Minimally cross-entropic conditional density [Elektronische Ressource] : a generalization of the GARCH model / Matthias Scherer. Betreuer: S. T. Rachev

Minimally cross-entropic conditional density [Elektronische Ressource] : a generalization of the GARCH model / Matthias Scherer. Betreuer: S. T. Rachev

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Minimally Cross-EntropicConditional Density:A Generalization of the GARCH ModelZur Erlangung des akademischen Gradeseines Doktors der Wirtschaftswissenschaften(Dr. rer. pol.)von der Fakult¨at fu¨rWirtschaftswissenschaftendes Karlsruher Instituts fu¨r Technologie (KIT)angenommeneDissertationvonDipl. Wi.-Ing. Matthias SchererTag der mu¨ndlichen Pru¨fung: 11. Mai 2011Referent: Prof. Dr. S.T. RachevKorreferent: Prof. Dr. M. Uhrig-Homburg2011, KarlsruheTo ElavAcknowledgementsI would like to express my deeply felt gratitude to my supervisor Prof. Dr.Svetlozar. T. Rachev and his assistant Dr. Young Shin Kim for their veryenthusiastic and inspiring support during my work. I am very grateful forthe possibilities they gave me and the trust they had in me. I am alsoindebted to my co-supervisor Prof. Dr. Marliese Uhrig-Homburg and toProf. Dr. Frank J. Fabozzi for their support in writing and finishing thisdissertation.Special thanks go to my family and my friends without whom this effortwould have been impossible. In particular, I would like to dedicate a thank˙you to Ms. Elz˙bieta Zuk and Mr. Simon Notheis for their great suggestionsand valuable advices.viContentsIntroduction 11 Probability theory 51.1 Distributions and random variables . . . . . . . . . . . . . . . 51.2 Skewness and heavy-tails . . . . . . . . . . . . . . . . . . . . 101.3 Likelihood and entropy . . . . . . . . . . . . . . . . . . . . . . 152 Econometric models 192.

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MinimallyCross-Entropic
ConditionalDensity:
AGeneralizationoftheGARCHModel

ZurErlangungdesakademischenGrades
einesDoktorsderWirtschaftswissenschaften

(Dr.rer.pol.)

vonderFakulta¨tfu¨r
Wirtschaftswissenschaften
desKarlsruherInstitutsfu¨rTechnologie(KIT)
angenommene

Dissertation

nov

Dipl.Wi.-Ing.MatthiasScherer

Tagdermu¨ndlichenPru¨fung:
Referent:
Korreferent:

11.Mai2011
Prof.Dr.S.T.Rachev
Prof.Dr.M.Uhrig-Homburg
2011,Karlsruhe

oT

alE

Acknowledgements

v

IwouldliketoexpressmydeeplyfeltgratitudetomysupervisorProf.Dr.
Svetlozar.T.RachevandhisassistantDr.YoungShinKimfortheirvery
enthusiasticandinspiringsupportduringmywork.Iamverygratefulfor
thepossibilitiestheygavemeandthetrusttheyhadinme.Iamalso
indebtedtomyco-supervisorProf.Dr.MarlieseUhrig-Homburgandto
Prof.Dr.FrankJ.Fabozzifortheirsupportinwritingandnishingthis
dissertation.

Specialthanksgotomyfamilyandmyfriendswithoutwhomthiseort
wouldhavebeenimpossible.Inparticular,Iwouldliketodedicateathank
youtoMs.Elz_bietaZ_ukandMr.SimonNotheisfortheirgreatsuggestions
andvaluableadvices.

iv

Contents

Introduction1
1Probabilitytheory5
1.1Distributionsandrandomvariables...............5
1.2Skewnessandheavy-tails....................10
1.3Likelihoodandentropy......................15
2Econometricmodels19
2.1Stochasticprocesses.......................19
2.2TheARMAmodel........................22
2.3TheGARCHmodel.......................24
2.4TheARMA-GARCHmodel...................27
2.5Modelinference..........................28
3TheMCECDmodel33
3.1Conditionaldensity........................33
3.2Minimumcross-entropy.....................34
3.3TheMCECDdenition.....................36
3.4Stationarity............................39
4Conditionalvolatility41
4.1TheVola-MCECDmodel....................41
4.1.1Vola-MCECDandGARCH...............41
4.1.2Non-Gaussianmodels..................42
4.2TheMean-Vola-MCECDmodel.................48
4.3Simulationandempiricalresults................49
4.3.1Time-varyingmeanandvolatility............50
4.3.2Thetime-varyingproperty................52
4.3.3Qualityofone-dayforecasting..............53

iiv

iiiv

CONTENTS

5Conditionalskewness57
5.1Skewnessmodels.........................58
5.1.1TheARCDmodel....................60
5.1.2Skewnessestimation...................62
5.2TheSkew-MCECDmodel....................66
5.2.1Thedenition.......................66
5.2.2ExplicitSkew-MCECDdynamics............67
5.3TheVola-Skew-MCECDmodel.................68
5.4Simulationandempiricalresults................71
5.4.1Thetime-varyingproperty................71
5.4.2Empiricalskewness....................73

Conclusion

Bibliography

81

83

AProofs91
A.1Theiterativeformula.......................91
A.2Predictability...........................92
A.3Convergenceofweightedgeometricseries...........93
A.4Stationarity............................94
A.5EquivalenceofVola-MCECDandGARCH...........97
A.6ExplicitMean-Vola-MCECDdynamics.............100
A.7ExplicitSkew-MCECDdynamics................103

BTables

107

ListofFigures

4.1Log-elasticityofPDFofastableParetiandistribution....
4.2Conditionalmeantrajectoriesofsimulateddata........
4.3Conditionalvolatilitytrajectoriesofsimulateddata......
4.4ConditionalmeanofS&P500index..............
4.5ConditionalvolatilityofS&P500index............

5.1Eectsofmeantransformation.................
5.2Eectsofvariancetransformation................
5.3Eectsofskewnesstransformation...............
5.4ComparisonofMMandMLskewnessestimators.......
5.5ComparisonofMLskewnessestimators............
5.6Eectsofskewnessparameteronvolatilityestimator.....
5.7Estimatedtrajectoriesfortime-varyingskewness.......
5.8Estimatedtrajectoriesforconstantskewness..........
5.9Time-varyingskewnessofARCDforU.S.stockindices....
5.10Time-varyingskewnessofMCECDforU.S.stockindices...
5.11TripletofMCECDmodelforDowJones............

xi

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8595955666073747778797

x

LITSOFIFUGERS

ListofTables

4.1Goodness-of-tresultsforsimulateddata...........51
4.2One-dayVaRbacktestingresultsforstockindices.......55

5.1Estimatesforsimulated,time-varyingskewness........72
5.2Goodness-of-tforsimulated,time-varyingskewness.....72
5.3Estimatesforsimulated,constantskewness..........73
5.4Goodness-of-tforsimulated,constantskewness.......74

B.1ParameterestimatesforS&P500................108
B.2ParameterestimatesforDowJones...............109
B.3ParameterestimatesforNasdaq100..............110
B.4Goodness-of-tforS&P500...................111
B.5Goodness-of-tforDowJones..................112
B.6Goodness-of-tforNasdaq....................113
B.7One-dayCDFforecastingresults................114
B.8Goodness-of-