System Matrices - Forecast of backward-looking variables

Topics: Kalman Filtering, Models
May 19, 2015 at 4:14 PM
Hello, folks. I have a DSGE model with 'nb' number of backward-looking variables and 'nf' number of forward-looking variables. I've been digging into IRIS filter algos and was wondering if you could help me clarify how the forecast of backward-looking variables is generated.

From the quasi upper triangular matrix T (of size [nb + nf] by nb) one could conclude that backward-looking variables dynamics are not being affected by forward-looking variables, and that forward-looking variables dynamics are affected by backward-looking variables ONLY.

Could you confirm this is the case? In other words, could one use the upper triangular portion of T to forecast backward-looking variables without the need to forecast forward-looking variables? The source of my confusion is that when we write the model equations in the model file, we write such equations so that backward-looking variables ARE affected by forward-looking variables.

Any enlightening words will be appreciated!
May 19, 2015 at 4:26 PM

You're absolutely correct. You can decouple the dynamics of bkw-looking variables (xb) from frw-looking variables (xf). Note that in IRIS solution matrices. the bkw-looking variables are placed in the lower part.

You're also right that in the original model equations, everything is interdependent. But this interdepency is sorted out when the below final solution is computed using the generalized Schur decomposition. Then, the effect of frw-looking variables is integrated away, and effectively replaced directly with expectations of future exogenous things, which means anticipated shocks in IRIS.

The IRIS transition equation (in its vector/matrix form) is, in fact, given by

[ xf; xb ] = [ Tf; Tb ] * xb(-1) + [ Rf0; Rb0 ] * e + [ Rf1; Rb1 ] * E[e(+1)] + ... [ Rfk; Rbk ] * E[e(+k)] + ...

where Tf, Tb, Rf0, Rb0, ... are matrices (only Tb is square, all other matrices can be rectangular in general), and e the vector of current shocks, and E[e(+k)] means the expectations of shocks at t+k (which is implicitly assumed zero if not specified otherwise by the user, through anticipated shocks).

Note that the first term on the RHS refers to xb(-1) only, and not to xf(-1).

Hope this helps.
Marked as answer by jaromirbenes on 5/19/2015 at 9:26 AM
May 19, 2015 at 4:53 PM
Thank you!