I'm trying to estimate a state space model of the form
x(t) = Ax(t-1) + Gz(t-1) + Bu(t)
y(t) = Cx(t) + De(t)
where z(t) is a vector of exogenous (predictor) variables. I'm using the econometrics toolbox. The toolbox documentation states that if you require predictor variables in the state equation, you can either
– Expand the states by including the constant 1 state or
– Expand the states by including predictors.
If z(t) is a constant vector,e.g. a vector of ones, then I suppose the state vector can be "expanded" as follows:
[x(t);z(t)] = [A G;0 1] *[x(t-1);z(t-1)] + [B;0]*u(t)
But what if z(t) is not a vector of constants, so z(t) is not equal to z(t-1)? If z(t) is comprised of random draws from some distribution for example, I'm not clear on how to include it in the state variable.
Thank you.
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