A Principal Components Approach to Cross-Section Dependence in Panels

geneity and differences in the response of cross-section units to the common
influence by letting β_i ~ U[0.5,1.5], λ_i ~ U[0.5,1.5] and γ_i ~ U[0.5,1.5]. The
mean slope estimate over replications for RI(RII) is 1.492(1.081), 1.493(1.082)
and 1.517(1.118) for the POLS, FE and MG estimators, respectively. These
results support our conjecture that the proposed approach is suitable for
heterogeneous panels also.

There remain two unresolved questions regarding the standard errors.
One is to ascertain why SE₃ are incorrect in RII while they work quite well
in RI for our baseline DGP. The other is to derive the theoretical covariance
matrix for the estimators of RII in a DGP with autocorrelation. One possible
solution is to implement a bootstrap technique. However, the consistency (or
otherwise) of the bootstrap s.e. in our setup may well depend on the answer
to the first question.

We investigate the number of factors problem via simulations. We ex-
amine the rule of thumb (known as Kaiser criterion) that only those factors
whose associated eigenvalues λi > 1 are retained. The intuition for the latter
is that — since the factors are extracted from standardized residuals — un-
less a factor extracts as much variation as one original variable it is dropped.
For our baseline DGP the number of factors thus chosen ranges between 9
and 12 (with mean 10.61 and standard deviation .027) for the annual data
panel and between 7 and 11 (mean 9.07 and standard deviation .028) for
the monthly panel over 500 replications. Hence the Kaiser criterion is too
conservative and this raises issues of interpretation in an economic context.¹¹

Next we explore the performance of the recent Bai and Ng (2002) ap-
proach for selecting the number of factors in approximate factor models.
They formulate this non-standard problem as a model selection problem and
propose minimizing, inter alios, the following two information criteria

N + T NT

ICpi (τ ) = ln V (τ ,IWτ )+ τ (ɔɪ-j ln ÇN+TJ      (16)

N+T

ICp2 (τ ) = ln V (τ ,Wτ )+ τ NN _- ,   ∣ ln CN τ              (17)

where CNNT = min{N, —} and V(τ, IW_τ) = ^e/N— is the average residual
variance of a factor model where τ factors are assumed for each cross-section

¹¹ Nevertheless, including too many factors does not seem to have an adverse effect on
bias reduction. This is borne out by the sample mean of b at some 1.069 (POLS) and
1.072 (FE) for the annual panel and 1.096 (POLS, FE) for the monthly panel.

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