Finite-sample refinement of GMM approach to nonlinear models under heteroskedasticity of unknown form

ABSTRACT

It is quite common to observe heteroskedasticity in real data, in particular, cross-sectional or micro data. Previous studies concentrate on improving the finite-sample properties of tests under heteroskedasticity of unknown forms in linear models. The advantage of a heteroskedasticity consistent covariance matrix estimator (HCCME)-type small-sample improvement for linear models does not carry over to the nonlinear model specifications since there is no obvious counterpart for the diagonal element of the projection matrix in linear models, which is crucial for implementing the finite-sample refinement. Within the framework of nonlinear models, we develop a straightforward approach by extending the applicability of HCCME-type corrections to the two-step GMM method. The Monte Carlo experiments show that the proposed method not only refines the testing procedure in terms of the error of rejection probability, but also improves the coefficient estimation based on the mean squared error (MSE) and the mean absolute error (MAE). The estimation of a constant elasticity of substitution (CES)-type production function is also provided to illustrate how to implement the proposed method empirically.

KEYWORDS:

• Eicker-White HCCME
• GMM
• nonlinear model

JEL CLASSIFICATION:

• C12
• C13

Acknowledgments

A debt of gratitude is conveyed to Stephen G. Donald for inspiring conversations in the early stage of this research. We would like to thank Cheng Hsiao, Chihwa Kao, John Chao, Fathali Firoozi, Hui-Lin Lin, Chien-Ho Wang, Chor-Yiu Sin, Ming-Tui Huang, and Jau-Lian Jeng for many useful comments and suggestions on earlier drafts of this paper. We would also like to acknowledge very constructive comments from the editor, the associate editor, an anonymous referee, and participants from the Department of Public Policy and Administration, American University in Cairo, Egypt, the Department of Economics, National Taiwan University in Taipei, Taiwan, and at the Western Economic Association International 86th Annual Conference in San Diego, California. Any remaining errors are our own.

Notes

In what follows, we adopt HCj naming convention in the modern literature to denote various versions of HCCME.

It is noted that, recently, Cribari-Neto and Silva (2011) further propose a modified HC4 (known as HC4 m) by fine-tuning the leverage-based discounting factor. Their numerical results show that the inference in terms of HC4 m is more reliable than that of using HC4.

The bootstrap method is also an alternative to using HCCMEs to obtain more reliable inferences. Two popular bootstrap methods in the literature include the pairs bootstrap and wild bootstrap. Please refer to MacKinnon (2011) for an excellent exposition.

Although GMM is consistent and asymptotically efficient under some regularity conditions, its finite-sample properties may not be satisfactory. Several studies have attempted to analyze the GMM bias and size distortion, e.g., Kocherlakota (1990), Hansen et al. (1996), and Newey and Smith (2004).

For example, Clark (1996) utilizes Monte Carlo experiments to study the small sample performance of the covariance structure for three nonlinear models, but the issue of heteroskedasticity has not yet been discussed.

We are aware that the bootstrap method has the potential to improve HCCME-based testing in linear regression models, e.g., MacKinnon (2011). It is worth making comparison and investigating the possible combination between our proposed method and bootstrap techniques in nonlinear GMM models.

It is noted that estimating a linear model with unknown heteroskedasticity in terms of Cragg’s (1983) GMM-type estimator using various HCCMEs also leads to alternative forms of the coefficient estimator, which accordingly displays efficiency gain on the coefficient.

Under the case of homoskedasticity, we note that the objective function in (4) implies a nonlinear instrument variable estimator (NLIV) and includes the nonlinear least squares estimator (NLS) as a special case when Z = I_n. In addition, for a linear model, is simply the two-stage least squares estimator (2SLS).

If all column vectors in are in the column space of Z, which can be achieved by construction, the matrix is symmetric and idempotent such that it is easy to show that is in the interval [0, 1].

Cribari-Neto et al. (2007, p. 1881) suggest the setting of κ = 0.7 for reliable HC5-based quasi-t tests in finite samples. Indeed, their simulation study shows that values of κ between 0.6 and 0.8 may deliver reliable inference.

For the benchmark derived under the hypothetical homoskedasticity, the correct asymptotic variance is: . However, under the assumption of homoskedasticity, the asymptotic variance may be conceived as .

The asymptotic variance of is estimated by a wrong formula: .

We also specify the simulation data generating process to include additional parameter, . The results are quite similar to those in terms of (30). Please refer to an online appendix for more details.

Because this example is used by Dominguez and Lobato (2004) to study the problem of identification, one may wonder whether this model specification which is not identified with an exactly-identified optimal instrumental variable can be identified in our simulation. Regarding this point, we have implemented several numerical experiments to confirm that the parameter β_o can indeed be identified in this Monte Carlo experiment with an overidentified setting for the instrumental variables. More specifically, by using an q-vector function to approximate with large S and q ≥ 2, one may evaluate at supports with sufficiently large ranges, and find that the intersection of all solutions of β for will determine a unique value.

More specifically, we collect the estimates () and calculate the squared estimation errors at each iteration, that is, . After all iterations (i.e., i = 1, 2,…, 10000), we define the loss difference for j = 2, 3, 4, 5, and construct t-ratios by , where and s_j are the average and standard deviation of , respectively.

For cases with more instrumental variables, i.e., q = 4, 5, the main findings are similar, and we thus skip those graphs.

Note that the results using other sample sizes (e.g., n = 50 and 100) are quite similar and are available upon request from the authors.

Please refer to an online supplementary appendix for the additional simulation results.

Those simulation results can be found in an online supplementary appendix.

We are grateful to the Associate Editor and an anonymous referee for directing us to the discussion in this subsection.

In addition to the GMM method, there are several econometric strategies for estimating the CES production function, such as nonlinear two-step least squares (Tsurumi, 1970), the Bayesian approach using the Metropolis–Hastings algorithm (Koop, 2003), etc.

In this data generating process, we set v = 15, φ_L = 1.88, φ_K = 1.8, and φ₀ = −5.43139.

We owe the discussion on the possible extension of time series data to an associate editor and anonymous referee.

Additional information

Funding

Financial support from the National Science Council of Taiwan in the form of grants NSC 99-2410-H-007-022 (Lin) and NSC 102-2410-H-033-005 (Chou) is greatly appreciated.