# Joint estimation of GARCH models with ARMA terms in the conditional mean: a necessity?

Supposing I am using the following models to forecast conditional volatility of index returns, whereby In-sample data is 1996-2007 and out of sample data is 2007-2012, using GARCH type models.

I have 3 indexes for South Africa (JSETop40), Australia (ASX200) and United States (S&P500 Composite).

Starting with South Africa, as I seem to have less doubts with this: 1) If I plot the squared returns of this index, I get a plot highly suggestive of AR(1) for the in-sample period, and out of sample period. See below.  2) Using the Information criteria for the returns, suggests AR(1) as having the lowest SIC. The only other possible option from analysing the information criteria and plots is an AR(3). 3) Next, having done the OLS, I can proceed to do my ARCH test. 4) I next proceed to see if my AR(1) will be ok in doing the joint estimation of my GARCH models. I find AR(1) when used in the mean specification gives significant terms in the mean and variance. I try some other combinations (as a confirmation check) which give significance and note down their AIC's and SIC's.

I find an AR(1) to have one of the lowest SIC's and the third best AIC, therefore I conclude I can model the JSEtop40 Index returns with an AR(1)-GARCH(1,1) model.

Does this sound reasonale?

The problem comes with Australia (ASX200) and the United States (S&P500)

Australia I repeat the steps in much the same way as South Africa. 1) Plot the ACF's and PACF's of squared returns - In-sample period and Full sample. The plots seem to suggest something in the range of 0<=p,q<=3  2) I then calculate the AIC's and SIC's of the returns series for the full sample and in-sample period. The SIC suggests an ARMA (0,0) specification both for the in-sample and full-sample periods. However the ARMA(1,1) model does perform reasonably well across the board. 3) Proceed to do the OLS and ARCH-LM tests 4) I check if indeed an ARMA(1,1)-GARCH(1,1) is appropriate. It is found that it isn't. I try all other different combinations and none yield significant terms execept an ARMA(0,0)-GARCH(1,1). Therefore I am forced to settle with this model.

Does this sound reasonable?

United States I repeat the steps in much the same way as South Africa and Australia. 1) Plot the ACF's and PACF's of squared returns - In-sample period and Full sample. The plots are harder to interpret and not really suggestive of a pattern. See below.  2) I then calculate the AIC's and SIC's of the returns series for the full sample and in-sample period. The SIC suggests an ARMA (1,1) specification for the full sample period and an ARMA(0,0) for the in-sample period. However, The ARMA(1,1) does perform really well across the board for both AIC and SIC and for both full sample and in-sample. 3) Proceed to do the OLS and ARCH-LM tests 4) I check if indeed an ARMA(1,1)-GARCH(1,1) is appropriate. It is found that it isn't. I try all other different combinations and none yield significant terms execept an ARMA(0,0)-GARCH(1,1) and an ARMA(2,2)-GARCH(1,1). Therefore I choose the ARMA(2,2) for GARCH(1,1) and for GARCH-M. I am forced to choose ARMA(0,0) for EGARCH and GJR-GARCH.

Does this sound reasonable?

I am aware SIC tends to underestimate while AIC will generally overestimate. But in this case it does appear SIC is more accurate. Using AR and MA terms when doing joint estimation of my GARCH models does not give me significant terms, therefore surely it can't be correct, and I am left with no option but to use ARMA(0,0) for the mean specification for both Australia and the United States. However, is this not inconsistent with what the ACF and PACF plots show at the very beginning?

Can one model stock index returns with no AR and MA terms? If so what is the logic. A comment would be appreciated as to what would be best/what ARMA terms are usually selected for the S&P500 and ASX200.

• Albe, you keep posting the same question both on Cross Validated and here. This is not recommended, so I think you should avoid that in the future. – Richard Hardy Apr 4 '17 at 18:30

## 1 Answer

You are confusing the cond. mean process and cond. variance process : the autocorrelation plot of the squared returns gives you information about the cond. variance process (not the ARMA part !) . So you can't draw conclusion on the mean process. The squared returns are almost always autocorrelated since volatility is know to be time-varying.

You need to observe the autocorrelations of the raw logs returns, if there are significant you need to model the cond mean, otherwise you can directly model the variance process.

You can read the following book for more information :

• Tsay, R. S. (2005). Analysis of Financial Time Series Second Edition (Second Edition).
• Thanks a lot. I finally got it now by looking at that book. I was reading all over but could not understand this, seems I missed a basic point. That makes sense now especially since we actually use squared returns as a volatility proxy. Given log returns seem to have no significant spikes at all, it would appear that the I do not need to use ARMA terms for the conditional mean, and that a constant is sufficient. Save for South Africa which can modelled with an AR(1) but that is alluded to in the book as a possibility. – Albe Apr 4 '17 at 14:35