# Download Edward W. Frees - Regression Modeling With Actuarial and Financial Applications - 2009 PDF

Title Edward W. Frees - Regression Modeling With Actuarial and Financial Applications - 2009 4.3 MB 585
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Series-title
Title
Contents
Preface
Who Is This Book For?
Acknowledgments
How Does This Book Deliver Its Message?
Chapter Development
Real Data
Statistical Software and Data
Technical Supplements
Suggested Courses
Acknowledgments
1 Regression and the Normal Distribution
1.1 What Is Regression Analysis?
1.2 Fitting Data to a Normal Distribution
Normal Curve Approximation
Box Plot
Quantile-Quantile Plots
1.3 Power Transforms
1.4 Sampling and the Role of Normality
1.5 Regression and Sampling Designs
1.6 Actuarial Applications of Regression
Reserving and Solvency Testing
Other Risk Management Applications
Chapter References
1.8 Exercises
1.9 Technical Supplement - Central Limit Theorem
Part I Linear Regression
2 Basic Linear Regression
2.1 Correlations and Least Squares
Scatter Plot and Correlation Coefficients – Basic Summary Tools
Method of Least Squares
2.2 Basic Linear Regression Model
2.3 Is the Model Useful? Some Basic Summary Measures
2.3.1 Partitioning the Variability
2.3.2 The Size of a Typical Deviation: s
2.4 Properties of Regression Coefficient Estimators
2.5 Statistical Inference
2.5.1 Is the Explanatory Variable Important? The t-Test
2.5.2 Confidence Intervals
2.5.3 Prediction Intervals
2.6 Building a Better Model: Residual Analysis
2.7 Application: Capital Asset Pricing Model
Data
Unusual Points
Sensitivity Analysis
2.8 Illustrative Regression Computer Output
Legend Annotation Definition, Symbol
Chapter References
2.10 Exercises
2.11 Technical Supplement - Elements of Matrix Algebra
2.11.1 Basic Definitions
2.11.2 Some Special Matrices
2.11.3 Basic Operations
Scalar Multiplication
Matrix Multiplication
Matrix Inverses
2.11.4 Random Matrices
3 Multiple Linear Regression - I
3.1 Method of Least Squares
Summarizing the Data
Method of Least Squares
Matrix Notation
3.2 Linear Regression Model and Properties of Estimators
3.2.1 Regression Function
3.2.2 Regression Coefficient Interpretation
3.2.3 Model Assumptions
3.2.4 Properties of Regression Coefficient Estimators
3.3 Estimation and Goodness of Fit
Residual Standard Deviation
The Coefficient of Determination: R2
3.4 Statistical Inference for a Single Coefficient
3.4.1 The t-Test
3.4.2 Confidence Intervals
3.4.4 Partial Correlation Coefficients
3.5 Some Special Explanatory Variables
3.5.1 Binary Variables
3.5.2 Transforming Explanatory Variables
3.5.3 Interaction Terms
Chapter References
3.7 Exercises
4 Multiple Linear Regression - II
4.1 The Role of Binary Variables
4.2 Statistical Inference for Several Coefficients
4.2.1 Sets of Regression Coefficients
4.2.2 The General Linear Hypothesis
Some Special Cases
4.2.3 Estimating and Predicting Several Coefficients
Estimating Linear Combinations of Regression Coefficients
Prediction Intervals
4.3 One Factor ANOVA Model
Model Assumptions and Analysis
Reparameterization
4.4 Combining Categorical and Continuous Explanatory Variables
Combining a Factor and Covariate
Combining Two Factors
General Linear Model
Chapter References
4.6 Exercises
4.7 Technical Supplement - Matrix Expressions
4.7.1 Expressing Models with Categorical Variables in Matrix Form
One Categorical Variable Model
One Categorical and One Continuous Variable Model
