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 yit. Suppose the data are generated by
yit = x0itβi + z0tγi + εit (2)
where the idiosyncratic error εit is zero-mean white noise distributed inde-
pendently over units and zt is a J -vector of unobserved common shocks or
random variables which may be correlated with each country regressor. The
auxiliary regression z0t = x0itDi + η0it where Di is a K × J matrix, decomposes
z0t into two terms, one which is correlated with the ith country regressors
and another which is orthogonal to them. Substituting for z0t in (2) gives
yit = x0it(βi + Diγi) + η0itγi + εit which implies that the omitted z0t creates
two problems in (1). First uit measures uit = η0itγi + εit = z'tγi — x0itDiγi + εit
which contains η0it , the part of z0t orthogonal to x0it .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 θi inefficient and its standard
error biased, even if Di is the null matrix. Second, uit is correlated with the
included explanatory variables unless Di =0and this makes θi , and also the
GLS estimator, biased for βi (since E(θi) = βi + Diγi) and inconsistent.
2.1 Principal components approach
Equation (1) for group i can be written for the N groups as
yt0 = x0tΘ + u0t (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 θ2 . ,
... ... 0
0 ∙ ∙ ∙ 0 Θn