The aleatory‐variability component (standard deviation) of a ground motion has a large influence on results of a probabilistic seismic hazard assessment. kappa, a measure of high‐frequency attenuation, has site‐ and record‐specific effects that have been suggested as reasons for observing heteroscedastic aleatory variability within earthquake ground motions. Specifically, kappa has been proposed as a reason why ground motions from small earthquakes are more variable than those from large earthquakes, which is modeled by magnitude‐dependent within‐event standard deviations in ground‐motion prediction equations (GMPEs). In this study, we use ground motions simulated using the stochastic method to examine the influence of the site‐specific component of kappa on aleatory variability of earthquake ground motions and examine the hypothesis that this could be a cause of the observed heteroscedasticity in this variability. We consider simulations with both fixed and continuous stress drop distributions and the site‐specific component of kappa to demonstrate that variation in the stress drop parameter contributes minimally to magnitude‐dependency, unlike the site‐specific component of kappa, which causes significant magnitude‐dependency. Variation in the site‐specific component of kappa is, therefore, proposed to be at least partially responsible for the magnitude‐dependency captured in the aleatory‐variability components of some recent GMPEs. It is found, however, that the expected impact of the site‐specific component of kappa on aleatory variability is much greater than modeled in these GMPEs, which suggests that there could be a mitigating effect that is not captured within the simulations (e.g., correlated inputs to the simulations).