A vector autoregressive model of order 
, denoted 
, is a multivariate time series
model in which each component is a linear function
of every component at earlier lags. It has the
form
 |
(1)
|
where 
and 
are vectors, the 
are coefficient matrices,
and 
is a vector of innovations.
If the innovations form a stationary
time series with mean zero and the model is stable, its unconditional mean is
 |
(2)
|
provided the inverse exists.
The model can be written as a first-order system by stacking the present value and 
lags into 
. In this companion
matrix representation,
 |
(3)
|
where 
is a block matrix whose first row of blocks is 
and whose first subdiagonal consists of identity
matrices.
The model is stable, and hence has a causal time series solution that is a stationary time
series, when the spectral radius of 
is less than 1. Equivalently, the roots
of the autoregressive polynomial
 |
(4)
|
all lie outside the unit circle. In a stable model, deterministic forecasts obtained by setting future innovations to their mean satisfy
 |
(5)
|
A univariate autoregressive model is the one-component special case.
