by cfp » Wed Feb 08, 2012 10:44 am
Linearity means you can simulate a (zero-mean) model without bounds by the following method:
1) Generate one s.d. infinite length IRFs to all shocks. Denote these by z(t,v,e) where t>=0 is the time since the shock hit, v is the variable being examined and e is the shock.
2) Let x(t,v) denote the simulated value of the variable v at t, and let s(t,e) be a draw from an NIID(0,1) variable for all times t and shocks e. Then:
x(t,v) = sum( sum( s(t-k,e) * z(k,v,e) , e in the set of shocks ), k = 0 .. infinity )
Therefore:
E[x(t,v) * x(t,v)] = sum( sum( z(k,v,e) * z(k,v,e), e in the set of shocks ), k = 0 .. infinity )
So the standard deviation of a variable in a model without bounds is the square-root of the sum of the squared IRF coefficients.
Since you don't have linearity in the presence of bounds this formula no longer holds exactly, but it may still be reasonable.