We propose an indirect inference strategy for estimating heterogeneous-agent business cycle models with micro data. At its heart is a first-order vector autoregression that is grounded in linear filtering theory as the cross-section grows large. The result is a fast, simple and robust algorithm for computing an approximate likelihood that can be easily paired with standard classical or Bayesian methods. Importantly, our method is compatible with the popular sequence-space solution method, unlike existing state-of-the-art approaches. We test-drive our method by estimating a canonical HANK model with shocks in both the aggregate and cross-section. Not only do simulation results demonstrate the appeal of our method, they also emphasize the important information contained in the entire micro-level distribution over and above simple moments.