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I have financial data (return) and use the following model:

returnmodel

where $R_t$ is the return at time t, $\mu$ is set to zero, $\sigma$ is the volatility and $\epsilon$ is an innovation process.

The $\sigma^2$ is estimated with a volatility model. The distribution of the innovations is mostly Gaussian, more advanced models use e.g. generalized hyperbolic distribution.

My question is now, how to estimate the parameters of the models? For example, consider the easiest combination: The volatility at timepoint t is estimated with a simple moving average, which uses exponentially weights. The innovation process is said to be Gaussian. I do the estimation as follows:

  1. I calculate the volatility at each time point using the original return series.
  2. I calculate the standardized residuals ($R_t/\sigma_t)$ and use these values to estimate a normal distribution using ML.
  3. I look at the estimates of ML, if the model is correct specified, the mean should be zero and the standard deviation/variance equal to zero, because the standard model assumes the innovation process to be Gaussian with mean zero and Variance of one.
  4. Likely this will not be the case, so I use another model, e.g. the generalized hyperbolic distribution and hopefully see, that the generalized hyperbolic distribution seems to match the first four empirical moments better than the Gaussian distribution

So in this case, the procedure would be the same

but

If I assume the volatility process to follow a GARCH process and the innovations to follow a certain distribution, e.g. Gaussian, generalized hyperbolic etc. do I have to do jointly estimation? So that I use a method of R which does the estimation of the volatility process and the innovation process?

Or

Can I do the following, according to the steps above: 1. Calculate the volatility using a GARCH process. 2. Use the standardized residuals ($R_t/\sigma_t$) to do ML estimation of the distribution

Will this "single step" estimation procedure lead to the same estimates as the "jointly" estimation procedure, which most GARCH procedures do?

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See page 162 of Quantitative Risk Management by McNeil et al.:

  1. You can do a jointly ARMA-GARCH estimation. This would be the "most correct" way. So any futher parameters of the innovation distribution would be estimated simultaneously int he model fitting.

  2. Or you can use a QML approach: Estimate the volatility using e.g. gaussian assumption and then use the residuals to estimate an appropriate innovation distribution.

If you doing EWMA and single step estimation, this would be like the second approach.

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