Introduction to time series (2008) 1

Chapter 5

Heterocedastic Models

Introduction to time series (2008) 2

Chapter 5. Contents.5.1. The ARCH model.

5.2. The GARCH model.

5.3. The exponential GARCH model.

5.4. The CHARMA model.

5.5. Random coefficient autoregressive (RCA) model.

5.6. Stochastic volatility.

Introduction to time series (2008) 3

• In options trading and in the foreignexchange rate market, volatility plays animportant role.

• Here volatility means conditional variance ofthe underlaying asset return.

• In this chapter we discuss econometric andstatistical models available in the literature tomodel the evolution of volatility over time.

Introduction to time series (2008) 4

• Althoug volatility is not directly measura-ble, it has some properties that are commonin asset returns:– 1. There are volatility clusters, that is volatility

may be high for certain time periods.– 2. Volatility evolves over time in a continuous

manner –there are no volatility jumps.– 3. Volatility does not diverge to infinity.– 4. Volatility seems to react differently to a big

positive return and a big negative return.

Introduction to time series (2008) 5

• Assumptions and notation . Let be thereturn series of an asset, specifically,

• If the asset is a stock with dividend payment

tz

)ln()ln( 1 ttt ppz

)ln()ln( 11 ttttt dpdpz

Introduction to time series (2008) 6

• It is informative to consider the conditionalmean and variance

• For simplicity, we will assume

• All volatility models we will mention are models for

]/)[()/()/( 2ttttttttt ZzEZzVarhZzE

0t

th

Introduction to time series (2008) 7

Chapter 5. Heterocedastic models.

5.1. The ARCH model.

Introduction to time series (2008) 8

The ARCH model.

• Engle (1982). The basic idea of ARCH models is that 1) the asset return is seriallyuncorrelated but dependent and 2) the de-pendence can be described with a cuadraticfunction. An ARCH(r) model is

• where

22110 ..., rtrttttt zzhhz

00)1,0( 0 it iid

Introduction to time series (2008) 9

The ARCH model.

• Interpretation. Large past square returnsimply a large conditional variance for thereturn. Consequently, the return tends toassume a large value (in modulus).

• Under the ARCH framework, large returnstend to be followed by another large return. This is similar to the volatility clusteringproperty.

Introduction to time series (2008) 10

The ARCH model.

• Properties. Consider the ARCH(1) model

• (1) the conditional mean remains zero

• (2) unconditional variance

2110, ttttt zhhz

0)]/([)( ttt ZzEEzE

)()()]/([)()(

2110

2110

22

tt

tttt

zEzEZzEEzEzVar

Introduction to time series (2008) 11

The ARCH model.

• Because is a stationary process,

• and, therefore,

• because variance must be positive, we need

tz

)()()( 21

2ttt zVarzEzE

)1/()()()( 1010 ttt zVarzVarzVar

10 1

Introduction to time series (2008) 12

The ARCH model.

• Under the ARCH assumptions, the taildistribution of is heavier than that of a normal distribution. Therefore, the probabi-lity of outliers is higher.

• This is in agreement ment with the empiri-cal finding that outliers appear more oftenin asset returns than that implied by an iidsequence of normal random variates.

tz

Introduction to time series (2008) 13

The ARCH model.

• Weakness of ARCH model.

– (1) treats positive and negative returns in thesame way (by past square returns)

– (2) is very restrictive (in parameters)– (3) does not provide any new insight for un-

derstanding financial time series (just mecha-nical way to describe behaviour)

– (4) often over-predicts the volatility, because itrespond slowly to large shocks.

Introduction to time series (2008) 14

The ARCH model.

• Building ARCH models:

– (1) an ARIMA model is built for the observedtime series to remove any serial correlation in the data.

– (2) examine the squared residuals to check forconditional heterocedasticity.

– (3) use the pacf of squared residuals to determi-ne the ARCH order and to perform the MLE ofthe specified model.

Introduction to time series (2008) 15

Chapter 5. Heterocedastic models.

5.2. The GARCH model.

Introduction to time series (2008) 16

The GARCH model.

• Bollerslev (1986). ARCH models oftenrequire to many parameters to describe theevolution of volatility. Al alternative is theGARCH(r,s) model

• where

r

i

s

jjtjititttt hzhhz

1 1

20,

),max(

10 1)(,0,0,0

sr

ijiji

Introduction to time series (2008) 17

The GARCH model.

