I still haven’t gotten the likelihood function to behave properly. The problem is the first parameter. It’s essentially the one representing average or “base” volatility. The problem is that the likelihood function as I have it is an increasing function of the first parameter.

This means that the higher I make the first parameter (unbounded), the better the model fits, which doesn’t seem right.

It took a while to realise what’s the underlying problem, tha the distriubtion function of the standard normal has its only maxima at 0. So in one sense, the smaller closer the value, the less likely. We have y_t = e^{\frac{h_t}{2}}\mu_t. If h_t is horrifically big, and y_t is not too big, we need \mu_t to be quite close to 0. So the bigger h_t is, the closer \mu_t has to be to 0, the higher the likelihood.

Something is wrong here.

P.S. Found the page describing how to incorporate \TeX into WordPress!

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