Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris
Introduction
Zellner’s (1962) pioneering paper considered the estimation and testing of seemingly unrelated regressions (SUR) with correlated error terms. SUR has been applied in many research areas in economics and other fields, see Srivastava and Giles, 1987, Fiebig, 2001 for excellent surveys. It is by now clear that SUR achieves gains in efficiency by estimating a set of equations simultaneously rather than estimating each equation separately. Common factors affecting these equations allow such gains in efficiency and has been demonstrated in economics, for e.g., in studying demand systems and translog cost functions, to mention a few important applications.
Avery, 1977, Baltagi, 1980 extended the SUR model to panel data models with error components. This extension allows one to take advantage of panel data which pools regions, counties, countries, neighborhoods over time. Besides the larger variation in the data across these regions, one is able to control for unobserved heterogeneity across these units of observation.
Anselin (1988) extended the SUR model to allow for spatial correlation in the data. This extension allows one to take advantage of spillover effects across regions. Here, we focus on combining the spatial and panel aspects of the data in a SUR context. In fact, Anselin, 1988, Elhorst, 2003 among others provided maximum likelihood (ML) methods that combine panel data with spatial analysis, while Kapoor et al. (2007) provided a generalized moments estimators (GM) approach for estimating a spatial random effects panel model with SAR disturbances. Fingleton (2008a) extended the GM approach of Kapoor, Kelejian and Prucha to allow for spatial moving average disturbances, see Anselin et al. (2008) for a recent survey.
This paper follows Wang and Kockelman (2007) who applied ML methods to a SUR model with spatial effects incorporated via autocorrelation in the spatial error terms and heterogeneity in the panel incorporated via random effects. However, this paper extends the ML approach developed by Wang and Kockelman (2007) to the general case where spatial effects are incorporated via spatial error terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification.
We propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. We show that ignoring these spatial effects and/or heterogeneity can lead to misleading inference.
An empirical application to hedonic housing prices in Paris illustrates these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990–2003.1 One of the main contributions of the paper is that it pays special attention to the heterogeneity and spatial variation in housing prices across districts and it tests for their existence.2 We find significant spatial effects and heterogeneity across the Paris districts, and we show that ML methods that incorporate these effects lead to reasonable estimates of the shadow prices of housing attributes.
Section 2 sets up the panel SUR model with spatial lag and spatial error components. In Section 3, we present the ML estimation under normality of the disturbances. Section 4 considers the problem of jointly testing for random effects as well as spatial correlation in the context of this spatial SUR panel model. This extends earlier work on testing in spatial panel models by Baltagi et al. (2007) from the single equation case to the SUR case. Section 5 performs Monte Carlo experiments which compare the small sample properties of the proposed ML estimators and LM tests. Section 6 provides an empirical application of these methods to the problem of estimating hedonic housing prices in Paris, while Section 7 concludes. We recognize that there is a large literature on hedonic housing and that our application is only meant to illustrate our spatial panel ML methods and the associated LM test statistics.
Section snippets
The panel SUR with spatial lag and spatial error components
We consider a spatial system of equations model viewed as an extension of the single equation spatial model introduced by Cliff and Ord, 1973, Cliff and Ord, 1981. In particular, we specify a system of spatially interrelated panel equations corresponding to N cross-sectional units over T time periods. The spatial SUR model for panel data is composed of M equations (each potentially having a different set of explanatory variables) for N regions which are observed over T time periods. Consider
Maximum likelihood estimation
The log-likelihood function (13) can also be written as follows:Using the results in Baltagi, 1980, Magnus, 1982,we can express the log-likelihood function as follows:Generalizing the Wang and Kockelman (2007) approach, the model can be estimated using a
Joint and conditional LM tests
Testing for spatial dependence has been surveyed by Anselin, 1988, Anselin and Bera, 1998. This has been extended to single equation spatial panels by Baltagi et al. (2007). Here we extend this to SUR spatial panels. Let us partition θ as follows: where θ1 pertains to the parameters included in the null hypothesis and θ2 to the remainder parameters. The Lagrange multiplier (LM) or score test statistic for testing, H0: θ1 = 0, may be written as:where is the
The data generating process
Consider the spatial SUR panel data model composed of M = 2 equations for N individuals (cities, regions, countries, …) and T time periods:Let Xj = [Xj1, Xj2] and βj = [β j1, βj2]′. We fix the spatial lag coefficients as γ1 = 0.8, γ2 = 0.8, the spatial error coefficients as λ1 = 0.5, λ2 = 0.5, the βj coefficients as β11 = β12 = β21 = β21 = 1. Following Nerlove (1970), we consider two explanatory variables [Xj1, Xj2] generated by:
An application to hedonic housing prices in Paris
We illustrate our spatial panel methods by estimating a system of three SUR equations for hedonic housing prices in Paris. As the capital of France, Paris represents one of the most important real estate markets. The city of Paris is divided into 20 arrondissements (administrative districts) which in turn are divided into 4 quartiers (quarters). Our units of observation are the 80 quartiers.
In France, the housing classification used for flats by real estate agencies and notaries is the
Conclusion
This paper proposed ML estimators for a panel SUR with both spatial lag and spatial error components. It extends the MLE approach developed by Wang and Kockelman (2007) to the general case where spatial effects are incorporated via spatial error terms and via a spatial lag on the dependent variables and where the heterogeneity in the panel is incorporated via an error component specification. This panel SUR model can be estimated using an iterative three-step method.
We also considered the
Acknowledgment
We would like to thank Stuart Rosenthal, Dan Black and anonymous referees for their helpful comments and suggestions. Also, numerous colleagues and conference participants at the Latin American Meeting of the Econometric Society, Buenos Aires, October 1–3, 2009, the First French Econometrics Conference, Toulouse School of Economics, December, 14–15, 2009 and the 16th International Conference on Panel Data, Amsterdam, July, 2–4, 2010. Many thanks to Annick Vignes for providing us with the
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