In the equation in Exercise 3.3, assume that \(\epsilon_{t}\) follows a standardized Student\(t\) distribution with \(v\) degrees of freedom. Derive the conditional log-likelihood function of the data.

Exercise 3.3:

Suppose that \(r_{1}, \ldots, r_{n}\) are observations of a return series that follows the AR(1)-GARCH(1,1) model

\[ r_{t}=\mu+\phi_{1} r_{t-1}+a_{t}, \quad a_{t}=\sigma_{t} \epsilon_{t}, \quad \sigma_{t}^{2}=\alpha_{0}+\alpha_{1} a_{t-1}^{2}+\beta_{1} \sigma_{t-1}^{2} \]

where \(\epsilon_{t}\) is a standard Gaussian white noise series. Derive the conditional \(\log\)-likelihood function of the data.