Color of Noise: Comparative Analysis of Sub-Periodic Variation in Empirical Hurst Exponent Across Foreign Currency Changes and Their Pairwise Differences
Posted: 18 Nov 2018
Date Written: April 1, 2018
This is a pioneering effort to estimate empirical Hurst exponents (H) and their sub-periodic variation for ten foreign currencies using predictive regressions with volatility input for T=250 different time horizons for the period January 1999 to February 2016, starting from the official introduction of Euro into international money circulation. Empirical Hurst exponent is a fully data driven, efficient estimator with low numerical complexity and computational cost; helping a large and diverse group of financial market participants easily access, understand and apply a key concept of econophysics and phynance in a discrete time domain. Sub-period 1 Hurst exponents are significantly different from0.5 for all currencies, except GBF and CHF, and range between 0.4403 and 0.5491; while ranging between 0.4593and 0.5678, and significantly different from 0.5 for all currencies, except CAD, in sub-period 2. The largest pairwise Hurst difference in sub-period 1 is 0.1110, and between JPY and NZD; being the latter higher. Average difference is0.0456 with a standard deviation of 0.0305. The largest difference in sub-period 2 is 0.1084, and between CHF and SEK; being the latter higher. Average difference is 0.0354 with a standard deviation of 0.0281. Six currencies are associated with a statistically significant increase in their Hurst exponents from sub-period 1 to sub-period 2: JPY (by 0.1104), NOK (0.0828), GBP (0.0474), CAD (0.0373), SEK (0.0333), and AUD (0.0280); all significant at 1% level.Four currencies show a significant decrease in their Hurst exponents: CHF (by 0.0337 and significant at 1% level), NZD (0.0284, 5%), DKK (0.0195, 5%), and EUR (0.0159, 10% level). It is found that the statistically significant sub-periodic changes exhibit six clusters with respect to both the direction of change and the range of Hurst exponent in terms of H<0.5 (anti-persistence/short memory/pink noise); H=0.5 (√T rule/Brownian motion/Brown or red noise); and H>0.5 (persistence/long memory/black noise) where H∈(0,1). Within cluster 1 and 6, empirical Hurst exponents exhibit persistency in both sub-periods; however, with a decreasing H for EUR, NZD and DKK, and with an increasing H for AUD and SEK, from sub-period 1 to sub-period 2. Cluster 2 and 3 are reverse of each other: CHF becomes anti-persistent in sub-period 2, changing from Brownian motion in sub-period 1; while CAD behaves exactly in the opposite direction. Under cluster 4, GBP becomes persistent, changing from Brownian motion.Cluster 5 shows that JPY and NOK become persistent in sub-period 2; while both are anti-persistent in sub-period1. These results provide evidence against the weak-form informational efficiency. Given that a difference in H by0.01 generates a volatility-multiplier effect (T^H) of 1.0568 for a 250-day investment horizon, the reported statistically significant results can also be considered economically significant.
Keywords: Econophysics and phynance, Scaling laws, Hurst exponent, Risk scaling, Realized volatility term structure, Fractal market hypothesis
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