A Principal Components Approach to Cross-Section Dependence in Panels

unit. They demonstrate the consistency of these and other criteria and their
simulations show that they are fairly robust provided min{N, T} > 40. One
important difference between our setup and that in Bai and Ng is that our
factors are extracted from residuals rather than observed variables such as
returns. Accordingly, we considered two cases for V (τ , W_τ). One is based on
the equation-by-equation OLS residuals of the τ -factor model u = WτA⁰_τ+E.
A second version which is more in the spirit of our approach is based on the
equation-by-equation OLS residuals of RII. Setting as τ_max the number of
factors dictated by Kaiser criterion and using the two DGPs described in the
foregoing analysis (where the true number of factors is J=1) the outcome
of these criteria is averaged over 500 replications. In all cases the criteria
systematically choose one factor for both the small and large panels.

Next our DGP is generalized to J ≥ 1 omitted global variables ceteris
paribus by letting

x_it = d_it +     λ_ij zj_t; y_it = β_ix_it +     γ_ijzj_t + ε_it

j=1

where zjt for j = 1, ..., J are orthogonal to each other and λij and γ_ij are
N(1,1) variates. Setting J = 2 the average of τ over replications for the
annual panel dimensions is 2.052 (with standard deviation of 0.4536) for the
baseline DGP and 4.146 (2.375) for the DGP with autocorrelation and het-
eroskedasticity. For the monthly panel these figures are 2 and 2.889 (1.881),
respectively.¹² Notwithstanding that min{N, T} < 40 for both the annual
and monthly panels and despite extracting the factors from residuals, these
criteria appear to perform reasonably well in this context.

5 Application to PPP equations

The issue of cross section dependence in empirical tests of PPP was recently
hightlighted by O’Connell (1998). He showed that assuming one (stationary)
common shock reversed the positive verdict on PPP (from tests ignoring such
dependence) for three out of four interational panels. We employ monthly

¹²Using Xit = √Jdit + PJ=1 λijZjt instead so that the idiosyncratic and common com-
ponent of x_it have the same variance, the results for the annual and monthly panel are
2.296 (1.013) and 2.0 for the baseline DGP.

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