4.7.2 Calculating Least Squares Recursively
Partitioned Matrix Results
Reparameterized Model
Extra Sum of Squares
4.7.3 General Linear Model
Normal Equations
Unique Fitted Values
Generalized Inverses
Estimable Functions
Testable Hypotheses
General Linear Hypothesis
One Categorical Variable Model
5 Variable Selection
5.1 An Iterative Approach to Data Analysis and Modeling
5.2 Automatic Variable Selection Procedures
5.3 Residual Analysis
5.3.1 Residuals
5.3.2 Using Residuals to Identify Outliers
5.3.3 Using Residuals to Select Explanatory Variables
5.4 Influential Points
5.4.1 Leverage
5.4.2 Cook’s Distance
5.5 Collinearity
5.5.1 What Is Collinearity?
5.5.2 Variance Inflation Factors
5.5.3 Collinearity and Leverage
5.5.4 Suppressor Variables
5.5.5 Orthogonal Variables
5.6 Selection Criteria
5.6.1 Goodness of Fit
5.6.2 Model Validation
5.6.3 Cross-Validation
5.7 Heteroscedasticity
5.7.1 Detecting Heteroscedasticity
5.7.2 Heteroscedasticity-Consistent Standard Errors
5.7.3 Weighted Least Squares
5.7.4 Transformations
Chapter References
5.9 Exercises
5.10 Technical Supplements for Chapter 5
5.10.1 Projection Matrix
5.10.2 Leave-One-Out Statistics
5.10.3 Omitting Variables
6 Interpreting Regression Results
6.1 What the Modeling Process Tells Us
6.1.1 Interpreting Individual Effects
Substantive Significance
Statistical Significance
Causal Effects
6.2 The Importance of Variable Selection
6.2.1 Overfitting the Model
6.2.2 Underfitting the Model
6.3 The Importance of Data Collection
6.3.1 Sampling Frame Error and Adverse Selection
6.3.2 Limited Sampling Regions
6.3.3 Limited Dependent Variables, Censoring, and Truncation
6.3.4 Omitted and Endogenous Variables
6.3.5 Missing Data
6.4 Missing Data Models
6.4.1 Missing at Random
6.4.2 Nonignorable Missing Data
Heckman Two-Stage Procedure
EM Algorithm
6.5 Application: Risk Managers' Cost-Effectiveness
Introduction
Preliminary Analysis
Back to the Basics
Some New Models
Chapter References
6.7 Exercises
6.8 Technical Supplements for Chapter 6
6.8.1 Effects of Model Misspecification
Part II Topics in Time Series
7 Modeling Trends
7.1 Introduction
Time Series and Stochastic Processes
Time Series versus Causal Models
7.2 Fitting Trends in Time
Understanding Patterns over Time
Fitting Trends in Time
Fitting Seasonal Trends
Reliability of Time Series Forecasts
7.3 Stationarity and Random Walk Models
White Noise
Random Walk
7.4 Inference Using Random Walk Models
Model Properties
Forecasting
Identifying Stationarity
Identifying Random Walks
Random Walk versus Linear Trend in Time Models
7.5 Filtering to Achieve Stationarity
Transformations
7.6 Forecast Evaluation
Chapter References
7.8 Exercises
8 Autocorrelations and Autoregressive Models
8.1 Autocorrelations
Application: Inflation Bond Returns
Autocorrelations
8.2 Autoregressive Models of Order One
Model Definition and Properties
Model Selection
Meandering Process
8.3 Estimation and Diagnostic Checking
8.4 Smoothing and Prediction
8.5 Box-Jenkins Modeling and Forecasting
8.5.1 Models
AR(p) Models
MA(q) Models
ARMA and ARIMA Models
8.5.2 Forecasting
Optimal Point Forecasts
Forecast Interval
8.6 Application: Hong Kong Exchange Rates
Model Selection and Partial Autocorrelations
Residual Checking
Residual Autocorrelation
Testing Several Lags
Chapter References
8.8 Exercises
9 Forecasting and Time Series Models
9.1 Smoothing with Moving Averages
Application: Medical Component of the CPI
