Palgrave Handbook of Econometrics: Applied Econometrics

S.G.B. Henry 931

themselves cointegrate, though tests in this case show they clearly do (see Henry,
Kirby and Riley 2007, for further details). The right-hand-side variables should be
weakly exogenous too, which, as discussed later, they are not. However, the prob-
lems with the model are not simply due to ARDL estimation, and the difficulty of
treating the equation (18.18) as a long-run unemployment equation is not resolved
by using an alternative such as the Johansen ML method to estimate a single co-
integrating vector, normalized on unemployment, which is then treated as “the”
unemployment relationship as in Nickell and Bell (1995). We review problems with
the use of Johansen estimation with this dataset next. The purpose of this is not
estimation, but to describe what would be required to estimate (18.18) so that it
had a behavioral interpretation. This review, incidentally, provides on explanation
for the parameter instability noted in Table 18.1.
As the data are mainly non-stationary (see below), the dynamic model under-
lying equation (18.18) can be written as a vector error correction model (VECM),
with eight equations, one for each of the variables in (18.18), as illustrated next:

zt=

p∑− 1

j= 1

jzt−j+γα′zt− 1 +εt. (18.19)

Here z is a column vector ofnvariables,nbeing the eight variables (including the
unemployment rate) from equation (18.18). Thej(j=1,...,(p− 1 ))are a set of
( 8 × 8 )matrices of parameters on the dynamic terms of the model, where the preset
lag-length of the model isp. Attention is focused on the long-run part of the VECM,
whereγandα′are the loading weights and cointegrating vectors respectively, and
γisnxrto reflect the reduced rank of the system, where it is implicitly assumed that
there arer<ncointegrating vectors in the model, andεtis a vector of white-noise
error terms, withεt∼N(0,).
Tests of orders of integration of the eight variables reported Table 18.2 reveal
that one,IT,isI( 0 )while the others areI( 1 ). In this set it appears there could be
up to three cointegrating vectors. Tests forr≤3 give 32.8 (34.4) for the Johansen
eigenvalue test (λ)but 75.2 (75.9) for the Johansen trace test (95% significance levels
in brackets).^22 Tests of weak causality show that only the terms of trade (TT) and
the tax wedge (T)are weakly exogenous (Table 18.3). In the light of these first-stage
results on orders of integration and exogeneity tests, the implied model appears to
be a five-equation conditional model plus a two-equation marginal model forTT
andT, that is,

yt=

p∑− 1

j= 1

 1 jzt−j+γα′zt− 1 +ηt (18.20)

xt=

p∑− 1

j= 1

 2 jzt−j+υt (18.21)

whereytandxtare( 5 × 1 )and( 2 × 1 )column vectors ofI( 1 )endogenous (lnu,
Skill,RR,UPandR) andI( 1 )weakly exogenous variables (TTandT), respectively.