Applied Econometrics

Seminar 4GARCH models

(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

ARIMA fitting methodology on PX 2004-2009 (repetition is the mother of wisdom)

ARIMA forecasting

problems of ARIMA forecasts

empirical strategy of fitting ARIMA-GARCH models to the data

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2 Seminar4.nb

ARIMA

What do we need to care about? stationarity (visual, ADF), difference if nonstationary, check ACF and PACF for any dependen-cies, check Q statistics, do regression estimates for p and q lags until you find no improvementin information criteria and residuals seem to have no dependencies (not always possible as wewill see later)

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Today's data - Prague Stock Exchange analysis

Today we will test Prague Stock Exchange data again including last "crisis" data:

PX_2004_2009.txt

Load the dataset and let's do the ARIMA analysis

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ARIMA cont. - PX 2004 - 2009Plot of index level and returns

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ARIMA cont. - PX 2004 - 2009

ACF/PACF and residuals after ARIMA(2,1,0) - do not forget to check Portmentau statistics tobe sure !

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PX forecast with ARIMACan you see anything wrong about it?

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Problems with ARIMA forecast of PX?

ARIMA reveals linear dependencies, and as you can see from the residuals, it really did not helpus in PX returns modelling, as variance is not constant in time.

Homoscedasticity of residuals - not at all

If we use this model for forecasting, we could see that it is of no use, so one have to really becarefull !!!

How can we deal with this problem? - ARCH/GARCH models|

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ARCH-LM test

Now we know that ARIMA might be useless for forecasting of PX, let's test, if the assumptionof homoscedasticity holds. We can do it by allowing heteroscedasticity in the model - useARCH or GARCH. Before that we might use simple formal test to find, if there are such depen-dencies - ARCH LM test

H0 : b1 =. .. = bq = 0

H1 : b1 0 or ... or bq 0,

where b's are estimates of ARCH(q)

LM~c2HqL if the null hypothesis of no conditional heteroskedasticity holds|

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ARCH-LM test on PX data

ARCH-LM test strongly rejects the null hypothesis of no conditional heteroskedasticity in PXresiduals from ARIMA(2,1,0), Let's have a look at SQUARED RESIDUALS ACF ANDPACF !!!

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10 Seminar4.nb

GARCH

from ACF and PACF of squared residuals from ARIMA(2,1,0) and from ARCH-LM test wecan see, that there are further dependencies in the data left, thus we will model them by allowingfor heteroskedasticity: ARCH, and GARCH models.

please note that ARCH and GARCH is able to model all the empirically found properties ofstock market returns (stylized facts) as excess volatility, volatility clusters, also fat tails which tellsus that there is greater probability of unexpected events

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

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GARCH cont.

We will fit the ARCH, GARCH until there is no dependencies left in the residuals: ARIMA(2,1,0)-GARCH(1,1) best describes the data - is our model OK now?

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GARCH cont. - standard deviation process

So we managed to model the standard deviation process by GARCH. We have captured theexplosive volatility in last years of financial crisis.

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