Heteroskedasticity-consistent covariance matrix estimators for spatial autoregressive models

ABSTRACT

In the presence of heteroskedasticity, conventional test statistics based on the ordinary least squares (OLS) estimator lead to incorrect inference results for the linear regression model. Given that heteroskedasticity is common in cross-sectional data, the test statistics based on various forms of heteroskedasticity-consistent covariance matrices (HCCMs) have been developed in the literature. In contrast to the standard linear regression model, heteroskedasticity is a more serious problem for spatial econometric models, generally causing inconsistent extremum estimators of model coefficients. This paper investigates the finite sample properties of the heteroskedasticity-robust generalized method of moments estimator (RGMME) for a spatial econometric model with an unknown form of heteroskedasticity. In particular, it develops various HCCM-type corrections to improve the finite sample properties of the RGMME and the conventional Wald test. The Monte Carlo results indicate that the HCCM-type corrections can produce more accurate results for inference on model parameters and the impact effects estimates in small samples.

KEYWORDS:

• spatial autoregressive models
• generalized method of moments (GMM)
• heteroskedasticity
• heteroskedasticity-consistent covariance matrix estimator (HCCME)
• asymptotic variance
• efficiency
• standard errors
• inference

JEL:

• C13
• C21
• C31

ACKNOWLEDGEMENT

We thank the editor and two anonymous referees for their insightful comments that lead to improvements of this paper. We also thank the conference participants at the 26th (EC)² Conference on Theory and Practice of Spatial Econometrics at Heriot-Watt University, Edinburgh, UK, for helpful comments. Any remaining errors and omissions are, of course, ours.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1 For a comprehensive review of HCCMEs for non-spatial linear regression models, see MacKinnon (2013).

2 We assumed a parametric spatial autoregressive process for the disturbance terms in deriving various HCCMEs. Non-parametric and semi-parametric approaches that do not require a parametric spatial autoregressive process for the disturbance terms have also been considered in the literature (e.g., Driscoll & Kraay, 1998; Conley, 1999; Pinkse, Slade, & Brett, 2002; Conley & Molinari, 2007; Kelejian & Prucha, 2007; Kim & Sun, 2011).

3 On the specification of parameter space for autoregressive parameters, see Kelejian and Prucha (2010), Elhorst, Lacombe, and Piras (2012) and Elhorst (2014).

4 An initial RGMME can be based on $P_{1n}=W_n^{{}^{\mathrm{\prime }}}{W}_n-D\left(W_n^{{}^{\mathrm{\prime }}}{W}_n\right)$, $P_{2n}=M_n^{{}^{\mathrm{\prime }}}{M}_n-D\left(M_n^{{}^{\mathrm{\prime }}}{M}_n\right)$ and $Q_n=\left[W_n{M}_n{X}_n,W_n{X}_n,M_n{X}_n,X_n\right]$. Let $g_n\left({\theta }_0\right)=({\epsilon }_n^{{}^{\mathrm{\prime }}}{P}_{1n}{\epsilon }_n,{\epsilon }_n^{{}^{\mathrm{\prime }}}{P}_{2n}{\epsilon }_n,{\epsilon }_n^{{}^{\mathrm{\prime }}}{Q}_n{)}^{{}^{\mathrm{\prime }}}$. The initial RGMME is then defined by ${\hat{\theta }}_{1n}=\underset{\theta \in \mathrm{\Theta }}{\mathrm{argmin}}{g}_n^{{}^{\mathrm{\prime }}}\left(\theta \right)g_n\left(\theta \right)$.

5 For simplicity, we assume that the true parameter vector is identified. The asymptotic results in this section are proved by Dogan and Taspinar (2013) along the asymptotic argument given by Lin and Lee (2010).

6 For example, Anselin (1988, ch. 6) assumes a parametric specification for the variance terms and shows how a log-likelihood function can be formulated for the joint estimation of parameters.

7 For a non-spatial linear regression model, the hat matrix is given by $H=X(X^{\mathrm{\prime }}X{)}^{-1}{X}^{\mathrm{\prime }}$. A value of $H_{ii}$ greater than ${\displaystyle \frac{2}{n}\mathrm{tr}\left(H\right)=\frac{2k}{n}}$ or ${\displaystyle \frac{3}{n}\mathrm{tr}\left(H\right)=\frac{3k}{n}}$ is considered as a high leverage point (Judge, Hill, Griffiths, Lutkepohl, & Lee, 1988).

8 In terms of bias and RMSE performance measures, the results indicate that all correction methods produc similar results for the total effects. Therefore, we focus on the effect of correction methods on the finite sample size properties of the Wald statistic.

9 An estimation routine written in Matlab is available from the authors upon request.

Additional information

Funding

The CUNY HPCC is operated by the College of Staten Island and funded, in part, by grants from the City of New York, State of New York, CUNY Research Foundation, and National Science Foundation [grant numbers CNS-0958379, CNS-0855217 and ACI 1126113.