Contents

Part I DATA EXPLORATION, ESTIMATION AND SIMULATION

1 UNIVARIATE EXPLORATORY DATA ANALYSIS ............... 3

1.1 Data,RandomVariablesandTheirDistributions................. 3

1.1.1 ThePCSData....................................... 4

1.1.2 TheS&P500IndexandFinancialReturns ............... 5

1.1.3 RandomVariablesandTheirDistributions ............... 7

1.1.4 ExamplesofProbabilityDistributionFamilies............ 8

1.2 FirstExploratoryDataAnalysisTools ......................... 13

1.2.1 RandomSamples .................................... 13

1.2.2 Histograms ......................................... 14

1.3 MoreNonparametricDensityEstimation....................... 16

1.3.1 KernelDensityEstimation............................. 17

1.3.2 ComparisonwiththeHistogram........................ 19

1.3.3 S&PDailyReturns................................... 19

1.3.4 ImportanceoftheChoiceoftheBandwidth .............. 22

1.4 QuantilesandQ-QPlots..................................... 23

1.4.1 UnderstandingtheMeaningofQ-QPlots................ 24

1.4.2 ValueatRiskandExpectedShortfall.................... 25

1.5 EstimationfromEmpiricalData .............................. 28

1.5.1 TheEmpiricalDistributionFunction .................... 28

1.5.2 OrderStatistics...................................... 29

1.5.3 EmpiricalQ-QPlots.................................. 30

1.6 RandomGeneratorsandMonteCarloSamples.................. 31

1.7 ExtremesandHeavyTailDistributions......................... 35

1.7.1 S&PDailyReturns,OnceMore........................ 35

1.7.2 TheExampleofthePCSIndex......................... 37

1.7.3 TheExampleoftheWeeklyS&PReturns................ 41
Problems ...................................................... 43
Notes&Complements........................................... 46

2 MULTIVARIATE DATA EXPLORATION ....................... 49

2.1 MultivariateDataandFirstMeasureofDependence ............. 49

2.1.1 DensityEstimation................................... 51

2.1.2 TheCorrelationCoefficient............................ 53

2.2 TheMultivariateNormalDistribution.......................... 56

2.2.1 SimulationofRandomSamples ........................ 57

2.2.2 TheBivariateCase................................... 58

2.2.3 ASimulationExample................................ 59

2.2.4 Let’sHaveSomeCoffee .............................. 60

2.2.5 IstheJointDistributionNormal? ....................... 62

2.3 MarginalsandMoreMeasuresofDependence .................. 63

2.3.1 EstimationoftheCoffeeLog-ReturnDistributions ........ 64

2.3.2 MoreMeasuresofDependence......................... 68

2.4 CopulasandRandomSimulations............................. 70

2.4.1 Copulas ............................................ 71

2.4.2 FirstExamplesofCopulaFamilies...................... 72

2.4.3 CopulasandGeneralBivariateDistributions.............. 74

2.4.4 FittingCopulas...................................... 76

2.4.5 MonteCarloSimulationswithCopulas.................. 77

2.4.6 ARiskManagementExample ......................... 80

2.5 PrincipalComponentAnalysis ............................... 84

2.5.1 IdentificationofthePrincipalComponentsofaDataSet ... 84

2.5.2 PCAwithS-Plus .................................. 87

2.5.3 EffectiveDimensionoftheSpaceofYieldCurves......... 87

2.5.4 SwapRateCurves ................................... 90
Appendix1:CalculuswithRandomVectorsandMatrices ............. 92
Appendix2:FamiliesofCopulas .................................. 95
Problems ...................................................... 98
Notes&Complements........................................... 101

