EC3SLS Estimator for a Simultaneous System of Spatial Autoregressive Equations with Random Effects

Abstract

This article derives a 3SLS estimator for a simultaneous system of spatial autoregressive equations with random effects which can therefore handle endoegeneity, spatial lag dependence, heterogeneity as well as cross equation correlation. This is done by utilizing the Kelejian and Prucha (1998) and Lee (2003) type instruments from the cross-section spatial autoregressive literature and marrying them to the error components 3SLS estimator derived by Baltagi (1981) for a system of simultaneous panel data equations. Our Monte Carlo experiments indicate that, for the single equation spatial error components 2SLS estimators, there is a slight gain in efficiency when Lee (2003) type rather than Kelejian and Prucha (1998) instruments are used. However, there is not much difference in efficiency between these instruments for spatial error components 3SLS estimators.

Keywords:

• Error components
• Panel data
• Simultaneous equations
• Spatial model
• Three-stage least squares

JEL Classification:

• C13
• C33

ACKNOWLEDGMENT

This paper is dedicated to Aman Ullah who has made significant contributions to econometrics. A preliminary version was presented at the 18th International Conference on Panel Data held at the Banque de France, Paris, France, July 5-6, 2012.

Notes

¹For a survey of spatial panel models, see Elhorst (2003), Anselin et al. (2008), and Baltagi (2011), to mention a few.

²See Baltagi (1981), Balestra and Varadharajan-Krishnakumar (1987), Cornwell et al. (1992), and Baltagi and Li (1992) for panel data simultaneous equations estimators that do not account for spatial correlation.

³Some recent examples include Ho and Hite (2008) and Jeanty et al. (2010). In fact, Ho and Hite (2008) estimate a three equations simultaneous model relating housing prices, cancer mortality, and toxic chemical releases using a unique data set for 754 counties in the southeast region of the United States. Their findings suggest that housing prices respond negatively to toxic chemical releases and cancer mortality. Also, cancer death responds positively to toxic chemical releases. Similarly, Jeanty et al. (2010) estimate a spatial simultaneous equations model that jointly considers population change and housing values using 1789 Census tracts in Michigan. They find strong evidence of spatial spillovers in both population change and housing values across proximate neighborhoods. While these studies employ cross-section data, they can be easily extended to a richer panel data setting using our estimators.

⁴We also provide Monte Carlo results using different weight matrices for different equations. These results are compared with the simplified Monte Carlo design where the same weight matrix is assumed for both equations.

⁵Note that 2SLS and FE-S2SLS use KP1 type instruments.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2-9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.

^a In each cell of columns 2–9, the upper number denotes the bias, the middle one in parentheses denotes the s.d., and the bottom number denotes the RMSE. In each cell of column 10, the upper number denotes avgNOMAD, the middle one denotes medNOMAD, and the lower one in parentheses denotes NORMSQD.