Selecting GARCH Orders with Residual Diagnostics and Information Criteria
Summary
The document addresses whether GARCH(1,1) should be used for every stock market and how to select model orders when other specifications appear statistically significant. It says GARCH(1,1) performed best among 330 specifications in a cited study, while acknowledging that alternatives can perform better in some settings. ACF and PACF plots of squared errors are described as useful but insufficient on their own.
The proposed checks include examining residual autocorrelation to assess the mean equation, testing squared residuals with ACF/PACF or Ljung–Box statistics for remaining ARCH effects, checking residual normality, and comparing specifications with AIC or SBC. The response also points to Student-t errors for fat-tailed financial returns and suggests asymmetric or mean-in-variance variants when appropriate. These are diagnostic and model-comparison suggestions, not a universal order-selection rule; distributional assumptions, market, sample, and the purpose of the volatility forecast still matter.
Key ideas
- GARCH(1,1) performed strongly in a cited comparison, but it is not guaranteed to be best for every market or dataset.
- ACF and PACF of squared errors can inform specification, but they are not sufficient for choosing model orders.
- Residual autocorrelation diagnostics help assess whether the mean equation is adequately specified.
- Ljung–Box tests on squared residuals can identify remaining serial dependence in volatility.
- Normality checks and information criteria provide additional diagnostics for comparing GARCH specifications.
- Fat-tailed or asymmetric returns may motivate alternative error distributions or model forms.
Tags
Full text
# Define the order of GARCH(m.s)
# Define the order of GARCH(m.s)
- I know that if the order of Arch(m) is over 3, we should use GARCH and GARCH(1,1) was proved to be the best. But was GARCH(1,1) proved to be available for any country's stock market?
- My result show that GARCH(1,1) is not statistically significant (although i bsed on the result of ACF/PACF of Squared error). However, the Garch(2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1) are statistically significant.
- Consequently, i conflict that the method based on the ACF/PACF of the Squared return or Squared error to define the Order of GARCH are not available. How can we estimate the order of GARCH(m.s)?
## Answer by Übel Yildmar (score 3)
https://quant.stackexchange.com/a/27861
You're right. Hansen and Lunde ran 330 specifications, and found GARCH (1,1) the best fitting volatility model. However, in some cases other specifications can beat the results of GARCH (1,1).
Checking the ACF/PACF of the squared error term is necessary, although, not sufficient condition. Let's assume the following GARCH (m,s) model $$y_t=a_0+a(L)\varepsilon_t^2+b(L)y_t $$ $$\varepsilon_t=v_t\sqrt{a_0+a(L)\varepsilon_t^2+b(L)y_t }$$ where $v_t$ is a white-noise procedure. There're many other things to investigate:
- Checking the ACF/PACF of the error term. Residuals cannot be autocorrelated. The ARMA model ($y_t=a_0+a(L)\varepsilon_t^2+b(L)y_t $) part of the GARCH model is not correctly specified if the residuals are autocorrelated.
- Checking the ACF/PACF of the squared error term. You can test whether your procedure is a GARCH procedure or not. Equivalently, one can examine Ljung–Box Q-statistics. It can be used to test for groups of significant coefficients. Rejecting the null hypothesis that the $\varepsilon_t^2$ sequence is serially uncorrelated is equivalent to rejecting the null hypothesis of no ARCH or GARCH errors.
- Normality of the errors. When you estimate GARCH models with Maximum Likelihood method the default probability density function is usually standard normal in many packages (e.g. in R). You can easily perform a Jarque-Bera normality test on your residuals. (Note that: For most financial assets, the distribution function for the rate of return is fat tailed, like Student's t-distribution.)
- Information criterion. You can compare how well the estimated models perform using AIC or SBC. It's a fast strategy.
I'd recommend you to check ACFs and normality tests of different specifications. In some cases, there is a trade-off: some models do better at ACFs, some at normality tests. Maybe the residuals of yours are pretty asymmetric. For further improvement you can apply EGARCH, TARCH or ARCH-M.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.