Assuming we are calculating base (not conditional) forecasts, can we not modify the expectations as below:
E(y(t+1)) = v(t+1) + A(1)y(t) + ... + A(p)y(t-p+1)
E(y(t+2)) = v(t+2) + A(1)y(t+1) + ... + A(p)y(t-p+2)
For a VAR(1) process, instead of getting the below forecast:
E(y(t+h)) = [I(k) + A(1) + ... + A(1)^(h-1)]v + A(1)^h*y(t)
we would get:
E(y(t+h)) = v(t+1) + ... + v(t+h) + ... + A(1)[y(t) + y(t+1) + ...] + ... + A(p)[y(t) + y(t+1) + ...]
The distributions are still normal with same volatilities and correlations but a time dependent mean.
In other words, we would take the results from the forecast instruction and add or subtract a constant
at a particular timestep of interest. We would do the same when calculating forecast intervals. In that case:
y(t+h) - y(t)(h) ~ N[v(t+h)-v(t), sigma(y)(h)] so that [y(t+h) - y(t)(h) - (v(t+h)-v(t))] / sigma(k)(h) ~ N(0,1)
Doesn't this make (some) sense ? Thanks.