[GARCH model] Why do we need a "mean equation" ?

[Latiole]

Hi,

As i said in my first topic, I'm a beginner in financial statistics. I read a lot of things about GARCH, but I didn't find a precise response.

According to this document (pages 4, equation (1) ) :
http://www.eurojournals.com/jmib_7_01.pdf

We need a "mean equation" (certainly AR or ARMA model) to formulate a GARCH model. What should I do with this equation ?

Additional question : Are residuals the differences between the result of this equation and the observed values ?

Thanks in advance

[vinux]

Most of the modelling is based on mean equation. For example regression you could see as E[Y | X]. In stationary time series, mean equation is always AR or ARMA or MA. GARCH modelling is a relationship in variance equation.

We need a "mean equation" (certainly AR or ARMA model) to formulate a GARCH model. What should I do with this equation ?

Additional question : Are residuals the differences between the result of this equation and the observed values ?
Thanks in advance

Mean equation part gives forecast. Using the lag structure you can forecast the future periods.
You are right on the additional question part.

[Latiole]

Thank you very much.

In concrete terms,when should I use the mean equation ? Is the following process right ? :

- Returns modelling through an AR model
- Residuals and then squared residuals calculations
- Conditional variance following a GARCH model
- Log likehood function
- GARCH Parameters estimation by iteration.

... or it's totally wrong ?

[vinux]

The steps would be:
- First ensure the series is stationary.
- Then build mean equation to make residual is white noise. Use ACF and PACF of residual. At this stage you don't need to check any thing of squared residual.
- Once the mean equation is done, check the squared residuals ACF or PACF for evidence of GARCH. Fit the mean equation and GARCH parameters estimation by iteration.

I would suggest Tsay's book for the theory and details of the model building.

[Latiole]

So to sum up the first step,

- If the series is stationary (it's the case I verified this with the aide of a KPSS test in Matlab), I can use an AR model.
- I determine residuals by calculating the difference between the results of the AR model and the real values.

Even if I understand the function of ACF and PACF, I have some trouble to interpret them. So I will work on these tools.

Thank you

[Latiole]

Thank you for the book. I got it and it's awesome.

[Latiole]

Hi,

In the Tsay's book (for the GARCH model), it says

"let a= r - u be the innovation at time t"

Is it the difference between the results of the AR model and the observed value ? If no, what's u ?

Thank you in advance.

[vinux]

Hi,

In the Tsay's book (for the GARCH model), it says

"let a= r - u be the innovation at time t"

Is it the difference between the results of the AR model and the observed value ? If no, what's u ?

Thank you in advance.

If your mean equation is AR then you are right.

[Latiole]

Ok Thx, so innovation = residuals ?

[vinux]

Ok Thx, so innovation = residuals ?

You can say that.

[Latiole]

Ok thank you for your answers a sunday

[derksheng]

The mean equation can take the form:



where are the AR(i) coefficients, is the coefficient of the garch-in-mean, is a row vector of coefficients to some exogenous variables represented by column vector , .

Any of the parameters can be set to zero (this is done by not entering the data into the GARCH estimator).

EDIT: Also you can have moving average terms in the mean equation.

The mean equation is simply the hypothesized data generating process of the actual variable that you're modelling. The variance equation is the hypothesized DGP of the mean equation's residual's variance. Without a mean equation the variance equation makes no sense. The reason it makes no sense is because it's the variance equation of the residuals of a specific mean equation (and if this doesn't exist then the variance equation doesn't exist).