Cross sectional dependence for firm panel data

[Harsh Tuli]

Do we conduct a cross sectional dependence test for panel data with N being listed firms and T being financial years. Is it only conducted when the data has geographies involved? Please share a reference for either answer.

[Maxence Morlet]

Professor Wooldridge provided a very informative answer on your other post, I would follow his advice.

[Harsh Tuli]

Ya maxence doing that thanks. Had put this post between the time of replies.

[Maxence Morlet]

That being said, you should give this paper a read, it is very interesting:

https://econpapers.repec.org/article..._3a281-312.htm

[Harsh Tuli]

I had actually used this paper as my reference point as the example used in this were stocks....and thus do we think of cross sectional dependence in stocks and not firms.... thus the confusion

[JanDitzen]

The definition of cross-section dependence depends on the type of your data and your model. There are several things to carefully consider:

In case of N large, T fixed, cross-section dependence is mainly seen in a "weak" form. That means the coefficient estimates are asymptotically unbiased and dependence remains in the residuals. Therefore a CSD robust variance estimator is required. An example could be the SHAC estimator the the Driscoll and Kraay standard errors in xtscc.

The case of large N and large T is a bit more complicated. It further depends on the structure of the error components and the assumptions on the coefficients. Lets assume the following model:

y(it) = a(i) + b(i) x(it) + u(it)

with u(it) = gamma_u(i)*f(t) + err(it) and x(it) = gamma_x(i)*f(t) + xi(it), where err(it) and xi(it) are both IIDN and independent from each other. f(t) is a common factor and gamma_u(i) and gamma_x(i) its loading. The common factor introduces strong cross-sectional dependence. If x(it) and u(it) are correlated via f(t), that is gamma_u != 0 and gamma_x !=0, then not accounting for the common factors introduces an omitted variable bias. Therefore ignoring this term would lead to a biased estimate of b(i) [respectively of b_mg or b_p, the pooled estimator] and the residuals would be strongly cross-section dependent. Using CSD robust variance estimators does not help in this case. We need to approximate the factor structure. If we account for it by, for example approximating using common correlated effects (CCE, Pesaran (2006)) or principal components (Bai, Ng, 2002), we obtain an unbiased estimate for b(i) and the residuals should be free of any CSD. This means we can use the variance estimators of the CCE-MG or CCE-Pooled estimators.

If we have a common factor in both x(it) and u(it), but say the model is:

x(it) = gamma_x(i) * f1(t) + xi(it) and u(it) = gamma_u(i) * f2(t) + err(it) and f1(t) and f2(t) are uncorrelated, then the estimator depends on the assumptions on the coefficients. If we assume heterogeneous coefficients with b(i) = b + vu(i), where vu(i) is N(0,sigma^2), then we can still use a MG estimator. However we do not need to account for cross-section dependence to obtain unbiased estimated. Since the variance estimator of the MG estimator relies on the variance of b(i) [i.e. it is essentially sigma^2], we can even do inference on the coefficients. However the residuals still may contain strong dependence. Thus any inference (or analysis) on the residuals is invalid.
If we have a homogeneous slope model, the variance estimator depends on the residuals and therefore we need to make sure the residuals are free of cross-section dependence. This mean we use the CCE-Pooled estimator and use as cross-section averages y and x. Programs such as xtdcce2 offer several variance estimators which can account for autocorrelation and heterosekdasticity. How do you make sure that your residuals are free of CSD? You do the test for cross-section dependence on the residuals or estimate the exponent of CSD.

In a nutshell: your dataset with firms being observed over time is likely to be large N, large T. Thus I would advice to test for CSD in the variables, compare a model with and without cross-section averages using the MG or pooled estimator and make sure that the remaining residuals are free of strong CSD.

[Harsh Tuli]

Thanks for sharing your views my N is 157 companies and T is 15years....I don't think both N and T are large in this case and thus have been advised to use xtreg, cluster for the same since data hetrogenous and autocorrelated.

[JanDitzen]

Agreed, in your case I would strongly recommend xtreg with CSD robust standard errors. Given the small T of your model, I think determining strong CSD using the CD test will lead to incorrect results.