3.1 Post-estimation phase

• Diagnostic checking is post–estimation phase commonly includes checking if all assumptions are met and how well model fits the data:

1. no autocorrelation in standardized residuals (if Ljung–Box test on standardized residuals inidcates significant autocorrelation we should include autoregressive terms $r_{t-1}$, $r_{t-2}$, $\dots$, $r_{t-k}$ in the mean equation and re–estimate the model)

2. no autocorrelation in squared standardized residuals, i.e. no remaining ARCH effects (if Ljung–Box test on squared standardized residuals or ARCH LM test are significant, we should use a higher order GARCH model – including more lags with respect to ARCH and/or GARCH parameters can better capture volatility clustering and heteroscedasticity)

3. no leverage effect (if sign–bias test indicates the presence of the leverage effects, i.e. significant reaction of squared standardized residuals on lagged negative and/or positive shocks, we should consider one of the asymmetric GARCH models)

4. parameters stability (if Nyblom stability test indicates that all or some parameters are not constant over time we should consider GARCH model with time–varying parameters, such as Markov–switching GARCH with two–regimes or rolling window re–estimation of the GARCH)

5. empirical distribution of the standardized residuals fits the theoretical distribution (if $\chi^2$ goodness–of–fit test is significant we should re–estimate the model considering theoretical distribution other than initially assumed or ignore this kind of misspecification and use robust standard errors)

• Along with formal diagnostic tests some informative plots can be drawn from GARCH objects in R

Note: regarding non–normal innovations: Quasi maximum likelihood estimates (QMLE) are consistent and asymptotically normal estimates even if standardized residuals are not–normal

• If different GARCH($p,q$) specifications are estimated, the model with the lowest AIC or BIC is prefered (those criteria, along with Shibata and Hannan–Quinn, penalize the model’s goodness–of–fit with it’s complexity to avoid the overfitting issue)

• Based on standard GARCH($1,1$) a one–step ahead volatility forecasting is obtained by

$$\begin{equation} \sigma^2_{t+1}=\omega+\alpha_1 u_{t}^2+ \beta_1 \sigma_{t}^2 \tag{3.6} \end{equation}$$

• For two or more steps ahead (horizons $h$) forecasting is given be recursion

$$\begin{equation} \sigma^2_{t+h}=\omega+(\alpha_1+\beta_1)\sigma^2_{t+h-1} \tag{3.7} \end{equation}$$

• The Sequential Quadratic Programming (SQP) algorithm solnp is used by default when fitting GARCH models in R. This quasi–Newton iterative algorithm is similar to BFGS which uses approximation of the Hessian matrix of the second derivatives. Other algorithms may be used, like BHHH with major advantage that Hessian matrix is approximated by the outer product of gradients, and thus is positive definite which ensures positive increments of the log–likelihood function in each iteration, particularly in regions where the log–likelihood function is convex