How can difference equations with an infinite summation be represented in matrix form? I have derived the microeconomic foundations of a dsge model and I've obtained the IS and NKPC. I would like to represent them in matrix form to study the system. However the problem is that both equation have in infinite summation. How can i proceed? Do I need to approximate, if possible, the infinite sum to a finite term?
\begin{equation}
 \tilde \pi_t = E_t \sum^{\infty}_{T=t} (A)^{T-t} \left[ B  x_{T+1} + C \tilde \pi_{T+1} \right]. 
 \end{equation}
\begin{equation}
{x}_{t}=\tilde{E}_{t} \sum_{T=t}^{\infty} D^{T-t} 
 \left[F{x}_{T+1}-\sigma\left(\tilde{\imath}_{T}-\tilde{\pi}_{T+1}\right) \right] 
\end{equation}
Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (If you're not familiar with the lag operator I suggest you to read the first section of the page linked above.):
We have the optimal price level equation (equivalent to your NKPC but less developed):
$$\tilde{P_{t}}=\theta\tilde{P_{t-1}}+(1-\theta)(1-\beta\theta)E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$$
Now, we can see that the difference between one period of time might be represented by:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k(\tilde{MC}_{t+k}-\beta\theta\tilde{MC}_{t+k+1})$$
In terms of lag operator we could factor $\tilde{MC_t}$:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k\tilde{MC_t}(1-\beta\theta L^{-1})$$
What this tells us is that we can use the same series lagged by $-1$ (same as forward by 1) to eliminate future periods from the original one. Let $S_t=E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$, then (omitting expectations operator):
$$S_t-S_{t+1}=
$$
\begin{bmatrix}
\tilde{MC}_{t}+(\beta\theta)\tilde{MC}_{t+1}+(\beta\theta)^2\tilde{MC}_{t+2}+(\beta\theta)^3\tilde{MC}_{t+3}+...\\
-\beta\theta\tilde{MC}_{t+1}-(\beta\theta)^2\tilde{MC}_{t+2}-(\beta\theta)^3\tilde{MC}_{t+3}-(\beta\theta)^4\tilde{MC}_{t+4}-...
\end{bmatrix}
$$=\tilde{MC}_t$$
Look that we managed to reduce the infinite sum into a single expression, now to replicate this in our prices equation, we need to get something similar to $S_t-S_{t+1}$ from $S_t$ only, taking advantage of lags because if we just sum $-S_{t+1}$ in both sides of the equation we would've done nothing. Then you can check that:
$$S_t-S_{t+1}=(1-\beta\theta L^{-1})S_t=\tilde{MC}_t$$
With that in mind, we just need to multiply both sides of our prices equation by $(1-\beta\theta L^{-1})$, and develop the corresponding algebra for arriving to:
$$\tilde{P_t}-\beta\theta E_t\tilde{P_{t+1}}=(1-\theta)(1-\beta\theta)\tilde{MC}_t-\theta^2\beta\tilde{P}_t+\theta\tilde{P}_{t-1}$$
Which allows us to make better analysis by getting rid of the series, and therefore being able to write in terms of matrices and vectors that are more &quot;readable&quot;. I'm pretty sure you can do this same process with your equation that is in term of inflation (instead of prices). Hope this works.