Starting values in numerical algorithms?

I am estimating an ARCH(1) model, not to difficult apart from one small problems, which starting value should I use for the estimation. I am estimating it long hand, so understanding the minor details are important here.

Basically, I have read that,

"All numerical algorithms require starting values $$\theta_{(0)}$$""

I have no idea how to find (or estimate) or pluck out of thin air, the starting values for the coefficients.

Any suggestions would be greatly appreciated?

• while $\theta_0$ theoretically should be close to the true parameters $\theta$ you have to take an educated guess in practice, since $\theta$ is unknown. as a naive approach if you don't have any prior informations: why don't you just try a range of different starting values and see if the sequence of estimators from the optimization algorithm always converge to the "same" final estimator? (keep in mind that if you know that the log likelihood function is strictly concave, then the maximum likelihood estimator will be unique.) Jan 30, 2019 at 12:44