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

1 School of Management, University of Chinese Academy of Sciences, Beijing 100190, China 2 Institute of China’s Economic Reform and Development, Renmin University of China, Beijing 100892, China 3 School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China 4 School of Economics, Central University of Finance and Economics, Beijing 100081, China


Introduction
It is a stylized fact that plenty of relationships are dynamic and nonlinear in nature and society, especially in economic and financial systems [1][2][3][4][5][6][7][8].These relationships are usually depicted by nonlinear models.Generalized method of moments (GMM) has been widely applied for analysis for these nonlinear models since it was first introduced by Hansen [9] and gradually became a fundamental estimation method in econometrics [10].Nevertheless, although GMM has good asymptotic properties under fairly moderate regularity conditions, its finite sample performance is not very well [11][12][13].Similar to the maximum likelihood estimation (MLE), GMM does not have an exact finite sample distribution.In practice, we generally use the asymptotic distribution to approximate this finite sample distribution, but many applications of GMM reveal that this approximation has low precision [14].
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 [15].Nagar [16] studied the small sample properties of the general -class estimators of simultaneous equations and gave the higher-order asymptotic expansion of the first-and secondorder moments for two-stage least squares (2SLS) estimator.Donald and Newey [17] gave a theoretical derivation of the higher-order MSE for 2SLS based on Nagar [16]; however, their MSE formula applied to the case where the number of instruments grows with but at a smaller rate than the sample size, while Nagar [16] considered the cases where the number of instruments is fixed.Kuersteiner [18] derived the higherorder asymptotic properties of GMM estimators for linear time series models using many lags as instruments.
Besides the linear models, Rilstone et al. [19] derived and examined the second-order bias and MSE of a fairly wide class of nonlinear estimators, which included nonlinear least squares, maximum likelihood, and GMM estimators as special cases.Bao and Ullah [20] extended the secondorder bias and MSE results of Rilstone et al. [19] for time series dependent observations.In addition, Bao and Ullah [21] derived the higher-order bias and mean squared error of a large class of nonlinear estimators to order ( −5/2 ) and ( −3 ), respectively.However, although these papers gave the high-order bias and MSE for nonlinear estimators, they were not suitable for two-step efficient GMM estimators.
Newey and Smith [22] studied the higher-order bias for two-step GMM estimators, empirical likelihood (EL) and generalized empirical likelihood (GEL) estimators through higher-order asymptotic expansions.But this paper needs to be improved in the following aspects.First, the data generating process considered in this paper was independently identically distributed.Second, the number of moments is fixed.Third, the MSE of GMM was not given.Anatolyev [23] extended Newey and Smith [22] to stationary time series models with serial correlation.Again, the number of moments in this paper was fixed, and this paper only gave the higher-order bias for the estimators, but not the MSE.Donald et al. [24] examined higher-order asymptotic MSE for conditional moment restriction models.Based on this MSE, they developed moment selection criteria for two-step GMM estimator, a bias corrected version, and GEL estimators.Donald et al. [24] allowed the number of instruments to grow with sample size.However, this paper constructed moment conditions through instrumental variables, which was not suitable for general moment restriction models.Thus, our paper tends to fill an important lacuna in the literature about higher-order asymptotic expansion of nonlinear estimators.Specially, we consider a general nonlinear regression model with endogeneity, and our theoretical results are suitable for general moment restriction models, which contain conditional moment restriction models as a special case.
The remainder of the paper proceeds as follows: Section 2 introduces the model and notations.Section 3 discusses the estimation for the threshold and slope coefficients.Section 4 concludes.

Model
Many economic and financial models can be written as nonlinear functions of data and parameters.Consider the following nonlinear regression model with endogeneity: where   is dependent variable,   is a  × 1 vector of explanatory variables, which contains endogenous variables,  0 ∈ B is a  × 1 vector of parameters, B is a compact subset of   , and ( 2  |   ) =  2  .For model (1), the usual way to estimate  0 is nonlinear least squares (NLS).However,   contains endogenous variables, which means that (  |   ) ̸ = 0.In this case, NLS estimator of  0 is not consistent.We have to look for another consistent estimator, such as GMM estimator.Let   be  × 1 ( ≥ ) instrumental variables of   , and   ( = 1, . . ., ) are independent random vectors.The orthogonality conditions can be written as  (    ) = 0. ( And the sample moments are Then, the two-step efficient GMM estimator of  0 is given by where Φ() =  −1 ∑  =1   ()  ()  , β = arg min ∈ ()  Ŵ−1 (), and Ŵ is a random weighting matrix that almost surely converges to a nonstochastic symmetric positive definite matrix .
Our goal is to obtain the MSE of β0 .However, formula (4) does not have an analytical solution.We have to obtain it through higher-order asymptotic theory.

Higher-Order MSE of GMM Estimators
This part uses the iterative idea [19] to derive the higher-order MSE of the GMM estimator, which can be seen as a generalization of Donald et al. [24] to the unconditional moment restriction models.For the convenience of discussion, we use the following notations: where ∇ is a derivation operator defined by the following recursive method: That is, ∇ +1 () can be seen as a block matrix whose entry in the th row and th column is a 1 ×  vector [∇  ()]  /  , in which [∇  ()]  is the th row and th column element of matrix ∇  ().

Assumption 3. 𝐸(𝜂
Assumption 1 is a necessary condition for a higher-order Taylor expansion.Assumption 2 is a common condition for the moments of remainder terms to bound (see also [19,20,[23][24][25]). Assumption 3 requires that the third moments are zero, which can simplify the MSE calculations (see also [24,26,27]).Assumption 4 is a further identification condition.The purpose of Assumption 5 is to control the remainder terms of higher-order expansions (see [25] for details).
Since the right-hand side of (11) contains terms β0 −  0 and λ, we use iterative techniques to remove these terms.For GMM estimator β0 under Assumptions 1-5, if  → ∞ and  2 ()/ → 0, then √( β0 −  0 ) can be decomposed as follows: where and, for  ℎ 3 , Before deriving the higher-order MSE of β0 , we need the following lemmas.
For the second term, Similarly, according to the independence assumption and Assumption 1, the mathematical expectation of the third, fourth, and fifth terms are zero.
Lemma 10.Consider Proof.By definition of  ℎ 4 and ℎ, Similar to the proof of Lemma 8, we have And according to the independence assumption and Assumption 1, To sum up, ( ℎ 4 ℎ  ) = ( 2 /).
Using these lemmas, then we can get the higher-order MSE of β0 as follows.

Conclusions
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.