E cient Two-Step Estimation via Targeting

The standard description of two-step extremum estimation amounts to plugging in a first step estimator of nuisance parameters in order to simplify the optimization problem and then to deduce a user friendly estimator for the parameters of interest. This two-step procedure often induces an e ciency loss with respect to estimation of the parameters of interest. In this paper, we consider a more general setting where we do not necessarily have such thing as nuisance parameters but rather awkward occurrences of the parameters of interest. By awkward, we mean that within the estimating equations for a vector of unknown parameters of interest ✓ , some occurrences of ✓, encapsulated by a vector ⌫(✓), may be computationally tricky. Then, it is still the case that prior knowledge of the unknown auxiliary parameters ⌫ = ⌫(✓) would make inference on ✓ much simpler, and it is this fact that motivates the two-step approach developed in this paper. The e ciency problem is more di cult than for the case of standard nuisance parameters since even the (infeasible) approach of plugging in the true unknown value of ⌫ = ⌫(✓) may not allow e ciency, since it overlooks the information about ✓ contained in the awkward occurrences ⌫(✓). Moreover, we stress that standard ways to restore e ciency for two-step procedures may not work due to a consistency issue; when setting the focus on a first step estimator for only some of the occurrences ⌫ = ⌫(✓) of the unknown parameters ✓, global identification may be lost. To alleviate this issue, we develop a targeting strategy that enforces consistency and achieves e ciency. Such di cult occurrences ⌫(✓) of the parameters, which are a nuisance when it comes to solving estimating equations, are present in many financial econometrics applications, often handled by indirect inference. Leading examples are asset pricing models with latent variables (their observation would make estimation much simpler), models where it is simpler to first set the focus of inference on marginal distributions (multivariate GARCH, copulas), models with highly nonlinear objective functions, etc. Based on targeting and penalization of the auxiliary parameters, we propose a new two-step estimation procedure that leads to stable and user-friendly computations. Moreover, estimators delivered in the second step of the estimation procedure are asymptotically e cient. We compare this new method with existing iterative methods in the framework of copula models and asset pricing models. Simulation results illustrate that this new method performs better than existing iterative procedures and is (nearly) computationally equivalent. ⇤Department of Econometrics and Business Statistics. email: [email protected] †Department of Economics, Brown University. email: eric [email protected] 1