Chapter Preview. This chapter considers regression methods where the analyst has the ability to follow a unit of analysis, such as a policyholder, over time. In the biomedical literature, this type of information is known as longitudinal data and, in the econometric literature, as panel data.

Introduction

7.1.1 What Are Longitudinal and Panel Data?

Regression is a statistical technique that serves to explain the distribution of an outcome of interest (y) in terms of other variables, often called “explanatory” or “predictor” (x's) variables. For example, in a typical ratemaking exercise, the analyst gathers a cross-section of policyholders and uses various rating variables (x's) to explain losses (y's).

In this chapter, we assume that the analyst has the ability to follow each policyholder over time. In many situations, a policyholder's past loss experience can provide an important supplement to the information gleaned from available rating variables. We use the notation i to represent the unit of analysis (e.g., policyholder) that we will follow over time t. Using double subscripts, the notation yit refers to a dependent variable y for policyholder i at time t, and similarly for explanatory variables xit.

Consider four applications that an actuary might face that fall into this framework:

1. (1) Personal lines insurance such as automobile and homeowners: Here, i represents the policyholder that we follow over time t, y represents the policy loss or claim, and the vector x represents a set of rating variables.

2. […]

Chapter Preview. Bayesian methods have grown rapidly in popularity because of their general applicability, structured and direct incorporation of expert opinion, and proper accounting of model and parameter uncertainty. This chapter outlines the basic process and describes the benefits and difficulties inherent in fitting Bayesian models.

Why Bayesian?

Although the theory underpinning Bayesian methods is about 350 years old (Sir Thomas Bayes' essay was read to the Royal Society after his death; see Bayes and Price 1763), their widespread use was limited by computational power. In models of reasonable size and complexity, large iterated integrals need to be numerically approximated, which can be computationally burdensome. In the early 1990s the development of randomized algorithms to approximate these integrals, including the Gibbs sampler (Gelfand and Smith 1990), and the exponential growth of computing power made Bayesian methods accessible. More recently, with the development and growth of statistical software such as R, WinBUGS, and now PROC MCMC in SAS, Bayesian methods are available and employed by practitioners in many fields (see Albert 2009, for some details and examples). Actuaries have been using Bayesian methods since Whitney (1918) approximated them in a credibility framework. Unfortunately, many have been slow to extend them for use in other natural contexts. Among other benefits, Bayesian methods properly account for model and parameter uncertainty, provide a structure to incorporate expert opinion, and are generally applicable to many methods in this book and throughout the industry.

Chapter Preview. Survival modeling focuses on the estimation of failure time distributions from observed data. Failure time random variables are defined on the non-negative real numbers and might represent time to death, time to policy termination, or hospital length of stay. There are two defining aspects to survival modeling. First, it is not unusual to encounter distributions incorporating both parametric and nonparametric components, as is seen with proportional hazard models. Second, the estimation techniques accommodate incomplete data (i.e., data that are only observed for a portion of the time exposed as a result of censoring or truncation). In this chapter, we apply R's survival modeling objects and methods to complete and incomplete data to estimate the distributional characteristics of the underlying failure time process. We explore parametric, nonparametric, and semi-parametric models; isolate the impact of fixed and time-varying covariates; and analyze model residuals.

Survival Distribution Notation

Frees (2010) provides an excellent summary of survival model basics. This chapter adopts the same notation.

Let y denote the failure time random variable defined on the non-negative real numbers. The distribution of y can be specified by any of the following functions:

1. • f(t) = the density of y

2. • F(t) = Pr(y ≤ t), the cumulative distribution of y

3. • S(t) = Pr(y > t) = 1 − F(t), the survival function

4. • h(t) = f(t)/S(t), the hazard function

5. • H(t) = − 1n(S(t)), the cumulative hazard function

Chapter Preview. The focus of this chapter is on various methods of unsupervised learning. Unsupervised learning is contrasted with supervised learning, and the role of unsupervised learning in a supervised analysis is also discussed. The concept of dimension reduction is presented first, followed by the common methods of dimension reduction, principal components/factor analysis, and clustering. More recent developments regarding classic techniques such as fuzzy clustering are then introduced. Illustrative examples that use publicly available databases are presented. At the end of the chapter there are exercises that use data supplied with the chapter. Free R code and datasets are available on the book's website.

Introduction

Even before any of us took a formal course in statistics, we were familiar with supervised learning, though it is not referred to as such. For instance, we may read in the newspaper that people who text while driving experience an increase in accidents. When the research about texting and driving was performed, there was a dependent variable (occurrence of accident or near accident) and independent variables or predictors (use of cell phone along with other variables that predict accidents).

In finance class, students may learn about the capital asset pricing model (CAPM)

R = α + βRM + ε,

where the return on an individual stock R is a constant α plus beta times the return for market RM plus an error ε.

