I have a simple transition model I am trying to use to predict the probability of two states.

$$ \begin{bmatrix} p_{1,t+1}\\ p_{2,t+1} \\ \end{bmatrix}= \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \\ \end{bmatrix} \begin{bmatrix} p_{1,t}\\ p_{2,t} \\ \end{bmatrix} $$

I compute $p_{1,t+1},p_{1,t+2},p_{1,t+3},p_{1,t+4...}$ the predicted probability of a state I am interested for $T$ periods out in the future. How do I compute the prediction confidence interval for such a probability $p_{1,t+1}$?

From what I understand the probability $p$ is binomial distributed and would have the following confidence interval. But I am unsure what $n$ would be in this confidence interval, if it indeed it was the correct confidence interval. My data is panel ($i$ individuals, and $T$ time periods), so I am not sure what $n$ would be.

$$\hat{p}\pm z_{1-\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$



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