  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 yield substantially
different results. Our theoretical results demonstrate that even though the
model is misspecified, under regularity conditions, the accept/reject ABC
approach concentrates posterior mass on an appropriately defined pseudo-true
parameter value. However, under model misspecification the ABC posterior does
not yield credible sets with valid frequentist coverage and has non-standard
asymptotic behavior. In addition, we examine the theoretical behavior of the
popular local regression adjustment to ABC under model misspecification and
demonstrate that this approach concentrates posterior mass on a completely
different pseudo-true value than accept/reject ABC. Using our theoretical
results, we suggest two approaches to diagnose model misspecification in ABC.
All theoretical results and diagnostics are illustrated in a simple running
example.
