Breusch and Pagan (1980) propose a Lagrange multiplier (LM) statistic to test the null hypothesis of zero cross-sectional error correlations. Let be the OLS estimate of the error term under the null hypothesis. Then the pairwise cross-sectional correlations can be estimated by the sample counterparts ,
where and are the lower bound and upper bound, respectively, which mark the overlap time periods for the cross sections i and j. If the panel is balanced, and . Let denote the number of overlapped time periods (). Then the Breusch-Pagan LM test statistic can be constructed as
When N is fixed and , . So the test is not applicable as .
Because , are asymptotically independent under the null hypothesis of zero cross-sectional correlation, . Then the following modified Breusch-Pagan LM statistic can be considered to test for cross-sectional dependence:
Under the null hypothesis, as , and then . But because is not correctly centered at zero for finite , the test is likely to exhibit substantial size distortion for large N and small .
Pesaran (2004) proposes a cross-sectional dependence test that is also based on the pairwise correlation coefficients ,
The test statistic has a zero mean for fixed N and under a wide class of panel data models, including stationary or unit root heterogeneous dynamic models that are subject to multiple breaks. For each , as , . Therefore, for N and tending to infinity in any order, .
To enhance the power against the alternative hypothesis of local dependence, Pesaran (2004) proposes the CDp test. Local dependence is defined with respect to a weight matrix, . Therefore, the test can be applied only if the cross-sectional units can be given an ordering that remains immutable over time. Under the alternative hypothesis of a pth-order local dependence, the CD statistic can be generalized to a local CD test, CDp,
where . When , CDp reduces to the original CD test. Under the null hypothesis of zero cross-sectional dependence, the CDp statistic is centered at zero for fixed N and , and CD as and .