I attempt to fit the following probit model to a time series where we observe the binary variable $R_{t}$ and another variable $X_{t}$, a latent unobserved variable $y^{*}_{t}$ and a state variable $s_{t}$: $$ y^{*}_{t} = -c_{0t} -c_{1t}\,X_{t-k} + u_{t} $$ where u_{t} is (0,1) normally distributed and $R_{t} = 1$ if $y^{*}_{t} \le 0$ and $R_{t} = 0$ otherwise. $c_{it}$ depends on the current state $s_{t}$ which can take either 0 or 1 values, such that $$ P(R_{t} = 1 | s_{t}, X_{t-k}) = N(c_{0t} +c_{1t}\,X_{t-k}) $$ See for example this paper page 47. Now my problem is that I do not find any derivation of how to estimate the parameters. It should be straightforward and easy to solve, but I do not find anything related to probit but only (vector) autoregressions with Markov switching. Any ideas?
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Estimating a hidden Markov model is not straightforward. Look for "particle filters" and "SMC methods" within the Bayesian literature.
This is a probit model with hidden Markov dependence: the distribution of $R_t$ conditional on $X_{1:t}$ is Bernoulli with probability $$\mathbb{P}(R_t=1|X_{1:t})= 1-\mathbb{P}(R_t=0|X_{1:t})=p(X_{1:t})=\Phi(c_{0t}+c_{1t}X_{t-k})$$
What you omitted in the definition of your model is the dependence on the hidden state $s_t$: it should read $$y^{*}_{t} = -c_{0t}(s_t) -c_{1t}(s_t)\,X_{t-k} + u_{t}$$resulting into the probit model$$\mathbb{P}(R_t=1|X_{1:t},s_t)= 1-\mathbb{P}(R_t=0|X_{1:t},s_t)=p(X_{1:t},s_t)=\Phi(c_{0t}(s_t)+c_{1t}(s_t)X_{t-k})$$ Once again, this is far from a trivial problem.
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$\begingroup$ My problem is that I need to somehow back out unconditional transition probabilities for each point in time. The paper just says to choose as transition value which is "appropriate" though. $\endgroup$ – user3673486 Apr 26 '15 at 14:26
