A Principal Components Approach to Cross-Section Dependence in Panels

observations for the nominal exchange rate and price index variables, 1972:1-
1998:12, giving T = 324 observations. Four panels are constructed which
represent different combinations of price indexes and numeraire currencies.
The CPI-DM and CPI-US$ panels use consumer price index data and nom-
inal exchange exchange rates against the German mark and US dollar, re-
spectively. The WPI-DM and WPI-US$ panels are based on wholesale price
indexes.¹³

The following two PPP equations are considered:

i) Augmented Dickey-Fuller (ADF) type regression:

k

∆q_it = α_0i + α_1it + ρ_iq_i,t-1 +    γ_j ∆q_i,t-j + u_it           (18)

j=1

where qit = ei_t — dit represents the real exchange rate, dit = pit — pt is the
price differential between domestic country i and the foreign country, and all
variables are in logarithms.

ii) Autoregressive distributed lag (ARDL) regression:

kk

∆^eit ^{= α}i + βi∆dt + δ1iqi,t-1 + γidi,t-1 +    γj¹i∆^ei,t-j +    γj²i∆di,t-j + ^uit

j=1              j=1             ₍₁₉₎

which is a reparameterization of an analogous equation with e_i,t-1 as regressor
instead of qi_,t-1, where the coefficients on the former and on di_,t-1 are δ1i and
δ2i , respectively, with γ_i = δ1i + δ2i .

The average correlations between the cross-section OLS (standardized)
residuals are shown in Table 3 together with the condition number and p-
values of a diagonality test. All these indicate a considerable degree of cross-
section dependence. This is especially so for the dollar panels where the
depdendence may well reflect the large swings in the dollar during the 1980s.
The number of retained factors (λi > 1) is also reported in Table 3.

Panel regressions are estimated with and without factors. Table 4 reports
the estimated coefficients and t-ratios on the factors. The t-ratio on the single

¹³The CPI panels comprise N = 17 countries, Austria, Canada, Denmark, Finland,
Germany, Greece, Japan, Netherlands, Sweden, Spain, Switzerland, United Kingdom,
United States, Belgium, France, Italy, Norway, Portugal. The WPI panels exclude the
latter five countries and instead include Ireland to make N =13.

16

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