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I am looking at how well GARCH, GJR-GARCH and EGARCH capture the volatility dynamics before, during and after the financial crisis. I also compute out-of-sample forecasts and examine if there is one model that does well in all samples or not.

However, I'm kind of stuck in my search for a benchmark. I do the same analysis for stock returns and I use $y_t = y_{t-1} + \varepsilon_t$ as my benchmark. However, for volatility and I do not know what to use because I have daily data and I would like to stick with this frequency. I know that realized volatility is $$ RV=\sum_{i=1}^tr_t^2 $$ but I do not know how to create a plot/benchmark (like a random walk for stock returns) to compare with the models that I picked.

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It seems you want to measure the true underlying variance (volatility) to be able to assess how well this volatility is captured by your models. The problem is that volatility is not directly observed and you do not have other (high-frequency) data from which you could create a good proxy for the daily volatility (such as using realized variance from the high-frequency data to proxy for the true variance of any given day in your daily data).

As far as I know, there is no solution to your problem. That is, technically it is impossible to measure the underlying daily variance when you only have one observation per day. You have a daily series of $T$ days where the variance may be different on each day and there may be no connection between the variance on day $t$ and the variance on day $s$ for any $(t,s)\in 1, \dots, T$ where $t\neq s$. Thus effectively you can treat your series as $T$ separate random variables, unless you are willing to assume some structure that connects them. And it is clearly impossible to measure the variance of a random variable from just one realization of that variable, without making further assumptions.

If you were willing to make some assumptions, you could start from assuming that the expectation is zero on each day. Then you would be technically able to calculate the sample variance which would be, on any day $t$, $$ \widehat{Var}_t(x_t) = (x_t-0)^2 = x_t^2. $$ This would be a very noisy measure of variance, so it would not be a "good" proxy for the true variance. Unfortunately, you can only get this far without making further assumptions or collecting data at a higher frequency.

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  • $\begingroup$ Indeed I know calculating the daily volatility this way will result in a noisy measure of variance, but is the best I could do with the data I had on my hands. The procedure I followed was to regress my series on a constant and square the residuals from this regression. After fitting the volatility equation they look similar which sometimes is just enough. $\endgroup$ – Alex Aug 7 '17 at 11:14
  • $\begingroup$ I see. I would just be careful with interpretations of modelling results which might be affected by the noisy measure. Just use a good dose of common sense there. $\endgroup$ – Richard Hardy Aug 7 '17 at 13:37
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  1. You could have a look at this publication, the models are compared by AIC and loss functions. Seems to me the author did pretty much what you are doing with different data / time frames.

  2. Also Tsay's (2010) Analysis of Financial Time Series is worth checking out when applying GARCH models. In a few of the examples Tsay uses in his book he plots the conditional volatility from various GARCH models vs a windowed or also known as rolling standard deviation of the return series. He uses 23 or 69 day Windows for an approximate trade month / quarter. He also gives code examples on how to estimate and plot the models.

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