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In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. If we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function?  (my guess is that empirical innovations does not affect the likelihood function, but any reference or explanation will be highly welcome)

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. If we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function?  (my guess is that empirical innovations does not affect the likelihood function, but any reference or explanation will be highly welcome)

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. If we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that empirical innovations does not affect the likelihood function, but any reference or explanation will be highly welcome)
5 deleted 12 characters in body
source | link

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. ifIf we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that using empirical innovations does not affect the likelihood function, but again any reference or explanation will be highly welcome)

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. if we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that using empirical innovations does not affect the likelihood function, but again any reference or explanation will be highly welcome)

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):

  1. Can someone provide a reference/explanation to justify this?
  2. If we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that empirical innovations does not affect the likelihood function, but any reference or explanation will be highly welcome)
4 added 411 characters in body
source | link

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

My questions areQuestion: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):  

  1. Is thereCan someone provide a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovationsreference/explanation to justify this? 
  2. if there are no closed-form solutions, how can I implementwe use empirical innovations instead of normal ones (e.g: a MLEFiltered Historical Simulation/FHS approach for), would this particular modelchange the functional form of the likelihood function? (my guess is that using empirical innovations does not affect the likelihood function, but again any reference or explanation will be highly welcome)

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

My questions are:

  1. Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?
  2. if there are no closed-form solutions, how can I implement a MLE approach for this particular model?

In the standard GARCH(1,1) model with normal innovations

$\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

the likelihood of $m$ observations occurring in the order in which they are observed is

$\sum_{t=1}^m\left[-\ln(\sigma^2_{t})-{\left(\frac{\epsilon^2_{t}}{\sigma^2_{t}}\right)}\right] $

This expression, with the usual caveats of optimization, allows us to obtain the MLE estimates of the GARCH(1,1) parameters.

However, in the GJR-GARCH(1,1) model by Glosten et al. (1993), the conditional variance is

$ \sigma^2_t=\omega+(\alpha+\gamma I_{t-1})\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $

where $ I_{t-1} $ is the indicator function:
$I_{t-1}(\epsilon_{t-1})=1 $ when $\epsilon_{t-1}<0$ and
$I_{t-1}(\epsilon_{t-1})=0$ otherwise.

Question: Is there a closed-form expression for the likelihood function in the GJR-GARCH(1,1) with normal innovations?

EDIT: Per comments, the likelihood function in the GJR-GARCH(1,1) model is the same than in the standard GARCH(1,1):  

  1. Can someone provide a reference/explanation to justify this? 
  2. if we use empirical innovations instead of normal ones (e.g: a Filtered Historical Simulation/FHS approach), would this change the functional form of the likelihood function? (my guess is that using empirical innovations does not affect the likelihood function, but again any reference or explanation will be highly welcome)
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