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

Question

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

Examlex · Accepted Answer

The Answer is

\begin{aligned}\sigma _ { t } ^ { 2 } & = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 1 } ^ { 2 } = \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 1 } ^ { 2 } + \phi _ { 1 } \left( \alpha _ { 0 } + \alpha _ { 1 } u _ { t - 2 } ^ { 2 } + \phi _ { 1 } \sigma _ { t - 2 } ^ { 2 } ight) \& = \alpha _ { 0 } \left( 1 + \phi _ { 1 } ight) + \alpha _ { 1 } \left( u _ { t - 1 } ^ { 2 } + \phi _ { 1 } u _ { t - 2 } ^ { 2 } ight) + \phi _ { 1 } ^ { 2 } \sigma _ { t - 2 } ^ { 2 } \& = \alpha _ { 0 } \left( 1 + \phi _ { 1 } + \phi _ { 1 } ^ { 2 } ight) + \alpha _ { 1 } \left( u _ { t - 1 } ^ { 2 } + \phi _ { 1 } u _ { t - 2 } ^ { 2 } + \phi _ { 1 } ^ { 2 } u _ { t - 3 } ^ { 2 } ight) + \phi _ { 1 } ^ { 3 } \sigma _ { t - 3 } ^ { 2 } ext {. Continuing with the }\end{aligned}

substitutions infinitely and noting that the sum of the geometric series is

1 + \phi _ { 1 } + \phi _ { 1 } ^ { 2 } + \phi _ { 1 } ^ { 3 } + \ldots = \frac { 1 } { 1 - \phi _ { 1 } }

you finally arrive at

\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 ight)

. This expression states that the variances depend on a weighted average of past squared residuals, where the distant past receives a smaller weight than more recently observed squared residuals.

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

Definitions:

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​

Definitions:

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