• the GARCH model implies that the uncon-ditional variance of is finite whereas itsconditional variance evolves over time. Let

• We can rewrite the GARCH model as

tzth

tttttt zhhz 22

),max(

1 1

20

2 )(sr

i

s

jjtjtitiit zz

Introduction to time series (2008) 18

The GARCH model.

• The equation before is an ARMA form ofthe squared series . It is then clear

• And therefore, the unconditional variance isfinite whereas the conditional one evolvesover time.

2tz

),max(

1

02

)(1)()( sr

iii

tt zEzVar

Introduction to time series (2008) 19

The GARCH model.

• Some properties.

– (1) recreates the clustering behavior.– (2) heavier tail than that of a normal dist.– (3) simple parametric form that can be used to

describe the evolution of volatility but notunderstanding.

– (4) do not reflect asimetric behavior.– (5) tails are still too short.

Introduction to time series (2008) 20

The GARCH model.

• Example: Monthly returns of S&P500 for792 observations, starting from 1926.

• An AR (3) (or MA(3)) model is suggestedby the correlogram.

• AR(3) estimation:

• but none of the parameters is significant.

0033.ˆ123.023.088.0066. 2321 attttt azzzz

Introduction to time series (2008) 21

The GARCH model.

• .

Introduction to time series (2008) 22

The GARCH model.

• .

Introduction to time series (2008) 23

The GARCH model.

• Joint estimation of AR(3)-ARCH(1,1). TheAR(3) parameters are still non significant, and therefore, dropped,

• the unconditional variance is

211 1221.8470.0001.

0085.

ttt

tt

ahhaz

0036.1221.847.1

0001.)( 2

tzE

Introduction to time series (2008) 24

Chapter 5. Heterocedastic models.

5.3. The exponential GARCH model.

Introduction to time series (2008) 25

The exponential GARCH model.

• To overcome some weakness of the GARCH models in handling financial data, Nelson (1991) proposed the EGARCH. To allow forasymmetric effects, he considers the weigh-ted innovation:

• where are real constants, andare zero mean iid sequences with continuousdistributions.

)]([)( tttt Eg

, )(, ttt E

Introduction to time series (2008) 26

The exponential GARCH model.

• An EGARCH (r,s) model can be written as,

• Differences with GARCH:– Uses logged conditional variance to relax the

postiveness constraint of model coefficients.– The model can respond asymmetrically to

positive and negative values of the returns.

)(...1...1)ln( 1

1

10

trr

ss

tttt gBBBBhz

Introduction to time series (2008) 27

Chapter 5. Heterocedastic models.

5.4. The CHARMA model.

Introduction to time series (2008) 28

The CHARMA model.

• Conditional Heterocedastic ARIMA model. Uses random coefficients to produce conditional heterocedasticity. Have similar second order conditional properties thanGARCH.

• Wherettttt aBaBzB )()()(

),0( 2 Niidt

Introduction to time series (2008) 29

The CHARMA model.

• and

• is a purely random coefficient polynomialin B. The random coefficient vector is a sequence of iid random vectors with mean zero and nonnegative definite covariancematrix .The conditional variance of

rtrtt BBB ,,1 ...1)(

ta

)',...,(),...,( 112

rttrttt aaaah

Introduction to time series (2008) 30

The CHARMA model.

• The conditional variance is equivalent tothat of an ARCH(r) model if the matrixis diagonal. The CHARMA model uses cross-products of the lagged values of

• For example, in modelling asset returns, thecross product terms denote interactions bet-ween previous returns.

tz

Introduction to time series (2008) 31

Chapter 5. Heterocedastic models.

5.5. Random Coefficient Autoregressive(RCA) model.

Introduction to time series (2008) 32

The RCA model.

• Is introduced to account for variability amongdifferent subjects under study, similar to pa-nel data in econometrics.Is a condional hete-rocedastic model, but it is used to obtain a better description of the conditional mean ofthe process by allowing for the parameters toevolve over time. A RCA(p) model is

p

itititit azz

1)(

Introduction to time series (2008) 33

The RCA model.

