# Maximum likelihood estimation, how to derive the hessian

I am reading a paper and trying to understand how the authors estimated the standard errors of a set of parameter estimates $[\delta \ \ \phi \ \ \Sigma]$. Below is the loglikelihood function (sorry I could not typeset): enter image description here where,

Using the trace trick and the rules of differentiation I managed to derive the first order conditions (after reading this). The authors provided the final results, which helped to verify my results. But I have been unable to derive the second order partial derivatives matrix i.e. Hessian. The authors do not derive the Hessian. I want to estimate the standard errors of the parameter estimates using the Hessian.

Will anyone spare a few minutes to help?

If you know any other method for estimating the standard errors of the parameter estimates, do please explain or provide reference.

• +1. But are you sure you need to do this derivation? Isn't a numerical approximation when optimizing adequate? Also what is the "this" that helped you derive first order conditions after reading it? Did you mean to link something? – usεr11852 Jul 23 '16 at 16:40
• @usεr11852 If a Quasi-Newton method, e.g., BFGS, is used for the optimization, the final Hessian (or inverse) Quasi-Newton Hessian could be complete rubbish (as the British say) as an estimate of the true Hessian, even though it may work quite well for purposes of the optimization algorithm. – Mark L. Stone Jul 23 '16 at 17:59
• Yes you are right, I've used iterative process to estimate the parameters. However, I want to calculate the standard errors of the parameters. But I do not know how to derive the Hessian. I updated the "this" it is a link. – mr.rox Jul 23 '16 at 19:35
• If you can afford the computation, use of bootstrap or (leave one out) jackknife to estimate covariance matrix (or standard errors) of estimated parameters may be more accurate, due to avoiding linearization inherent in use of (inverse) Hessian. – Mark L. Stone Jul 23 '16 at 20:40
• @MarkL.Stone, great, many thanks! I think I can afford a few days of waiting. I will try bootstrapping data with replacement. I only need to standard errors of each coefficient estimates. Thanks! – mr.rox Jul 24 '16 at 18:10