Context. Some recent observational results impose significant constraints on all the models that have been proposed to explain the Galactic black-hole X-ray sources in the hard state. In particular, it has been found that during the hard state of Cyg X-1 the power-law photon number spectral index, Γ, is correlated with the average time lag, $\langle t_{\rm lag}\rangle$, between hard and soft X-rays. Furthermore, the peak frequencies of the four Lorentzians that fit the observed power spectra are correlated with both Γ and $\langle t_{\rm lag}\rangle$. Aims. We have investigated whether our jet model can reproduce these correlations. Methods. We performed Monte Carlo simulations of Compton upscattering of soft, accretion-disk photons in the jet and computed the time lag between hard and soft photons and the power-law index Γ of the resulting photon number spectra. Results. We demonstrate that our jet model naturally explains the above correlations, with no additional requirements and no additional parameters.