Abstract Specialist predators with oscillating dynamics are often strongly affected by the population dynamics of their prey, yet they are not always the cause of prey cycling. Only those that exert strong (delayed) regulation of their prey can be. Inferring predator–prey coupling from time series therefore requires contrasting models with top‐down versus bottom‐up predator–prey dynamics. We study here the joint dynamics of population densities of the Icelandic gyrfalcon Falco rusticolus , and its prey, the rock ptarmigan Lagopus muta . The dynamics of both species are likely not only linked to each other but also to stochastic weather variables acting as confounding factors. We infer the degree of coupling between populations, as well as forcing by abiotic variables, using multivariate autoregressive models MAR (p), with p = 1 and 2 time lags. MAR (2) models, allowing for species to cycle independently from each other, further suggest alternative scenarios where a cyclic prey influences its predator but not the other way around (i.e., bottom‐up scenarios). The classical MAR (1) model predicts that the time series exhibit predator–prey feedback (i.e., reciprocal dynamic influence between prey and predator), and that weather effects are weak and only affecting the gyrfalcon population. Bottom‐up MAR (2) models produced a better fit but less realistic cross‐correlation patterns. Simulations of MAR (1) and MAR (2) models further demonstrate that the top‐down MAR (1) models are more likely to be misidentified as bottom‐up dynamics than vice versa. We therefore conclude that predator–prey feedback in the gyrfalcon–ptarmigan system is likely the main cause of observed oscillations, though bottom‐up dynamics cannot yet be excluded with certainty. Overall, we showed how to make more out of ecological time series by using simulations to gauge the quality of model identification, and paved the way for more mechanistic modeling of this system by narrowing the set of important biotic and abiotic drivers.