Estimating ARMA–GARCH Models with an EGB2 Error Distribution

Summary

The document asks how to estimate an ARMA–GARCH model when innovations follow an EGB2 distribution rather than a normal distribution. It gives an ARMA specification with nonconsecutive autoregressive lags, a standard GARCH(1,1) variance equation, and a proposed log-likelihood expression and implementation. The stated parameter restrictions require positive variance and distribution parameters, with the GARCH coefficients summing to less than one.

The author reports that optimization produces a very large objective value instead of the expected likelihood and asks whether additional restrictions are needed. The document does not include a resolution, diagnostic evidence, or validation of the likelihood formula. It therefore serves mainly as a troubleshooting question: checking the likelihood's parameterization, signs, scaling, and numerical stability would be necessary before drawing conclusions from the optimizer's output.

Key ideas

• The model combines an ARMA mean equation with a GARCH(1,1) conditional variance.
• The author replaces normally distributed innovations with an EGB2 distribution.
• The proposed constraints require positive parameters and a sum of GARCH coefficients below one.
• The reported optimization result differs greatly from the expected likelihood, but the document gives no solution.

Tags

# ARMA-GARCH estimation with EGB2 distribution


# ARMA-GARCH estimation with EGB2 distribution












I want to estimate a ARMA-GARCH model by using the EGB2 distribution instead of the normal distribution. The model I want to estimate is: $$y_t = \mu + \phi_1 y_{t-6} + \phi_2 y_{t-8} + \theta_1 \epsilon_{t-1} + \epsilon_t $$ $$\sigma^2_t = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$$ The loglikelihood for the EGB2 is: $$ \text{log L} = T[log(\sqrt{\Omega} - log(B(p,q)) + p\Delta] + \sum \Bigg[p\Bigg(\frac{\sqrt{\Omega}\epsilon_t}{\sqrt{\sigma_t^2}}\Bigg)\\ - 0.5 log(\sigma^2_t) - (p+q)log\Bigg(1+exp\frac{\sqrt{\Omega}\epsilon_t}{\sqrt{\sigma_t^2}} + \Delta\Bigg) \Bigg].$$

Now, I am going to minimize this function:

```
loglike = 0;
for i = 1:length(R)-8
    loglike = loglike + (p*sqrt(OMEGA)*eps(i)/sqrt(sigma_2(i)) - 0.5*log(sigma_2(i)) - (p+q)*log(1+exp(sqrt(OMEGA)*eps(i)/sqrt(sigma_2(i))) + DELTA));
end 

logL = -(length(eps)*(log(sqrt(OMEGA))-log(betaFunc(p,q))+p*DELTA) + loglike);
```

And the constraints where that $\omega, \alpha, \beta, p, q>0$ and that $\alpha+ \beta <1$. The answer is that I must get a loglikelihood value around -3000. But I get 2.5+e07. What is going wrong? Must I have more restrictions?

I really hope someone can help me Thanks in advance

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.