Table Of Content
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Regression Modeling with Actuarial
and Financial Applications
Statistical techniques can be used to address new situations. This is important in
a rapidly evolving risk management and financial world. Analysts with a strong
statisticalbackgroundunderstandthatalargedatasetcanrepresentatreasuretrove
ofinformationtobeminedandcanyieldastrongcompetitiveadvantage.
This book provides budding actuaries and financial analysts with a foundation
in multiple regression and time series. Readers will learn about these statistical
techniques using data on the demand for insurance, lottery sales, foreign exchange
rates, and other applications. Although no specific knowledge of risk management
or finance is presumed, the approach introduces applications in which statistical
techniquescanbeusedtoanalyzerealdataofinterest.Inadditiontothefundamentals,
this book describes several advanced statistical topics that are particularly relevant
to actuarial and financial practice, including the analysis of longitudinal, two-part
(frequency/severity),andfat-taileddata.
Datasets with detailed descriptions, sample statistical software scripts in R and
SAS,andtipsonwritingastatisticalreport,includingsampleprojects,canbefound
onthebook’sWebsite:http://research.bus.wisc.edu/RegActuaries.
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INTERNATIONAL SERIES ON ACTUARIAL SCIENCE
ChristopherDaykin,IndependentConsultantandActuary
AngusMacdonald,Heriot-WattUniversity
The International Series on Actuarial Science, published by Cambridge University
PressinconjunctionwiththeInstituteofActuariesandtheFacultyofActuaries,will
contain textbooks for students taking courses in or related to actuarial science, as
wellasmoreadvancedworksdesignedforcontinuingprofessionaldevelopmentor
for describing and synthesizing research. The series will be a vehicle for publish-
ing books that reflect changes and developments in the curriculum, that encourage
the introduction of courses on actuarial science in universities, and that show how
actuarialsciencecanbeusedinallareasinwhichthereislong-termfinancialrisk.
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Thereisanoldsaying,attributedtoSirIssacNewton:
“IfIhaveseenfar,itisbystandingontheshouldersofgiants.”
I dedicate this book to the memory of two giants who helped me, and
everyonewhoknewthem,seefartherandlivebetterlives:
JamesC.Hickman
and
JosephP.Sullivan.
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Regression Modeling with
Actuarial and Financial
Applications
EDWARD W. FREES
UniversityofWisconsin,Madison
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521760119
© Edward W. Frees 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-67528-7 eBook (NetLibrary)
ISBN-13 978-0-521-76011-9 Hardback
ISBN-13 978-0-521-13596-2 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
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Contents
Preface pagexiii
1 RegressionandtheNormalDistribution 1
1.1 WhatIsRegressionAnalysis? 1
1.2 FittingDatatoaNormalDistribution 3
1.3 PowerTransforms 7
1.4 SamplingandtheRoleofNormality 8
1.5 RegressionandSamplingDesigns 10
1.6 ActuarialApplicationsofRegression 12
1.7 FurtherReadingandReferences 13
1.8 Exercises 14
1.9 TechnicalSupplement–CentralLimitTheorem 18
PartI LinearRegression
2 BasicLinearRegression 23
2.1 CorrelationsandLeastSquares 23
2.2 BasicLinearRegressionModel 29
2.3 IstheModelUseful?SomeBasicSummaryMeasures 32
2.4 PropertiesofRegressionCoefficientEstimators 35
2.5 StatisticalInference 37
2.6 BuildingaBetterModel:ResidualAnalysis 41
2.7 Application:CapitalAssetPricingModel 46
2.8 IllustrativeRegressionComputerOutput 51
2.9 FurtherReadingandReferences 54
2.10 Exercises 54
2.11 TechnicalSupplement–ElementsofMatrixAlgebra 62
3 MultipleLinearRegression–I 70
3.1 MethodofLeastSquares 70
3.2 LinearRegressionModelandPropertiesofEstimators 76
3.3 EstimationandGoodnessofFit 81
3.4 StatisticalInferenceforaSingleCoefficient 85
3.5 SomeSpecialExplanatoryVariables 92
3.6 FurtherReadingandReferences 100
3.7 Exercises 101
vii
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viii Contents
4 MultipleLinearRegression–II 107
4.1 TheRoleofBinaryVariables 107
4.2 StatisticalInferenceforSeveralCoefficients 113
4.3 OneFactorANOVAModel 120
4.4 CombiningCategoricalandContinuousExplanatoryVariables 126
4.5 FurtherReadingandReferences 133
4.6 Exercises 133
4.7 TechnicalSupplement–MatrixExpressions 138
5 VariableSelection 148
5.1 AnIterativeApproachtoDataAnalysisandModeling 148
5.2 AutomaticVariableSelectionProcedures 149
5.3 ResidualAnalysis 153
5.4 InfluentialPoints 160
5.5 Collinearity 165
5.6 SelectionCriteria 171
5.7 Heteroscedasticity 175
5.8 FurtherReadingandReferences 179
5.9 Exercises 180
5.10 TechnicalSupplementsforChapter5 182
6 InterpretingRegressionResults 189
6.1 WhattheModelingProcessTellsUs 190
6.2 TheImportanceofVariableSelection 196
6.3 TheImportanceofDataCollection 198
6.4 MissingDataModels 205
6.5 Application:RiskManagers’Cost-Effectiveness 209
6.6 FurtherReadingandReferences 218
6.7 Exercises 219
6.8 TechnicalSupplementsforChapter6 222
PartII TopicsinTimeSeries
7 ModelingTrends 227
7.1 Introduction 227
7.2 FittingTrendsinTime 229
7.3 StationarityandRandomWalkModels 236
7.4 InferenceUsingRandomWalkModels 238
7.5 FilteringtoAchieveStationarity 243
7.6 ForecastEvaluation 245
7.7 FurtherReadingandReferences 248
7.8 Exercises 249
8 AutocorrelationsandAutoregressiveModels 251
8.1 Autocorrelations 251
8.2 AutoregressiveModelsofOrderOne 254
