Trouble understanding hamilton 1996 specification testing in MS
In my question, I rewrite the joint probability as the product of the conditionals probabilities but I wrongly implemented the markov assumption of this model. The markov assumption is:
$$
p(s_{t}=j|s_{t-1}=i,\Omega_{t-1})=p(s_{t}=j|s_{t-1}=i)=p_{ij}
$$
But, when conditioned to information available at date t
$$
p(s_{t}=j|s_{t-1}=i,\Omega_{t})\neq p(s_{t}=j|s_{t-1}=i)=p_{ij}
$$
Meaning that

$$
p(s_{t}=j,s_{t-1}=i|\Omega_{t})\neq p(s_{t}=j|s_{t-1}=i)p(s_{t-1}=i|\Omega_{t})
$$