Hi all,
I am trying to code a simulation and estimation of the Smets Wouters '07 model with adaptive learning. I have thus far worked on the following model of Woodford 03, which is used in Milani 05 as a benchmark model.
\begin{align} \tilde{x}_t &= E_t \tilde{x}_{t+1} - (1-\beta \eta) \sigma [i_t - E_t \pi_{t+1} - r_t^n ] \\ \tilde{\pi}_t &= \xi_p [\omega x_t + [(1-\eta \beta)\sigma]^{-1}\tilde{x_t}]+\beta E_t \tilde{\pi}_{t+1} + u_t \\ \tilde{\pi}_t &= \pi_t - \gamma \pi_{t-1} \\ \tilde{x}_t &= (x_t-\eta x_{t-1}) - \beta \eta E(x_{t+1}-\eta x_t) \\ i_t &= \rho i_{t-1} + (1-\rho)[\psi_{\pi}\pi_t+\psi_x x_t] + \varepsilon_t \\ r_t^n &= \phi^r r^n_{t-1} + v^r_t \\ u_t &= \phi^u u_{t-1} + v^u_t \end{align}
One particular challenge of adaptive learning in this setup, however, is the presence of the redundant state variables \tilde{x}, \tilde{\pi} that require me to compute expectations at time-t of variables at time-t, t+1, and t+2, namely E_t \tilde{x}_{t+1} requires me to compute E_t x_{t+2} and E_tx_t and E_t \tilde{\pi}_{t+1} requires me to compute E_t \pi_t
As I am trying to embed AL into the Smets Wouters model, my question is: Does this multiplicity of timing of expectations arise in the Smets Wouters model, and if so for which variables must I compute E_t x_{t+n} for z \in \mathbb{Z}?
One solution that seems to work in the Giannoni & Woodford model is to solve out the \tilde{x} and \tilde{\pi} and use the minimum state variable solution but I’m not sure this is is feasible for a model as large as the Smets Wouters model.