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General Autoregressive Conditional Heteroskedastistic Model (GARCH)

This model differs to the ARCH model in that it incorporates squared conditional

variance terms as additional explanatory variables. This allows the conditional

variance to follow an ARMA process. If we write the residual as

ttttt hvvu == !

"here !t

is written as htand vthas a #ero mean and variance of one. "e can them

write the conditional variance as

=

=

++=p

iiti

q

iitit huh

$$

!%

&or instance a 'ARCH($)$* process would be

$!$ +.%,.%-.%

++= ttt huh

T/statistics and the usual dia0nostic tests could also be added to this model. If there

was only the one la0 on the hvariable) we would have a 'ARCH($)%* process which

is the same as an ARCH($* process.

Model Restrictions

The principal restriction of this model is that all the explanatory variables in a

'ARCH and therefore ARCH model must be positive) this is 1nown as the non/

ne0ativity constraint) clearly it is impossible to have a ne0ative variance) as it consists

of squared variables.

Advantages of GARCH models compared to ARCH models

The main problem with an ARCH model is that it requires a lar0e number of la0s to

catch the nature of the volatility) this can be problematic as it is difficult to decide

how many la0s to include and produces a non/parsimonious model where the non/

ne0ativity constraint could be failed. The 'ARCH model is usually much more

parsimonious and often a 'ARCH($)$* model is sufficient) this is because the

'ARCH model incorporates much of the information that a much lar0er ARCH

model with lar0e numbers of la0s would contain.

ML Estimation of ARCH models in practice:$. 2pecify the model and its li1elihood function

!. 3se 452 re0ression to 0et initial estimates (starting values* for$ )$ etc.

,. Choose initial estimates for the parameters of the conditional variance

function. Microfit (and other software* offers you #eros as startin0 values for

these. In practice it is better to choose small positive values.

6. 2pecify a conver0ence criteria (usually the software has a default value for

this*.

+. Maximise the li1elihood by iteration until no further improvement in the

model coefficients can be obtained (and the conver0ence criteria in step 6 is

met*.

Asymmetric GARCH

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The 'ARCH model can be extended to include any number of la0s on the squared

error term and conditional variance term. The 'ARCH (p)q* model has p la0s on the

conditional variance term and q on the squared error term. However in 0eneral a

'ARCH($)$* model is sufficient.

Asymmetric 'ARCH models due to the levera0e effect with asset prices) where apositive shoc1 has less effect on the conditional variance compared to a ne0ative

shoc1. This can be incorporated into the 'ARCH model usin0 a dummy variable.

This was introduced by 'losten) 7an0athann and Run1le ('7R*) and showed that

asymmetric ad8ustment was an important consideration with asset prices. The model

is of the form

$!$

!$

!$$%

! +++= ttttt Iuu

"hereIis a dummy variable that ta1es the value of $ when the shoc1 is less than %

(ne0ative* and % otherwise. To determine if there is asymmetric ad8ustment) depends

on the si0nificance of the last term) which can be determined usin0 the t/statistic.

'iven the followin0 set of results

$

!

$

!

$

!

$

!$-.%!.%-.%+.%

+++=

ttttt Iuu

If we assume that !$t

9%.- and +.% $ =tu ) when the shoc1 is positive)!

t

9( %.+: %.$-+:%.$6*9 %.;$+) if the shoc1 is ne0ative !t

9(%.+:%.$-+:%.$6

:%.%6!+*9%.;+-+.

The alternative to the above model is to use *ln(*ln(!$

$

!$

$!$

!

+++=

t

t

t

ttt

uu

GARCHinmean

In this class of models) the conditional variance enters into the conditional mean

equation as well as the usual error variance part.

!$

!$$%

!

$

++=

++=

ttt

ttt

u

uy

Ifytis assumed to be an asset return) then in effect the first equation su00ests that the

mean return is dependent on the ris1) if the parameter ? is positive and si0nificant)

then it means that the mean return increases when there is 0reater ris1) in effect ? can

be interpreted as a ris1 premium.