# Hidden Markov model MS GARCH

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?

• you haven't described what $\sigma_{\Delta_n,n}$ is – Taylor Mar 21 '18 at 21:37
• Yes, it is the square root of one of the elements of the vector $\boldsymbol{\sigma}_n$. We have k elements which can take the value of $\sigma_{\Delta_n,n}$, and it is chosen from these k elements depending on which regime we are in at time $n$. – Anna Mar 21 '18 at 21:43
• okay, I understand now – Taylor Mar 21 '18 at 21:46

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.
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.
• I do not understand why removing the GARCH part helps, since actually there is still the dependence on $\epsilon_{n-1}$ which depends on the regime at time $n-1$. and what do you mean treat the $\sigma_t^2$ as being part of the state? – Anna Mar 22 '18 at 6:34