How to implement dummy variables into GARCH(1,1) model from structural breaks (ICSS)

Question

Hello everybody,

I was already searching a lot of forums and read a huge amount of different papers. But I guess I am to stupid or I am at a loss. Hopefully some of you are able to help me out. Here is the description.

What I have so far

The aim is to perform a volatility analysis on daily stock prices by incorporating possible structural breaks into a GARCH(1,1) model This is already performed several times in the past (see e.g. Volatility in Emerging Stock Markets or Sudden changes in variance and volatility persistence in foreign exchange markets).

The approach is identical and as follows:

• At first the log returns are calculated from the daily stock prices (no problem)
• Then the ICSS approach from Inclan[I am not able to post more than two links - ] is applied to find structural break points. (Implementation from Matlab is used, is there a faster way or other program to use? Matlab is on daily data so slow that some series are not able to be completely calculated.)
• The next step would be the integration into the GARCH model, but I do not know how:

GARCH(1,1)

The combined model with GARCH(1,1) and dummy variables is given by $ Y_t= \mu +e_t, e_t|I_{t-1} approx. N(0,h_t)\\$

$ h_t=\omega +d_1D_1+\cdots+d_nD_n+\alpha e_{t-1}^2+\beta h_{t-1}$

where $D_1,\cdots,D_n $ are the dummy variables taking a value of one from each point of sudden change of variance onwards, zero elsewhere

I do understand what $D_1$ is but I habe no idea what $d_1$ is? Further I want to incorporate different return series to analyse the dependance (like precious metal returns to oil and stock indices (Volatility Analysis of Precious Metals Returns and Oil Returns - Lucia Morales 2011).

So I would have $ Y_t = \mu +\delta_1X_{t-1}+\delta_2 Z_{t-1}+e_t $

where $ Y_t = $ Precious Metals Returns (Gold, Silver and Platinum),

$X_t=$ Stock Markets Returns and

$Z_t=$ Crude Oil Brent.

When I am right, this is a multiple regression which must be performed in first place, no? But how do I do this?

So can anybody help me?

I calculated the normal GARCH(1,1) with one return series already in Matlab, but have no idea how to continue. Can anybody help me? Because I am not able to incorporate my breakpoint results into the GARCH model. I really appreciate any help. Thanks in advance!

Answer 1

$D_{n}$ are the dummy variables taking a value of one from each point of sudden change of variance onwards, and $d_{n}$ are the estimated coefficients in relation to these breaks.

So to apply your model:

1- Find the breaks using the ICSS algorithm.

2- Apply a garch model to your data by including dummy variables obtained in (1) in the conditional variance process and by including explanatory variables in the mean process regarding the precious metal returns (no need to perform a multiple regression).

PS: Note that the dummies should look like 00001111 and not like 00001000.

Practically, you should probably use R or a similar programming language because I don't think you'll find a package in matlab allowing to incorporate exogenous variable in both the mean and variance processes.