A Principal Components Approach to Cross-Section Dependence in Panels

We conjecture that the unbiasedness and consistency of b, the (POLS) es-
timator of RII slope coefficient, carry over to more complex setups. However,
an appropriate asymptotic var(b) is needed. The latter has to account not
only for contemporaneous dependences — such as cov(^_it, v_j∙_t) and cov(^_it, x_jt)
which arise in finite samples to the extent that wt does not perfectly measure
z_t — but also serial dependence and groupwise heteroskedasticity.

4MonteCarloAnalysis

Monte Carlo simulations are employed to assess the finite sample properties
of the proposed estimation approach. The following DGP is used

dit = Pdidi,t-1 ^{+ ε}di,t; ^εdi,t ~ NID(^{0, σ}d,_i)

Zt = PzZt-1 + εzt; εzt ~ NID(0, σ2)
xi_t = di_t + λiz_t

yit = βiXit + YiZt + εit; εit ~ NID(0, σ2i)

In the first set of simulations, reported in Table 1, it is assumed P_di = P_z =
0, σ_z² = σ²_d,i = σ_ε²_,i =1and λi = β_i = γ_i =1. In a second set, reported
in Table 2, we assume σ2_i ~ U[0.5,1.5] and γ_i ~ U[0.5,1.5] to introduce
groupwise heteroskedasticity and P_di = P_z =0.9 to introduce serial correla-
tion. The panel dimensions N, T = {30, 300} and N, T = {30, 25} which are
typical of monthly and annual PPP studies, respectively, are used. R = 500
replications are employed throughout. We estimate RI and RII — where wt
is the first factor extracted from the equation-by-equation OLS residuals of
RI — using the POLS, FE and MG estimators.⁶

The comparison focuses on the bias of the slope estimator and on the
difference between the true standard errors (s.e.) and various estimates of
them. The former is measured by the sample standard deviation (SSD) of the
slope estimates over replications while the latter is measured by the sample
mean (SM) of the estimated s.e. For comparative purposes, Table 1 reports
three s.e. estimates for the pooled estimators. These are SE₁ based on the
conventional formula s²(X⁰X)^-1 ,SE2 using (10) and SE3 based on (13).⁷

⁶All computations were performed in GAUSS version 3.2.32.

7∏            M M ^mg^a^mg∖

⁷For the MG estimator SE(θ ) = σ(θi)/ N where σ(θi) is the standard deviation
of the individual OLS estimates.

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