The Mundlak Approach in the Spatial Durbin Panel
Data Model

Abstract

This paper extends the Mundlak approach to the spatial Durbin panel data model (SDM) to help the applied researcher to determine the adequacy of the random effects specification in this setup. We propose a likelihood ratio (LR) test that assesses the significance of the correlation between regressors and individual effects. By contrast to the Hausman test, the Mundlak approach identifies (to some extent) the regressors correlated with individual effects. The second advantage is that once the correlation with individual effects has been modelled through an auxiliary regression, the random effects specification provides consistent estimators and the effect of time-constant variables can be estimated. Some Monte Carlo simulations study the properties of this proposed LR test in small samples and show that in some cases, it has a better behaviour than the Hausman test. We finally illustrate the usefulness of the extended Mundlak approach by estimating a house price model where some of the price determinants are time-constant. We show that ignoring the endogeneity of regressors with respect to individual effects leads to unreliable estimated parameters while results obtained using the Mundlak approach and the fixed effects specification are similar (concerning time-varying variables), implying that correlation between regressors and individual effects is well captured.

RÉSUMÉ la présente communication applique l'approche de Mundlak au modèle de données spatiales de Durbin pour aider le chercheur appliqué à déterminer dans quelle mesure la spécification des effets aléatoires est adéquate dans cette configuration. Nous proposons un test de ratio de vraisemblance évaluant l'importance de la corrélation entre régresseurs et effets individuels. Contrairement au test de Hausman, l'approche de Mundlak identifie (dans une certaine mesure) les régresseurs corrélés à des effets individuels. Le deuxième avantage est que lorsque la corrélation avec les effets individuels a été modélisée via une régression auxiliaire, la spécification des effets aléatoires fournit des estimateurs convergents, et il est alors possible d’évaluer l'effet de variables constantes dans le temps. Des simulations Monte Carlo étudient les propriétés de ce test de ratio de vraisemblance proposé dans des échantillons de taille finie, et indiquent que, dans certains cas, il présente un meilleur comportement que le test de Hausman. Nous illustrons enfin l'utilité de l'approche étendue de Mundlak en évaluant un modèle de prix des maisons, dans lequel certains déterminants des prix sont constants dans le temps. Nous montrons que si on ne prend pas en compte l'endogénéité des régresseurs par rapport aux effets individuels, on obtient des paramétres estimés non fiables, alors que les résultats obtenus avec l'approche de Mundlak et la spécification des effets fixes sont similaires (sur le plan des variables variant dans le temps), ce qui implique que la corrélation entre régresseurs et effets individuels est bien captée.

EXTRACTO Este estudio extiende el planteamiento Mundlak al modelo espacial de datos de panel (SDM) Durbin para ayudar al investigador aplicado a determinar la idoneidad de la especificación de efectos aleatorios dentro de esta configuración. Proponemos una prueba de relación de la probabilidad (LR) que evalúa la significancia de la correlación entre regresores y efectos individuales. En contraste con la prueba Hausman, el planteamiento Mundlak identifica (hasta cierto punto) los regresores correlacionados con efectos individuales. La segunda ventaja es que, una vez modelada la correlación con efectos individuales a través de una regresión auxiliar, la especificación de efectos aleatorios proporciona estimadores consistentes y puede estimarse el efecto de las variables constantes en el tiempo. Algunas simulaciones de Monte Carlo estudian las propiedades de esta prueba LR propuesta en muestras pequeñas y demuestran que, en algunos casos, se comporta mejor que la prueba Hausman. Finalmente, ilustramos la utilidad del planteamiento Mundlak ampliado estimando el precio de una vivienda donde varios determinantes del precio son constantes en el tiempo. Mostramos que ignorar la endogeneidad de los regresores con respecto a efectos individuales conduce a parámetros estimados no fiables, mientras que los resultados obtenidos mediante el planteamiento Mundlak y la especificación de efectos fijos son similares (en lo concerniente a variables que varan en el tiempo), sugiriendo que la correlación entre regresores y efectos individuales se ha capturado satisfactoriamente

Keywords:

• spatial autocorrelation
• panel data model
• random effects
• Mundlak approach
• house price model

JEL CLASSIFICATION:

• C12
• C21
• C23
• C52

Acknowledgements

Nicolas Debarsy is Fellow researcher of the F.R.S.-FNRS and gratefully acknowledges their financial support. The author would like to thank the editor and two anonymous referees for helpful comments and suggestions. This paper has received the award for the best young researcher spatial econometrics paper by the Spatial Econometric Association, Toulouse 2011.

Notes

1. Interactive heterogeneity should not be confused with what literature labels spatial heterogeneity and which refers to standard individual heterogeneity coming from spatial structural instability in coefficients or residual variance.

2. The weight matrix W and all variables should be indexed by N since they form triangular arrays. However, to keep some clarity in the notations, we omitted it.

3. In the context of cross-sectional models, several papers have also shown that the SDM was theoretically grounded. Among others, Ertur & Koch (2007, 2008) revisited growth theory between countries and Pfaffermayr (2009) studied the convergence process among European regions.

4. As a reminder, the random effects specification assumes independence between individual effects and the regressors while the fixed effects approach does not impose such an hypothesis. The Hausman statistic assesses the relevance of this assumption of independence by comparing the estimated parameters of two specifications.

5. See Kelejian & Prucha (2010, p. 55) for a discussion on the parameter space for the spatial autoregressive parameter.

6. Belgium counts 589 municipalities but we were obliged to disregard Herstappe since this municipality is so small that we could not get house price data due to confidentiality issues.

7. The dataset comes from the Belgian Directorate-general Statistics and Economic information and all relevant data are expressed in constant 2004 Euro prices.

8. For further details, the interested reader may consult Denil et al. (2004).

9. Relaxing the normality assumption is possible as long as we estimate the model with quasi-maximum likelihood.

10. Other interaction matrices considered are a 10-nearest neighbours and a distance based with a threshold, for different values of the threshold but gave all similar results.

11. LeSage & Pace (2009, chap. 2) present a comprehensive analysis of those effects along with some useful summary measures in the cross-section setting. Their extension to our panel data setting is straightforward.

12. Let us note that all except the time mean of the surf variable are individually significant at the 1% level.

13. The interested reader may consult Lesage (1997) and LeSage & Pace (2009, chap. 5) for further details.