Assuming that $x_t\sim N(0,\sigma_t^2)$, where $\sigma_t^2=\mu+\beta\sigma_t^2+\alpha x_{t-1}^2$

In this case, what is the likelihood function given the sample data $(x_1,x_2,\ldots,x_T)$?

I understand how to solve the typical MLE for a normal distribution by referring to this https://www.statlect.com/fundamentals-of-statistics/normal-distribution-maximum-likelihood but I cannot seem to get the idea on how to solve one with a $\sigma_t^2$ related to $x_{t-1}^2$.

Any help would be appreciated. Thank you.

  • $\begingroup$ Here is a MathJax tutorial for typesetting. $\endgroup$ – StubbornAtom Jul 2 at 7:32
  • $\begingroup$ It looks like there must be a typo at the outset, because you attempt to define $\sigma_t^2$ in terms of itself. Please confirm or fix that. $\endgroup$ – whuber Jul 2 at 13:15

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