Weighted Least Squares
9.2 Exponential Smoothing
Weighted Least Squares
9.3 Seasonal Time Series Models
Fixed Seasonal Effects
Seasonal Autoregressive Models
Seasonal Exponential Smoothing
9.4 Unit Root Tests
9.5 ARCH/GARCH Models
ARCH Model
GARCH Model
Chapter References
10 Longitudinal and Panel Data Models
10.1 What Are Longitudinal and Panel Data?
10.2 Visualizing Longitudinal and Panel Data
Trellis Plot
10.3 Basic Fixed Effects Models
Data
Model
Estimation
10.4 Extended Fixed Effects Models
Analysis of Covariance Models
Variable Coefficients Models
Models with Serial Correlation
10.5 Random Effects Models
Basic Random Effects Model
Estimation
Extended Random Effects Models
Chapter References
Part III Topics in Nonlinear Regression
11 Categorical Dependent Variables
11.1 Binary Dependent Variables
Linear Probability Models
11.2 Logistic and Probit Regression Models
11.2.1 Using Nonlinear Functions of Explanatory Variables
11.2.2 Threshold Interpretation
11.2.3 Random Utility Interpretation
11.2.4 Logistic Regression
Odds Interpretation
11.3 Inference for Logistic and Probit Regression Models
11.3.1 Parameter Estimation
11.4 Application: Medical Expenditures
Dependent Variable
Explanatory Variables
Summary Statistics
11.5 Nominal Dependent Variables
11.5.1 Generalized Logit
Parameter Interpretations
11.5.2 Multinomial Logit
11.5.3 Nested Logit
11.6 Ordinal Dependent Variables
11.6.1 Cumulative Logit
11.6.2 Cumulative Probit
Chapter References
11.8 Exercises
11.9 Technical Supplements - Likelihood-Based Inference
11.9.1 Properties of Likelihood Functions
11.9.2 Maximum Likelihood Estimators
11.9.3 Hypothesis Tests
11.9.4 Information Criteria
12 Count Dependent Variables
12.1 Poisson Regression
12.1.1 Poisson Distribution
12.1.2 Regression Model
12.1.3 Estimation
12.2 Application: Singapore Automobile Insurance
12.3 Overdispersion and Negative Binomial Models
Adjusting Standard Errors for Data Not Equidispersed
Negative Binomial
12.4 Other Count Models
12.4.1 Zero-Inflated Models
12.4.2 Hurdle Models
12.4.3 Heterogeneity Models
12.4.4 Latent Class Models
Chapter References
12.6 Exercises
13 Generalized Linear Models
13.1 Introduction
13.2 GLM Model
13.2.1 Linear Exponential Family of Distributions
13.3 Estimation
13.3.1 Maximum Likelihood Estimation for Canonical Links
13.3.2 Overdispersion
13.3.3 Goodness-of-Fit Statistics
13.4 Application: Medical Expenditures
13.5 Residuals
13.6 Tweedie Distribution
Chapter References
13.8 Exercises
13.9 Technical Supplements - Exponential Family
13.9.1 Linear Exponential Family of Distributions
13.9.2 Moments
13.9.3 Maximum Likelihood Estimation for General Links
13.9.4 Iterated Reweighted Least Squares
14 Survival Models
14.1 Introduction
14.2 Censoring and Truncation
14.2.1 Definitions and Examples
14.2.2 Likelihood Inference
14.2.3 Product-Limit Estimator
14.3 Accelerated Failure Time Model
Location-Scale Distributions
Inference for AFT Models
14.4 Proportional Hazards Model
14.4.1 Proportional Hazards
14.4.2 Inference
14.5 Recurrent Events
Chapter References
15 Miscellaneous Regression Topics
15.1 Mixed Linear Models
15.1.1 Weighted Least Squares
15.1.2 Variance Components Estimation
15.1.3 Best Linear Unbiased Prediction
15.2 Bayesian Regression
15.3 Density Estimation and Scatter plot Smoothing
15.5 Bootstrapping
Chapter References
Part IV Actuarial Applications
16 Frequency-Severity Models
16.1 Introduction
