We analyze the behavior of approximate Bayesian computation (ABC) when the model generating the simulated data differs from the actual data generating process; i.e., when the data simulator in ABC is misspecified. We demonstrate both theoretically and in simple, but practically relevant, examples that when the model is misspecified different versions of ABC can lead to substantially different results. Our theoretical results demonstrate that under regularity a version of the ABC accept-reject approach concentrates posterior mass on an appropriately defined pseudo-true parameter value, while the popular linear regression adjustment to ABC concentrates posterior mass on a completely different pseudo-true value. Our results suggest two diagnostic approaches to diagnose model misspecification in ABC.
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