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I am implementing a program to fit an ARMA-GARCH model to given data.

My model parameters are optimised by maximizing the Maximum Likelihood function using a nonlinear algorithm.

The algorithm requires an initial set of parameter values to start from, and I noticed by looking at other GARCH tools out there that those initial parameters have a huge impact on the result.

How can I choose those initial parameter values for my model?

I know for instance that for an AR model we can use Yule-Walker equations or OLS, but I am not sure about an ARMA-GARCH process.

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    $\begingroup$ What do you mean by "GARCH/ARMA"? Is it an ARMA-GARCH model where cond. mean is ARMA and cond. variance is GARCH? Or is it GARCH and/or ARMA models? If it is ARMA-GARCH, you may consider changing to that. $\endgroup$ – Richard Hardy Nov 24 '15 at 18:21
  • $\begingroup$ Have you tried looking at the documentation of statistical software for ARMA-GARCH modelling? There should be some references there. Or if there are no specific references, you would be able to check how the procedure actually works (at least R should be transparent, maybe not the commercial software). If I remember correctly, for a GARCH(1,1) model one of the R packages (perhaps "fGarch") used the following: $\alpha_1$=0.1, $\beta_1=0.8$, $intercept=\frac{\hat\sigma^2}{1-\alpha_1-\beta_1}$ or something similar; here $\hat\sigma^2$ is the empirical unconditional variance. $\endgroup$ – Richard Hardy Nov 24 '15 at 18:48
  • $\begingroup$ Thanks @RichardHardy. I have looked into the docs of rugarch but there is no mention of it. I have looked at fgarch but they say "ar" and "ma" are set to 0, which is from simple tests not a good idea at all. Rugarch seems to work better, and I can definitely look at the source but I am more interested in the theoretical work that this has been built on. The goal is not to mimick the behavior or rugarch (or other libraries) but to provide a good implementation of a garch algorithm that I understand and can maintain because I understand the theory behind. $\endgroup$ – Ihab Nov 25 '15 at 9:10
  • $\begingroup$ OK, then academic references could be useful for you. Unfortunately, I do not remember any good ones now. Good luck in your search! $\endgroup$ – Richard Hardy Nov 25 '15 at 10:01
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You can set the starting values for the conditional variance/mean values to their theoretical unconditional values. For errors you could set them to zero (because their mean = 0). For other AR,MA parameters you can solve the Yule-Walker equation, for garch parameters to the best of my knowledge you need to fix them to some arbitrarily values BUT they should respect the theoretical constraints ( positivity constraints, stationarity constraints...). Ex for Garch(1,1) , constraints: alpha + beta <= 1 ...


EDIT : For Initial values of the ARMA part you can have a look to the following article : (paragraph "16.2 Initial values" ):

A Package for Estimating, Forecasting and Simulating Arfima Models: Arfima package 1.0 for Ox / By Jurgen A. Doornik and Marius Ooms

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