arch model.ppt

7/22/2019 ARCH Model.ppt

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Time Series Econometrics:

ARCH/GARCH Models

Measuring volatility: Conditional

heteroscedastic ModelsK.R. Shanmugam, MSE


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Features of Financial Data (e.g

Stock returns)

1. Volatility Clustering: Some periods are highlyvolatile while others are less. Big shocks(residuals) tend to follow big shocks in eitherdirection and small shocks follow small. Thisimplies strong AC. In addition, constant varianceassumption is inappropriate

(If unconditional (or long-run) variance isconstant but there are periods in which thevariance is relatively high. Such series are calledconditionally heteroscedastic).


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Features.

2. Leverage Effect: Volatility is higher in a

falling market than it is in a rising market

(there is a tendency for volatility to rise

more following a large price fall than

following a price rise of the same

magnitude).


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Features

3. Leptokurtosis: They have distributionswhich exhibit flatter tails and excess peakedness

(due to a large number of excessive values).

(a normal distribution is skewed (3rd moment)and has a coefficient of kurtosis of 3 (it is

symmetric and said to be meso-kurtic)

Leptokurtic is one which has flatter tails and ismore peaked at mean than normal with same

mean and variances.


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NSE (2/1/1996 to 20/12/2002)

600

800

1000

1200

1400

1600

1800

1996 1997 1998 1999 2000 2001 2002

CLOSE

-.12

-.08

-.04

.00

.04

.08

.12

1996 1997 1998 1999 2000 2001 2002

RETURNS


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Impact of special characteristics

They lead to violations ofhomoscedasticity as well as auto-correlation assumptions of OLS

(consequences of heteroscedasticity:estimates of parameters are unbiased, butSEs are large, CIs are very narrow, andprecision is affected)

Linear Models are unable to explain thesespecial features


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Tasks of the Asset Holder

He may be interested in forecasting the rate ofreturn and the variance of the stock asset overthe holding period.

ARMA models are useful to forecast meanreturns. But it ignores the risk factor (variance orSD is a measure of risk or volatility)

Some series are subject to fat tails, volatility

clustering and leptokurtic, the task is to specifyand forecast both mean and variance of theseries conditional on past information.


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Historical Volatility

Calculating the variance of the series in theusual way over some historical period

(e.g. In option pricing model, historical average

variance is used as volatility measure) Rolling standard deviation (RSD): Studies such

as Tauchen and Pitts (1987) used this. Theycalculate standard deviation using fixed number

of observations. That is, first calculate it usingmost recent (say) 22 days data. Then drop firstday and add 23rd day data. This is also calledOfficiers approach.


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Problems with RSD

It uses equal weight for all cases (more

recent observations should be more

relevant and be given higher weight)

Use of overlapping observations lead to

correlation issues

Zero weight for other observations are

unattractive


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Summary of Problems

unconditional forecast has a greater

variance than the conditional forecast, plus

overlapping problem and usage of equal

weighting.

So conditional forecast (since they take

into account the known current and past

realization of series) are preferable.


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Engle (1982) ARCH Model

It says that the variance of the error term at time

t depends on the squared error terms from

previous periods:

Rt =mt + et and et = N(0, t2) .(1)(or et = vtt ; vt ~ N(0, 1)) where

2t =0 + 1 et-12 + 2 et-2

2++ p et-p2..(2)

(here the AC in volatility is modeled) this is an ARCH (p) model

ARCH(1) Model: 2t =0 + 1 et-12


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ARCH TESTARCH (joint) TEST:To choose number of lagged terms.

If we start with one lag, this test will tell us whether weneed to add any additional lag term.

Steps:

(i) Run mean regression Rt =mt + et and save residuals et

(ii) Squared the residuals and regress it on its own laggedterm to test for ARCH (e2t =0 + 1 et-12 + 2 et-2

2++p et-p

2

(iii) Define TR2 ~ 2 (p), where p is number of lagged termon the right has side of second equation

(iv) Null Hypothesis: H0: 1=0; 2=0. p =0Alternate Hypothesis: H1 : 10; 2 0. p 0

(v) If test value is greater than critical value, reject the null


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Properties of ARCH (1) process

et = vtt and 2t =0 + 1 et-1

2

t = (0 + 1 et-12)1/2

Since E vt =0; E et = 0

Since E vt vt-i = 0, E et et-i =0

Unconditional variance:

E et2 = E [vt

2 (0 + 1 et-12)]

Since E vt2 =1, E et

2 = 0 + 1Eet-12

2 = 0 + 12 = 0 /(1- 1)


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Conditional Mean and Variance

E [et | et-1, et-2..] =0

E [et2 | et-1, et-2..] = E [vt

2 (0 + 1et-12)]

2

t = 1. (0 + 1et-12

) Thus, the conditional variance depends on

realized value of et-12.

