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As suggested in Calculating log-likelihood for given MLE (Markov Chains) I want to perform a likelihood ratio test for two fitted models (i.e., first and second order markov chains). Simply comparing the resulting log-likelihood values is as suggested in the other thread not enough. I know how to calculate the likelihood ratio, but I am unsure about how to determine the statistical significane.

I need the degree of freedoms for both models. What exactly is this in the case of my Markov Chain MLEs. The number of non-zero probabilities in the MLE, or the number of states?

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Is there any structure to the model or are you estimating the distribution of $Y_{t}| Y_{t-1} = i, Y_{t-2} = j$ independently for every combination of $i,j$.? – Macro Jan 18 at 16:06
@Macro What do you mean exactly by structure? Could you give an example? – ph_singer Jan 19 at 15:10
I am basically asking whether you are building a model for $Y_t | Y_{t-1} = i, Y_{t-2} = j$, e.g. $P(Y_t) = f(Y_{t-1}, Y_{t-2})$, where you estimate $f$; a simple example would be a series of ordinal (if $Y_t$ is ordinal) or multinomial logit regression models. Or, are you just estimating each of the probabilities independently? In that case, if there are $m$ states, I think you would be estimating $m^2 (m-1)$ parameters, since there are $m^2$ possible combinations of the $t-1,t-2$ states and $m-1$ cell probabilities to estimate (since the $m$'th one is determined by the others). – Macro Jan 19 at 20:56
@Macro Sorry for the late response. I am estimating each of the probabilities independently. So it should be the latter case. I am reducing a higher order MC to a k=1 MC, so this would be the case as well, right? But, $m$ states would always refer to the number of states of the input data, or would it change for the new states I create for my reduced model? Because, the reduced higher order MC has a higher states space $S'$. – ph_singer Jan 29 at 10:43

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