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Why Random-Walk Metropolis Sampling Struggles with GARCH(1,1) Parameters

Article Quant Q&A · Author: Fly_back
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Summary

The document raises a question about estimating the constant, ARCH, and GARCH parameters of a GARCH(1,1) model with Markov chain Monte Carlo. It asks why a cited approach samples parameters using an auxiliary proposal distribution instead of a normal random-walk Metropolis–Hastings proposal when the likelihood can be evaluated directly.

The author reports that a random-walk sampler did not converge when all parameters were estimated together, but appeared to work when two parameters were fixed and only one was sampled. This observation suggests that joint parameter dependence or proposal design may make the multi-parameter problem harder, though the document does not establish the cause. It provides no diagnostics, parameter constraints, tuning details, or simulation evidence, and presents the issue as an open question rather than a worked solution.

Key ideas

  • The question concerns joint MCMC estimation of the constant, ARCH, and GARCH terms in a GARCH(1,1) model.
  • A normal random-walk Metropolis–Hastings proposal may behave differently from an auxiliary proposal distribution.
  • The author reports difficulty sampling all parameters jointly but success when only one parameter is unknown.
  • The document does not diagnose the convergence problem or provide a tested remedy.

Tags

StatisticsVolatilityMachine learning
Full text
# Why random walk Metropolis Hasting algorithm works bad on GARCH(1,1) parameters estimation


# Why random walk Metropolis Hasting algorithm works bad on GARCH(1,1) parameters estimation












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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