On the Existence of the Moments of the Asymptotic Trace Statistic

The existence of the first two moments of the asymptotic trace statistic is claimed in
Larsson et al. (2001), but their proof is incorrect as explained in Section 3. Therefore we
provide a corrected version of the proof. Moreover, the asymptotic moments are usually
approximated by simulating a certain statistic which converges weakly to the asymptotic
trace statistic. To justify this approach we show that the first two moments of the mentioned
statistic converge to those of the asymptotic trace statistic.

To be more specific, we consider, as Larsson et al. (2001), a sample ofN cross-sections (in-
dividuals) observed over T time periods and suppose that for each individual i (i = 1, . . . , N)
the K-dimensional time series yit is generated by the following heterogeneous VAR(pi) model:

pi

yi_t =      Aij yi_,t-j + ei_t,       i = 1, . . . , N ; t = 1, . . . , T,                        (1)

j=1

where the initial values yi,-p_i+1 , . . . , yi0 are fixed, Aij are (K × K) coefficient matrices and
the errors eit are stochastically independent across i and t with e_it ~ NK(0, Ω_i) for some
nonsingular covariance matrices Ω_i. The components of the process y_it are assumed to be
integrated at most of order one and cointegrated with cointegrating rank ri with 0 ≤ ri ≤ K .
The error correction representation of model (1) is

pi-1

∆yit = Πiyi,t-1 +     Γij∆yi,t-j+eit,      i= 1,...,N; t= 1,...,T,

j=1

where the (K × K) parameter matrices Γij = -(Ai,j+1 + . . . + Ai,p_i) describe the short-run
dynamics, and the (K × K) matrix Πi = -(IK - Ai1 - . . . - Ai,p_i) can be written as Πi = αiβ_i⁰
with (K × ri) matrices αi and βi of full column rank.

Interest is in testing whether in all of the N cross-sections there are at most r cointe-
grating relations among the K variables. Thus, the null hypothesis

H0(r) : rank(Πi) = ri ≤ r, for all i = 1, . . . , N,

is tested against the alternative

H1 : rank(Πi) = K, for all i = 1, . . . , N.

According to Johansen (1988), the cointegrating rank of the process may be determined by a
sequential procedure. First, H0(0) is tested, and if this null hypothesis is rejected then H0(1)
is tested. The procedure continues until the null hypothesis is not rejected or H0(K - 1) is
rejected.

The standardized LR-bar statistic for the panel cointegrating rank test is defined by

-T ^P_j^K_=r+1 ln(1 - λ^bij) - E(Z_d)

√Var( Zd )

where λ^bij is the jth largest eigenvalue to a suitable eigenvalue problem for the ith cross-
section defined in Johansen (1995). Moreover, E(Z_d) and Var(Z_d) denote the mean and the
variance, respectively, of the asymptotic trace statistic

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