Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$

question 25

Essay

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

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Definitions:

Conditioned Stimulus

A previously neutral stimulus that, after becoming associated with the unconditioned stimulus, eventually comes to trigger a conditioned response.

Fear Conditioning

A behavioral paradigm in which organisms learn to predict aversive events, linking a neutral stimulus with a fear-inducing situation.

Aversive CS

A conditioned stimulus that predicts an unpleasant outcome, leading to aversive learning.

Neutral US

Refers to a stimulus that does not initially elicit a strong emotional or physiological response, used in behavioral conditioning experiments.