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I am studying a New Keynesian model with a hybrid Phillips curve which reads \begin{align*} \pi_t=\beta E_t \pi_{t+1} + \varpi \pi_{t-1}+ \kappa x_t. \end{align*} The demand shock in the IS curve follows a two-state Markov process, i.e. I assume that there are two states, states $n$ and $z$. The transition probabilities are given as follows \begin{align*} P(s_{t+1}=n| s_t=n)&=p\\ P(s_{t+1}=z| s_t=n)&=1-p\\ P(s_{t+1}=z| s_t=z)&=q\\ P(s_{t+1}=n| s_t=z)&=1-q\\ \end{align*} For $\varpi=0$, it follows that the Phillips curve in state $n$ reads \begin{align*} \pi_n = \beta\left[ p\pi_n + (1-p)\pi_{z}\right]+ \kappa x_n. \end{align*} However, I am interested in the Phillips curve in state $n$ for the case where $\varpi\neq 0$. My idea is that, given the economy is in state $n$ today, this can be only because

  1. The economy was in state $n$ in $t-1$ and remained in state $n$ with probability $p$, or
  2. The economy was in state $z$ in $t-1$ and switched to state $n$ with probability $1-q$.

Following this idea, for $\varpi\neq 0$, I would write down the Phillips curve in state $n$ as \begin{align*} \pi_n = \beta\left[ p\pi_n + (1-p)\pi_{z}\right]+ \varpi\left[p \pi_n + (1-q)\pi_z\right]+ \kappa \pi_n. \end{align*} Is this approach correct? If not, how do I add $\pi_{t-1}$ given the economy is in state $n$ in period $t$?

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  • $\begingroup$ Hi: what does $x_t$ represent ? is that the demand shock ? $\endgroup$
    – mark leeds
    Feb 17, 2021 at 17:51
  • $\begingroup$ No, x_n is the output gap which is determined via another equation (IS curve). I left it out as it (in my view) does not matter for the question at hand, although the shock r_t appears in the IS curve. $\endgroup$
    – chopschoc
    Feb 17, 2021 at 18:07
  • $\begingroup$ but where did you get the $\kappa \pi_n$ as the last term ? $\endgroup$
    – mark leeds
    Feb 18, 2021 at 4:16

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