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Interpreting spatial regression models with multiplicative interaction explanatory variables

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Abstract

Use of multiplicative interaction of explanatory variables has been a standard practice in the regression modeling literature, and estimation of the parameters of such a model in the case of spatial autoregressive (SAR) or spatial Durbin (SDM) models can be accomplished using existing software for spatial regression estimation. However, use of the conventional scalar summary estimates of direct and indirect effects reflecting the own- and other-region impacts on the dependent variable associated with changes in the explanatory variables will not produce valid inferences. We discuss the issues that arise and introduce new methods for interpretation of own- and other-region impacts based on estimates from this type of model.

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Notes

  1. In the case of spatial probit models, the nonlinear normal probability transformation leads to what have been labeled “marginal effects” whose impact on the probability outcomes vary with the level of the explanatory variable observations. For this reason, the label “observation-level” effects is perhaps a more appropriate label than marginal effects.

  2. We do not include fixed effects for simplicity in the equations presented. These would play no part in our presentation, but should be part of the data generating process for any static panel data model with fixed effects. The usual demeaning transformations would be used during estimation to capture variation over regions and time periods represented by these effects. See Lee and Yu (2010) regarding some issues arising from traditional demeaning transformations.

  3. We note that the partial derivatives for the SAC model from Miguelez and Moreno (2015) take the same form at those for the SAR model specification set forth here. (While LeSage and Pace 2009 use the label SAC, others have used the label SARAR(1,1), for the specification that contains a spatial lag of the dependent variable as well as a spatial autoregressive disturbance process.)

  4. This also means that the matrix \(M=[I_{nT}-\rho (I_T \otimes W)]^{-1}\) would be block diagonal, consisting of \(n \times n\) blocks for each time period. Weight matrices constructed to reflect non-spatial relationships between regions might vary over time periods, leading to different \(S_t\) across time periods, a case we do not consider here.

  5. We note that partial derivatives for the specification that Elhorst (2014) labels GNS (which contains a spatial lag of the dependent variable, a spatial lag of the explanatory variable as well as a spatial autoregressive disturbance process) take the same form at those for the SDM model specification set forth here. In addition, the partial derivatives for the spatial lag of X model (SLX) and spatial Durbin error model (SDEM) model take a similar form of (18) with \(\rho =0, V_t = D_{1,t} + W D_{2,t}\).

  6. Since this is a static panel data model, there is no spillover/cross-derivative impact across time periods. This results from use of the block diagonal matrix \(I_T \otimes W\).

  7. Of course, there are analogous expressions for the derivatives with respect to the \(Z_t\) variable, where \(U_t\) replaces \(V_t\) in these expressions.

  8. State taxes are from the Census Bureau, Annual Survey of State Government Tax Collections, and state capital stocks are from El-Shagi and Yamarik (2019).

  9. Specifically, the MCMC draws for both \(\rho\) and the parameters \(\beta ,\theta ,\gamma\) were used to construct P posterior distributions which could be used to calculate a standard deviation, \(t-\)statistic and corresponding \(t-\)probability for the set of P different effects estimates.

  10. In the table, \(X, p= 0.80Z\) reflects a higher level of capital stock (Z) than \(X, p= 0.20Z\).

  11. In the table, \(Z,p= 0.80X\) reflects a higher level of taxes (X) than \(Z,p= 0.20X\).

  12. A related point is that the very popular SARAR(1,1) (or SAC) specification has the same partial derivatives as the SAR specification.

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Acknowledgements

The first author acknowledges Grant support from the National Natural Science Foundation of China (No. 71803034).

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Authors and Affiliations

  1. Business School of Guangxi University, No.100 Daxue Rd., Nanning, 530004, China

    Yuxue Sheng

  2. The University of Toledo, Toledo, OH, USA

    James Paul LeSage

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  1. Yuxue Sheng
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  2. James Paul LeSage
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Correspondence to James Paul LeSage.

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Sheng, Y., LeSage, J.P. Interpreting spatial regression models with multiplicative interaction explanatory variables. J Geogr Syst 23, 333–360 (2021). https://doi.org/10.1007/s10109-021-00356-4

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