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:

Perfectly Competitive Firm

A business operating in a market where it has no power to influence the price of its product; it is a price taker.

Marginal Cost

The financial outlay for producing a further unit of a product or service.

Perfectly Competitive Firm

A business that operates in a market where there are many buyers and sellers, each selling identical products, and no single buyer or seller can influence market prices.

Short Run

In economics, it denotes a period in which at least one input (e.g., capital) is fixed and cannot be altered.