A Principal Components Approach to Cross-Section Dependence in Panels



Similarly, (2) can be written for the N groups as yt0 = x0tβ + z0tΓ + ε0t where
Γ is a J × N matrix. Stacking the K × J matrices Di for the N groups gives
the NK
× J matrix D = [D01 D02 ∙ ∙ ∙ DN]0 and z't = xtD + η0it. We have

yt0 =x0t[β+DΓ]+(z0t-x0tD)Γ+ε0t

and it follows that if we estimate (3) by OLS, then E(Θ) = Θ = β + DΓ.
The residuals measure
u0t = (z0t - x0tD)Γ + ε0t which can be written as u =
ZΓ-XDΓ+ε for the T time periods, where Z is T ×J, X is T ×NKand ε and
u are T × N matrices. Post-multiplying the latter by U = diag (uliui)-1/2 gives
the T
× N matrix of standardized errors u = ZΓ XDΓ + ε where Γ = ΓU
and ε = εU. Since Z is unobserved, one needs to impose J2 normalising
restrictions on
Γ to provide estimates of Z. For this purpose assume ΓΓ0 = I.
With this normalisation one can write u
Γ0 = Z XD + εΓ0 or, equivalently

z = uΓ0 + XD — εΓ0                     (4)

This suggests measuring Z by W = uA, the N principal components of
u obtained via the spectral decomposition of R
= u0u = AΛA0, where A
is the orthogonal matrix of eigenvectors and
Λ is the diagonal eigenvalue
matrix. If a few random factors, W
J , account for most of the disturbances
covariation then the cross-section dependence can be characterized by means
of a factor model u
= WJ A0J + E. The N × J matrix AJ (non-random
factor loadings) contains the J < N eigenvectors associated with the largest
eigenvalues and E is a T
× N idiosyncratic error matrix. This suggests an
augmented regression (RII) for handling the cross-section dependence that
biases the estimators of the regression of interest (RI). This is

yit = x0itbi + wt0 ci + vit                            (5)

where wt is a J vector of principal components from the RI errors.

One issue is how to determine J and, relatedly, how well the factors WJ
proxy the unobserved variables Z. Another issue regards interpreting the
factors because the identifying assumptions
ΓΓ0 = I need not be meaningful
from an economic viewpoint. However, for a reasonable small J it may be
possible to give them an economic or
financial interpretation.

Our approach has some commonalities with SURE-GLS. First, it re-
sembles the latter in that (5) includes linear combinations of OLS residu-



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