Efficient minimum distance estimation with multiple rates of convergence

Abstract

Abstract:This paper extends the asymptotic theory of GMM inference to allow sample counterparts of the estimating equations to converge at (multiple) rates, different from the usual square-root of the sample size. In this setting, we provide consistent estimation of the structural parameters. In addition, we define a convenient rotation in the parameter space (or reparametrization) which permits to disentangle the different rates of convergence. More precisely, we identify special linear combinations of the structural parameters associated with a specific rate of convergence. Finally, we demonstrate the validity of usual inference procedures, like the overidentification test and Wald test, with standard formulas. It is important to stress that both estimation and testing work without requiring the knowledge of the various rates. However, the assessment of these rates is crucial for (asymptotic) power considerations. Possible applications include econometric problems with two dimensions of asymptotics, due to trimming, tail estimation, infill asymptotic, social interactions, kernel smoothing or any kind of regularization. JEL classification: C32; C12; C13; C51.

Structural estimation of behavioral heterogeneity

“…Fortunately, Gagliardini et al (2011) and Antoine and Renault (2012) have developed XMM, an extension of GMM, to explicitly incorporate moments with different rates of convergence. Fortunately, Gagliardini et al (2011) and Antoine and Renault (2012) have developed XMM, an extension of GMM, to explicitly incorporate moments with different rates of convergence.…”

“…Following Gagliardini et al (2011) and Antoine and Renault (2012), we assume local identification in the sense of Rothenberg (1971). That is, 0 is locally identified if there exists an open neighborhood of 0 containing no other that can generate the same distribution.…”

“…With such a mixture of sample moments converging at various rates, the standard asymptotic theory of GMM is inapplicable. Fortunately, Gagliardini et al (2011) and Antoine and Renault (2012) have developed XMM, an extension of GMM, to explicitly incorporate moments with different rates of convergence. This latest methodological advancement makes the following empirical estimation possible.…”

Averaging of an Increasing Number of Moment Condition Estimators

“…We suppose that J may have infinite cardinality, or it may have a cardinality that is increasing with sample size n. Han and Phillips (2006) consider a similar setup except that they combine the moment conditions in the classical GMM way, i.e., (2.2), using identity weighting; they also allow for the possibility that some of their moment conditions provide only weak identification. Lee (2010) considers the case where τ is fixed but each estimator may have a different rate of convergence (see also Antoine and Renault, 2012). We work with a situation where each estimator is √ n-consistent, however, in our case the asymptotic variance V j j of the j th estimator may increase to infinity with j , which essentially reflects the same phenomenon of different precision across the estimators.…”

Estimation and inference with a (nearly) singular Jacobian

“…The rotation technique we use in our asymptotic derivations has many antecedents in the literature. For example, Sargan (), Phillips () and Choi and Phillips () used similar rotations to derive limit theory for estimators under identification failure; Antoine and Renault (, ) used similar rotations to derive limit theory for estimators under “nearly‐weak” identification; Andrews and Cheng () (AC14 hereafter) used similar rotations to find the asymptotic distributions of Wald statistics under weak and nearly‐strong identification; and recently Phillips () used similar rotations to find limit theory for regression estimators in the presence of near‐multicollinearity in regressors. However, unlike their predecessors used for specific linear models, our nonlinear reparameterizations are not generally equivalent to the rotations we use to derive asymptotic theory.…”