16.2 Tobit Model
16.3 Application: Medical Expenditures
Summary Statistics
16.4 Two-Part Model
Tobit Type II Model
16.5 Aggregate Loss Model
Frequency of Claims
Property and Casualty
Health Care
Chapter References
16.7 Exercises
17 Fat-Tailed Regression Models
17.1 Introduction
17.2 Transformations
17.3 Generalized Linear Models
17.3.1 What Is Fat Tailed?
17.3.2 Application: Wisconsin Nursing Homes
Summarizing the Data
Fitting Generalized Linear Models
17.4 Generalized Distributions
Application: Wisconsin Nursing Homes
17.5 Quantile Regression
17.6 Extreme Value Models
Chapter References
17.8 Exercises
18 Credibility and Bonus-Malus
18.1 Risk Classification and Experience Rating
18.2 Credibility
18.2.1 Limited Fluctuation Credibility
18.2.2 Greatest Accuracy Credibility
18.3 Credibility and Regression
18.3.1 One-Way Random Effects Model
18.3.2 Longitudinal Models
18.4 Bonus-Malus
Chapter References
19 Claims Triangles
19.1 Introduction
19.1.1 Claims Evolution
19.1.2 Claims Triangles
19.2 Regression Using Functions of Time as Explanatory Variables
19.2.1 Lognormal Model
19.2.2 Hoerl Curve
19.2.3 Poisson Models
19.3 Using Past Developments
19.3.1 Mack Model
19.3.2 Distributional Models
Chapter References
19.5 Exercises
20 Report Writing: Communicating Data Analysis Results
20.1 Overview
20.2 Methods for Communicating Data
Within-Text Data
Graphs
20.3 How to Organize
Title and Abstract
Introduction
Data Characteristics
Model Selection and Interpretation
Summary and Concluding Remarks
References and Appendix
20.4 Further Suggestions for Report Writing
20.5 Case Study: Swedish Automobile Claims
Determinants of Swedish Automobile Claims
Abstract
Section 1. Introduction
Section 2. Data Characteristics
Section 3. Model Selection and Interpretation
Discussion of the Frequency Model
Discussion of the Severity Model
Section 4. Summary and Concluding Remarks
Appendix
A1. References
A2. Variable Definitions
A3. Basic Summary Statistics for Frequency
A4. Final Fitted Frequency Regression Model: R Output
A5. Checking Significance of Factors in the Final Fitted Frequency Regression Model: R Output
A6. Final Fitted Severity Regression Model: R Output
A7. Checking Significance of Factors in the Final Fitted Severity Regression Model: R Output
Chapter References
20.7 Exercises
21 Designing Effective Graphs
21.1 Introduction
21.2 Graphic Design Choices Make a Difference
21.3 Design Guidelines
Guideline 1: Avoid Chartjunk
Guideline 2: Use Small Multiples to Promote Comparisons and Assess Change
Guideline 3: Use Complex Graphs to Portray Complex Patterns
Guideline 4: Relate Graph Size to Information Content
Guideline 5: Use Graphical Forms That Promote Comparisons
Guideline 6: Integrate Graphs and Text
Guideline 7: Demonstrate an Important Message
21.4 Empirical Foundations for Guidelines
21.4.1 Viewers as Units of Study
21.4.2 Graphs as Units of Study
21.5 Concluding Remarks
Chapter References
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 11
Chapter 12
Appendix 1: Basic Statistical Inference
A1.1 Distributions of Functions of Random Variables
A1.2 Estimation and Prediction
A1.3 Testing Hypotheses
Appendix 2: Matrix Algebra
A2.1 Basic Definitions
A2.2 Review of Basic Operations
A2.3 Further Definitions
Appendix 3: Probability Tables
A3.1 Normal Distribution
A3.2 Chi-Square Distribution
A3.3 t-Distribution
A3.4 F-Distribution
Index
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