Forecasting Large Datasets with Reduced Rank Multivariate Models

fp
i and p denote the vector of covariances between yi;t and X p
p;t
to OLS estimation. Let
and the covariance matrix of X p
p;t , respectively and ^fp
Further, denote by p
, ^fp the i; j-th elements
of
fp
ij
and ^fp
ij
ij
, ^p
ij
fp
ij
ij
i and ^p their sample counterparts.
p
ij
and ^p
ij
, respectively. Then, by (25) of An et al. (1982)
p ^ A i A i = ^p p ^ A i A i
^ fp
i
fp
i
and the j-th elements
^p p A i
Since each yi;t is part of a stationary VAR process by assumption 1(a), and, also taking
into account assumption 1(b)-(c), it follows that yi;t satis…es the assumptions of Theorem
5 of An et al. (1982). De…ne A i = (A i 1 ; :::; Ai Np )0 and ^ A i = ( ^ A i 1 ; :::; ^ A i Np )0 . Then, by the
proof of Theorem 5 of An et al. (1982) (see the last 3 equations of page 935 of An et al.
(1982)), we have
and
^p p ^ A i A i
^ fp
i
fp
i
2
XNp
= op(1)
j=1
2
= op (ln T=T ) 1=2
^A i j
^p p i
A 2
= op (ln T=T ) 1=2
Note that (28)-(30) follow from Theorem 5 of An et al. (1982), if further,
and
sup
i;j
sup
j
^ p
ij
^ fp
ij
p
ij = Op (ln T=T ) 1=2
fp
ij = Op (ln T=T ) 1=2
But this follows easily by minor modi…cations of the proof of Theorem 7.4.3 of Deistler
and Hannan (1988) and the uniformity assumption on the fourth and second moments
of et given in Assumption 1(c). Hence,
(1 + op(1)) ^ A i 2
i
A = op (ln T=T ) 1=2
which implies (26) and completes the proof of the theorem.
Proof of Theorem 2. We de…ne formally the functions gO(:) and gK(:) such that
A i j
2
(27)
(28)
(29)
(30)
(31)
vec( ^ K 0 ) = gK vec( ^ A) (32)
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