Part II REGRESSION

3 PARAMETRIC REGRESSION ................................105

3.1 SimpleLinearRegression.................................... 105

3.1.1 GettingtheData..................................... 106

3.1.2 FirstPlots .......................................... 107

3.1.3 RegressionSet-up.................................... 108

3.1.4 SimpleLinearRegression ............................. 111

3.1.5 CostMinimizations .................................. 114

3.1.6 RegressionasaMinimizationProblem .................. 114

3.2 RegressionforPrediction&Sensitivities....................... 116

3.2.1 Prediction .......................................... 116

3.2.2 IntroductoryDiscussionofSensitivityandRobustness ..... 118

3.2.3 ComparingL2andL1Regressions ..................... 119

3.2.4 TakingAnotherLookattheCoffeeData................. 121

3.3 SmoothingversusDistributionTheory......................... 123

3.3.1 RegressionandConditionalExpectation................. 123

3.3.2 MaximumLikelihoodApproach........................ 124

3.4 MultipleRegression ........................................ 129

3.4.1 Notation............................................ 129

3.4.2 TheS-Plus Functionlm ............................ 130

3.4.3 R² asaRegressionDiagnostic ......................... 131

3.5 MatrixFormulationandLinearModels ........................ 133

3.5.1 LinearModels....................................... 134

3.5.2 LeastSquares(Linear)RegressionRevisited ............. 134

3.5.3 FirstExtensions ..................................... 139

3.5.4 TestingtheCAPM ................................... 142

3.6 PolynomialRegression...................................... 145

3.6.1 PolynomialRegressionasaLinearModel ............... 146

3.6.2 ExampleofS-Plus Commands....................... 146

3.6.3 ImportantRemark ................................... 148

3.6.4 PredictionwithPolynomialRegression.................. 148

3.6.5 ChoiceoftheDegreep ............................... 150

3.7 NonlinearRegression ....................................... 150

3.8 TermStructureofInterestRates:ACrashCourse................ 154

3.9 ParametricYieldCurveEstimation............................ 160

3.9.1 EstimationProcedures................................ 160

3.9.2 PracticalImplementation.............................. 161

3.9.3 S-Plus Experiments ................................ 163

3.9.4 ConcludingRemarks ................................. 165
Appendix:CautionaryNotesonSomeS-Plus Idiosyncracies ......... 166
Problems ...................................................... 169
Notes&Complements........................................... 172

4 LOCAL & NONPARAMETRIC REGRESSION ..................175

4.1 ReviewoftheRegressionSetup .............................. 175

4.2 NaturalSplinesasLocalSmoothers ........................... 177

4.3 NonparametricScatterplotSmoothers.......................... 178

4.3.1 SmoothingSplines ................................... 179

4.3.2 LocallyWeightedRegression .......................... 181

4.3.3 ARobustSmoother .................................. 182

4.3.4 TheSuperSmoother.................................. 183

4.3.5 TheKernelSmoother................................. 183

4.4 MoreYieldCurveEstimation ................................ 186

4.4.1 AFirstEstimationMethod ............................ 186

4.4.2 ADirectApplicationofSmoothingSplines .............. 188

4.4.3 USandJapaneseInstantaneousForwardRates............ 188

4.5 MultivariateKernelRegression............................... 189

4.5.1 RunningtheKernelinS-Plus ........................ 192

4.5.2 AnExampleInvolvingtheJune1998S&PFuturesContract 193

4.6 ProjectionPursuitRegression ................................ 197

4.6.1 TheS-Plus Functionppreg ......................... 198

4.6.2 ppreg PredictionoftheS&PIndicators................. 200

4.7 NonparametricOptionPricing................................ 205

4.7.1 GeneralitiesonOptionPricing ......................... 205

4.7.2 NonparametricPricingAlternatives..................... 212

4.7.3 DescriptionoftheData ............................... 213

4.7.4 TheActualExperiment ............................... 214

4.7.5 NumericalResults ................................... 220
Appendix:KernelDensityEstimation&KernelRegression............ 222
Problems ...................................................... 225
Notes&Complements........................................... 233

