ABC and Indirect Inference

Abstract

Indirect inference (II) is a classical method for estimating the parameter of a complex model when the likelihood is unavailable or too expensive to evaluate. The idea was popularised several years prior to the main developments in ABC by Gourieroux et al. (1993); Smith (1993), where interest was in calibrating complex time series models used in financial applications. The II method became a very popular approach in the econometrics literature (e.g. Smith (1993); Monfardini (1998); Dridi et al. (2007)) in a similar way to the ubiquitous application of ABC to models in population genetics. However, the articles by Jiang and Turnbull (2004) and Heggland and Frigessi (2004) have allowed the II approach to be known and appreciated by the wider statistical community. In its full generality, the II approach can be viewed as a classical method to estimate the parameter of a statistical model on the basis of a so-called indirect or auxiliary summary of the observed data (Jiang and Turnbull, 2004). A special case of II is the simulated method of moments (McFadden, 1989), where the auxiliary statistic is a set of sample moments. In this spirit, the traditional ABC method may be viewed as a Bayesian version of II, where prior information about the parameter may be incorporated and updated using the information about the parameter contained in the summary statistic. However, much of the II literature has concentrated on developing the summary statistic from an alternative parametric auxiliary model that is analytically and/or computationally more tractable. The major focus of this book chapter is on approximate Bayesian methods that harness such an auxiliary model. These are referred to as parametric Bayesian indirect inference (pBII) methods by Drovandi et al. (2014a)...