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 uΓ0 = 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, WJ , 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-