A Principal Components Approach to Cross-Section Dependence in Panels

2 Cross section dependence

Consider the following baseline regression (RI) model

yit =x⁰itθi+uit,i=1,2,...,N,t=1,2,...,T,                (1)

where xit is a K-vector of explanatory variables which would typically in-
clude an intercept and lags of yi_t. Suppose the data are generated by

yit = x⁰_itβi + z⁰_tγi + εit                            (2)

where the idiosyncratic error ε_it is zero-mean white noise distributed inde-
pendently over units and z_t is a J -vector of unobserved common shocks or
random variables which may be correlated with each country regressor. The
auxiliary regression z⁰_t = x⁰_itDi + η⁰_it where Di is a K × J matrix, decomposes
z⁰_t into two terms, one which is correlated with the ith country regressors
and another which is orthogonal to them. Substituting for z⁰_t in (2) gives
y_it = x⁰_it(β_i + D_iγ_i) + η⁰_itγ_i + ε_it which implies that the omitted z⁰_t creates
two problems in (1). First u_it measures u_it = η⁰_itγ_i + ε_it = z'_tγ_i — x⁰_itD_iγ_i + ε_it
which contains η⁰_it , the part of z⁰_t orthogonal to x⁰_it .Thistermmaybese-
rially correlated and will certainly be correlated across units. Both of these
will induce a nondiagonal residual covariance matrix or non-spherical dis-
turbances. This will make the OLS estimator θ_i inefficient and its standard
error biased, even if Di is the null matrix. Second, ui_t is correlated with the
included explanatory variables unless Di =0and this makes θi , and also the
GLS estimator, biased for β_i (since E(θi) = β_i + Diγ_i) and inconsistent.

2.1 Principal components approach

Equation (1) for group i can be written for the N groups as

y_t⁰ = x⁰_tΘ + u⁰_t                               (3)

where yt =[y1t,y2t, ...,yNt]⁰ and xt =[x1t,x2t,...,xNt]⁰ are an N —vector and
an NK —vector, respectively, and Θ is the block-diagonal NK × N matrix

^ θ₁ 0     ∙∙∙ 0 ’

Θ = ^{0   θ}2         .      ,

^._..                              ^.._{. 0}

0    ∙ ∙ ∙ 0     Θn

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