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I'm trying to estimate daily volatility of an index I created. I do not wish to forecast, simply estimate the volatility for a period of about 500 days. My problem is that the sometimes, my results/estimates with a GARCH(1,1) model is wildly different than an EWMA model with lamda=0.94

Is there a way of knowing which estimates are better?

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The problem is non-trivial because volatility is not perfectly observable and you cannot simply compare the value of your estimator to another, "true" value. However, you can engage a general method of seeing whether a particular complex model captures the distribution of the data correctly. This method is related to Fisher's test used in meta-analysis.

Suppose your model is saying: conditional on the set of predictors $\{X_{1t},...,X_{pt}\}$, random variable $Y_t$ is distributed with cumulative distribution function $F(y\ |\ X_{1t},...,X_{pt})$, for any $t \geq 1$. Then it must be the case that

$\ \ \ \ \ \ \ \ \ \ \ F(Y_t\ |\ X_{1t},...,X_{pt})\ \ $ ~ $\ \ U([0,1])$.

For each candidate model, you can calculate the transformed values $F(Y_t\ |\ X_{1t},...,X_{pt})$ and see how close their distribution is to the uniform distribution. A good thing about this type of diagnostics is that it allows you to identify problems in different parts of the model. For example, you could see the model doing a good job in the middle but not in the tails, or the other way around.

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  • $\begingroup$ Hey thanks for the answer! This might be a stupid question, but why is volatility not observable? $\endgroup$ Mar 29 '18 at 2:16

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