Hidden Markov model MS GARCH

Question

I am currently working on the MS GARCH model proposed by Haas paper given below

$\epsilon_n=Z_n \sigma_{\Delta_n,n}$ where $\epsilon_n$ is a time-series of residuals, $\left\{Z_n ,n\in \mathbb{Z}\right\}$ is a sequence of normal independent and identically distributed random variables with mean zero and unit variance, and $\left\{ \Delta_n,n\in \mathbb{Z}\right\}$ is a Markov chain.

The $k \times 1$ vector $\boldsymbol{\sigma}_n^2=[\sigma_{1n}^2,\sigma_{2n}^2,\ldots,\sigma_{kn}^2]'$ of regime variances follows the multivariate GARCH(1,1) equation \begin{equation} \label{2.2} \boldsymbol{\sigma}_n^2=\boldsymbol{\alpha_0}+\boldsymbol{\alpha_1}\epsilon_{n-1}^2+\boldsymbol{\beta}\boldsymbol{\sigma}_{n-1}^2\,, \end{equation} where $\boldsymbol{\alpha_i}=[\alpha_{i1},\alpha_{i2},\ldots,\alpha_{ik}]'$, $i=0,1$; $\boldsymbol{\beta}=diag(\beta_1,\beta_2,\ldots,\beta_k)$.

Now I am trying to link the underlying latent Markov chain to the theory of hiddem markov models. In the theory of HMM I am finding that the observations sequence should be independent. In my case the observation sequence is the residuals $\epsilon_n$ which even though they are uncorrelated, they are not independent.

Can this assumption of independence be eliminated in some way? Or am I making some mistake by joining HMM and MS GARCH?

Answer 1

In the theory of HMM I am finding that the observations sequence should be independent.

Typically for HMMs, conditioning on the states, the observations are conditionally independent. In the case of your model, it is not true, as $$ p(\epsilon_1, \ldots, \epsilon_n \mid \Delta_1, \ldots, \Delta_n) \neq p(\epsilon_1 \mid \Delta_1) \times \cdots \times p(\epsilon_n \mid \Delta_n). $$

For your model, the distribution of $\epsilon_t$ depends on the previous values, as well as the contemporaneous hidden state. So this isn't a typical example of a HMM, although I wouldn't object to it being called that.

Can this assumption of independence be eliminated in some way?

If you changed the GARCH part of your model to an ARCH part, and if you treated the $\sigma^2_t$ process as being part of the state, there would be conditional independence. The $k+1 \times 1$ state vector could be $$ \left[ \begin{array}{c} \Delta_t \\ \boldsymbol{\sigma}_t^2 \end{array}\right]. $$ This would give you a latent first-order Markov process, although its state space is not finite, which is another typical assumption of what are called Hidden Markov Models. Also note that the $\boldsymbol{\sigma}_t^2$ is totally deterministic, which isn't typical either, for these sorts of models.