We show how a heterogeneous-agent New-Keynesian (HANK) model with incomplete markets and portfolio choice can be estimated in state space using a Bayesian approach. To render estimation feasible, the structure of the economy can be exploited and the dimensionality of the model automatically reduced based on the Bayesian priors. We apply this approach to analyze how much inequality matters for the business cycle and vice versa. Even when the model is estimated on aggregate data alone and with a set of shocks and frictions designed to match aggregate data, it broadly reproduces observed US inequality dynamics.