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# Hall ARCH and GARCH

## 1. Lecture 8 Stephen G. Hall ARCH and GARCH

## 2.

REFSA thorough introduction

‘ARCH Models’ Bollerslev T, Engle R F and Nelson D B

Handbook of Econometrics vol 4. or UCSD Discussion paper

no 93.49. (available on my web site)

A quick survey

Cuthbertson Hall and Taylor

## 3.

Until the early 80s econometrics had focused almost solelyon modelling the means of series, ie their actual values.

Recently however we have focused increasingly on the

importance of volatility, its determinates and its effects on

mean values.

A key distinction is

unconditional variance.

between

the

conditional

and

the unconditional variance is just the standard measure of

the variance

var(x) =E(x -E(x))2

## 4.

the conditional variance is the measure of our uncertaintyabout a variable given a model and an information set.

cond var(x) =E(x-E(x| ))2

Conditional

variance

this is the true measure of uncertainty

variance

mean

## 5.

Stylised Facts of asset returnsi) Thick tails, they tend to be leptokurtic

ii)Volatility clustering, Mandelbrot, ‘large changes tend to be

followed by large changes of either sign’

iii)Leverage Effects, refers to the tendency for changes in stock

prices to be negatively correlated with changes in volatility.

iv)Non-trading period effects. when a market is closed information

seems to accumulate at a different rate to when it is open. eg stock

price volatility on Monday is not three times the volatility on

Tuesday.

v) Forcastable events, volatility is high at regular times such

as news announcements or other expected events, or even at

certain times of day, eg less volatile in the early afternoon.

## 6.

vi)Volatility and serial correlation. There is a suggestion of aninverse relationship between the two.

vii)

Co-movements in volatility. There is considerable evidence

that volatility is positively correlated across assets in a market and

even across markets

## 7.

## 8.

## 9.

Engle(1982) ARCH ModelAuto-Regressive Conditional Heteroscedasticity

Yt X t t

2

t

t

q

i 2 t i ( L) 2

define

2

t ~ N (0, t 2 )

i 1

t t t

2

2

( L) t

2

an AR(q) model for squared innovations.

## 10.

note as we are dealing with a variance0 i 0 all i

even though the errors may be serially uncorrelated they are not

independent, there will be volatility clustering and fat tails.

if the standardised residuals

zt t / t

are normal then the fourth moment for an ARCH(1) is

E ( t ) / E ( t ) 3(1 ) /(1 3 ) if 3 1

4

2

2

2

2

2

## 11.

GARCH (Bollerslev(1986))In empirical work with ARCH models high q is often required, a

more parsimonious representation is the Generalised ARCH model

q

p

i 1

j 1

2 t i 2 t i j 2 t j

( L) ( L)

2

define t t

2

2

2

t

t ( ( L) ( L)) ( L) t

2

2

which is an ARMA(max(p,q),p) model for the squared

innovations.

## 12.

This is covariance stationary if all the roots of( L) ( L) 1

lie outside the unit circle, this often amounts to

(1) (1) 1

If this becomes an equality then we have an Integrated GARCH

model (IGARCH)

## 13.

Nelsons’ EGARCH modelthis captures both size and sign effects in a non-linear

formulation

q

p

i 1

j 1

log( 2 t ) i ( zt i (| zt i | E | zt i |)) j log( 2 t j )

## 14.

Non-linear ARCH model NARCHthis then makes the variance depend on both the size and the

sign of the variance which helps to capture leverage type

effects.

q

t

p

i | t i | j

i 1

j 1

t j

## 15.

Threshold ARCH (TARCH)Large events to have an effect but no effect from small events

q

t ( i I ( t i 0) i ( t i 0))

2

i 1

p

2

t i

) j

Many other versions are possible by adding minor

asymmetries or non-linearities in a variety of ways.

j 1

2

t j

## 16.

All of these are simply estimated by maximum likelihoodusing the same basic likelihood function, assuming

normality,

T

log( L) ( log( t ) t / t )

2

i 1

2

2

## 17.

ARCH in MEAN (G)ARCH-MMany classic areas of finance suggest that the mean of a

relationship will be affected by the volatility or uncertainty of

a series. Engle Lilien and Robins(1987) allow for this

explicitly using an ARCH framework.

yt xt t t

2

q

t i

2

i 1

p

2

t i

j

2

t j

j 1

typically either the variance or the standard deviation are

included in the mean relationship.

## 18.

often finance stresses the importance of covariance terms.The above model can handle this if y is a vector and we

interpret the variance term as a complete covariance matrix.

The whole analysis carries over into a system framework

## 19.

Non normality assumptionsWhile the basic GARCH model allows a certain amount of

leptokurtic behaviour this is often insufficient to explain real

world data. Some authors therefore assume a range of

distributions other than normality which help to allow for the

v

fat tails in the distribution.

t Distribution

The t distribution has a degrees of freedom parameter which

allows greater kurtosis. The t likelihood function is

lt ln( (0.5(v 1)) (0.5v) 1 (v 2) 1 / 2 (1 zt (v 2) 1 ) ( v 1) / 2 ) 0.5 ln( 2 t )

where F is the gamma function and v is the degrees of

freedom as v this tends to the normal distribution

## 20.

IGARCH.The standard GARCH model

t ( L) ( L)

2

2

2

is covariance stationary if

(1) (1) 1

But Strict stationarity does not require such a stringent

restriction (That is that the unconditional variance does not

depend on t),in fact we often find in estimation that

(1) (1) 1

## 21.

this is then termed an Integrated GARCH model (IGARCH),Nelson has established that as this satisfies the requirement

for strict stationarity it is a well defined model.

However we may suspect that IGARCH is more a product of

omitted structural breaks than the result of true IGARCH

behavior.

## 22.

Multivariate ModelsIn general the Garch modelling framework may be easily

extended to a multivariate framework where

Et ( t ' t ) t

however there are some practical problems in the choice of

the parameterisation of the variance process.

## 23.

A direct extension of the GARCH model would involve a verylarge number of parameters.

The conditional variance could easily become negative even

when all the parameters are positive.

The chosen parameterisation should allow causality between

variances.

## 24.

Vector ARCHlet vech denote the matrix stacking operation

a b

(a b d )

vech

b d

a general extension of the GARCH model would then be

vech( t ) W A( L)vech( t 1 't 1 ) B( L)vech( t 1 )

this quickly produces huge numbers of parameters, for

p=q=1 and n=5 there are 465 parameters to estimate here.

## 25.

One simplification used is the Diagonal GARCH model whereA and B are taken to be diagonal, but this assumes away

causality in variances and co-persistence. We need still

further complex restrictions to ensure positive definiteness in

the covariance matrix.

A more tractable alternative is to state

q

p

i 1

j 1

t V 'V A'i t i 't i Ai B' j t j B j

we can further reduce the parameterisation by making A and

B diagonal.

## 26.

Factor ARCHSuppose a vector of N series has a common factor structure.

Such as;

yt B t t

where

are the common factors and

~ N (0, )

E ( 't ) t

then the conditional covariance matrix of y is given by

Covt 1 ( yt ) t t B t B'

Or

k

t t i 'i k

i 1

## 27.

So given a set of factors we may estimate a parsimoniousmodel for the covariance matrix once we have parameterized

One assumption is that we observe a set of factors which

cause the variance, then we can simply use these. E.G. GDP,

interest rates, exchange rates, etc.

another assumption is that each factor has a univariate

GARCH representation.

K

K

k 1

k 1

t k ( k t 1 't 1 'k ) k ( k t 1 'k )