2023MNRAS.523..111C

The Granger causality is an econometric test for determining whether one time series is useful for forecasting another one with a certain Granger lag. Here, the light curves in the 0.3-1 keV (reflection dominated, soft) and 1.2-5 keV (continuum dominated, hard) bands of active galactic nuclei are produced, and the Granger lags are estimated and compared to the traditional lag-frequency spectra. We find that the light curves in the hard band Granger-cause (lead) those in the soft band, whereas the obtained Granger lags could be interpreted as the intrinsic reverberation lags. Then, we extract the Granger-lag profiles from 14 XMM-Newton observations of IRAS 13224-3809, and find that the lags are significant in 12 observations. The majority of the obtained Granger (intrinsic) lags are ∼200-500 s. With the IRAS 13224-3809 mass of 2 × 10⁶ M_⊙, these lags correspond to the true light travel distance of ∼20-50 r_g. Hence, by assuming a lamp-post geometry and a face-on disc, this places the corona at ∼10-25 r_g above the central black hole. Moreover, multiple Granger lags consisting of the small and large lags of <500 s and >1000 s are detected in four observations. This means that the corona height can significantly evolve from ∼10-25 r_g to ∼55r_g, or vice versa, towards the end of the observation. Granger lags are a promising way to measure the intrinsic lags, and provide evidence of coronal height variability within each individual observation.