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A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same stepsteps can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same step can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same steps can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

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A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) - -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}\begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same step can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) - -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same step can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same step can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.

1
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A conditional volatility model such as the GARCH model is defined by the mean equation

\begin{equation} r_t = \mu + \sigma_t z_t = \mu + \varepsilon_t z_t \end{equation}

and the GARCH equation (this is for the simple GARCH)

\begin{equation} \sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \end{equation}

To perform maximum-likelihood estimation, we must make distributional assumptions on $z_t$. It is typically assumed to be i.i.d. $N(0,1)$.

Conditional on the informationset at time t, we have that

\begin{equation} r_t \sim N(\mu, \sigma_t^2) \end{equation}

or

\begin{equation} \varepsilon_t = r_t - \mu \sim N(0, \sigma_t^2) \end{equation}

However when we perform maximum-likelihood estimation, we are interested in the joint distribution \begin{equation} f(\varepsilon_1,...,\varepsilon_T; \theta) \end{equation} where $\theta$ is the parameter vector. Using iteratively that the joint distribution is equal to the product of the conditional and the marginal density, we obtain \begin{eqnarray} f(\varepsilon_0,...,\varepsilon_T; \theta) &=& f(\varepsilon_0;\theta)f(\varepsilon_1,...,\varepsilon_T\vert \varepsilon_1 ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1},...,\varepsilon_{0}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T f(\varepsilon_t \vert \varepsilon_{t-1}, ;\theta) \\ &=& f(\varepsilon_0;\theta) \prod_{t=1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}}\exp\left(-\frac{\varepsilon_t^2}{2\sigma_t^2}\right) \end{eqnarray} Dropping $f(\varepsilon_0;\theta)$ and taking logs, we obtain the (conditional) log-likelihood function \begin{equation} L(\theta) = \sum_{t=1}^T \frac{1}{2} \left[-\log2\pi-\log(\sigma_t^2) - -\frac{\varepsilon_t^2}{\sigma_t^2}\right] \end{equation}

To question 1): The exact same step can be followed for the GJR-GARCH model. The log-likelihood functions are similar but not the same due to the different specification for $\sigma_t^2$.

To question 2): One is free to use whatever assumption about the distribution of the innovations, but the calculations will become more tedious. As far as I know, Filtered Historical Simulation is used to performe e.g. VaR forecast. The "fitted" innovations are bootstrapped to better fit the actual empirical distribution. Estimation is still performed using Gaussian (quasi) maximum-likelihood.