Fig. 5

Performance of a predictive model as a function of simulated correlations. From left to right and top to bottom, the plots show for each asset the normalized performance of an ARMA model (\(p=2,q=0\)), defined as \(\pi = \left( \frac{1}{T}\sum _t\left( \tilde{X}_i(t) - M_{\omega _1}\left[ \tilde{X}_i\right] (t)\right) ^2 \right) / \sigma \left[ \tilde{X}_i \right] ^2\) (95% confidence intervals computed by \(\pi = \bar{\pi } \pm (1.96\cdot \sigma [\pi ])/\sqrt{T}\), local polynomial smoothing to ease reading). It is interesting to note the U-shape for EUR/USD, due to interference between components at different scales. Correlation between simulated noises deteriorates predictive power. The study of lagged correlations (here \(\rho [\Delta X_{\text {EURUSD}}(t),\Delta X_{\text {EURGBP}}(t-\tau )]\)) on real data clarifies this phenomenon: the fourth graph shows an asymmetry in curves at any scale compared to zero lag \((\tau = 0)\) what leads fundamental components to increase predictive power for the dollar, amelioration then perturbed by correlations between simulated components. Dashed lines show time steps (in equivalent \(\tau\) units) used by the ARMA at each scale, what allows to read the corresponding lagged correlation on fundamental component