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Spatial Panel Data Models

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Spatial Econometrics

Part of the book series: SpringerBriefs in Regional Science ((BRIEFSREGION))

Abstract

This chapter provides a survey of the specification and estimation of spatial panel data models. Five panel data models commonly used in applied research are considered: the fixed effects model, the random effects model, the fixed coefficients model, the random coefficients model, and the multilevel model. Today a (spatial) econometric researcher has the choice of many models. First, he should ask himself whether or not, and, if so, which type of spatial interaction effects should be accounted for. Second, he should ask himself whether or not spatial-specific and/or time-specific effects should be accounted for and, if so, whether they should be treated as fixed or as random effects. A selection framework is demonstrated to determine which of the first two types of spatial panel data models considered in this chapter best describes the data. The well-known Baltagi and Li (2004) panel dataset, explaining cigarette demand for 46 US states over the period 1963 to 1992, is used to illustrate this framework in an empirical setting.

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Notes

  1. 1.

    Baltagi et al. (2003) are the first to consider the testing of spatial interaction effects in a spatial panel data model. They derive a joint LM test which tests for spatial error autocorrelation and spatial random effects, as well as two conditional tests which test for one of these extensions assuming the presence of the other.

  2. 2.

    \( \phi = 1 \) implies \( \sigma_{\mu}^{2} = 0 \), since \( \sigma_{\mu}^{2} \) may be calculated from \( \phi \) by \( \sigma_{\mu}^{2} = \frac{{1 - \phi^{2}}}{{\phi^{2}}}\frac{{\sigma^{2}}}{T} \).

  3. 3.

    Note that \( \varphi = {{\sigma_{\mu}^{2}} \mathord{\left/{\vphantom {{\sigma_{\mu}^{2}} {\sigma^{2}}}} \right. \kern-0pt} {\sigma^{2}}} \) is different from \( \phi \) in the non-spatial random effects model and in the random effects spatial lag model.

  4. 4.

    Note that the matrix Z 0 in Baltagi et al. (2007, pp. 39–40) has been replaced by \( \varvec{Z}_{0} = \left({T\sigma_{\mu}^{2} \varvec{I}_{N} + \sigma^{2} \left({\varvec{B}^{'} \varvec{B}} \right)^{- 1}} \right)^{- 1} =\, \frac{1}{{\sigma^{2}}}\left({T\varphi \varvec{I}_{N} + (\varvec{B}^{'} \varvec{B})^{- 1}} \right)^{- 1} =\, \frac{1}{{\sigma^{2}}}\varvec{V}^{- 1} \).

  5. 5.

    This remark through Balestra and Nerlove is striking especially since they are the devisers of the random effects model (Balestra and Nerlove 1966).

  6. 6.

    Mutl and Pfaffermayr (2011) derive the Hausman test when the fixed and random effects models are estimated by 2SLS instead of ML.

  7. 7.

    Software programs, such as Spacestat and Geoda, have built-in routines that automatically report the results of these tests. Matlab routines have been made available at http://oak.cats.ohiou.edu/~lacombe/research.html by Donald Lacombe and at www.regroningen.nl by Paul Elhorst.

  8. 8.

    See the routine “sar” posted at LeSage's website <www.spatial-econometrics.com>.

  9. 9.

    Note that the test results satisfy the condition that LM spatial lag + robust LM spatial error = LM spatial error + robust LM spatial lag (Anselin et al. 1996).

  10. 10.

    These tests are based on the log-likelihood function values of the different models. Table 3.2 shows that these values are positive, even though the log-likelihood functions only contain terms with a minus sign. However, since σ2 < 1, we have –log(σ2) > 0. Furthermore, since this positive term dominates the negative terms in the log-likelihood function, we eventually have LogL > 0.

  11. 11.

    One application of this model in the literature is of Sampson et al. (1999), but this paper does not describe the estimation procedure in detail.

  12. 12.

    Bowden and Turkington start from the regression equation \( \varvec{Y} = \varvec{X\beta} +\varvec{\mu} \), where \( {\text{E}}\left({\varvec{\mu \mu}^{\text{T}}} \right) =\varvec{\Upomega} \), and some of the X variables are endogenous. Let Z denote the set of instrumental variables. Then, the GLS analog instrumental variables estimator is \( \varvec{b} = \left({\varvec{X}^{pT}\varvec{\Upomega}^{- 1} \varvec{X}^{P}} \right)^{- 1} \varvec{X}^{pT}\varvec{\Upomega}^{- 1} \varvec{Y} \), where \( \varvec{X}^{p} = \varvec{Z}(\varvec{Z}^{T}\varvec{\Upomega}^{- 1} \varvec{Z})^{- 1} \varvec{Z}^{T}\varvec{\Upomega}^{- 1} \varvec{X} \).

  13. 13.

    Applications based on the multilevel approach in regional economic research are Jones (1991), Ward and Dale (1992), Gould and Fieldhouse (1997), McCall (1998), Elhorst and Zeilstra (2007), Chasco and Lopez (2009), and Zeilstra and Elhorst (2012).

  14. 14.

    The linear expenditure system in its basic form is linear in the variables but nonlinear in the parameters. However, Barnum and Squire (1979) have shown that it can be rewritten in such a way that linear estimation techniques can still be used to estimate the parameters. Since the linear expenditure system extended to include interaction effects is also nonlinear in its variables, linear estimation techniques can no longer be used. The same applies to the techniques spelled out in Anselin (1988), which are partially linear.

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    J. Paul Elhorst

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Elhorst, J.P. (2014). Spatial Panel Data Models. In: Spatial Econometrics. SpringerBriefs in Regional Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40340-8_3

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