Spectrum-based estimators of the bivariate Hurst exponent

We discuss two alternate spectrum-based estimators of the bivariate Hurst exponent in the power-law cross-correlations setting, the cross-periodogram and local $X$-Whittle estimators, as generalizations of their univariate counterparts. As the spectrum-based estimators are dependent on a part of the spectrum taken into consideration during estimation, a simulation study showing performance of the estimators under varying bandwidth parameter as well as correlation between processes and their specification is provided as well. These estimators are less biased than the already existent averaged periodogram estimator, which, however, has slightly lower variance. The spectrum-based estimators can serve as a good complement to the popular time domain estimators.

Figure 1

Mean values of APE, LXW, and XPE estimators dependent on $m$ and correlation between error terms I. Values are based on 1000 simulations of ARFIMA(0,$d$,0) processes with correlated error terms and $d_1=d_2=0.4$. Correlation between error terms ranges between 0.2 and 1 with a step of 0.2 and is represented by different shades of gray (the lightest for 0.2 and black for 1). Red line represents the true value of $H_{xy}=0.9$.

Figure 2

Mean values of APE, LXW, and XPE estimators dependent on $m$ and correlation between error terms II. Values are based on 1000 simulations of mixed-correlated ARFIMA(0,$d$,0) processes with $d_1=d_4=0.4$ and $d_2=d_3=0.2$. Correlation between error terms ranges between 0.2 and 1 with a step of 0.2 and is represented by different shades of gray (the lightest for 0.2 and black for 1). Red line represents the true value of $H_{xy}=0.7$.

Figure 3

Variance of APE, LXW, and XPE estimators dependent on $m$ and correlation between error terms I. Values are based on 1000 simulations of ARFIMA(0,$d$,0) processes with correlated error terms and $d_1=d_2=0.4$. Correlation between error terms ranges between 0.2 and 1 with a step of 0.2 and is represented by different shades of gray (the lightest for 0.2 and black for 1). Red line represents the asymptotical values for the univariate case (for LXW and XPE only).

Figure 4

Variance of APE, LXW, and XPE estimators dependent on $m$ and correlation between error terms II. Values are based on 1000 simulations of mixed-correlated ARFIMA(0,$d$,0) processes with $d_1=d_4=0.4$ and $d_2=d_3=0.2$. Correlation between error terms ranges between 0.2 and 1 with a step of 0.2 and is represented by different shades of gray (the lightest for 0.2 and black for 1).