# Constraints, bounds, and initialization variables in the GARCH / ARMA-GARCH models

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.

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.