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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.

It is a stylized fact that plenty of relationships are dynamic and nonlinear in nature and society, especially in economic and financial systems [

When traditional asymptotic theory cannot precisely approximate the finite sample distributions of estimators or tests, we need higher-order asymptotic expansion for these estimators or tests to get more accurate approximation [

Besides the linear models, Rilstone et al. [

Newey and Smith [

Donald et al. [

The remainder of the paper proceeds as follows: Section

Many economic and financial models can be written as nonlinear functions of data and parameters. Consider the following nonlinear regression model with endogeneity:

For model (

Our goal is to obtain the MSE of

This part uses the iterative idea [

To derive the higher-order expansion of the GMM estimator

For some neighborhood of

For some neighborhood of

The smallest eigenvalues of

There is

Assumption

The first order condition for the optimization problem in (

For (

From (

Since the right-hand side of (

Before deriving the higher-order MSE of

Consider

Consider

Similarly,

Consider

By definition,

Consider

By definition of

Consider

By definition of

For the first term,

Since

According to the independence assumption,

For the second term,

For the third term,

For the fourth term,

For the fifth term,

Similarly, according to the independence assumption and Assumption

To sum up,

Consider

By definition of

Using these lemmas, then we can get the higher-order MSE of

For GMM estimator

By (

In (

In this paper, we consider a general nonlinear regression model with endogeneity and derive the higher-order mean square error of two-step efficient generalized method of moments estimators for this nonlinear model. The theoretical results in this paper allow the number of moments to grow with but at smaller rate than the sample size. And the derivations are suitable for general moment restriction models, which contain conditional moment restriction models and linear models as special cases. The higher-order mean squared error got in this paper has many uses. For example, it can be used to compare among different estimators or to construct the selection criteria of moments for improving the finite sample performance of GMM estimators. This paper considered a restrictive condition in which the data generating process is independent. It would be valuable to extend the results to the dynamic panel data models, in which the moments are going with the time dimension. It is saved for future research.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 71301160, 71203224, and 71301173), Beijing Planning Office of Philosophy and Social Science (13JGB018), and Program for Innovation Research in Central University of Finance and Economics.