A Principal Components Approach to Cross-Section Dependence in Panels

are averages of RI disturbances, wt = _i aiuit where uit = γη_it + εit . Hence,
there is correlation between w_t and RII disturbances which also contain ε_it .
However, the weight of ε_it in the former (a_i) falls as N increases and so
does the endogeneity bias. Second, since corr(x_it, Xj_t) = σ2∕(σ2 + σ²_d), if we
let σ_d² → 0 then the regressors are identical across equations. In this case
plim_T,N→∞ b = β + γδ and there are no gains from RII over RI. Third, al-
though the foregoing theoretical results (large T, large N consistency of b) are
derived for a homogeneous panel with identical β and γ across units, we con-
jecture that similar results apply to heterogenous panels. This is investigated
via Monte Carlo simulations in §4.

The variance of the POLS estimator of a =[bc]⁰ in RII is

var(a) ≡ E [(^ - α)(^ - a)⁰] = (X⁰X)^-1E (X⁰vv⁰X) (X⁰X)^-1      (15)

where X is an NT × 2 matrix. As shown above ρ_zw converges in probability
to 1 as T →∞and N →∞. However, to the extent that w_t is not a perfect
measure of the unobserved z_t in finite samples, there will be cross-equation
dependence in the disturbances, E(vi_t,vj_t) 6=0, and between disturbances
and regressors E(vit,xjt) 6=0for i 6= j.HenceE(X⁰vv⁰X) 6= X⁰ E (vv⁰)X.
This suggests estimating var(a) using an analogous formula to (13).

3.3 Serial dependence and heteroskedasticity

The assumption that dit and zt are iid processes is relaxed by letting

dit = Pddi,t-1 + εd,it, εd,it ~ iid(0, σd)
zt = pzzt—i + εz,t, εz,t ~ iid(0, σ2)

with -1 < ρ_d, ρ_z < 1. This introduces the serial correlation typical of eco-
nomic variables in the disturbances of RI (and RII). In particular, the matrix
E (X⁰ vv⁰X) in (15) will have also nonzero elements E (x_itv_itxj_svj_s) for t 6= s.
By letting ε_d,_it ~ iid(0, σ_di) the heterogeneous variance of the idiosyncratic
influence in xit introduces groupwise heteroskedasticity in RI and RII errors.
In particular, for RI we have

E(ui²t) = γ²(1 - δ)²σz² + γ²δ²σ²d,i

By rewriting (6) as yit = βxit + γ_iwt + γ_i(zt - wt) + εit it follows that vit =
γ_i (zt - wt) + εit in RII. Hence to the extent that wt 6= zt , letting γ_i 6= γ_j will
also introduce heteroskedasticity in RII residuals.

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