Description

# Key Takeaways: Financial Time Series and Related Models

1. **DAX log-returns** exhibit stylized facts: stationarity (KPSS p>0.1), fat tails, left skewness, and significant ARCH effects from squared returns ACF.

2. **ARMA(0,1) model**: \(r_t = \epsilon_t + \theta_1 \epsilon_{t-1}\) with \(\hat{\theta}_1 = -0.006\), but residuals still show GARCH effects.

3. **GARCH(1,1) specification**: \(\sigma_t^2 = \omega + \alpha_1 \epsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2\) with estimates \(\hat{\alpha}_1=0.062\), \(\hat{\beta}_1=0.925\).

4. **High volatility persistence**: \(\hat{\alpha}_1 + \hat{\beta}_1 = 0.987\), but residuals remain non-normal (skewness -0.359, kurtosis 5.438, JB test p<0.001).

5. **Theoretical ARCH properties**: \(\eta_t = \sigma_t^2(\epsilon_t^2-1)\) is white noise (mean 0, constant variance, zero autocorrelation).

6. **Excess kurtosis proof**: For ARCH(1) with \(\alpha_1=0.5\), theoretical kurtosis \(\kappa = 3(1-\alpha_1^2)/(1-3\alpha_1^2) = 3.74 > 3\), confirmed empirically (mean 3.69).

7. **GARCH(2,1) simulation example**: \(X_t = \sigma_t\epsilon_t\) with \(\sigma_t^2 = 0.1 + 0.2X_{t-1}^2 + 0.1X_{t-2}^2 + 0.6\sigma_{t-1}^2\).

8. **Estimated GARCH(2,1) results**: \(\hat{\omega}=0.131\), \(\hat{\alpha}_1=0.152\), \(\hat{\alpha}_2=0.107\), \(\hat{\beta}_1=0.551\) (all statistically significant).

9. **Temperature series application**: Southern Hemisphere monthly temperatures require harmonic regression + SARIMA + GARCH(6,8) to remove all ARCH effects.

10. **Practical implication**: Financial returns need GARCH-type models for volatility clustering, but normal innovations often insufficient—consider t-distributions or asymmetric variants.