On the Existence of the Moments of the Asymptotic Trace Statistic

Finally, to bound E[tr(AT¹)]⁴ uniformly, we recall U ~ W_ds(m,I_d) and use results of von
Rosen (1988, 1997) on moments for the inverted Wishart distribution. In particular it is
known that the qth moments of U^T1 exist if m - d - 2q + 1 > 0. Consequently,

E[tr(U ^T1)]⁴ ≤ c2 < ∞ for m ≥ d + 8,                      (14)

so that an application of inequality (10) for m = d+ 8 (assuming T > m) together with (11),
(14) and Lemma 1 yields the desired result.

The proof of (ii) is analogous to that of (i) and thus only sketched. First, the Cauchy-
Schwarz inequality provides, using (7),

E(ZT,_d) ≤ E £tr(AT ¹)tr(BtB'_t)]4 ≤ ©E[tr(AT ¹)]8E[tr(BτBT)]⁸}1 /2 .

It is easy to see that E[tr(BT B_T⁰ )]⁸ is uniformly bounded in T, since sup_T E(α_i¹_j⁶) < ∞.
Finally, E[tr(A_T^T1)]⁸ is uniformly bounded by choosing m = d+ 16 and applying (10) together
with (11), because then E[tr(U ^T1)]⁸ ≤ c3 < ∞. This completes the proof.                 ¥

Theorem. It holds that E(Z_d²) < ∞ and lim E(Z_T^q _d) = E(Z_d^q) for q = 1, 2.

T→∞    ^,

Proof. Recalling that Z_T,d converges weakly to the asymptotic trace statistic Z_d (Jo-
hansen, 1995), the result follows if {Z_T²_,d} is uniformly integrable (see Theorem A on p.14 in
Serfling, 1980). A sufficient condition for the uniform integrability of {ZT_d} is that E∖Zτ,d∖2+^δ
is uniformly bounded for some δ > 0, i.e sup_τ E∣Z_τ,_d∣2+^δ < ∞. But this is an immediate
consequence of Lemma 2 (ii), completing the proof.                                    ¥

3 Discussion

Several authors have used the first two moments of the asymptotic trace statistic to base
panel cointegration tests on a standardized average of individual cointegration test statistics;
see, for instance, Larsson et al. (2001), Groen & Kleibergen (2003) and Breitung (2005).
Our Theorem provides a theoretical justification for such an approach. To the best of our
knowledge, the only attempt to establish this result is due to Larsson et al. (2001). However,
the proof of their Lemma 1, which coincides with our Lemma 2, is incorrect and has thus
initiated this note. In what follows, we comment in more detail on the proof by Larsson et
al. (2001).

In our notation, Larsson et al. (2001) assumed ε_t ~ N_d(0, Ω) i.i.d for defining Z_τ,_d in (3).
This seems to be unnecssary, but would not lead to complications in our proof. Moreover,
they used the spectral decomposition of the (random) positive definite (d × d) matrix AT (see
(4)), i.e.

1^T

At = t^ ∑X-1 Xt-1 = GΓG,
T
t=1

where G is an orthogonal (d × d) matrix and Γ = diag(γ ₁, ...,γ_d), and defined ε by ε = εG.
Then they rewrote (3) as

d

ZT,d = tr(BT⁰ G⁰Γ^-1GBT) = ^XHiiγ_i^-1,

i=1

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