Abstract
The Approximate Bayesian Computation (ABC) filter extends the particle filtering methodology to general state-space models in which the density of the observation conditional on the state is intractable. We provide an exact upper bound for the mean squared error of the ABC filter, and derive sufficient conditions on the bandwidth and kernel under which the ABC filter converges to the target distribution as the number of particles goes to infinity. The optimal convergence rate decreases with the dimension of the observation space but is invariant to the complexity of the state space. We show that the adaptive bandwidth commonly used in the ABC literature can lead to an inconsistent filter. We develop a plug-in bandwidth guaranteeing convergence at the optimal rate, and demonstrate the powerful estimation, model selection, and forecasting performance of the resulting filter in a variety of examples.