692 Panel Methods to Test for Unit Roots and Cointegration
where:
ZmtNT=∑N
i= 1tαmi,0; m=c,τZγtNT(λ)=∑N
i= 1tαγi,0(λ),and, asT→∞,N→∞in sequence,
N−^1 /^2 ZtmNT−√
Nm 2 ⇒N(0,! 2 m),m=c,τ
N−^1 /^2 ZtγNT(λ)−√
Nm 2 (λ)⇒N(
0,! 2 m(λ))
.The moments for the pooled tests whenm=c,τare the same as those derived by
Bai and Ng (2004), while form=γ, the moments (which depend on the break
fractions) are presented by Banerjee and Carrion-i-Silvestre (2007).
The analysis described above generalizes both the results of Bai and Carrion-
i-Silvestre (2007) and Bai and Ng (2004) and also of Pedroni (1999, 2004). We
have proposed here an encompassing framework that allows investigators to study
dependence and breaks within the context of testing for cointegration and unit
roots in macro-panels.
Two further extensions may be considered to allow exogenous regressors and for
unknown break dates. The former is relatively straightforward and uses dynamic
OLS (D-OLS) regressions to extract the residuals to be tested for integration. The
methods are as rehearsed in section 13.3.2, albeit for the case of systems estimators
without breaks, but no conceptual novelties are involved. It is shown by Banerjee
and Carrion-i-Silvestre (2007) that using D-OLS leads to the asymptotic distribu-
tions being the same as those given above, even in the presence of endogeneity of
regressors.
The case of unknown break dates is somewhat more complicated but relies on
consistent estimation of the break fractions, so that wherever knowledge of the
break date is necessary (as in the case where there are changes in trend) the true
break date can be replaced by a consistent estimate. Since the estimation algorithms
must allow for the breaks to occur at every point in time (within the closed interval),
the densities depend on computing the infimum of the standardizedt-statistics
and are therefore non-standard. The critical values for this case are also given by
Banerjee and Carrion-i-Silvestre (2007).
13.3.1.5 Empirical illustration with exchange rate pass-through in the euro-area
This sub-section is based on material from Banerjee and Carrion-i-Silvestre (2007)
and de Bandt, Banerjee and Kozluk (2008) and provides an illustration of single-
equation panel cointegration techniques when both cross-sectional dependence
and structural breaks are allowed to be present.
Prominent in the exchange rate pass-through literature are the papers by Campa
and González-Minguez (2006) (henceforth CM), Campa, Goldberg and González-
Minguez (2005) (henceforth CGM), and Frankel, Parsley and Wei (2005),inter alia,
who have investigated the issue of exchange rate pass-through (ERPT) of foreign to
domestic prices – that is, how changes in prices of imported goods are transmitted