Let's say I have a $GARCH(2, 3)$ model with $$\nu_i = \sigma_i\epsilon_i$$ where $\epsilon_i \sim N(0, 1)$ and $$\sigma_i^2 = a_0 + \sum\limits_{k = 1}^{2} a_k\sigma_{i - k}^2 + \sum\limits_{l = 1}^{3} b_l\nu_{i - l}^2.$$ I know that there's the bayesGARCH package in R that can handle GARCH(1, 1) models with student-t innovations but this is $GARCH(2, 3)$ and, although for implementation, I also want to understand it.

I collected 250 data points of the process and now want estimate the parameters $a_0, a_1, a_2, b_1, b_2, b_3$ with the help of the Metropolis-Hastings algorithm. In other words, I want to draw samples from the posterior distribution of each parameter $P(\cdot \vert y)$, either with independence kernel or random walk kernel. With the random walk kernel one would in the first iteration , e.g. for $a_0$, calculate $a_0^{proposal} = a_0^{start} + x$ where $x$ is drawn from some symmetric distribution around the origin, e.g. normally distributed with mean zero and $\sigma_{a_0} = 0.003$.

In the next step, I would calculate the acceptance ratio which in this case equals the ratios of the target densities evaluated at the proposal and current value respectively and compare it to the drawn value of a uniformly distributed random variable - if it is smaller than the ratio I would accept $a_0^{proposal}$ or stay with the previous value otherwise. This is how I understood the MH algorithm with random walk kernel.

The problem I have is the actual implementation in the GARCH setting. The two most important questions I have are the following:

  • What is the actual posterior (target) density necessary to compute the acceptance ratio? As posterior it should be proportional to $P(y\vert a_0) \cdot P(a_0)$ yet what is the prior for $a_0$? Or is it just the likelihood?
  • Am I allowed to do the algorithm for each parameter in turn or do I have to do it as one entire parameter vector, i.e. $A = (a_0, a_1, a_2, b_1, b_2, b_3)$ and $A^{proposal} = A^{start} + x$ where $x$ now is multivariate normally distributed, having a covariance matrix and $\bf{\mu} = 0$?

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