Part III TIME SERIES & STATE SPACE MODELS

5 TIME SERIES MODELS: AR, MA, ARMA, & ALL THAT .........239

5.1 NotationandFirstDefinitions ................................ 239

5.1.1 Notation............................................ 239

5.1.2 RegularTimeSeriesandSignals ....................... 240

5.1.3 CalendarandIrregularTimeSeries ..................... 241

5.1.4 ExampleofDailyS&P500FuturesContracts ............ 243

5.2 HighFrequencyData ....................................... 245

5.2.1 TimeDate Manipulations ............................ 248

5.3 TimeDependentStatisticsandStationarity ..................... 253

5.3.1 StatisticalMoments .................................. 253

5.3.2 TheNotionofStationarity............................. 254

5.3.3 TheSearchforStationarity............................ 258

5.3.4 TheExampleoftheCO2 Concentrations ................ 261

5.4 FirstExamplesofModels.................................... 263

5.4.1 WhiteNoise ........................................ 264

5.4.2 RandomWalk....................................... 267

5.4.3 AutoRegressiveTimeSeries .......................... 268

5.4.4 MovingAverageTimeSeries .......................... 272

5.4.5 UsingtheBackwardShiftOperatorB................... 275

5.4.6 LinearProcesses..................................... 276

5.4.7 Causality,StationarityandInvertibility .................. 277

5.4.8 ARMATimeSeries.................................. 281

5.4.9 ARIMAModels ..................................... 282

5.5 FittingModelstoData ...................................... 282

5.5.1 PracticalSteps....................................... 282

5.5.2 S-Plus Implementation ............................. 284

5.6 PuttingaPriceonTemperature ............................... 289

5.6.1 GeneralitiesonDegreeDays........................... 290

5.6.2 TemperatureOptions ................................. 291

5.6.3 StatisticalAnalysisofTemperatureHistoricalData........ 294
Appendix:MoreS-Plus Idiosyncracies ........................... 301
Problems ...................................................... 304
Notes&Complements........................................... 308

6 MULTIVARIATE TIME SERIES, LINEAR SYSTEMS & KALMAN FILTERING .......................................311

6.1 MultivariateTimeSeries..................................... 311

6.1.1 StationarityandAuto-CovarianceFunctions.............. 312

6.1.2 MultivariateWhiteNoise.............................. 312

6.1.3 MultivariateARModels .............................. 313

6.1.4 BacktoTemperatureOptions .......................... 316

6.1.5 MultivariateMA&ARIMAModels .................... 318

6.1.6 Cointegration ....................................... 319

6.2 StateSpaceModels......................................... 321

6.3 FactorModelsasHiddenMarkovProcesses .................... 323

6.4 KalmanFilteringofLinearSystems........................... 326

6.4.1 One-Step-AheadPrediction............................ 326

6.4.2 DerivationoftheRecursiveFilteringEquations........... 327

6.4.3 WritinganS FunctionforKalmanPrediction............. 329

6.4.4 Filtering............................................ 331

6.4.5 MorePredictions .................................... 332

6.4.6 EstimationoftheParameters........................... 333

6.5 ApplicationstoLinearModels................................ 335

6.5.1 StateSpaceRepresentationofLinearModels............. 335

6.5.2 LinearModelswithTimeVaryingCoefficients ........... 336

6.5.3 CAPMwithTimeVaryingβ’s ......................... 337

6.6 StateSpaceRepresentationofTimeSeries...................... 338

6.6.1 TheCaseofARSeries................................ 339

6.6.2 TheGeneralCaseofARMASeries..................... 341

6.6.3 FittingARMAModelsbyMaximumLikelihood.......... 342

6.7 Example:PredictionofQuarterlyEarnings ..................... 343
Problems ...................................................... 346
Notes&Complements........................................... 351

NONLINEAR TIME SERIES: MODELS AND SIMULATION ......353

7.1 FirstNonlinearTimeSeriesModels ........................... 353

7.1.1 FractionalTimeSeries................................ 354

7.1.2 NonlinearAuto-RegressiveSeries ...................... 355

7.1.3 StatisticalEstimation ................................. 356

7.2 MoreNonlinearModels:ARCH,GARCH&AllThat............ 358

7.2.1 Motivation.......................................... 358

7.2.2 ARCHModels ...................................... 359

7.2.3 GARCHModels..................................... 361

7.2.4 S-Plus Commands ................................. 362

7.2.5 FittingaGARCHModeltoRealData................... 363

7.2.6 Generalizations...................................... 371

7.3 StochasticVolatilityModels.................................. 373

7.4 DiscretizationofStochasticDifferentialEquations............... 378

7.4.1 DiscretizationSchemes ............................... 379

7.4.2 MonteCarloSimulations:AFirstExample............... 381

7.5 RandomSimulationandScenarioGeneration................... 383

7.5.1 ASimpleModelfortheS&P500Index ................. 383

7.5.2 ModelingtheShortInterestRate ....................... 386

7.5.3 ModelingtheSpread ................................. 388

7.5.4 PuttingEverythingTogether........................... 389

7.6 FilteringofNonlinearSystems ............................... 391

7.6.1 HiddenMarkovModels............................... 391

7.6.2 GeneralFilteringApproach............................ 392

7.6.3 ParticleFilterApproximations ......................... 393

7.6.4 FilteringinFinance?StatisticalIssues................... 396

7.6.5 Application:TrackingVolatility........................ 397
Appendix:PreparingIndexData................................... 403
Problems ...................................................... 404
Notes&Complements........................................... 408

APPENDIX: AN INTRODUCTION TO S AND S-Plus ...............411

References ......................................................429

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