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I want to test whether cross-sectional dependence in one specific variable (y) in panel data format follows a known structure (W) (e.g. network, spatial dependence), after controlling for individual fixed effects and time fixed effects. The aim is simply to show that this structure is describing cross-sectional dependence in a statistically significant way.

My first idea was to look into the spatial econometrics literature. Here, W would be the spatial weight matrix. For purely cross-sectional data, Moran's I would probably be the relevant test but I am unsure how and whether this measure can be applied to panel data.

What would be the correct way of testing a known cross-sectional dependence structure on a panel of one variable, after controlling for individual fixed effects and time fixed effects? Any guidance and help would be very much appreciated! Thank you very much in advance.

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Maybe you a looking for the local variant of Pesaran's CD (cross-sectional dependence) test?

Have a look at the literature cited in the R package plm for function pcdtest (pcdtest has an argument W to define a spatial property), e.g., on the web here: http://rdocumentation.org/packages/plm/versions/2.4-1/topics/pcdtest:

Baltagi BH, Feng Q, Kao C (2012). “A Lagrange Multiplier test for cross-sectional dependence in a fixed effects panel data model.” Journal of Econometrics, 170(1), 164 - 177. ISSN 0304-4076, https://www.sciencedirect.com/science/article/pii/S030440761200098X.

Breusch TS, Pagan AR (1980). “The Lagrange Multiplier Test and Its Applications to Model Specification in Econometrics.” Review of Economic Studies, 47, 239–253.

Pesaran MH (2004). “General Diagnostic Tests for Cross Section Dependence in Panels.” CESifo Working Paper Series, 1229.

Pesaran MH (2015). “Testing Weak Cross-Sectional Dependence in Large Panels.” Econometric Reviews, 34(6-10), 1089-1117. doi: 10.1080/07474938.2014.956623, https://doi.org/10.1080/07474938.2014.956623.

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