668 Panel Methods to Test for Unit Roots and Cointegration
Then (13.15) can be thought of as a model with intercept breaks, where the num-
ber and timing of the breaks can be calculated (unit by unit) using the Bai–Perron
dynamic programming algorithm. Thus, due to the fast convergence of the esti-
mated break points to their “true” values, theaimatrix can be assumed to be
known upon substitutingaibyaˆi. Asymptotically, given consistency, the theoret-
ical results are unaffected by replacing the true break dates with their estimated
values. The accuracy of the estimation of the break dates and its impact on the
properties of the testing for unit roots using the MSB statistic may well be a prag-
matic issue in cases whereTis relatively small. This can again be addressed within
the framework of simulation studies, an example of which is contained in Bai and
Carrion-i-Silvestre’s paper but for which further work is clearly justifiable.
13.2.4 Two empirical examples
In this section, we illustrate the arguments and tests developed above by means
of two examples, analyzing in particular the effect of incorporating cross-sectional
dependence (via factors) and structural breaks into the testing procedures. The two
examples are based on Wagner (2008a) and Wagner (2008b) respectively.
13.2.4.1 Purchasing power parity
The empirical analysis of (weak) purchasing power parity (PPP), in anI(1) model-
ing framework, typically formulated as stationarity of the real exchange rate (RER),
is a prime application of both time series and panel unit root (respectively co-
integration) testing. We have already referred to the influential paper by O’Connell
(1998) when discussing the consequences of overlooking the impact of cross-
sectional dependence.
In logarithms, the RER for countryiis given by:
qi,t=ei,t+pi,t−p∗t, (13.16)whereqi,tis the RER,ei,tis the nominal exchange rate,pi,tis the price level in
countryiandp∗tis the price level of the base country (all in logarithms). In our
application below the base country is the United States, the nominal exchange
rates are thus vis-à-vis the US dollar (per unit of local currency) and the prices are
given by the consumer price indices, which implies that, like almost all of the
literature, we do not study real exchange rates but real exchange rate indices.
Early panel studies of PPP, including Coakley and Fuertes (1997), Frankel and
Rose (1996), Lothian (1997) and Wu (1996), have used first-generation panel unit
root tests to test stationarity (respectively unit root behavior) of RER panels. Panel
methods have been used to overcome the deficiencies of time series unit root
tests, as highlighted in the PPP context by Engel (2000) and as discussed in the
introductory sections to this chapter.
Such studies have often “found support for PPP,” that is, rejections of the unit
root null hypothesis, although it is presumably well understood that rejection of
the null hypothesis (that is, a unit root in RER) is not acceptance of the alternative.
Much more seriously, the use of first-generation tests designed for cross-sectionally
independent panels appears to be troublesome in the PPP context.