Predicting the response of nonlinear dynamical systems subject to random, broadband excitation is important across a range of scientific disciplines, such as structural dynamics and neuroscience. Building data-driven models requires experimental measurements of the system input and output, but it can be difficult to determine whether inaccuracies in the model stem from modelling errors or noise. This influences the investment of resources into building more accurate models. There are currently no solutions to this problem in the absence of a complete benchmark model. This paper presents a novel method to identify the causal relationship between the input and output of a system in the presence of output noise. Using this method, researchers could collect experimental input and output data for a nonlinear dynamical system, and identify how much of the output is caused by the input as a function of frequency.