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Applied Econometrics

Seminar 5GARCH models cont.

(Empirical modeling)

Please note that for interactive manipulation you need Mathematica 6 version of this .pdf. Mathematica 6 will be available soon at all Lab's Computers at IES

http://staff.utia.cas.cz/barunikJozef Barunik ( barunik @ utia. cas . cz )

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Outline

We went through most of the theory during last 2 lectures and 1 seminar

So we know empirical strategy of fitting ARIMA-GARCH models

We will introduce TGARCH modeling and see how it can improve our results

We will introduce other forms of GARCH - EGARCH, IGARCH, GARCH-M

We will also test Multivariate GARCH model

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ARIMA-GARCH on SAX

Today we will begin with following data:SAX_1998_2008.txt - Slovak index

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ARIMA-GARCH on SAX cont.

ACF and PACF does not show any significant dependencies:

If we fit ARIMA, we can see, that ARIMA(1,1,1) is not notably better than ARIMA(0,1,0) Forecasting??? model is useless !!!

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ARIMA-GARCH on SAX cont.

ARCH-LM test strongly rejects the null hypothesis of no conditional heteroskedasticity in SAX residuals from ARIMA(1,1,1), Let's have a look at SQUARED RESIDUALS ACF AND PACF !!!

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GARCH

from ACF and PACF of squared residuals from ARIMA(1,1,1) and from ARCH-LM test we can see, that there are further dependencies in the data, thus we will model them by allowing for heteroskedasticity: ARCH, and GARCH models.

ARCH and GARCH is able to model all the empirically found properties of stock market returns as excess volatility, volatility clusters, also fat tails which tells us that there is greater probability of unexpected events

BUT these effects are much weaker then AR dependencies, so we will not expect high degree of variance explained !

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ARIMA-GARCH on SAX cont.

We will fit the ARCH, GARCH until there is no dependencies left in residuals: ARIMA(1,1,1)-GARCH(1,1) best describes the data. BUT ARCH-LM test still shows some degree of dependencies... so maybe TGARCH ?Let's have a look at how we modeled volatility of the SAX index: (resp. residuals - second plot)

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We will fit the ARCH, GARCH until there is no dependencies left in residuals: ARIMA(1,1,1)-GARCH(1,1) best describes the data. BUT ARCH-LM test still shows some degree of dependencies... so maybe TGARCH ?Let's have a look at how we modeled volatility of the SAX index: (resp. residuals - second plot)

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TGARCH

We will find that certain assymetries might govern the financial time series

leverage effect:What happends when bad news / good news arrive to the market?Does it have the same effect? Or it's assymetric?How does it affect our model?

Commonly, returns are assymetric, and if we do not allow for such assymetry, we have assymetric residuals.

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TGARCH on SAX index

Following model allows for assymetries in resudials: TGARCH - Treshold GARCH:

st2 = w+g1 ut-12 +g1- ut-12 IIut-1<0M+ b1 st-12 ,

where IH.L denotes indicator function which is 1 for past innovations with negative effect. Assymetric effect is covered by the TGARCH model if g1->0

Back to SAX index:

We can see that g1- is not significant, thus TGARCH does not improve our estimate, and data does not seem to have significant asymmetry effect

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Other Extensions of GARCH - More Theory

You can see, that GARCH is just special case of TGARCH, it is TGARCH without assymetries.

We also know other models which can deal with different observations in data. Most important extensions to GARCH are following:

GARCH-M: GARCH in mean, when the returns are dependent directly on their volatilityEGARCH: Exponential GARCH - leverage effect is exponentialIGARCH: unit-root GARCH, the key is that past squared shocks are persistent

Let's have a look at demonstrations on how does these forms look like (using power of Mathematica 6)(Unfortunatelly we are not able to estimate these using JMulti, so we will have to use another software, i.e. Eviews

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Example of AR(1)-TAR-GARCH(1,1)

Let's consider another form of AR(1)-TAR-GARCH(1,1) model, which is AR(1), Two regime GARCH(1,1) model:

rt = f0+f1 rt-1+ at ,at = st et

st2 = :a0+a1 at-12 + b1 st-12 at-1 § 0

a2+a3 at-12 + b2 st-12 at-1 > 0

we choose Treshold, which will switch between two processes, or two regimes, and deal differently with assymetries

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Example of AR(1)-TGARCH(1,1) cont.

ARH1L-TAR-GARCHH1,1LSimulated series Simulated Volatility

treshold - k -1.3

f0 0.05

f1 0.5

a0 0.5

a1 0.3

b1 0.2

a2 0.5

a3 0.3

b2 0.5

New Random Case

ExportSimulated Series

100 200 300 400 500

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20

30

40

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GARCH-M

The return of a security may sometimes depend directly on volatility. To model this, we use GARCH in mean (GARCH-M) model. GARCH(1,1) - M is formalized as:

rt = m+ cst2+ atat = st et ,st2 = a0+ai at-12 + b1 st-12 ,

where m and c are constant. c is also called risk premium

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Example GARCH(1,1)-M artificial processes

Note, that with positive risk premium c, returns are positively skewed, as they are positively related to its past volatility

Sample GARCHH1,1L-M ACF function of at2 PACF function of at

2

risk premium c -0.025

a0 0.552

a1 0.324

b1 0.578

New Random Case

ExportSimulated Series

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IGARCHIGARCH models are unit-root GARCH models, their key feature is, that past squared shocks is persistent. IGARCH(1,1) is formalized as:at = st et ,st2 = a0+ b1 st-12 + H1- b1L at-12

Example IGARCH(1,1) artificial processes

IGARCHH1,1LSimulated series Simulated Volatility

a0 0.808

b1 0.308

New Random Case

ExportSimulated Series

100 200 300 400 500

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EGARCH

more complicated (only for interested students as reference)

logHst2L = w+⁄i=1q b j logIst- j2 M+⁄i=1

p aiet-ist-i

+⁄k=1r gk

et-kst-k

,

while left side is log of the conditional variance, leverage effect is expected to be exponential, rather than quadratic, and forecasts will be nonnegative.

We can test for presence of leverage effect by testing the null hypothesis that gi < 0, the impact will be asymmetric if g ¹≠ 0.

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Multivariate GARCH

Straighforward generalization of univariate models.

Modeling of covariances and correlations - forecasting.We allow covariances and correlations to be time-varying.

Problem? very large number of parameters to be estimated.

In JMulTi - BEKK form, quasi maximum likelihood estimator (you should know other forms from lecture).

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Multivariate GARCH on PX and WIG

load PX_WIG_2005_2009.txt dataset - Prague and Warsaw indices in 2005-2009 years

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Multivariate GARCH on PX and WIG cont.

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Multivariate GARCH on PX and WIG cont.

and look at the residuals left, do all the necessary testing...

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Questions

Now you know how to do your Term paper !

Good Luck :-)

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