Diagnosing Bimodal Standardized Residuals in GARCH Models
Summary
The document considers a GARCH(1,1) model fitted to the spread between two correlated assets. Although the reported coefficient estimates appear unremarkable, the standardized residuals show a bimodal distribution, prompting the question of whether such a shape is expected and whether it can support two-tailed trading signals.
The accepted explanation is that bimodality suggests model misspecification when the innovation distribution has not been specified to allow multiple peaks. The proposed diagnostic direction is to examine whether the distinct regimes belong in the conditional mean, and possibly variance, equations instead of forcing them into a single innovation distribution. A population split by a relevant group characteristic illustrates how pooled data can become bimodal. The discussion is conceptual: it does not identify the cause of the observed modes, test alternative specifications, or establish that the residuals should be normal or Student-t under every GARCH setup.
Key ideas
- Bimodal standardized residuals can indicate that a GARCH model is misspecified.
- A single-peaked innovation distribution may not capture residuals generated by distinct regimes.
- Consider whether the conditional mean or variance equations should represent the source of the bimodality.
- The example of pooled groups illustrates how combining different conditional distributions can create multiple peaks.
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Full text
# Standardized residual by GARCH model shows bimodal distribution, is it normal?
# Standardized residual by GARCH model shows bimodal distribution, is it normal?
I fit a GARCH(1,1) model on the spread of 2 correlated assets :
the GARCH model shows this summary:
```
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 0.2066 5.839e-02 3.537 4.042e-04 [9.211e-02, 0.321]
alpha[1] 0.6416 5.479e-02 11.712 1.107e-31 [ 0.534, 0.749]
beta[1] 0.3584 6.020e-02 5.953 2.640e-09 [ 0.240, 0.476]
==========================================================================
```
From this point, nothing weird, but then when I plot the standardized residual by their conditional volatility :
In order to retrieve entry/exit signals for my strategy, I'm doing a 2-tails test on the distribution of these standardized residual. However, as you can see, the distribution is very weird :
Is it normal to have such bimodal distribution for a standardized residual by a GARCH model ? I'm asking because this is definitely not something I was expecting (standard normal distribution, or at least t-student with fatter tails), neither something I found on the Internet as what we can expect for a GARCH std residual.. What did I miss here ?
## Answer by Richard Hardy (score 2, accepted)
https://quant.stackexchange.com/a/75245
This is not normal. I bet that you have not specified the distribution of the standardized innovations to be a bimodal one when specifying the GARCH model. If so, your model is misspecified. And even if there was a possibility to specify a bimodal distribution for standardized innovations, you would probably rather prefer to account for the bimodality in the conditional mean (and perhaps the conditional variance) equation instead.
To give an unrelated, hypothetical example, consider the wage distribution of a population. If the distribution for males has a different peak than the one for females, you might end up with a bimodal distribution for the total population. Why not use a sex dummy for the conditional mean (and probably for some higher-order moments) to account for that instead of trying to find a suitable bimodal distribution? The sex dummy approach makes matters more transparent.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.