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YouTube T2-26: Maximum likelihood estimation of GARCH parameters

Nicole Seaman

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GARCH(1,1) is the popular approach to estimating volatility, but its disadvantage (compared to STDDEV or EWMA) is that you need to fit three parameters. Maximum likelihood estimation, MLE, is an immensely useful statistical approach that can be used to find "best fit" parameters. In this video, I replicate John Hull's example (the data is S&P 500 index values) to find the best fit alpha (α), beta (β), and omega (ω). Keep in mind that the long-run variance = ω/(1 - α - β), such that indirectly this is solving for a long-run (aka, unconditional) variance.

David's XLS is here: https://trtl.bz/2NlLn7d

In practice I tried to find the best lambda parameter for EWMA using MLE, but it turned out that the function I wanted to maximize didn’t have an extremum on any lambda less than 1. It was simply growing as lambda increased. So as my Excel solver just kept the solution for lambda as close to 1 as possible which didn’t look too logical. Does this example show that MLE can sometimes don’t work?