• where is a sequence of independentrandom vectors with mean zero and cova-riance

• The conditional mean and variance are

)',...,(),...,(

),(

112

11

pttpttat

p

iitittt

zzzzh

zZzE

t

Introduction to time series (2008) 34

Chapter 5. Heterocedastic models.

5.6. Stochastic Volatility model.

Introduction to time series (2008) 35

The stochastic volatility model.

• An alternative approach to describe theevolutionnof volatility is to introduce aninnovation to the conditional varianceequation of . A simple SV model is

• Introducing the innovation makes the SV model more flexible in describing theevolution of , but also increases the

tz

ttr

rttt hBBhz 01 )ln()...1(

th

Introduction to time series (2008) 36

The stochastic volatility model.

• difficulty in parameter estimation. A quasi-likelihood method with Kalman filter isneeded.

Introduction to time series (2008) 37

Example

Daily price observations of the Deutschemark/British Pound foreignexchange rate.

Introduction to time series (2008) 38

Example

Convert the prices to a return series.

Introduction to time series (2008) 39

Example

. Check for correlation in the return series

Introduction to time series (2008) 40

Example

.

Introduction to time series (2008) 41

Example

. Check for correlation in the squared returns

Introduction to time series (2008) 42

Example

This figure shows that, although the returns themselves arelargely uncorrelated, the variance process exhibits somecorrelation. Note that the ACF shown in this figure appears todie out slowly, indicating the possibility of a variance processclose to being nonstationary.

Introduction to time series (2008) 43

Example

• Ljung-Box-Pierce Q-Test. no significant correlation is present in the raw returns when tested for up to 10, 15, and 20 lags of theACF at the 0.05 level of significance.

[H,pValue,Stat,CriticalValue] = 0 0.7278 6.9747 18.3070 0 0.2109 19.0628 24.9958 0 0.1131 27.8445 31.4104

• However, there is significant serial correlation in the squaredreturns when you test them with the same inputs.

[H,pValue,Stat,CriticalValue] = 1.0000 0 392.9790 18.30701.0000 0 452.8923 24.99581.0000 0 507.5858 31.4104

Introduction to time series (2008) 44

Example

. Engle's ARCH Test.[H,pValue,Stat,CriticalValue] = 1.0000 0 192.3783 18.3070 1.0000 0 201.4652 24.9958 1.0000 0 203.3018 31.4104

*Given sample residuals obtained from a curve fit, tests forthe presence of h-th order ARCH effects by regressingthe squared residuals on a constant and the laggedvalues of the previous h squared residuals. Under thenull hypothesis, the asymptotic test statistic (h*R^2), isasymptotically chi-square distributed with h degrees offreedom.

2* Rh

Introduction to time series (2008) 45

Example. Examine the Estimated GARCH Model.

Mean: ARMAX(0,0,0); Variance: GARCH(1,1)Conditional Probability Distribution: GaussianNumber of Parameters Estimated: 4

Standard T Parameter Value Error Statistic----------- ----------- ------------ -----------

C -6.1919e-005 8.4331e-005 -0.7342 K 1.0761e-006 1.323e-007 8.1341

GARCH(1) 0.80598 0.016561 48.6685ARCH(1) 0.15313 0.013974 10.9586

21-t1-t z 0.15313h 0.80598 006-1.0761e th

Introduction to time series (2008) 46

Example

. Compare the Residuals, Conditional Standard Deviations, andReturns

Introduction to time series (2008) 47

Example

. Plot and Compare the Correlation of the Standardized Innovations

Introduction to time series (2008) 48

Example

.

Introduction to time series (2008) 49

Example

. Quantify and Compare Correlation of the Standardized Innovations

[H,pValue,Stat,CriticalValue] = 0 0.5262 9.0626 18.3070 0 0.3769 16.0777 24.9958 0 0.6198 17.5072 1.4104

[H,pValue,Stat,CriticalValue] = 0 0.5625 8.6823 18.30700 0.4408 15.1478 24.99580 0.6943 16.3557 31.4104

Introduction to time series (2008) 50

Example

.MODELOS NO LINEALES EN LA MEDIA

Introduction to time series (2008) 51

Example

.

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Example

.

Introduction to time series (2008) 53

Example.MODELOS BILINEALES

Introduction to time series (2008) 54

Example

.

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Example

.

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Example

.MODELOS TAR

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Example

.

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Example

.

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Example

.

Introduction to time series (2008) 60

Example

.