The use of artificial regressions to compute the variance of the difference of pairs of panel data estimators that cannot be ranked in terms of efficiency is considered. It is illustrated how it is possible to get (asymtotically) valid estimators of covariance matrices for differences between estimators when the assumption that the error term in the auxiliary model is IID is violated. We distinguish two possible deviations, one leading only to a non-spherical-within groups covariance matrix and the second leading to a non-spherical-between-groups covariance matrix also. It is shown to what extent the use of an artificial regression with panel data can lead to a robust estimator of the covariance matrix in the first case whereas it leads to a non valid estimator in the second. An alternative step by step procedure is presented. Keywords; artificial regression models, panel data, covariance matrices estimates, hypothesis testing. JEL Classification: C12, C13, C23
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