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In this paper, stock market comovements are examined using cointegration, Granger causality tests and nonlinear approaches in context of mutual information and correlations. Since underlying data sets are affected by non-stationarities and trends, we also apply Adaptive Multifractal
Detrended Fluctuation Analysis (AMF-DFA) and Adaptive Multifractal Detrended
Cross-Correlation Analysis (AMF-DXA). We find only 170 pair
of stock markets cointegrated, and according to the Granger
causality and mutual information, we realize that the strongest
relations lies between emerging markets, and between emerging and
frontier markets. According to scaling exponent given by AMF-DFA,
$h(q=2)>1$, we found that all underlying data belong to
non-stationary process. According to Efficient Market Hypothesis (EMH), only 8 markets are classified in uncorrelated processes at $2\sigma$ confidence interval. 6 stoke markets belong to anticorrelated class and dominant part of markets have memory in corresponding daily index prices during January 1995 to February 2014. New-Zealand with $H=0.457\pm0.004$ and Jordan with $H=0.602\pm 0.006$ are far from EMH. The nature of cross-correlation exponent based on AMF-DXA is almost
multifractal for all pair of stock markets. The empirical relation,
$h_{xy}(q)=[h_{xx}(q)+h_{yy}(q)]/2$, was confirmed just for $q>0$,
while for $q<0$ there is a deviation from this relation confirming behavior of markets for small fluctuations is affected by contribution of major pair. For larger fluctuations, the cross-correlation contains information from both local (internal) and global (external) conditions. Width of singularity spectrum for auto-correlation and cross-correlation are $\Delta \alpha_{xx}\in [0.304,0.905]$ and $\Delta \alpha_{xy}\in [0.246,1.178]$, respectively. The wide range of singularity spectrum for cross-correlation confirms that the bilateral relation between stock markets is complex. The value
of $\sigma_{DCCA}$ indicates that all pairs of stock market studied in this time
interval belong to cross-correlated processes.
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