Forecasting Large Datasets with Reduced Rank Multivariate Models

and j max(:)j denotes the maximum eigenvalue of a matrix in absolute value.
(b) cmax(A) < 1; rmax(A) < 1 where cmax(:) and rmax() denote the maximum
column and row sum norm of a matrix.
1.
(c) et is an i.i.d. (0; e) sequence with uniformly …nite fourth moments and cmax( e) <
Denote the transpose of the i-th row of (A1; A2; :::; Ap) by A i . We then have the
following Theorem on the consistency of the in…nite dimensional VAR coe¢ cients:
Theorem 1 As N and T diverge, and under assumption 1 , A^ i Ai 2
= op(T a ) for
all i = 1; :::; N; and for all a < 1=2, as long as N = o (T= ln(T )) 1=2 .
Proof. See Appendix.
Next, we consider a reduced rank approximation to the VAR model. To keep things
general, we consider the case where a singular value decomposition is used to decompose
(A1; A2; :::; Ap) as OK where O and K 0 are N r and Np r matrices respectively, for
some r < N. The sample counterpart of this decomposition is given by ( ^ A1; ^ A2; :::; ^ Ap) =
^O ^ K. Then, we have the following Theorem on the consistency of the in…nite dimensional
RR coe¢ cients:
Theorem 2 As N and T diverge, and under assumption 1 , each element of O and K 0
is op(T a+2b )-consistent for O and K 0 , for all 0 < a < 1=2, and 0 < b < 1=4, 2b < a, as
long as N = o T b .
Proof. See Appendix.
Note that the above analysis straightforwardly implies that a lag order, p = pT ,
that tends to in…nity is acceptable. In this case, the above result holds as long as
NpT = o (T= ln(T )) 1=2 , which means that our results extend to in…nite dimensional
VAR and RR possibly with in…nite lags too. The above consistency result opens the
grounds to using large dimensional RR models for empirical analysis.
2.2 Bayesian VAR (BVAR)
Bayesian methods allow to impose restrictions on the data, but also to let the data
speak. The exclusion restrictions are imposed as priors, so if some a-priori excluded
variable turns out to be relevant in the data, the posterior estimate would contain such
information. This provides a way of solving the curse of dimensionality problem without
resorting to ad-hoc exclusion of some variables.
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