The PANEL Procedure

Panel Data Cross-Sectional Dependence Test

Breusch-Pagan LM Test

Breusch and Pagan (1980) propose a Lagrange multiplier (LM) statistic to test the null hypothesis of zero cross-sectional error correlations. Let $e_{it}$ be the OLS estimate of the error term $u_{it}$ under the null hypothesis. Then the pairwise cross-sectional correlations can be estimated by the sample counterparts $\hat{\rho }_{ij}$,

\begin{equation*}  \hat{\rho }_{ij}=\hat{\rho }_{ji}=\frac{\sum _{t=\underline{T}_{ij}}^{\overline{T}_{ij}}e_{it}e_{jt}}{\sqrt {\sum _{t=\underline{T}_{ij}}^{\overline{T}_{ij}}e_{it}^{2}}\sqrt {\sum _{t=\underline{T}_{ij}}^{\overline{T}_{ij}}e_{jt}^{2}}} \end{equation*}

where $\underline{T}_{ij}$ and $\overline{T}_{ij}$ 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, $\underline{T}_{ij}=1$ and $\overline{T}_{ij}=T$. Let $T_{ij}$ denote the number of overlapped time periods ($T_{ij}=\overline{T}_{ij}-\underline{T}_{ij}+1$). Then the Breusch-Pagan LM test statistic can be constructed as

\begin{equation*}  \mr{BP}=\sum _{i=1}^{N}\sum _{j=i+1}^{N}T_{ij}\hat{\rho }_{ij}^{2} \end{equation*}

When N is fixed and $T_{ij}\to \infty $, $\mr{BP}\rightarrow \chi ^{2}\left(N\left(N-1\right)/2\right)$. So the test is not applicable as $N\to \infty $.

Because $\hat{\rho }_{ij}^{2},i=1,\ldots ,N-1,j=i+1,\ldots ,N$, are asymptotically independent under the null hypothesis of zero cross-sectional correlation, $T_{ij}\hat{\rho }_{ij}^{2}\rightarrow \chi ^{2}\left(1\right)$. Then the following modified Breusch-Pagan LM statistic can be considered to test for cross-sectional dependence:

\begin{equation*}  \mr{BP}s=\sqrt {\frac{1}{N\left(N-1\right)}}\sum _{i=1}^{N}\sum _{j=i+1}^{N}\left(T_{ij}\hat{\rho }_{ij}^{2}-1\right) \end{equation*}

Under the null hypothesis, $\mr{BPs}\rightarrow \mathcal{N}\left(0,1\right)$ as $T_{ij}\to \infty $, and then $N\to \infty $. But because $E\left(T_{ij}\hat{\rho }_{ij}^{2}-1\right)$ is not correctly centered at zero for finite $T_{ij}$, the test is likely to exhibit substantial size distortion for large N and small $T_{ij}$.

Pesaran CD and CDp Test

Pesaran (2004) proposes a cross-sectional dependence test that is also based on the pairwise correlation coefficients $\hat{\rho }_{ij}$,

\begin{equation*}  \mr{CD}=\sqrt {\frac{2}{N\left(N-1\right)}}\sum _{i=1}^{N}\sum _{j=i+1}^{N}\sqrt {T_{ij}}\hat{\rho }_{ij} \end{equation*}

The test statistic has a zero mean for fixed N and $T_{ij}$ under a wide class of panel data models, including stationary or unit root heterogeneous dynamic models that are subject to multiple breaks. For each $i\neq j$, as $T_{ij}\to \infty $, $\sqrt {T_{ij}}\hat{\rho }_{ij}\Longrightarrow \mathcal{N}\left(0,1\right)$. Therefore, for N and $T_{ij}$ tending to infinity in any order, $\mr{CD}\Longrightarrow \mathcal{N}\left(0,1\right)$.

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, $\mb{W}=\left(w_{ij}\right)$. 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,

\begin{equation*} \begin{array}{l l l} \mr{CD}p &  = &  \sqrt {\frac{2}{p\left(2N-p-1\right)}}\left(\sum _{s=1}^{p}\sum _{i=s+1}^{N}\sqrt {T_{i,i-s}}\hat{\rho }_{i,i-s}\right)\\ &  = &  \sqrt {\frac{2}{p\left(2N-p-1\right)}}\left(\sum _{s=1}^{p}\sum _{i=1}^{N-s}\sqrt {T_{i,i+s}}\hat{\rho }_{i,i+s}\right) \end{array}\end{equation*}

where $p=1,\ldots ,N-1$. When $p=N-1$, 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 $T_{i,i-s}>k+1$, and CD$p\Longrightarrow \mathcal{N}\left(0,1\right)$ as $N\to \infty $ and $T_{i,i+s}\to \infty $.