Volume 2014, Issue 1 324904

Research Article

Open Access

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

Yi Hu,

Yi Hu

orcid.org/0000-0002-2000-6464

School of Management, University of Chinese Academy of Sciences, Beijing 100190, China ucas.ac.cn

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Xiaohua Xia,

Xiaohua Xia

Institute of China’s Economic Reform and Development, Renmin University of China, Beijing 100892, China ruc.edu.cn

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Ying Deng,

Ying Deng

School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China uibe.edu.cn

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Dongmei Guo,

Corresponding Author

Dongmei Guo

[email protected]

School of Economics, Central University of Finance and Economics, Beijing 100081, China cufe.edu.cn

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Yi Hu,

Yi Hu

orcid.org/0000-0002-2000-6464

School of Management, University of Chinese Academy of Sciences, Beijing 100190, China ucas.ac.cn

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Xiaohua Xia,

Xiaohua Xia

Institute of China’s Economic Reform and Development, Renmin University of China, Beijing 100892, China ruc.edu.cn

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Ying Deng,

Ying Deng

School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China uibe.edu.cn

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Dongmei Guo,

Corresponding Author

Dongmei Guo

[email protected]

School of Economics, Central University of Finance and Economics, Beijing 100081, China cufe.edu.cn

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First published: 28 April 2014

https://doi.org/10.1155/2014/324904

Citations: 1

Academic Editor: Chuangxia Huang

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Abstract

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.

1. Introduction

It is a stylized fact that plenty of relationships are dynamic and nonlinear in nature and society, especially in economic and financial systems [1–8]. 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 [11–13]. 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 k-class estimators of simultaneous equations and gave the higher-order asymptotic expansion of the first- and second-order 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 higher-order 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 second-order 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 O(n^−5/2) and O(n⁻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.

2. Model

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

(1)

where y_i is dependent variable, X_i is a p × 1 vector of explanatory variables, which contains endogenous variables, β₀ ∈ ℬ is a p × 1 vector of parameters, ℬ is a compact subset of R^p, and

For model (1), the usual way to estimate β₀ is nonlinear least squares (NLS). However, X_i contains endogenous variables, which means that E(ε_i∣X_i) ≠ 0. In this case, NLS estimator of β₀ is not consistent. We have to look for another consistent estimator, such as GMM estimator. Let Z_i be L × 1 (L ≥ p) instrumental variables of X_i, and Z_i (i = 1, …, n) are independent random vectors. The orthogonality conditions can be written as

(2)

And the sample moments are

(3)

Then, the two-step efficient GMM estimator of β₀ is given by

(4)

where

, and

is a random weighting matrix that almost surely converges to a nonstochastic symmetric positive definite matrix W.

Our goal is to obtain the MSE of . However, formula (4) does not have an analytical solution. We have to obtain it through higher-order asymptotic theory.

3. 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:

(5)

where ∇ is a derivation operator defined by the following recursive method:

(6)

That is, ∇^r+1g(β) can be seen as a block matrix whose entry in the ith row and jth column is a 1 × p vector

, in which

is the ith row and jth column element of matrix ∇^rg(β).

To derive the higher-order expansion of the GMM estimator , the following assumptions for the moment function are required.

Assumption 1. For some neighborhood of β₀, f_i is, at least, three times continuously differentiable and , r = 0,1, 2,3, i = 1,2, …, and j = 1, …, L.

Assumption 2. For some neighborhood of β₀, ∥∇^rg_i,j(β) − ∇^rg_i,j(β₀)∥≤∥β − β₀∥M_i,j, in which E(M_i,j) < ∞, r = 0,1, 2,3, and j = 1, …, L, i = 1,2, ….

Assumption 3. , in which τ₁ + τ₂ = 3, τ₁ and τ₂ are nonnegative integers, j = 1, …, L, k = 1, …, p, r = 1, …, L, and i = 1,2, ….

Assumption 4. The smallest eigenvalues of and are bounded away from zero, in which belongs to the neighborhood of β₀.

Assumption 5. There is ζ(L) and for some finite constant C, such that ∥g_i∥ < Cζ(L).

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, 23–25]). 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).

The first order condition for the optimization problem in (4) is given by

(7)

Define an auxiliary vector

; the above first order condition can be rewritten as

(8)

Let K = p + L,

, and

. Equation (8) can be written as

(9)

For (9), the number of parameters is equal to the number of equations. A second-order Taylor expansion of (9) at

yields

(10)

where m = m(θ₀),

, and

, t ∈ (0,1), in which

and θ_j are the jth element of

and θ₀, respectively.

From (10), we have

(11)

where

and

Since the right-hand side of (11) contains terms

and

, we use iterative techniques to remove these terms. For GMM estimator

under Assumptions 1–5, if L → ∞ and ζ²(L)L/n → 0, then

can be decomposed as follows:

(12)

where

(13)

and, for

(14)

Before deriving the higher-order MSE of , we need the following lemmas.

Lemma 6. Consider

(15)

Proof. Consider

(16)

in which

(17)

Similarly, ,

(18)

Lemma 7. Consider

(19)

Proof. By definition, and ; then

(20)

and according to the independence assumption,

(21)

Let

, and

; then

. Now, we examine the jth row and kth column element of C, C_jk:

. By Assumption 3,

. Then we have E(C) = 0; that is,

. Finally, we have

. So,

Lemma 8. Consider

(22)

Proof. By definition of and h, . And by the definition of , . Take mathematical expectation for this formula,

(23)

According to the independence assumption,

(24)

And by

, we have

. To sum up,

(25)

Lemma 9. Consider

(26)

Proof. By definition of and h, .

For the first term, .

Since and , we have .

According to the independence assumption,

(27)

According to the assumption for

and Assumption 1, the mathematical expectation of the first term is zero.

For the second term, . By Lemma 6, .

For the third term, .

For the fourth term, .

For the fifth term, .

Similarly, according to the independence assumption and Assumption 1, the mathematical expectation of the third, fourth, and fifth terms are zero.

To sum up, .

Lemma 10. Consider

(28)

Proof. By definition of and h,

(29)

Similar to the proof of Lemma 8, we have

(30)

And according to the independence assumption and Assumption 1,

(31)

To sum up,

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

Theorem 11. For GMM estimator in (4), under Assumptions 1–5, if L → ∞ and ζ²(L)L/n → 0, then the higher-order MSE of is given by

(32)

in which

(33)

Proof. By (12),

(34)

By Lemma 6,

, the above formula can be rewritten as

(35)

By Lemmas 7–10,

(36)

In addition, by definition of

. Consider

(37)

Denote u_i = (∂f_i/∂β) − E((∂f_i/∂β)∣Z_i); then

. Then,

(38)

According to the independence assumption,

(39)

In (32), Ω⁻¹ is asymptotic variance of , and Ω⁻¹(ΠΠ^′/n)Ω⁻¹ can be seen as the asymptotic bias terms. In practice, Ω and Π can be substituted by their consistent estimators.

4. 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.

Conflict of Interests

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

Acknowledgments

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.

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

Abstract

1. Introduction

2. Model

3. Higher-Order MSE of GMM Estimators

4. Conclusions

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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

Abstract

1. Introduction

2. Model

3. Higher-Order MSE of GMM Estimators

4. Conclusions

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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