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Abstract
This paper proposes a new generalized method of moments (GMM) estimator for spatial panel models with spatial moving average errors combined with a spatially autoregressive dependent variable. Monte Carlo results are given suggesting that the GMM estimator is consistent. The estimator is applied to English real estate price data.
Abstract
Une méthode généralisée d'estimateur de moments pour un modèle de panel spatial avec un décalage endogène spatial et des erreurs spatiales de type moyenne mobile
Cette étude propose un nouvel estimateur GMM pour des modèles de panel spatial avec des erreurs spatiales de type moyenne mobile combiné à une variable de dépendance spatiale autorégressive. Les résultats de Monte Carlo fournis suggèrent que l'estimateur GMM est cohérent. L'estimateur s'applique à des données sur des prix d'immobilier anglais.
Abstract
Un estimador de método generalizado de momentos para un modelo de panel espacial con retardo espacial endógeno y errores espaciales de media móvil
Este estudio propone un nuevo estimador GMM para modelos de panel espacial con errores espaciales de media móvil combinado con una variable dependiente autorregresiva. Se indican los resultados de Monte Carlo que revelan la coherencia del estimador GMM. El estimador se aplica a los datos de precios en inmobiliarias inglesas.
Acknowledgements
The author is grateful to Michael Pfaffermayr and the other participants at the International Workshop on Spatial Econometrics and Statistics, Rome, 25–27 May 2006, and participants at the 13th International Conference on Panel Data, University of Cambridge, 7–9 July 2006, for their contributions to the discussion of this paper.
Notes
1. An early detailed account of the MA spatial process is given by Haining (Citation1978).
2. Pre-multiplication of a TN×1 vector θ by Q 0 creates a TN×1 vector of deviations from the mean, where the mean is obtained by averaging θ over time. Pre-multiplication of a TN×1 vector θ by Q 1 creates a TN×1 vector, comprising N across time area-specific means stacked for each T.
3. Note that Tr(W′WW)=0 for the Rook's case contiguity matrix.
4. So that we can use equation (Equation49) in both stage 1 and stage 3, it is assumed that and
(so that Ω
ξ
is a diagonal matrix of 1s) and that ρ=0. The result is that at stage 1 we simply obtain IV estimates.
5. Using unconstrained non-linear least squares estimation. The method is a modified Newton–Raphson method which is suitable for minimizing any non-linear function, and which depends on numerical differences rather than derivatives.
6. In contrast, at stage 1, ρ is assumed to equal 0, so that in that case Y * =Y, X * =X.
7. Moore–Penrose generalized inverses are used to avoid singularities.
8. In the main body of the text W is a Rook's case contiguity matrix based on a 15×15 lattice. In the Monte Carlo simulations described in Appendix B, the lattice size is varied and an irregular spatial partitioning is also considered. Also, alternative contiguity definitions, namely the Queen's case and torus, are also implemented.
9. Appendix B gives the results obtained using K=1,000 replications.
10. These procedures were carried out using the DISTRIBUTION directive of the programming language GENSTAT. The DISTRIBUTION directive is used to fit an observed sample of data to a theoretical distribution function, in order to obtain maximum-likelihood estimates of the parameters of the distribution and test the goodness of fit.
11. To save space these results have not been reported here.
12. With the MA error process the C-O transform involves the inverse. See Smirnov & Anselin (Citation2001) for a discussion of the use of the power expansion to approximate the matrix inverse with large matrices.
13. Unitary authority and local authority districts, or UALADs.
14. Appendix A gives details of the sources and construction of these variables.
15. Small administrative areas, with median area equal to 250.77 km2.
16. Available on the NOMIS website (the ONS online labour market statistics database).
17. 1991 Census of Population—Special Workplace Statistics, available from NOMIS.
18. Total employees and self-employed with a workplace coded, tabulated by residents in each zone (10% sample).
19. Minimum of the sum of the squared deviations of the observed proportions in each distance band up to 40 km and the proportions of the sum of the function exp(−δ i d ij ) calculated using the upper limit of each distance band.