If et-12

is larger, the conditional variance in twill be larger as well.


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ARCH (1) Model

Unconditional variance:

E et2 = 0 /(1- 1) ; we restrict that 0>0

and |1|


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Conditional Forecast Vs

Unconditional Forecast

Engle (1982): conditional is better thanunconditional

Example: Consider an AR(1) Model:

yt = a0 + a1 yt-1 + et Conditional Forecast of yt+1:

E (yt+1|t) = a0 + a1 yt

Forecast Error Variance is:

E [(yt+1- a0 - a1 yt)2] = E et+1

2 =2


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Unconditional Forecast of y

yt= a0 + a1 yt-1 + e1

yt (1-a1L)= a0 + et

yt = a0/( 1-a1L) + et/(1-a1L)

yt+1 = a0 /(1-a1L) + et+1 / (1-a1L)

E yt+1 = a0/ (1-a1L) = a0 / (1-a1): Long run mean

Unconditional Forecast of Error Variance

E [yt+1- Eyt+1]]2 = E (et+1/(1-a1)) =

2/ (1-a1)


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Estimation

ML Estimation Method

Log-Likelihood function using a normality assumption forthe disturbance is:

ln L = -T/2 ln (2)-(1/2) ln t2 (1/2)[(yt

a0-a1 yt-1)2 (1/t2)] It is an updating formula: Observed variance of the

residuals is taken for 1st observation; then it calculatesvariance for the second and so on for any given set ofparameter values; thus the entire time series of varianceforecast is constructed; then the likelihood functionprovides a systematic way to adjust the parameter togive the best fit.


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Generalized ARCH

Developed independently by Bollerslev (1986) andTaylor (1986)

Bollerslaves GARCH Specification:

Rt =mt + et, where et ~ N(0, t2)

2

t =0 + 1 et-12

+ 2 et-22

++ 1t-12

+ It allows conditional variance to be dependent on

previous own lags

It is a weighted average of long term average (variance),information about volatility during previous years and

fitted variance in previous years. Such an updating formula is a simple description of

adaptive or learning behaviour.

Estimation is same as ARCH

However, we use parsimonious model; GARCH (1,1) is

sufficient


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GARCH (1,1) Vs. ARCH

Proof: Let 2t =0 + 1 et-12 + t-1

2......(1)

2t-1 =0 + 1 et-22 + t-2

2.................(2)

2t-2 =0 + 1 et-32 + t-3

2(3)

Substitute (2) in (1) and then (3) in 2t =0 (1+ +

2) + 1 et-12 (1+ L+ 2L2) + 3t-3

2

An infinite number of successive substitutions leads to

2t =0 (1+ + 2+.) + 1 et-1

2 (1+ L+ 2L2+.) + 02

The last term is almost zero

The rest is similar to ARCH infinite specification

Although GARCH (1,1) has 3 parameters in conditional varianceequation (parsimonious), it allows an infinite number of past squarederrors


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(1) GARCH (1,1)

2

2 2 2

0 1 1 2 1

t t t

t t t

r

where t-1

and t-1

are ARCH and GARCH

terms respectively.


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The (G)ARCH-M Model

2 2

2 2 2

0 1 1 2 1

t t t t

t t t

r

In this application, the dependent variable in the mean

equation is determined by its conditional variance.

For instance, the expected return on an asset is related

to expected asset risk.


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The Asymmetric ARCH Models

Often, we notice that the downward movement in the

market are highly volatile than upward movement of the

same magnitude. In such case, symmetric ARCH model

undermine the true variance process. Engle and Ng

(1993) provide a news impact curve with asymmetricresponse to good and bad news.

Good newsBad News

Volatility


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two types of asymmetric ARCH

models: TARCH and EGARCH

2

2 2 2 2

0 1 1 2 1 1 1

t t

t t

1 1

where d 1if < 0 and 0 otherwise

0 good news, 0 bad news

They have differential effects on conditional var.Good (bad) news has an effect of ( )

If 0,

t t t

t t t t t

r

d

there is leverage effect (bad news increases volatility).

if 0, the news impact is asymmetric.

The TARCH or Threshold ARCH due to Zakoian

(1990) can be defined as:


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The EGARCH Model (by Nelson

1991)

2 2 1 10 1 1 3

1 1

log log t tt t

t t

If 0, there is leverage effect.

if 0, the impact is asymmetric

Log of conditional variance equation. This means that leverage effect is

exponential than quadratic. This guarantees that forecasts of conditional

variances are positive;


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Power GARCH (PARCH)

Taylor (1986) and Schwert (1989) developedS.D GARCH model. The conditional variance isnot a linear function in lagged squared residuals

where >0.

For symmetric model =0; if0, asymmetric

effects are present If=2 and =1 for all i, then the model is

standard GARCH

2 20 1( ) ( ) ( )t i t i t i j t

arch model.ppt

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