Multifractal Quantile Detrending Moving-average Cross-correlation Analysis

Abstract We introduce multifractal quantile detrending moving-average cross-correlation analysis (MF-X-QDMA), the bivariate extension of the quantile detrending moving average algorithm of a jointly submitted companion paper. The local trends of the two profiles are moving quantiles at levels $(\lam_x,\lam_y)$, in one and two dimensions. For pairs of non-negative measures, the product of the two quantile residuals is exactly a product of gliding-box masses, so MF-X-QDMA measures the joint partition function. This yields the exact cross exponent of two binomial measures, $\tau_{xy}(q)=-\log_2[(p_xp_y)^{q/2}+((1-p_x)(1-p_y))^{q/2}]$. A Cauchy--Schwarz bound, $\tau_{xy}\ge(\tau_{xx}+\tau_{yy})/2$, shows that the widely used relation $h_{xy}=(h_{xx}+h_{yy})/2$ holds only for identical multipliers. Measured against the exact expression, the error of MF-X-DMA for $(p_x,p_y)=(0.1,0.45)$ falls from 0.094 to 0.017. For signals, quantile levels $\lam\neq1/2$ leave a residual offset that creates spurious cross-correlation (0.44 for independent series at $\lam=0.1$); we give three remedies. On benchmarks with known exponents, MF-X-QDMA with mixed levels $(0.1,0.9)$ is the most accurate estimator for binomial pairs (mean error 0.007, versus 0.012 for MF-X-DMA). In two dimensions it lowers the $\tau_{xy}$ error of $256^2$ cascade pairs by 27\%--35\%. For monofractal pairs it is comparable to the best moving-average variant. We apply the method to 16 pairs of daily financial series (stock indices, exchange rates, commodities, VIX, Treasury yields, Bitcoin) and 29 stock--index pairs. Under injected bad ticks, the median-detrended cross-correlation coefficient is less sensitive than the detrending moving-average coefficient for 27--29 of the 29 stock pairs. Combining the median trend with first-order moments roughly halves the sensitivity to gross bad ticks and makes it 3.5--18 times smaller than that of Pearson's coefficient. A rolling analysis recovers the 2001 sign reversal of the stock--bond correlation under errors that distort the Pearson coefficient by 0.27 on average. Quantile detrending does not help against return errors that shift the price level, or against dense microstructure noise.