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I am interested in the correct way to estimate a GARCH/ARMA-GARCH model.

I will refer to the coefficients as:

ARMA-GARCH:    
ret = const + ar1 + ma1
residuals = y - ret
sigma_2 = alpha + arch1 + garch1

GARCH:    
residuals = y
sigma_2 = alpha + arch1 + garch1

First: Initialisation. There is a range of information on how to initialise variables in the GARCH framework. One common way is to set sigma_2 at time 0 to the long run variance: alpha / (1 - arch1 - garch1). Is this preferred? What is the preferred initialisation procedure for returns and residuals? For the optimisation procedure, is there a 'correct' way to choose the starting coefficients? I am currently pulling from a uniform distribution as follows: U(0.1) * 0.5. This seems to work OK.

Second: Constraints. I imagine the only constraint is that arch1 + garch1 <= 1? Are there any other constraints to be aware of?

Third: Bounds. I have set the bounds for arch1 and garch1 as (0.0001,1), in the format of (lower_bound,upper_bound), giving alpha a boundary of (0.0001,None). In the ARMA-GARCH specification, are there any bounds for the constant, AR term, or MA term?

Thank you.

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In a GARCH process, $\varepsilon_t = \sigma_t z_t$ with $z_t \sim iid(0,1)$ with

$$ \sigma_t^2 = \omega + \alpha \varepsilon^2_{t-1} + \beta \sigma^2_{t-1} $$

it is typically assumed $\omega\geq 0 (> 0)$ , $\alpha \geq 0$ and $\beta \geq 0$ to ensure a positive variance. The condition $\alpha + \beta < 1$ is imposed to ensure stationarity.

Regarding initialization, you may want to look at Initial value of the conditional variance in the GARCH process

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