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This paper proposes a bias‐adjusted version of Lagrange multiplier (LM) test statistic of error cross‐section independence, in the case of panel models with strictly exogenous regressors and normal errors. The exact mean and variance of the test indicator of the LM test statistic are provided for the purpose of the bias‐adjustments. It is shown that the centring of the LM statistic is correct for fixed T and N. Importantly, the proposed bias‐adjusted LM test is consistent even when the CD test is inconsistent. Also an alternative bias‐adjusted LM test, which is consistent under local error cross‐section dependence of any fixed order p, is proposed. The finite sample behaviour of the proposed tests is investigated and compared to that of the LM and CD tests. It is shown that the bias‐adjusted LM tests successfully control the size, maintaining satisfactory power in panel with exogenous regressors and normal errors …