Alternative GMM estimators for spatial regression models

ABSTRACT

Using approximations of the score of the log-likelihood function, we derive moment conditions for estimating spatial regression models, starting with the spatial error model. Our approach results in computationally simple and robust estimators, such as a new moment estimator derived from the first-order approximation obtained by solving a quadratic moment equation, and performs similarly to existing generalized method of moments (GMM) estimators. Our estimator based on the second-order approximation resembles the GMM estimator proposed by Kelejian and Prucha in 1999. Hence, we provide an intuitive interpretation of their estimator. Additionally, we provide a convenient framework for computing the weighting matrix of the optimal GMM estimator. Heteroskedasticity robust versions of our estimators are also proposed. Furthermore, a first-order approximation for the spatial autoregressive model is considered, resulting in a computationally simple method of moment estimator. The performance of the considered estimators is compared in a Monte Carlo study.

KEYWORDS:

• spatial econometrics
• spatial error correlation
• generalized method of moments (GMM) estimation

JEL:

• C01
• C13
• C31

ACKNOWLEDGEMENTS

The authors thank the editor-in-chief and two anonymous referees for their very helpful comments and suggestions.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1. For some recent applications, see Lin (2010), Piras, Postiglione, and Aroca (2012), de Dominicis, Florax, and de Groot (2013), Kelejian, Murrell, and Shepotylo (2013), Miguelez and Moreno (2013), and Brady (2014). For a list of applications from 1991 until 2007, see Kelejian and Prucha (2010b).

2. Important contributions to the development of the GMM framework include Kelejian and Prucha (2001, 2004, 2007, 2010a, 2010b), Lee (2002, 2003, 2004, 2007a, 2007b), Kelejian, Prucha, and Yuzefovich (2004), Kapoor et al. (2007), Fingleton (2008a, 2008b), Fingleton and Le Gallo (2008a, 2008b), Arnold and Wied (2010), Arraiz, Drukker, Kelejian, and Prucha (2010), Elhorst (2010b), Lee and Liu (2010), Lin and Lee (2010), Liu and Lee (2010, 2013), Liu et al. (2010), Baltagi and Liu (2011), Drukker et al. (2013), Wang and Lee (2013), Kelejian and Piras (2014), Lee and Yu (2014, 2016), Qu and Lee (2015), and Qu, Wang, and Lee (2016).

3. Elhorst (2010a) provides a comprehensive discussion and comparison of different types of spatial regression models, whereas spatial panel models are analyzed by Elhorst (2003), Kapoor et al. (2007) and Elhorst (2012).

4. Due to this symmetry, for any pair of values with $m_n\left({\rho }_1\right)=m_n\left({\rho }_2\right)$, it follows that the minimum is obtained as $m_n\left(\right({\rho }_1+{\rho }_2\left)/\right(2\left)\right)$. Since in our case ${\rho }_1$ and ${\rho }_2$ are complex conjugate roots with $m_n\left({\rho }_1\right)=m_n\left({\rho }_2\right)=0$, it follows that $({\rho }_1+{\rho }_2)/2$ is just the real part of the two solutions.

5. In the fourth section, we will split that linear combination into two moment conditions, yielding an over-identified GMM version of the MLAM2 estimator.

6. In principle all real roots of (12) may be out of the domain $(-1,1)$. While this occurs rarely and only for extreme values of ${\rho }_0$, a simple and tractable solution to this issue is to minimize $\left|{\hat{m}}_{2,n}\left(\rho \right)\right|$ under the constraint $\left|\rho \right|\le 1$ whenever all real roots lie outside the domain $[-1,1]$. In the even more exceptional case that the solution is not unique, because more than one real root lies in the domain $(-1,1)$, it is less straightforward to determine ${\hat{\rho }}_2$. However, unreported simulations suggest this is not a relevant issue in practice.

7. For convenience, we assume ${\sigma }^2$ to be known. As usual, a consistent estimator can be obtained from the residuals of the model.

8. We are grateful to an anonymous referee for pointing out this potential problem with GMM estimates in the SDM.

9. Since we consider a ‘circular world’, observations $1$ and $n$ are direct neighbours. Hence, for observation $i=n$, the $j$th ‘behind’-neighbour $i+j$ is observation $j$.

10. This is done by running a t-test for each estimate $\hat{\rho }$ for whether it equals the true parameter ${\rho }_0$. In doing so, we use the corresponding standard error estimate of $\rho$, which is based on its asymptotic distribution. We indicate a rejection if the (true) null hypothesis of equality is rejected at the 5% level. Hence, for each estimator we expect a (wrong) rejection to occur in 5% of the MC replications.

11. These estimators differ only in the second element of the moment vector given in (27). In the case of a symmetrical spatial weight matrix, they are equal, since then $M_n^{\mathrm{\prime }}{M}_n=M_n{M}_n$.

Additional information

Funding

Christoph Wigger gratefully acknowledges the financial support provided by the German Research Foundation  in the context of the Priority Program 1764 ‘The German Labor Market in a Globalized World - Challenges through Trade, Technology, and Demographics’.