Chapter 9: Fat-Tailed Regression Models

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1 Chapter 9: Fat-Tailed Regression Models Peng Shi Northern Illinois University 2013 Ratemaking and Product Management Seminar March 11-13, 2013 Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 1 / 12

2 Outline Introduction Regression Models Transformation Exponential Family Parametric Regression Quantile Regression Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 2 / 12

3 Introduction Introduction Claims data often presents fat-tails: auto, home, medical costs... Fat-tailed Distribution The frequency of extreme events is higher than that implied by the normal distribution Alternative term: heavy-tailed or long-tailed Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 3 / 12

4 Introduction Introduction Tail measure Pr(Y 1 > y) lim y + Pr(Y 2 > y) = lim F Y1 (y) y + F Y2 (y) = lim f Y1 (y) y + f Y2 (y) A limiting value of zero of the ratio indicates that the distribution of Y 2 has a heavier tail than that of Y 1 Example F Weibull (y) lim y + F Pareto (y) = lim exp( (y/λ) τ ) y + θ α (y + θ) α = 0 Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 4 / 12

5 Parametric Regression Parametric Regression Two special cases: Transformation: e.g. log Exponential family: gamma Motivating example: Medical care expenditure Look at individual medical care expenditure for office-based visit Covariates include social economic charateristics, health status, employment etc. Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 5 / 12

6 Parametric Regression Parametric Regression General distributions Consider transformation ln Y = µ + σ ln B(φ 1, φ 2 ) 1 B(φ 1, φ 2 ) Y is known to follow a GB2 distribution denoted by GB2(µ, σ, φ 1, φ 2 ) Many special cases, including GG and Burr XII Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 6 / 12

7 Parametric Regression Multivariate Regression Examples Claims of different type in auto insurance Claims in multi-perils in homeowner insurance Medical costs by physician and non-physician Copula regression A copula is a multivariate joint distribution with all marginal follows uniform distribution on interval [0, 1] C(u 1,, u d ) = Pr(U 1 u 1,, U d u d ) According to Sklar s theorem, for any multivariate distribution F with continuous marginals F 1,, F d, there exists a copula C such that F(y 1,, y d ) = C(F 1 (y 1 ),, F d (y d )) Allows for flexible distributions for the marginal models while accommodates association Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 7 / 12

8 Parametric Regression Medical Costs Example Again consider individual medical care costs of office-based visits Break down into two categories: doctor v.s. non-doctor Consider a bivariate copula regression Burr XII for doctor and generalized gamma for non-doctor Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 8 / 12

9 Parametric Regression Medical Costs Example Residuals Use Gaussian copula to join two marginals f (y 1, y 2 ) = c(f 1 (y 1 ), F 2 (y 2 ))f 1 (y 1 )f 2 (y 2 ) The estimated association parameter ρ is about 0.25 Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 9 / 12

10 Quantile Regression Quantile Regression Linear model y = x β + ε, ε F Consider the quantile function Q τ (y x) = x β(τ) with 0 < τ < 1, the regression coefficient β(τ) can be found by solving ˆβ(τ) = arg min β R p n ρ τ (y i x i β) i=1 ρ τ (u) = u(τ I(u 0)) is known as the check function Use for skewed data and data with heteroscedasticity Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 10 / 12

11 Quantile Regression Medical Costs Example: Revisit Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 11 / 12

12 Concluding Remarks Conclusion See Volume I of the predictive modeling book for details More case studies in the Volume II Peng Shi (Northern Illinois University) Fat-Tailed Regression 2013 CAS RPM Seminar 12 / 12

Financial Vulnerability Index (IMPACT)

Financial Vulnerability Index (IMPACT) Household s Life Insurance Demand - a Multivariate Two Part Model Edward (Jed) W. Frees School of Business, University of Wisconsin-Madison July 30, 1 / 19 Outline 1 2 3 4 2 / 19 Objective To understand