Tutorial (GARCH Toolbox)

GARCH Toolbox

Post-Estimation Analysis

The post_estimation analysis:

Compare the Residuals, Conditional Standard Deviations, and Returns

In addition to the parameter estimates and standard errors, garchfit also returns the optimized log-likelihood function value (LLF), the residuals (innovations), and conditional standard deviations (sigma). Use the function garchplot to inspect the relationship between the innovations (i.e., residuals) derived from the fitted model, the corresponding conditional standard deviations, and the observed returns. garchplot displays the tiered plot shown in Figure 2-11, Comparison of Innovations, Conditional Standard Deviations and Observed Returns.

```
garchplot(innovations, sigma, xyz)
```

Figure 2-11: Comparison of Innovations, Conditional Standard Deviations and Observed Returns

Notice in Figure 2-11, Comparison of Innovations, Conditional Standard Deviations and Observed Returns that both the innovations (top plot) and the returns (bottom plot) exhibit volatility clustering. Also, notice that the sum, G₁ + A₁ = 0.96283 + 0.03178, is 0.99461, which is close to the integrated, nonstationary boundary given by Eq. (2-6).

Plot and Compare Correlation of the Standardized Innovations

Although the fitted innovations exhibit volatility clustering (Figure 2-11, Comparison of Innovations, Conditional Standard Deviations and Observed Returns), if you plot of the standardized innovations (the innovations divided by their conditional standard deviation), they appear generally stable with little clustering.

plot(innovations./sigma)
ylabel('Innovation')
title('Standardized Innovations')

Figure 2-12: Standardized Innovations

If you plot the ACF of the squared standardized innovations (Figure 2-13, ACF of the Squared Standardized Innovations), they also show no correlation.

```
autocorr((innovations./sigma).^2)
```

Figure 2-13: ACF of the Squared Standardized Innovations

Now compare the ACF of the squared standardized innovations (Figure 2-13, ACF of the Squared Standardized Innovations) to the ACF of the squared returns prior to fitting the default model (Figure 2-9, ACF of the Squared Returns). The comparison shows that the default model explains sufficiently the heteroscedasticity in the raw returns.

Quantify and Compare Correlation of the Standardized Innovations

Compare the results below of the Q-test and the ARCH test with the results of these same tests in the pre-estimation analysis. In the pre-estimation analysis, both the Q-test and the ARCH test indicate a rejection (H = 1 with pValue = 0) of their respective null hypotheses, showing significant evidence in support of GARCH effects. In the post-estimate analysis, using standardized innovations based on the estimated model, these same tests indicate acceptance (H = 0 with highly significant pValues) of their respective null hypotheses and confirm the explanatory power of the default model.

[H, pValue, Stat, CriticalValue] = 
lbqtest((innovations./sigma).^2,[10 15 20]',0.05);
[H  pValue  Stat  CriticalValue]
ans =
         0    0.8632    5.3966   18.3070
         0    0.9328    7.7677   24.9958
         0    0.9819    9.0843   31.4104

[H, pValue, Stat, CriticalValue] = 
archtest(innovations./sigma,[10 15 20]',0.05);
[H  pValue  Stat  CriticalValue]
ans =
         0    0.8883    5.0428   18.3070
         0    0.8765    9.0200   24.9958
         0    0.9521   10.7657   31.4104

Parameter Estimation The GARCH Specification Structure