Copulas and time series with long-ranged dependencies

We review ideas on temporal dependencies and recurrences in discrete time series from several areas of natural and social sciences. We revisit existing studies and redefine the relevant observables in the language of copulas (joint laws of the ranks). We propose that copulas provide an appropriate mathematical framework to study nonlinear time dependencies and related concepts—like aftershocks, Omori law, recurrences, and waiting times. We also critically argue, using this global approach, that previous phenomenological attempts involving only a long-ranged autocorrelation function lacked complexity in that they were essentially monoscale.

Figure 1

Two possible definitions of events: (a) $p_{-}$ and $p_{+}$ are probabilities of extremes (negative and positive, respectively), so $F\left(X^{(+)}\right)=1-F\left(X^{(-)}\right)=q$; (b) $p_{+}$ is a probability of extreme and $p_{-}=1-p_{+}$, so $F\left(X^{(+)}\right)=F\left(X^{(-)}\right)=q$.

Figure 2

Conditional probabilities $p_{\pm \mp }^{\left(\ell \right)}$ (left panels) and $p_{\pm \pm }^{\left(\ell \right)}$ (right panels) for different values of $\rho \left(\ell \right)$ with thresholds at $p_{+}=p_{-}=1-q$. In the case of the independence copula (shown in bold red), the conditional probabilities are all equal: $p_{\pm \mp }=p_{\pm \pm }=1-q$.

Figure 3

Conditional extreme probabilities at $\ell =1$ (the white noise contribution $p_{\pm }=1-q$ has been subtracted, hence, the possibly negative values; compare with Fig. 2). Filled symbols are for persistence, and empty symbols for reversion. Upward triangles are conditioned on positive jumps, and downward triangles are conditioned on negative jumps.

Figure 4

The self-copula for close-by ($\ell =1$ day) and remote ($\ell =128$ days) lags, with the product copula subtracted. Top: Diagonal $C_{\ell }(u,u)-u^2$. Bottom: $C_{\ell }(u,1-u)-u(1-u)$. The value determined empirically on stock returns is in bold black, and a fit with the model of Ref. [42] is shown in thin red. Adapted from Ref. [42].

Figure 5

DAX index returns. Left: Tail probability $1-\Pi \left(\tau \right)$ of the recurrence intervals, at several thresholds $p_{+}=1/\langle \tau \rangle$, in log-log scale. Gray curves are best fits to Eq. (25) suggested in Ref. [44]. Right: Estimated parameters $a$ and $b$ of the best fit.

Figure 6

Conditional expected shortfall of a Gaussian pair $(X_0,X_{\ell })$ for different values of $\rho \left(\ell \right)$. Left: Lower tail; Right: Upper tail. The value at $q=0.5$ is $\sqrt{2/\pi }\rho \left(\ell \right)$.

Figure 7

Conditional extreme amplitudes, at lags $\ell =1,5,20$, from left to right. The upper-right and lower-left quadrants express persistence, while the upper-left and lower-right quadrants reveal reversion. For a scale-free dependence structure, one would expect the magnitudes to decrease with the lag $\ell$ but the global shape to be conserved. What we instead observe is important changes of configuration at different lags: For example, the strong reversion of negative tail events at $\ell =1$ vanishes at farther lags and even turns into strong persistence for the CAC and DAX indices. That is to say, these indices tend to mean-revert after a negative event at the daily frequency but to trend on the weekly scale. Similarly, the strong persistence of positive events at $\ell =1$ converts to a strong reversion in the tails at $\ell =20$ for the European indices (CAC, DAX); a weaker reversion is observed at intermediate scale ($\ell =5$) for most indices (including US and Korean).

Figure 8

Copula ${\mathcal{C}}_n\left(p\right)$ of the fractional Gaussian noise (FGN) with different Hurst exponents and at threshold $p_{+}=0.7$ (log-log scale).