Through the looking glass: Indirect inference via simple equilibria

Approximate maximum likelihood for complex structural models

“…As recently pointed out by Calvet and Czellar (2015), imposing potentially false equality constraints on a given structural model can be an attractive method for obtaining simple and rich auxiliary models for the purposes of I-I. For instance, in the context of a long-run risk model (Bansal and Yaron, 2004), Calvet and Czellar (2015) demonstrate that imposing specific equality constraints on certain parameters produces a simple auxiliary model for use in I-I (with a computationally tractable likelihood function) that closely resemble the structural model. The fact that this resulting auxiliary model may not deliver consistent estimates of the true structural parameters is immaterial insofar as matching a simulation-based approximation against the observation-based version will allow us to erase the misspecification bias.…”

“…The benefits of such an approach are two-fold: one, by using constraints to define the auxiliary model, we sketch a systematic strategy for the choice of an auxiliary model; two, this auxiliary model closely matches the structural model and so for issues of robustness and efficiency this auxiliary model is very useful. However, while highly-useful, the suggestion of Calvet and Czellar (2015) is incomplete, and does not allow for consistent estimation of the structural parameters on its own. That is, since the auxiliary model imposes a number of constraints on the structural model, by definition the auxiliary model can not consistently estimate all the structural parameters, except in the unlikely case where the constraints are satisfied at the true value of the structural parameters.…”

“…That is, since the auxiliary model imposes a number of constraints on the structural model, by definition the auxiliary model can not consistently estimate all the structural parameters, except in the unlikely case where the constraints are satisfied at the true value of the structural parameters. To circumvent this issue, Calvet and Czellar (2015) propose to add to the statistics obtained from the auxiliary model additional statistics so that, when considered jointly, this new vector can jointly identify the structural parameters when estimated by I-I.…”

“…Motivated by the above ideas and the approach to handling constraints within I-I proposed in Calzolari et al (2004) and Frazier and Renault (2019), we propose a novel inference approach based on constraining the structural model parameters to create a simple, but highly informative, proxy for the score vector that can be used to estimate the structural parameters. However, unlike the strategy put forward by Calvet and Czellar (2015), our approach provides an automatic, and nearly-efficient, method to identify the structural parameters.…”

Simultaneous Indirect Inference, Impulse Responses and ARMA Models

“…2 Despite the simplicity of ARMA models in time series, identification and boundary issues raise enduring complications for estimation and inference. First, Autoregression (AR) unit roots cause the so-called non-uniform convergence problem, which means that methods requiring stationarity 1 See, for example, Robins et al (2000); Ronchetti and Trojani (2001); Calzolari et al (2004); Dridi et al (2007); Gouriéroux et al (2010); Li (2010); Dominicy and Veredas (2013); Fuleky and Zivot (2014); Calvet and Czellar (2015); Chaudhuri et al (2018) and Forneron and Ng (2018).…”