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In the following article we consider approximate Bayesian parameter inferencefor observation driven time series models. Such statistical models appear in awide variety of applications, including econometrics and applied mathematics.This article considers the scenario where the likelihood function cannot beevaluated point-wise; in such cases, one cannot perform exact statisticalinference, including parameter estimation, which often requires advancedcomputational algorithms, such as Markov chain Monte Carlo (MCMC). We introducea new approximation based upon approximate Bayesian computation (ABC). Undersome conditions, we show that as $n\rightarrow\infty$, with $n$ the length ofthe time series, the ABC posterior has, almost surely, a maximum \emph{aposteriori} (MAP) estimator of the parameters which is different from the trueparameter. However, a noisy ABC MAP, which perturbs the original data,asymptotically converges to the true parameter, almost surely. In order to drawstatistical inference, for the ABC approximation adopted, standard MCMCalgorithms can have acceptance probabilities that fall at an exponential ratein $n$ and slightly more advanced algorithms can mix poorly. We develop a newand improved MCMC kernel, which is based upon an exact approximation of amarginal algorithm, whose cost per-iteration is random but the expected cost,for good performance, is shown to be $\mathcal{O}(n^2)$ per-iteration.

Approximate Inference for Observation Driven Time Series Models with Intractable Likelihoods