On the Existence of the Moments of the Asymptotic Trace Statistic

ZT,d = tr

1 ^T             1 ^{T               -1}

T ∑εtX⁰-1 T2 Σ^x<-1 Xt-1
t=1               t=1

1^T

T X Xt-1 ^εt

t=1

(3)

where ε_t ~ Nd(0, Id) i.i.d. and X_t = P^t =₁ ε_i for t = 1,..., T. This is motivated by the weak
convergence of Z_T,d to Z_d as T → ∞. Consequently, the proposed procedure relies crucially
on the fact that the first two moments of the asymptotic trace statistic exist and may be
obtained as limits of the corresponding moments of Z_T,d.

2 Results

On account of the weak convergence of ZT,d to the asymptotic trace statistic Zd , the first two
moments of Zd exist if the sequence {Z_T²_,d} is uniformly integrable. A sufficient condition for
this is established in Lemma 2, which states that the fourth moments of Z_T,d are uniformly
bounded in T. We start with showing that all moments of ZT,d exist. To ensure that the
inverted matrix appearing in (3) is nonsingular with probability one, we assume T > d.

Lemma 1. Assume that T > d. Then all moments of ZT,d defined by (3) exist.

Proof. As Larsson et al. (2001), we introduce the (T × d) matrices ε
and X = (X1, X2, ..., XT)⁰ as well as the (T × T) matrices

(ε1, ε2, ..., εT)⁰

10

and

..... 0 ^

..... 0

0   ■■■ 0

0  10

Then, X = Aε and the (d × d) matrices appearing in (3) can be rewritten as

1^T                  1

AT := T2 Σ<X^t-1 Xt-1 = T2^ε A B BA^ε,
t=1

1^T             1

Bt := T ΣX-1 ^t = TεA'B'ε.

t=1

(4)

Defining D = BA and Y = Dε, we obtain therefore

ZT,d = tr(B_T⁰ A_T^-1BT) = tr[ε⁰Dε(ε⁰D⁰Dε)^-1ε⁰D⁰ε]
= tr(ε⁰PYε) ≤ tr(ε⁰ε),

(5)

(6)

where PY = Y (Y⁰Y )^-1Y⁰ denotes the projection matrix onto the column space of Y . The
assumption ε_t ~ Nd (0 ,Id ) i.i.d. now implies tr( ^ε ) = P T_=x ε'_tε_t ~ χ Td, which completes the
proof, since all moments of a χ²-distributed random variable exist.                       ¥