A Principal Components Approach to Cross-Section Dependence in Panels

Similarly, (2) can be written for the N groups as y_t⁰ = x⁰_tβ + z⁰_tΓ + ε⁰_t where
Γ is a J × N matrix. Stacking the K × J matrices D_i for the N groups gives
the NK × J matrix D = [D⁰₁ D⁰₂ ∙ ∙ ∙ DN]⁰ and z'_t = xtD + η0_it. 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 u⁰_t = (z⁰_t - x⁰_tD)Γ + ε⁰_t 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 (u^l_iu_i)^-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 = W_J A⁰_J + E. The N × J matrix A_J (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 = x⁰_itbi + w_t⁰ 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 W_J
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-

<<   2   3   4   5   6   7   8   >>

More intriguing information