lobato, “Testing the martingale hypothesis,” in Palgrave handbook of econometrics. Springer, 2009, pp. 972–1003.
J. C. Escanciano and C. Velasco, “Generalized spectral tests for the martingale diﬀerence hypothesis,” Journal of Econometrics, vol. 134, no. 1, pp. 151–185, 2006.
L. Guan and Y. Wan, “A strong law of large numbers for set-valued negatively dependent random variables,” International Journal of Statistics and Probability, vol. 5, no. 3, p. 102, 2016.
K. Hamid, M. T. Suleman, S. Z. A. Shah, and R. S. I. Akash, “Testing the weak form of eﬃcient market hypothesis: Empirical evidence from asia-paciﬁc markets,” International Research Journal of Finance and Economics, no. 58, pp. 121–133, 2010.
Y. Hong and T.-H. Lee, “Inference on predictability of foreign exchange rates via generalized spectrum and nonlinear time series models,” Review of Economics and Statistics, vol. 85, no. 4, pp. 1048–1062, 2003.
Y. Hong and Y.-J. Lee, “Generalized spectral tests for conditional mean models in time series with conditional heteroscedasticity of unknown form,” The Review of Economic Studies, vol. 72, no. 2, pp. 499–541, 2005.
D. A. Hsieh, “Testing for nonlinear dependence in daily foreign exchange rates,” Journal of Business, pp. 339–368, 1989.
D. G. Kendall, Foundations of a Theory of Random Sets, In Stochastic Geometry. John Wiley & Sons, 1974.
M. G. Kendall and A. B. Hill, “The analysis of economic time-series-part i: Prices,” Journal of the Royal Statistical Society. Series A (General), vol. 116, no. 1, pp. 11–34, 1953.
J. H. Kim, “Wild bootstrapping variance ratio tests,” Economics letters, vol. 92, no. 1, pp.38–43, 2006.
M. Kisielewicz, “Martingale representation theorem for set-valued martingales,” Journal of Mathematical Analysis and Applications, vol. 409, no. 1, pp. 111–118, 2014.
H. L. Koul, W. Stute et al., “Nonparametric model checks for time series,” The Annals of Statistics, vol. 27, no. 1, pp. 204–236, 1999.
C.-M. Kuan and W.-M. Lee, “A new test of the martingale diﬀerence hypothesis,” Studies in Nonlinear Dynamics & Econometrics, vol. 8, no. 4, 2004.
J.-G. Li and S.-Q. Zheng, “Set-valued stochastic equation with set-valued square integrable martingale,” in ITM Web of Conferences, vol. 12. EDP Sciences, 2017, p. 03002.
S. Li, Y. Ogura, and V. Kreinovich, Limit theorems and applications of set-valued and fuzzy set-valued random variables. Springer Science & Business Media, 2013, vol. 43.
S. Li, Y. Ogura, F. N. Proske, and M. L. Puri, “Central limit theorems for generalized set-valued random variables,” Journal of mathematical analysis and applications, vol. 285, no. 1, pp. 250–263, 2003.
G. M. Ljung and G. E. Box, “On a measure of lack of ﬁt in time series models,” Biometrika, vol. 65, no. 2, pp. 297–303, 1978.
A. W. Lo and A. C. MacKinlay, “Stock market prices do not follow random walks: Evidence from a simple speciﬁcation test,” The review of ﬁnancial studies, vol. 1, no. 1, pp. 41–66, 1988.
I. Lobato, J. C. Nankervis, and N. E. Savin, “Testing for autocorrelation using a modiﬁed box-pierce q test,” International Economic Review, vol. 42, no. 1, pp. 187–205, 2001.
I. N. Lobato, J. C. Nankervis, and N. Savin, “Testing for zero autocorrelation in the presence of statistical dependence,” Econometric Theory, vol. 18, no. 3, pp. 730–743, 2002.
B. G. Malkiel, “Eﬃcient market hypothesis,” in Finance. Springer, 1989, pp. 127–134.
G. Matheron, Random Sets and Integral Geometry. John Wiley and Sons, 1975.
J. L. McCauley, K. E. Bassler, and G. H. Gunaratne, “Martingales, detrending data, and the eﬃcient market hypothesis,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 1, pp. 202–216, 2008.
P. M¨orters and Y. Peres, Brownian motion. Cambridge University Press, 2010, vol. 30.
M. F. Osborne, “Brownian motion in the stock market,” Operations research, vol. 7, no. 2, pp. 145–173, 1959.
P. C. Phillips and S. Jin, “Testing the martingale hypothesis,” Journal of Business & Economic Statistics, vol. 32, no. 4, pp. 537–554, 2014.
M. L. Puri and D. RALESCU, “Integration on fuzzy sets,” Time Series, Fuzzy Analysis and Miscellaneous Topics, vol. 3, p. 415, 2011.
S. Roberts, “Control chart tests based on geometric moving averages,” Technometrics, vol. 1, no. 3, pp. 239–250, 1959.
H. J. Skaug and D. Tjøstheim, “A nonparametric test of serial independence based on the empirical distribution function,” Biometrika, vol. 80, no. 3, pp. 591–602, 1993.
N. Van Quang and N. T. Thuan, “Strong laws of large numbers for adapted arrays of set-valued and fuzzy-valued random variables in banach space,” Fuzzy Sets and Systems, vol. 209, pp. 14–32, 2012.
J. H. Wright, “Alternative variance-ratio tests using ranks and signs,” Journal of Business & Economic Statistics, vol. 18, no. 1, pp. 1–9, 2000.
V. V. Yen, “On convergence of multiparameter multivalued martingales,” Acta Mathematica Vietnamica, vol. 29, pp. 177–184, 2004.
L. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338 – 353, 1965. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S001999586590241X http://www.sciencedirect.com/science/article/pii/S001999586590241X">
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