Efficient minimum distance estimation with multiple rates of convergence

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) 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.