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 modification of (5) can circumvent the
endogeneity problem. For each group i, the T × (N - 1) residual matrix
u_i = [u₁,..., u_i-1, u_i+₁, ...,uN] which excludes group i residuals is used to ex-
tract the first J factors, w_τt, and y_it = x⁰_itb_i+w'°_tc_i +v_it 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 x_it although
not necessarily to Xj_t for j = i — this suggests that the smaller the cross-
section correlations among regressors the closer the factors WJ will be to Z
and, hence, the more gains are expected from our approach in terms of bias
reduction. Conversely, if x_1t = x_2t = ... = x_Nt then the inclusion of W_J will
not improve the properties of the estimator b_i in RII (over θ_i in RI) which
will be still biased and inconsistent. This is similar to the situation where
for identical regressors there are no efficiency gains from SURE-GLS over
equation-by-equation OLS.

3 Analytical results

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

x_it  = d_it + z_t

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 z_t ~ iid(0, σZ), respectively, which are orthogonal
to each other and also to ε_it.

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 wt in (5).

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