Let us consider a RBC model like the one in Ireland’s A Method For Taking Models To The Data. We’ve taken the data Y_t(it’s a vector. I’ll be using a different and more simplified notation), and written the State-Space Model (SSM) representation with respect to \hat{y}_t (log-deviations from steady-state) after a log-linearization:
\begin{align*} s_{t+1}&=P(\theta)s_t+ \epsilon_{t+1}\\ \hat{y}_{t+1}&=M(\theta)s_t+ \eta_{t+1} \end{align*}
However, for each i-th component of \hat{y}_{t}, we have \hat{y}_{it}=\log(Y_{it})-t\log(f_i(\theta))-\log(y_i^*(\theta)), where y_i^*(\theta) is the steady-state of the i-th component which is also a function of the model’s vector of parameters \theta. The term t\log(f(\theta)) is present for detrending purposes.
I’m having doubts on how to link the observable data with the SSM, so that we can estimate the steady states and the parameters from the data alone…
Are we supposed to rewrite (with a slight abuse of notation in the term \log(Y_{t+1})) the SSM into
\begin{align*} s_{t+1}&=P(\theta)s_t+ \epsilon_{t+1}\\ \log(Y_{t+1})&=A(\theta)+M(\theta)s_t+ \eta_{t+1} \end{align*} ?
A(\theta) is a vector, where each component is t\log(f_i(\theta))+\log(y_i^*(\theta)).