Luis A. Gil-Alana and Javier Hualde 457version of PPP, so this is testable by the Wald statistic of (10.19). Using unit root
tests, Cheung and Lai (1993) failed to reject the hypothesis ofδ=1 and then, using
differenced OLS residuals, computed semiparametric log-periodogram estimates of
γ−1 and tested the non-cointegration null hypothesis ofδ−γ=0 against the
alternativeδ−γ>0, using critical values computed by simulation. They found
evidence of cointegration in a number of bivariate series, but did not testν=1.
We employ a step-by-step approach, first testing whether the integration orders,δx
andδy,ofxtandytare the same, then testing for the presence of cointegration,
then testing forδ−γ>0.5 and, finally, given that all these hurdles have been
crossed, testingν=1. In the first three steps we use semiparametric procedures (as
did Cheung and Lai, 1993; Marinucci and Robinson, 2001), while in the final step
we identify parametric models for the autocorrelation inutand hence compute
estimates ofνand Wald statistics.
The semiparametric estimates of integration orders were all Robinson’s (1995b)
versions of log-periodogram estimates, but without trimming, using first differ-
ences and then adding back 1. We estimatedδxandδyseparately, and then tested
δx=δy(=δ)by an adaptation of Robinson and Yajima’s (2002) statisticTˆabto log-
periodogram estimation, with their trimming sequenceh(n)chosen asb−^1 /(^5 +^2 i)
fori=1,..., 4, withbthe bandwidth used in the estimation. Givenδx=δyis not
rejected, we performed the Hausman test for no-cointegration of Marinucci and
Robinson (2001), comparing the estimateδ ̃xofδxwith the more efficient bivariate
one of Robinson (1995b), which uses the information thatδx=δy. Given cointe-
gration is not rejected, the nullδ−γ=0.5 was rejected in favor ofδ−γ>0.5 if
and only if a studentizedδ ̃x− ̃γ−0.5 was significantly large relative to the standard
normal distribution, whereγ ̃is the estimate ofγusing OLS residuals.
Using annual data (as is relevant to the long-run version of PPP) of Obstfeld and
Taylor (2002) for the period 1870–1992 (withn= 123 ), we applied the above
methodology to four bivariate series, the US (“domestic”) versus the “foreign”
countries, Australia, Canada, Italy and the UK. Strong evidence against equality
of integration orders was found in the case of Australia and Italy, and against coin-
tegration in the case of Canada. However, the UK “passed” all three initial tests.
Across the rangeb=10,..., 29, (δ ̃x,δ ̃y)varied between the extremes (1.341, 1.095)
and (1.572, 1.376), and acrossb=16,..., 25 and the fourh(n)choices,δx=δywas
rejected in only 9 out of 40 cases, and these were all at the 10% level. For the same
b, no-cointegration was rejected at the 10% level in all cases, at 5% in 4 cases, and
at 1% in 3 cases, whileδ−γ=0.5 was rejected againstδ−γ>0.5 at the 1% level
in all cases.
For the US–UK data, we identified parametric models forf(λ)as follows. We
consider:
ut=A(L)εt, (10.20)whereεtis considered to be an i.i.d. process. Throughout,A(L)in (10.20) was
diagonal, andu 1 t,u 2 ttreated separately. They were proxied byγ ̃(yt−ˆνOLSxt),
δ ̃x
xt, for each of the extremeγ ̃,δ ̃x, namelyγ ̃ =0.374, 0.698 andδ ̃x=1.572,