FEVD using bandpass_filter = [6 32] in stoch_simul, interpretation?


If I am using the bandpass_filter = [6 32] when computing a forecast error variance decomposition after simulating my model, am I getting the FEVD corresponding to 6 period ahead to 32 period ahead terms? Is this the correct interpretation of what the outputs are?

For example, simple state space model given by:

s_t = A s_{t-1} + B \epsilon_t

Recursively substituting yields,

s_t = \sum_{j=0}^{\infty} A^{j} B \epsilon_{t-j} \tag{0}, or
s_{t+k} = \sum_{j=0}^{\infty} A^{j} B \epsilon_{t+k-j} \tag{1}

While the expectation of the state space is all shocks occurring at and prior to the expectation iterated through. Which says

E_t s_{t+k} = \sum_{j=k}^{\infty} A^{j} B \epsilon_{t+k-j} \tag{2}

Then, for example, using (1) and (2) the thiry-two period ahead FEVD is given by:

(E_t s_{t+32} - s_{t+32})(E_t s_{t+32} - s_{t+32})' = (B \epsilon_{t+32} + AB \epsilon_{t+31} + \dots)(B \epsilon_{t+32} + AB \epsilon_{t+31} + \dots)'

So if I use bandpass_filter = [6 32] after a stochastic simulation, is this returning to me:

= BVB' + ABVB'A' + A^{2}B V B' A^{2'} \dots + A^{27} B V B' A^{27'} ?

That is, I would be dropping the 28-32 terms, or the first 5 forecast periods?


No, that’s not it. Rather, you use a transformation to the frequency domain and apply the ideal band-pass filter there. See https://www.sfu.ca/~kkasa/uhlig1.pdf