A Principal Components Approach to Cross-Section Dependence in Panels

The mean slope estimates for RI reported in Table 1(A-B) are in line with
the theoretical bias at 0.5 for the baseline DGP. Those for RII reveal that
the proposed approach succeeds in reducing bias. As expected for RI, SE₁
underestimates the true s.e. because of neglected non-zero cov(U_it,Uj_t) and
cov(u_it, Xj_t) caused by z_t.8 The SE₂ underestimate the true s.e. also because
of failing to account for cov(U_it, x_jt) = 0. By contrast, the SM(SE₃) matches
the SSD(θ) quite well. However, this is not the case for RII where there is
still some underestimation in SE3 particularly for the annual panel. The
reasons underlying this bias warrant further investigation.

If our baseline DGP is modified to have a common global influence or-
thogonal to the regressors by letting xit = dit + λi z2t and yit = β_ixit +
γ_iz_2t + ε_it where cov(z_2t, z_1t) = 0 ceteris paribus, it follows that cov(U_it,x_jt) =
cov(^_it, x_jt) = 0. Unsurprisingly, simulations show that SM(SE₂) and SM(SE₃)
are reasonably close for both RI and RII. Again these estimators match the
true s.e. for RI (and SE₁ underestimates it) and they are biased downwards
for RII. For a simpler DGP where λ_i =0 ceteris paribus, SE₁ ' SE₂ '
SE3 are correct in RI. The latter can be explained by the fact that, al-
though cov(U_it,Uj_t) = 0 the second term in (11) and (12) vanishes because
cov(x_it, xj_t) = 0. This result (correct SE₁) also emerges in RII.⁹ These exper-
iments suggest that there is an additional effect in RII (over RI) not captured
by any of the covariance matrices considered and which becomes apparent
when cov(xit , xjt) 6=0.

When the assumptions of groupwise homoskedastic and non autocorre-
lated errors are relaxed, the proposed approach continues to reduce bias
substantially as Table 2(A-B) shows. Unsurprisingly, for this DGP none
of the available covariance matrices provides accurate estimates of the true
standard errors, not even for RI. Formula (13) fails to account for the autocor-
relation pattern while the Newey-West covariance estimator for panels does
not account for the cross-equation correlations cov(u_it, Uj_t) and cov(u_it, Xj_t).¹⁰
Finally the baseline DGP is modified to introduce slope coefficient hetero-

⁸ The biased SE1 for the MG estimator may stem from non-zero covariances between
the coefficient estimates for each group driven by a common bias term δγ.

⁹ In the case where the unobservable global variables are uncorrelated with the regressors
our approach offers efficiency gains — borne out by SSD(b) ≪SSD(θ) — like SURE-GLS.
However, our approach has the additional advantage that no restriction is imposed on the
relation between N and T .

¹⁰The s.e. from (13) and Newey-West (L=2) are .0561(.0277) and .0408(.0266) for FE
and POLS, respectively, for RI and .0285(.0215) and .0197(.0148) for RII (annual panel).

13

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