Higher-order Mean Square Research Articles

REGULARIZED ESTIMATION OF DYNAMIC PANEL MODELS

In a dynamic panel data model, the number of moment conditions increases rapidly with the time dimension, resulting in a large dimensional covariance matrix of the instruments. As a consequence, the generalized method of moments (GMM) estimator exhibits a large bias in small samples, especially when the autoregressive parameter is close to unity. To address this issue, we propose a regularized version of the one-step GMM estimator using three regularization schemes based on three different ways of inverting the covariance matrix of the instruments. Under double asymptotics, we show that our regularized estimators are consistent and asymptotically normal. These regularization schemes involve a tuning or regularization parameter which needs to be chosen. We derive a data-driven selection of this regularization parameter based on an approximation of the higher-order mean square error and show its optimality. As an empirical application, we estimate a model of income dynamics.

Choosing a Set of Instruments among Many and Possibly Invalid Instruments

We consider the instrument selection problem in instrumental variable regression model when there is a large set of instruments with potential invalidity. Existing methods for instrument selection are based on a priori assumptions of an instrument's validity. Moreover, existing approaches based on consistent moment selection or first-order asymptotics can lead to a large finite sample bias with many instruments. We derive higher-order mean square error (MSE) approximation of a two-stage least squares (2SLS) estimator allowing locally invalid instruments. Based on the approximation to the higher-order MSE, we propose an invalidity-robust instrument selection criterion (IRC) that captures two sources of finite sample bias at the same time: bias from using many instruments and bias from invalid instruments. We show that our criterion is an asymptotically unbiased estimator of the higher-order MSE approximation. In various simulation experiments, our criterion performs well in terms of lower bias and better coverage properties of the estimator compare with other existing selection methods. We also apply our criterion to the empirical application in Card (1995).

Higher Order Mean Squared Error of Generalized Method of Moments Estimators for Nonlinear Models

Generalized method of moments (GMM) has been widely applied for estimation of nonlinear models in economics and finance. Although generalized method of moments has good asymptotic properties under fairly moderate regularity conditions, its finite sample performance is not very well. In order to improve the finite sample performance of generalized method of moments estimators, this paper studies higher-order mean squared error of two-step efficient generalized method of moments estimators for nonlinear models. Specially, we consider a general nonlinear regression model with endogeneity and derive the higher-order asymptotic mean square error for two-step efficient generalized method of moments estimator for this model using iterative techniques and higher-order asymptotic theories. Our theoretical results allow the number of moments to grow with sample size, and are suitable for general moment restriction models, which contains conditional moment restriction models as special cases. The higher-order mean square error can be used to compare different estimators and to construct the selection criteria for improving estimator’s finite sample performance.

Minimum Divergence, Generalized Empirical Likelihoods, and Higher Order Expansions

This article studies the minimum divergence (MD) class of estimators for econometric models specified through moment restrictions. We show that MD estimators can be obtained as solutions to a tractable lower dimensional optimization problem. This problem is similar to the one solved by the generalized empirical likelihood estimators of Newey and Smith (2004), but it is equivalent to it only for a subclass of divergences. The MD framework provides a coherent testing theory: tests for overidentification and parametric restrictions in this framework can be interpreted as semiparametric versions of Pearson-type goodness of fit tests. The higher order properties of MD estimators are also studied and it is shown that MD estimators that have the same higher order bias as the empirical likelihood (EL) estimator also share the same higher order mean square error and are all higher order efficient. We identify members of the MD class that are not only higher order efficient, but also, unlike the EL estimator, well behaved when the moment restrictions are misspecified.

A Note on Mean Square Successive Differences

This paper deals with the use of higher order mean square successive differences for estimating dispersion when a strong trend is present in the mean value and the method of first differences does not adequately eliminate the trend. The first four moments of the mean square successive second difference, δ22, are derived in samples from an arbitrary population with constant mean (or with linear trend in the means). Because the efficiency of δ22 relative to δ22 increases with β2, the approximate distribution of δ22 is discussed only for a leptokurtic population, specifically for the symmetrical two-tailed exponential population. For populations with varying mean the first two moments of δ22 are given and the effect of trend of the mean values on these moments is discussed. Short tables of the efficiency of the mean square successive second, third, and fourth differences, all relative to s2 are given.