Consider the GARCH (1,1) model $\sigma _ { t } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 }$ Show that this model can be
rewritten as $\sigma _ { t } ^ { 2 } = \frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } } + \alpha _ { 1 } \left( u _ { t - 1 } ^ { 2 } + \phi _ { 1 } u _ { t - 2 } ^ { 2 } + \phi _ { 1 } ^ { 2 } u _ { t - 3 } ^ { 2 } + \phi _ { 1 } ^ { 3 } u _ { t - 4 } ^ { 2 } + \ldots \right)$ (Hint: use the GARCH(1,1) model but specify it for $\sigma _ { t - 1 } ^ { 2 }$; substitute this expression into the original specification, and so on.) Explain intuitively the meaning of the resulting formulation.

Question

Consider the GARCH  (1,1)  model $\sigma _ { t } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 }$ Show that this model can be
rewritten as $\sigma _ { t } ^ { 2 } = \frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } } + \alpha _ { 1 } \left( u _ { t - 1 } ^ { 2 } + \phi _ { 1 } u _ { t - 2 } ^ { 2 } + \phi _ { 1 } ^ { 2 } u _ { t - 3 } ^ { 2 } + \phi _ { 1 } ^ { 3 } u _ { t - 4 } ^ { 2 } + \ldots \right)$  (Hint: use the GARCH(1,1) model but specify it for $\sigma _ { t - 1 } ^ { 2 }$; substitute this expression into the original specification, and so on.) Explain intuitively the meaning of the resulting formulation.

Quizplus · Accepted Answer

The Answer of Consider the GARCH  (1,1)  model $\sigma _ { t } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 }$ Show that this model can be
rewritten as $\sigma _ { t } ^ { 2 } = \frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } } + \alpha _ { 1 } \left( u _ { t - 1 } ^ { 2 } + \phi _ { 1 } u _ { t - 2 } ^ { 2 } + \phi _ { 1 } ^ { 2 } u _ { t - 3 } ^ { 2 } + \phi _ { 1 } ^ { 3 } u _ { t - 4 } ^ { 2 } + \ldots \right)$  (Hint: use the GARCH(1,1) model but specify it for $\sigma _ { t - 1 } ^ { 2 }$; substitute this expression into the original specification, and so on.) Explain intuitively the meaning of the resulting formulation.

Consider the GARCH (1,1) Model σt2=α0+α1ut−12+ϕ1σt−12\sigma _ { t } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 }σt2​=α0​+α1​ut−12​+ϕ1​σt−12​

Consider the GARCH (1,1) Model $\sigma _ { t } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 }$