Indirect Inference: Which Moments to Match?
- We argue that the first principle puts too much emphasis on the idea that “to implement the EMM estimator we require a score generator that fits these data well”. Gallant and Long (1997) show that (see their Lemma 1, p. 135) under convenient regularity conditions, asymptotic efficiency is reached if and only if the linear span of a “true score” (i.e., the score of a well-specified parametric model for the structural model) is asymptotically included (at the true value of the structural parameters) in the linear span of the score of the auxiliary model. This does not require in any way that the auxiliary model is (even asymptotically for an arbitrary large number of parameters) a well-specified model, in the sense that it is consistent with the Data Generating Process (DGP). Of course, as emphasized by GT, a sufficient condition for this score spanning property is the so-called smooth embedding of the score generator, which means that there is a one-to-one and twice continuously differentiable mapping between the two parametrizations (i.e., the auxiliary and structural). Then score spanning is just a consequence of computing compounded derivatives. However, this sufficient condition for score spanning is definitely not necessary and, thus, there is no logical argument to impose a model selection criterion, like AIC or BIC, to select an auxiliary model. We remind the reader that the purpose of the auxiliary model is not to describe the DGP, but to provide informative estimating equations. After all, it may be possible to satisfy the linear spanning condition by using a vector of moments that define well-suited auxiliary parameters but have no interpretation as a score function of a quasi-likelihood.
- The next point is the realization that what determines the efficiency of indirect inference estimators is the moments they match, and not necessarily the auxiliary parameters. Hence, and in contrast to the example given in Principle 2 above, one may well contemplate using a set of moment conditions that overidentify the vector of unknown auxiliary parameters. Of course, moment estimation of the auxiliary parameters will eventually resort to a just-identified set of moment conditions, through the choice of a particular linear combination of the (possibly) overidentified moment conditions. However, we argue that the choice of this just-identified set of moment conditions should not be guided by efficiency of the resulting estimator of auxiliary parameters (as would be the case with an efficient two-step Generalized Method of Moments (GMM) estimator) but, on the contrary, by our goal of obtaining an asymptotically efficient indirect estimator of the structural parameters. We demonstrate that this new focus of interest produces a novel way to devise a two-step GMM estimator of the auxiliary parameters that is, in general, different from standard efficient two-step GMM estimators.
2. Auxiliary versus Structural Models
- Identification of the true unknown value of the structural parameters , for a given value (the true unknown one) of the auxiliary parameters.
- Identification of the true unknown value of the auxiliary parameters , for a given value (the true unknown one ) of the structural parameters.
2.1. Auxiliary Model
- is full column rank.
- is a positive definite matrix.
2.2. The Structural Model
- is full column rank.
2.3. Moment Selection Criterion
3. Which Moments to Match?
3.1. Optimal Selection Matrix K
3.2. Efficient Two-Step Moment Matching
3.3. Efficient Two-Step Parameter Matching
- There exists a function such that, for all ,
- and , where is the solution of
- The Jacobian matrix has rank .
3.4. Interpretation of Results and Discussion
Conflicts of Interest
Appendix A. Proofs
Appendix A.1. Proof of Equation (6)
Appendix A.2. Proof of Equation (15)
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Similar to the selection matrix K, the weighting matrix W can be replaced by a consistent, data-dependent estimator, say such that as , without altering the first-order asymptotic theory of the II estimator.
See the Appendix A for more detailed derivations.
A discussion and interpretation of this surprising feasibility (in spite of the fact that is unknown) is given in Section 3.4.
Note that one may want to compute this derivative numerically and to choose very large (and possibly different from H in the rest of the procedure) to take advantage of the smoothness properties of the population moments.
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Frazier, D.T.; Renault, E. Indirect Inference: Which Moments to Match? Econometrics 2019, 7, 14. https://doi.org/10.3390/econometrics7010014
Frazier DT, Renault E. Indirect Inference: Which Moments to Match? Econometrics. 2019; 7(1):14. https://doi.org/10.3390/econometrics7010014Chicago/Turabian Style
Frazier, David T., and Eric Renault. 2019. "Indirect Inference: Which Moments to Match?" Econometrics 7, no. 1: 14. https://doi.org/10.3390/econometrics7010014