Indirect Inference With(Out) Constraints

Indirect Inference (I-I) estimation of structural parameters $\theta$ {{requires matching observed and simulated statistics, which are most often generated using an auxiliary model that depends on instrumental parameters $\beta$.}} {The estimators of the instrumental parameters will encapsulate} the statistical information used for inference about the structural parameters. As such, artificially constraining these parameters may restrict the ability of the auxiliary model to accurately replicate features in the structural data, which may lead to a range of issues, such as, a loss of identification. However, in certain situations the parameters $\beta$ naturally come with a set of $q$ restrictions. Examples include settings where $\beta$ must be estimated subject to $q$ possibly strict inequality constraints $g(\beta) > 0$, such as, when I-I is based on GARCH auxiliary models. In these settings we propose a novel I-I approach that uses appropriately modified unconstrained auxiliary statistics, which are simple to compute and always exists. We state the relevant asymptotic theory for this I-I approach without constraints and show that it can be reinterpreted as a standard implementation of I-I through a properly modified binding function. Several examples that have featured in the literature illustrate our approach.