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Calculate Transition Probabilities Interest rate data

I came across a paper by Rodda (2004), who simulates interest rates with a Markov sequence. To simulate changes in the interest rates, they used the historical transition probabilities. Their ...
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Moment conditions for MS GARCH (Haas et al.)

I am doing some research on MS GARCH, specifically the model proposed by Haas et al., however I am really stuck on the moment conditions derived in the appendix of the paper. Right before equation ($...