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  • What is the difference between Moran I and Pesaran and free's cross sectional dependence tests?

    Dear readers,
    I have a question about spatial econometrics:
    I run a Moran I test on my residuals from OLS estimation and I didn't find evidence of any spatial correlation. However, when I run Peseran and Free's cross sectional dependence tests I find that there is a spatial or say cross sectional dependence.
    The Moran I is only for cross sectional data however the Peseran and Free's test work on panel data.
    I am looking what are the differences between Moran I and Peseran and free's test beyond the fact that they work on different set of data.
    Finally, is there any Moran I for panel data...
    And Also, is make it any sense to find cross sectional dependence using Pesaran and Free's test but not finding spatial correlation using Moran I ?
    My data is 30 countries from 1995 to 2009. For the Moran I, I perform the test for each year and also using average over all the period.
    Finding that there is cross sectional dependence I rely on a Driskoll Kraay estimator..Without going on pur spatial econometrics.
    Thank you in advance for any help..

  • #2
    Hi Emma,
    a good reading which compares the two tests is De Hoys, Sarafids (2006, The Stata Journal; https://www.stata-journal.com/sjpdf....iclenum=st0113). They explain the difference between the Pesaran (2004/2015) test and the Free's test.

    Out of my head, the difference between the Moran's I and the two other tests is that Moran's I assumes a known spatial weights matrix. Pesaran and Free's test assume that the cross-sectional dependence is driven by unknown common factors, which depending on their strength introduce cross-sectional dependence. Free's test is a bit more restrictive to the type of panel, i.e. it only works for static panels. Pesaran's test works for dynamic panels as well.

    Important to note is that the Pesaran test in its published version (2015, Econometric Reviews; https://www.tandfonline.com/doi/abs/...38.2014.956623) tests for weak cross-sectional dependence. The alternative is strong cross-sectional dependence. Weak cross-sectional means that the dependence disappears or remains constant if further cross-sectional units are added. Therefore it can be interpreted as a type of spatial dependence (the weights of the weights matrix need to be bounded, which is the same assumption for weak cross-sectional dependence) and taken out by spatial interactions.
    The reason not to test for cross-sectional independence is, that the Pesaran (2015) test is designed for large panels, i.e. N and T are going to infinity. Therefore it is by construction likely that weak dependence occurs.

    Finally, I am not sure if you can run the Moran I over all periods. Implicitly you are assuming that all time periods are independent of each other, which might be rather restrictive.

    Hope this helps!

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    • #3
      Dear Janditzen,
      First thanks a lot for your answers. About the difference between Moran I and Pesaran test, I have some doubt about your explanation. If I am not wrong, the Moran I test runs on a model without weighted matrix (directly on OLS estimation without any matrix normally?). In a second step, if Moran I confirms spatial dependence therefore we correct for it using a weighted matrix.

      Finally, I try to understand why the Moran I confirms that there is no spatial correlation while the Peseran does.. Any idea on that? Are both testing the same thing? I mean if they test both for the presence of spatial dependence why they did not give the same results? Or they test for different things, if it is the case which of them.

      Emna

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      • #4
        Emma, which command do you use to calculate the Moran I? If you use estat moran after regress, you need to specify a spatial weights matrix - at least following the Stata handbook and the examples. It is true, that residuals are obtained from a OLS regression without any spatial dependence, but for the test itself you need to use a spatial weights matrix (which of course can be all zero). In a sense you are testing if this spatial weights matrix represents the spatial dependence in your variable or residuals. If so, then the residuals are uncorrelated, which is your null hypothesis.

        My take on why the two tests are different is that one is for cross-sectional data with a known weights matrix, the other one for panel data without known weights matrix. Finally, the CD test will suffer from the small sample size. It is a test designed for large N and large T data.

        Do you have any code examples?

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