## Estimating model for transition probabilities of a Markov Chain

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Suppose that I have a Markov chain with $$S$$ states evolving over time. I have $$S^2\times T$$ values of the transition matrix, where $$T$$ is the number of time periods. I also have $$K$$ matrices $$X$$ of $$T\times S$$ values of (independent) variables, where $$K$$ is the number of variables that I can use to explain the transition probabities ($$p_ij$$ are my dependent variables and the matrices $$X_k$$ are the independent variables).

Remember that $$\sum_j pij=1$$ for each $$t$$.

In the end, I am looking for panel models to explain the transition probabilities, where the parameters are constant over time and (with maybe exception the constant) the parameters are also constant over diferent transition probabilities.

Just to be clear... Consider the following example... Imagine that an animal prefers to stay in places that there are food and water. Let the $$T\times S$$ matrix (X_F) the matrix that tells the amount of food in each place $$s\in S$$ and in each time $$t\in T$$ and (X_W) the matrix that tells the amount of water in each place $$s\in S$$ and in each time $$t\in T$$.

I want to use $$X_F$$ and $$X_S$$ to explain the transition probabilities. I do have the values of the transition probabilities over time and I want to use these matrices to explain their values.

I think that I design a kind of fixed effect logit model for each state in $$S$$. However, I would have to estimate $$S$$ logit models. I believe that the probabilities $$p_{ij}$$ and $$p_{ji}$$ should not be estimated in different models, since they seem to be related.

Any hints? Are there solutions in the literature of such kind of problem?