Re: [Gretl-users] Estimation of a State Space Model with Exogenous Variables in the State Transition Equation

Am 21.08.2015 um 16:37 schrieb Fernando Fernandes Neto: Forget my remark about the constant, that was probably off-track. What I have in mind is more or less the following. Suppose you have a scalar state variable s_t, and your state transition equation is like this (in Latex format): $s_{t+1} = G s_t + D z_t$ where z_t is your exogenous variable and I have omitted the innovation because I'm lazy. If there was no exogenous variable (if D were zero) then the gretl manual would use F as the name of G. But here the time-varying transition matrix F_t is going to be something else. Now expand your state to a vector $\xi_t$ with an identity (like you said) to re-formulate the equation like this (mish-mash of Latex and a matrix format where semi-colons come between one row and the next): (s_{t+1}, 1)' = \xi_{t+1} = (G, D*z_t; 0, 1) \xi_t = F_t * (s_t, 1)' The 1 as the auxiliary second state element should of course be initialized to 1 and should not get a variance. The new 2x2 state transition matrix F_t = (G, Dz_t; 0, 1) is now time-varying as it depends on z_t. As explained in the gretl manual (ch. 30 about the Kalman filter) you can make the 'statemat' time-varying by writing a function that updates its value, so you would write: statemat F ; modify_F(&F, z) where z is a matrix (vector) holding your z_t values. (I guess it must be a matrix and cannot be a gretl series.) The function modify_F in this example would have to look something like this: <hansl> function void modify_F(matrix *F, matrix z) tstep = $kalman_t F[1,2] = F[1,2] * z[tstep] / z[tstep-1] end function </hansl> This function takes the upper-right element of the matrix which on entry will have the value D*z_{t-1} and turns it into D*z_t. (Obviously this will fail if z_t contains zero values, I guess one could think about workarounds for this problem.) So that's the idea, but I must stress that I have never tried it and I would be interested in hearing (a) your experience, and (b) jack's thoughts on this. good luck, sven