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I am trying to estimate the parameters of the GARCH(1,1) model with MCMC method, firstly, I read the paper:

http://mpra.ub.uni-muenchen.de/12985/1/MPRA_paper_12985.pdf

Metropolis Hasting method is used in the article, but the sampler of parameters, I mean the constant, arch parameter and garch parameter for the conditional variance, are sampled from distribution built by an auxiliary distribution. My question is that, as we could build the likelihood function, then why we need to sample from the auxiliary distribution but not sample from a normal distribution and implement the random walk Metropolis Hasting algorithm.

Actually, I have tried the random walk algorithm but all parameters can not convergent, i do not know the reason. But if I fixed two parameters, supposed we know them, then the unknown parameter can be estimated by random walk Metropolis Hasting algorithm well.

New in this field, thanks.

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  • $\begingroup$ The Metropolis-Hastings step is that they have to ensure that alpha and beta are positive. I can't speak much more to this particular paper. I usually fit Garch with MLE because I have sufficient data. MC Stan has a good example on fitting Stochastic Volatility models in its manual that you might check out. $\endgroup$ – John Jan 28 '15 at 15:05
  • $\begingroup$ @John, thanks so much. Actually, I can ensure that alpha and beta are positive in the random walk as they are sampled from a normal distribution and I can reject the negative ones. By the way, could you recommend me a method in MLE? I mean an algorithm to find the optimization point, I first tried Newton's method but I hate second partial, especially when it comes to the multivariate GARCH models, it is annoying to me. And I also tried gradient method, which only requires first order, do you have some method that can be useful in fit these model but do not need have second or high order? $\endgroup$ – Fly_back Jan 28 '15 at 19:22
  • $\begingroup$ Rejecting the negative ones is Metropolis-Hastings. For MLE, you might look at the source code for Kevin Sheppard's MFE toolbox for Matlab. You can look at his implementation of multivariate Garch there as well. Alternately, fGarch or rugarch for R. $\endgroup$ – John Jan 28 '15 at 19:30
  • $\begingroup$ Rejecting negative also could be used in the random walk, for example, alpha + x, x is sampled from a normal distribution, if alpha + x is negative, reject and re-sample till it is positive, true? $\endgroup$ – Fly_back Jan 28 '15 at 19:34
  • $\begingroup$ I would distinguish between Gibbs sampled MCMC and Metropolis-Hastings MCMC. Gibbs sampled MCMC (what I assumed you meant by random walk) does not do a rejection step the way that Metrpolis-Hastings does. $\endgroup$ – John Jan 28 '15 at 19:38

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