706 Panel Methods to Test for Unit Roots and Cointegration
that Breitung incorporates the homogeneity restriction on the cointegrating spaces
in the construction of the test statistics. We continue the discussion for the VAR(1)
model without deterministic components. Denote withγi∈Rm×(m−k)full column
rank matrices and consider:
Yi,t=αiβ′
Yi,t− 1 +γiβ′
⊥Yi,t− 1 +wi,t. (13.35)Under the null hypothesis of ak-dimensional cointegrating space it holds that
γi=0 fori=1,...,Nand, under the alternative of anm-dimensional cointegrating
space,γiis unrestricted to allow fori=αiβ
′
+γiβ′
⊥of full rank. For test consistency
the alternative has to comprise a non-vanishing fraction of cross-section members.
Pre-multiply (13.35) withα
′
i,⊥to obtain:α′
i,⊥Yi,t=α′
i,⊥γiβ′
⊥Yi,t− 1 +α′
i,⊥wi,t
α′
i,⊥Yi,t=φi(β′
⊥Yi,t− 1 )+w ̃i,t, (13.36)where the last equation defines the coefficient matrices and variables. Replacing
α
′
i,⊥andβ⊥by estimates allows us to estimate equations (13.36) separately for
each cross-section unit by OLS and to construct test statistics for the hypothesis
H 0 :φi=0,i=1,...,N.
Any of the classical testing principles, that is, the likelihood ratio, Wald or
Lagrange multiplier principle, can be used. Breitung discusses the Lagrange multi-
plier test statistic, which has the advantage that it only requires estimation under
the null hypothesis. Denoting byfˆi,t =ˆα
′
i,⊥Yi,tand withgˆi,t =
βˆ′
⊥Yi,t, the
Lagrange multiplier test statistic for unitiis given by:
LMi(k,m)=Ttr⎡
⎢
⎣∑Tt= 2fˆi,tgˆi′,t− 1⎛
⎝
∑Tt= 2gˆi,t− 1 gˆ′
i,t− 1⎞
⎠− (^1) T
∑
t= 2
gˆi,t− 1 fˆ
′
i,t
⎛
⎝
∑T
t= 2
fˆi,tfˆi′,t
⎞
⎠
− 1 ⎤
⎥
⎦,
which is sequentially computed for the different values ofk=0,...,m.
The panel test statistic is then, as usual, given by the corresponding centered and
scaled cross-sectional average, where we now use again the superscriptsto indicate
the dependence upon deterministic components. Under the null hypothesis we
hence arrive at:
Bs(k,m)=N−^1 /^2
∑N
i= 1
LMis(k,m)−μsLM(k,m)
(σLM2,s(k,m))^1 /^2
⇒N(0, 1). (13.37)
The correction factorsμsLM(k,m)andσLM2,s(k,m)coincide exactly with those of LLL
above. The method also shares with LLL the limitations concerning the availability
of proper asymptotic theory. Clearly, instead of basing the panel tests upon the
trace statistic, the max statistic of Johansen (1995) could be used as the underlying
time series test statistic in both the LLL and Breitung tests.