# Markov property of manifest variables in longitudinal ESM

I am working on a longitudinal ESM model were the indicators are (highly) autocorrelated. This means that the classic cross-lagged models of panel data analysis cannot be used directly. I have explained the main problem here (although my proposed approach to model the sequence was very inefficient.) And I am going to discuss a one-level model (capturing the within-person variability of one individual) here to avoid further complications.

I have seen several papers and tutorials on dynamic structural equation models (DSEM) like (Hamaker et al. 2018) and (Asparouhovet al. 2018) and Mplus' learning material on DSEM (here). However, mostly in order to implement it in R (using lavaan) and to avoid limitations of their approaches, I decided to assume Markov property of lag one ($$MP(1)$$) which allows significant simplification (while there is suggestive evidence supporting it for the type of data I'm analyzing).

To illustrate my model, suppose that at each time point $$t$$, I have a vector of manifest variables $$Y_{t_{n \times 1}} = [y^1_{t}, y^2_{t},..., y^n_{t}]^T$$ and a vector of latent variables $$\Theta_{t_{m \times 1}} = [\theta^1_{t}, \theta^2_{t}, ..., \theta^m_{t}]^T$$, which are related by a loading matrix $$\Lambda_{t_{n \times m}}$$ such that $$Y_t = \Lambda_t \Theta_t + \Delta_t$$

Where $$\Delta_{t_{n \times 1}} = [\delta^1_{t}, \delta^2_{t}, ..., \delta^n_{t}]^T$$ is the vector of measurement errors of items.

I assume that my model has $$MP(1)$$ at the factor level, meaning that $$P(\Theta_{t}|\Theta_{t-1}, \Theta_{t-2},..., \Theta_{1}) = P(\Theta_{t}|\Theta_{t-1})$$. However, the manifest variables (at the item level) are (highly) autocorrelated, meaning that I cannot basically assume error independence, so in general $$corr(\delta^i_{t}, \delta^j_{t}) \neq 0$$. As a result, I have to take the autocorrelation of errors into account.

My solution was to make a new time series of consecutive observations $$\Upsilon_{2n \times 1} = [Y_{t-1},Y_{t}]^T$$, and as a result, I will have a new vector of factors $$\Omega_{2m \times 1} = [\Theta_{t-1}, \Theta_{t}]^T$$. I further assume measurement invariance accross time (i.e., $$\Lambda_t = \Lambda$$). Hence, I will have a new loading matrix $$\Pi_{2n \times 2m} = \begin{bmatrix} \Lambda & 0 \\ 0 & \Lambda \end{bmatrix}$$.

However, this formulation relies on the $$MP(1)$$ assumption, both at the factor level and at the item level. In other words, I have generalized the $$MP(1)$$ assumption from factor level to the item level, which might not be true.

So, my question is, am I actually allowed to assume $$MP(1)$$ at the item level given that $$MP(1)$$ is the case at the factor level? How do I justify this choice and/or derive the independence of $$Y_{t-2}$$ and $$Y_{t}$$ implied by this assumption analytically?