# How to choose a GARCH model which delivers iid standardized residuals?

For my thesis I first need to examine nine financial time series and fit a conditional volatility model such that the obtained standardized residuals ($z_t = \epsilon_t / \sigma_t$) are approximately iid with mean 0 and variance 1.

Whereas GARCH(1,1) succeeds in delivering iid standardized residuals for five of these series, and GJR-GARCH(1,1) achieves iid standardized residuals for other two series, I've not been able to get iid $z_t$ for the remaining two series using GARCH, GJRGARCH, ThresholdGARCH, EGARCH, NAGARCH and CSGARCH.

When I shared my results with my thesis supervisor, he said I've probably done something wrong since GARCH, GJRGARCH and ThresholdGARCH usually succeed wrt this goal. The problem is I don't understand what I could have done wrong.

The mentioned series are three SPDR ETFs (XLF and XLU). Closing prices can be found here (this is my first question, so I'm sorry if this isn't the way you're supposed to share data):

After obtaining log returns and demeaning them, I use the first 1766 observations to estimate all parameters and obtain standardized residuals. I then conduct ARCH tests on standardized residuals (at lags 1, 5, and 10), which for these two series reject homoscedasticity. Therefore I don't obtain iid residuals and can't go on with my analysis.

Any help would be greatly appreciated.

PS: Is there any other test which is more adequate in testing iid residuals? I think I've read somewhere that, while most people still use ARCH tests on residuals, these are supposed to test raw data and should not be used for residuals.

• you can also use the Ljung–Box/Portmanteau Test. You should also check if there is a need to use an ARMA structure to de-mean your returns (maybe an AR1 is necessary) – Malick May 2 '16 at 12:41
• ARCH test is inappropriate when applied on standardized residuals, as it neglects the fact that the data is not raw but rather estimated (similarly as you need to adjust for the degrees of freedom when applying Ljung-Box test on model residuals rather than raw data). Li-Mak test should be used instead. – Richard Hardy May 2 '16 at 19:21
• (sorry guys, being the first time I'm using quant.SE I don't know whether I'm replying the right way) @Malick: I first tried your suggestion but the results coincided with those from ARCH test. Thanks anyway Richard: Thank you for the suggestion. This test actually delivers the results I was hoping for (except for one series for which the null is rejected at 10%, but I don't care that much). Thanks!! – Kondo May 3 '16 at 16:26