Consider a simple stochastic dynamic programming growth model Bellman,

$$V(K,Z)=\ln(C)+\beta\mathbb{E}[V(K',Z')|Z]$$ where $$C=ZK^\alpha+(1-\delta)K-K'$$ and $$Z''=\rho_1Z'+\rho_2Z+\phi\varepsilon'+\varepsilon'';\varepsilon''\sim N(0,\sigma^2).$$

Notice that the innovations/shocks follow a higher order stochastic process, with a moving average component. This could really be any structure, and my question remains the same.

If we use a standard process to solve for the Euler equation (forgive my abuses of time subscript notation), we have $$\frac{1}{C}=\beta\mathbb{E}_t[\frac{\alpha Z'K^{\alpha-1}}{Z'K'^\alpha-K''}].$$

And by the method of undetermined coefficients, we can conjecture the policy function to be $K'=sZK^\alpha \implies K''=sZ'K'^\alpha $. Substituting this into the EE we have, $$\frac{1}{C}=\beta\mathbb{E}_t[\frac{\alpha Z'K^{\alpha-1}}{Z'K'^\alpha-sZ'K'^\alpha}]$$

which simplifies to, $$\frac{1}{C}=\beta\mathbb{E}_t[\frac{\alpha Z'K^{\alpha-1}}{Z'K'^\alpha(1-s)}].$$ From which we can simplify to, $$\frac{1}{C}=\beta\mathbb{E}_t[\frac{\alpha}{K'(1-s)}]$$ which is non stochastic in the expectation. I won't continue, as I think I can ask my question now.

Question: Most if not all of the DP problems I have seen have assumed either AR1 or iid shocks. I know that with a higher-order Markov process (I think it's called this) that you need to redefine the state space such as the one from above: $$Z''=\rho_1Z'+\rho_2Z+\phi\varepsilon'+\varepsilon''$$ becomes some first order vector autoregressive process of the AR(2) process (again forgive the switching notation) $$A_{t+1}=HA_t+\phi E_{t+1}.$$ where $A_{t+1}=[Z_{t+1}\quad Z_t]'$ and $H$ contains the coefficients of past shocks.

I don't see in my short derivation above where I would actually need to complicate things with the VAR. We make no assumptions about the stochastic process before Z' cancels out and we would need to take an expected value of it. It seems to me that the standard approach may work with higher order processes, but my intuition (maybe false) is telling me that it can't be that simple. Am I making a false assumption about the stationarity of the value function? If I am fine to do it using the standard approach from advanced macro, then what assumptions must we make about the stochastic process for this to be valid? Any ideas would be great, Thanks!