Table Of ContentSpringerBriefs in Statistics
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Adrian Pizzinga
Restricted Kalman Filtering
Theory, Methods, and Application
123
AdrianPizzinga
DepartmentofStatistics
InstituteofMathematicsandStatistics
FluminenseFederalUniversity
RiodeJaneiro,Brazil
ISSN2191-544X ISSN2191-5458(electronic)
ISBN978-1-4614-4737-5 ISBN978-1-4614-4738-2(eBook)
DOI10.1007/978-1-4614-4738-2
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012941620
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Preface
In this book, I highlight the developments in Kalman filtering subject to general
linear constraints. Essentially, the material to be presented is almost entirely
based on the results andexamplesoriginallydevelopedin Pizzingaet al. (2008a),
Cerqueiraet al.(2009),Pizzinga(2009,2010),Souzaetal. (2011),Pizzingaetal.
(2011), and Pizzinga (2012). There are fundamentally three kinds of topics: (a)
new proofs for already established results within the restricted Kalman filtering
literature; (b) additional results that are should shed light on theoretical and
methodologicalframeworksforlinearstatespacemodelingunderlinearrestrictions;
and (c) applications in investment analysis and in macroeconomics, where the
proposedmethodsare illustrated andevaluated.At the end,I brieflydiscusssome
extensionsinthesubject,which,again,stepintotheory,methods,andapplications.
Itisimportanttomentionthatmydoctoralthesis,ofwhichthisbookisamajor
revision,wouldnothavebeencompletedwithoutthefinancialsupportfromCNPq
and FAPERJ. I would like to thank my friends and colleagues who have been
importantinmyprofessionalandpersonallife.I’drathernotlisteachoneofthem
here, because they know who they are – and I refuse to run the risk of failing to
mentionsomeone.
I am especially indebted to certain professorswho have greatly influenced and
furthered my education and professional success. Some of them have become
good friends over the years. In alphabetical order, I am grateful to Adherbal
Filho, Antonio Dias, Ali Messaoudi, Carlos Kubrusly, Claudio Landim, Cristiano
Fernandes, Edson Relvas, Eduardo Campos, Fernando Albuquerque, Frederico
Cavalcanti, Hedibert Lopes, Inez Costa Chaves, Kaizo Beltrao, Leonardo Rolla,
Luiz Carlos da Rocha, Marcelo Medeiros, Nei Carlos Rocha, Sergio Volchan,
WaldirLobao,andZeliaBianchini.
Very special thanks go to my friends and colleagues from the Institute of
MathematicsandStatisticsofFluminenseFederalUniversity.Thesefolksgaveme
suchawarmwelcome:::Honestly,whenIamatwork,IfeelasifIamhome.
Finally,Iamgratefultomyparents,RoseNanieHeringerdaSilvaandRodolfo
DomenicoPizzinga,fortheirsupport.Furthermore,Imustdedicatethisbooktomy
v
vi Preface
mother,towhomIameternallyindebtedforherremarkablehelpandunderstanding
throughout my life – and also for her invaluable support on proofreading my
scientifictexts,includingthisbook.
RiodeJaneiro,Brazil AdrianPizzinga
Contents
1 Introduction ................................................................... 1
1.1 Motivation................................................................ 1
1.2 AGlimpseattheLiterature.............................................. 2
1.2.1 StatisticsPapers.................................................. 2
1.2.2 EngineeringPapers .............................................. 3
1.3 TheBook’sContents..................................................... 4
1.4 Organization.............................................................. 5
2 LinearStateSpaceModelsandKalmanFiltering ........................ 7
2.1 TheModel................................................................ 7
2.2 KalmanEquations........................................................ 7
2.3 IntroducingLinearRestrictions ......................................... 8
3 RestrictedKalmanFiltering:TheoreticalIssues.......................... 11
3.1 AugmentedRestrictedKalmanFiltering:AlternativeProofs.......... 11
3.1.1 GeometricalProof ............................................... 11
3.1.2 ComputationalProof ............................................ 13
3.1.3 ConditionalExpectationProof.................................. 15
3.2 StatisticalEfficiency ..................................................... 16
3.3 RestrictedKalmanFilteringVersusRestrictedRecursive
LeastSquares............................................................. 18
3.4 Initialization.............................................................. 21
3.4.1 Motivation........................................................ 21
3.4.2 ReviewingtheInitialExactKalmanSmoother................. 21
3.4.3 CombiningExactInitializationwithLinearRestrictions...... 22
4 RestrictedKalmanFiltering:MethodologicalIssues ..................... 27
4.1 Random-WalkStateVectorsUnderTime-InvariantRestrictions...... 27
4.2 ReducedRestrictedKalmanFiltering................................... 28
4.2.1 Motivation........................................................ 28
4.2.2 TheMethod...................................................... 29
4.2.3 ReducingVersusAugmenting................................... 30
4.3 PredictionsfromaRestrictedStateSpaceModel ...................... 32
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viii Contents
5 Applications ................................................................... 35
5.1 CaseI:SemistrongDynamicStyleAnalysis ........................... 36
5.1.1 Motivation........................................................ 36
5.1.2 CompetingModels .............................................. 37
5.1.3 ModelSelection.................................................. 39
5.1.4 EmpiricalResults................................................ 40
5.2 CaseII:EstimationofDynamicExchange-RatePass-Through....... 44
5.2.1 Motivation........................................................ 44
5.2.2 EmpiricalResults................................................ 46
5.3 CaseIII:GDPBenchmarkingEstimationandPrediction.............. 51
5.3.1 Motivation........................................................ 51
5.3.2 ModelSetup...................................................... 51
5.3.3 EmpiricalResults................................................ 52
6 FurtherExtensions ........................................................... 53
References.......................................................................... 55
List of Figures
Fig.5.1 USdollar/realvolatilityundertheAR(1)-GARCH(1,1)model...... 41
Fig.5.2 SmoothedSTARexposuresandJensen’smeasurefor
HSBCFIFCambialwithrespective95%confidenceintervals....... 43
Fig.5.3 SmoothedSTAR exposuresandJensen’smeasure
forItauMatrixUSHedgeFIFwithrespective95%
confidenceintervals .................................................... 43
Fig.5.4 IPAsmoothedbetas..................................................... 46
Fig.5.5 IPAlong-unexchange-ratepass-through ............................. 47
Fig.5.6 IPAconsumption-smoothedbeta ...................................... 48
Fig.5.7 IPAproduction-smoothedbeta......................................... 49
Fig.5.8 IPCsmoothedbetas .................................................... 50
Fig.5.9 IPClong-runpass-through............................................. 50
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