A Principal Components Approach to Cross-Section Dependence in Panels



als.3 However, our procedure is distinctive to the extent that it includes
the own residual. This results in endogeneity bias which falls with N as
shown in §3.2. For small N a slight modi
fication of (5) can circumvent the
endogeneity problem. For each group i, the T
× (N - 1) residual matrix
ui = [u1,..., ui-1, ui+1, ...,uN] which excludes group i residuals is used to ex-
tract the
first J factors, wτt, and yit = x0itbi+w'°tci +vit is estimated.4 Second,
since our proxies for the common shocks are calculated as linear combina-
tions of OLS residuals — by construction U
it is orthogonal to xit although
not necessarily to
Xjt for j = i — this suggests that the smaller the cross-
section correlations among regressors the closer the factors W
J will be to Z
and, hence, the more gains are expected from our approach in terms of bias
reduction. Conversely, if
x1t = x2t = ... = xNt then the inclusion of WJ will
not improve the properties of the estimator
bi in RII (over θi in RI) which
will be still biased and inconsistent. This is similar to the situation where
for identical regressors there are no e
fficiency gains from SURE-GLS over
equation-by-equation OLS.

3 Analytical results

Consider a simple data generating process (DGP) comprising a country spe-
ci
fic regression, say a PPP equation, in which a global variable zt , such as
oil prices, is omitted but where it also in
fluences the country specific regres-
sors x
it. For instance, oil prices could influence inflation differentials because
country-speci
fic inflation differs in its response to oil prices depending on
whether the country imports or exports oil. Suppose data are generated by

xit  = dit + zt

(6)


yit  = βxit + γzt + εit, εit ~ iid(0, σ2)

where the innovations εit are uncorrelated across countries. We assume that
each regressor has an
idiosyncratic (or country-specific) and common influ-
ence, d
it ~ iid(0, σd) and zt ~ iid(0, σZ), respectively, which are orthogonal
to each other and also to
εit.

3 Telser (1964) suggested an iterative approach to account for the cross-equation resid-
ual correlation which converges to an estimator with the same asymptotic properties as
Zellner’s SURE-GLS estimator. This consists of including as additional variables in each
equation the OLS residuals of all other equations.

4 An alternative approach to abate the small N endogeneity bias would be to use the
IV method, that is, to instrument the factors w
t in (5).



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