The traditional implementation of Indirect Inference (I-I) is to perform inference on structural parameters $\theta$ by matching observed and simulated auxiliary statistics. These auxiliary statistics are consistent estimators of instrumental parameters whose value depends on the value of structural parameters through a binding function. Since instrumental parameters encapsulate the statistical information used for inference about the structural parameters, it sounds paradoxical to constrain these parameters, that is, to restrain the information used for inference. However, there are situations where the definition of instrumental parameters $\beta$ naturally comes with a set of $q$ restrictions. Such situations include: settings where the auxiliary parameters must be estimated subject to $q$ possibly binding strict inequality constraints $g(\cdot) > 0$; cases where the auxiliary model is obtained by imposing $q$ equality constraints $g(\theta) = 0$ on the structural model to define tractable auxiliary parameter estimates of $\beta$ that are seen as an approximation of the true $\theta$, since the simplifying constraints are misspecified; examples where the auxiliary parameters are defined by $q$ estimating equations that overidentify them. We demonstrate that the optimal solution in these settings is to disregard the constrained auxiliary statistics, and perform I-I without these constraints using appropriately modified unconstrained versions of the auxiliary statistics. In each of the above examples, we outline how such unconstrained auxiliary statistics can be constructed and demonstrate that this I-I approach without constraints can be reinterpreted as a standard implementation of I-I through a properly modified binding function.
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