Chapter Preview. This chapter deals with the analysis of measurements over time, called time series analysis. Examples of time series include inflation and unemployment indices, stock prices, currency cross rates, monthly sales, the quarterly number of claims made to an insurance company, outstanding liabilities of a company over time, internet traffic, temperature and rainfall, and the number of mortgage defaults. Time series analysis aims to explain and model the relationship between values of the time series at different points of time. Models include ARIMA, structural, and stochastic volatility models and their extensions. The first two classes of models explain the level and expected future level of a time series. The last class seeks to model the change over time in variability or volatility of a time series. Time series analysis is critical to prediction and forecasting. This chapter explains and summarizes modern time series modeling as used in insurance, actuarial studies, and related areas such as finance. Modeling is illustrated with examples, analyzed with the R statistical package.

Exploring Time Series Data

17.1.1 Time Series Data

A time series is a sequence of measurements y1, y2, …, yn made at consecutive, usually regular, points in time. Four time series are plotted in Figure 17.1 and explained in detail later. Each time series is “continuous,” meaning each yt can attain any value in some interval of the line.

Chapter Preview. Predictive modeling involves the use of data to forecast future events. It relies on capturing relationships between explanatory variables and the predicted variables from past occurrences and exploiting them to predict future outcomes. The goal of this two-volume set is to build on the training of actuaries by developing the fundamentals of predictive modeling and providing corresponding applications in actuarial science, risk management, and insurance. This introduction sets the stage for these volumes by describing the conditions that led to the need for predictive modeling in the insurance industry. It then traces the evolution of predictive modeling that led to the current statistical methodologies that prevail in actuarial science today.

Introduction

A classic definition of an actuary is “one who determines the current financial impact of future contingent events.” Actuaries are typically employed by insurance companies who job is to spread the cost of risk of these future contingent events.

The day-to-day work of an actuary has evolved over time. Initially, the work involved tabulating outcomes for “like” events and calculating the average outcome. For example, an actuary might be called on to estimate the cost of providing a death benefit to each member of a group of 45-year-old men. As a second example, an actuary might be called on to estimate the cost of damages that arise from an automobile accident for a 45-year-old driver living in Chicago.

Chapter Preview. This chapter presents statistical models that can handle spatial dependence among variables. Spatial dependence refers to the phenomenon that variables observed in areas close to each other are often related. Ignoring data heterogeneity due to such spatial dependence patterns may cause overdispersion and erroneous conclusions. In an actuarial context, it is important to take spatial information into account in many cases, such as in the insurance of buildings threatened by natural catastrophes; in health insurance, where diseases affect specific regions; and also in car insurance, as we discuss in an application.

In particular, we describe the most common spatial autoregressive models and show how to build a joint model for claim severity and claim frequency of individual policies based on generalized linear models with underlying spatial dependence. The results show the importance of explicitly considering spatial information in the ratemaking methodology.

Introduction

It is important to take spatial information related to insurance policies into account when predicting claims and ratemaking. The most prominent example is the modeling of natural catastrophes needed for the insurance of buildings. Another example is health insurance, where spatial information is relevant for an accurate assessment of the underlying risks, because frequencies of some diseases may vary by region. Frequencies in neighbor regions are often expected to be more closely related than those in regions far from each other. This phenomenon is usually referred to as spatial dependence.

Chapter Preview. This chapter considers the application of predictive models to insurance claims triangles and the associated prediction problem of loss reserving (Section 18.1). This is approached initially by reference to the chain ladder, a widely used heuristic reserving algorithm. Rigorous predictive models, in the form of Generalized linear models (GLMs) that reproduce this algorithm, are then explored (Section 18.2). The chain ladder has a restricted model structure and a number of embellishments are considered (Section 18.3). These include the incorporation in the model of accident year effects through the use of exposure data (e.g., accident year claim counts) (Section 18.3.2), and also the incorporation of claim closure data (Section 18.3.3). A subsequent section considers models that incorporate claim closure data on an operational time basis (Section 18.3.4). In each of these cases emphasis is placed on the ease of inclusion of these model features in the GLM structure. All models in Sections 18.1 to 18.3 relate to conventional claims triangles. These datasets are aggregate, as opposed to unit record claim datasets that record detail of individual claims. The chapter closes with a brief introduction to individual claim models (18.4). On occasion these models use survival analysis as well as GLMs.

Introduction to Loss Reserving

18.1.1 Meaning and Accounting Significance of Loss Reserving

Typically, property & casualty (P&C) insurance will indemnify the insured against events that occur within a defined period.

Chapter Preview. This chapter introduces the reader to credibility and related regression modeling. The first section provides a brief overview of credibility theory and regression-type credibility, and it discusses historical developments. The next section shows how some well-known credibility models can be embedded within the linear mixed model framework. Specific procedures on how such models can be used for prediction and standard ratemaking are given as well. Further, in Section 9.3, a step-by-step numerical example, based on the widely studied Hachemeister's data, is developed to illustrate the methodology. All computations are done using the statistical software package R. The fourth section identifies some practical issues with the standard methodology, in particular, its lack of robustness against various types of outliers. It also discusses possible solutions that have been proposed in the statistical and actuarial literatures. Performance of the most effective proposals is illustrated on the Hachemeister's dataset and compared to that of the standard methods. Suggestions for further reading are made in Section 9.5.

Introduction

9.1.1 Early Developments

Credibility theory is one of the oldest but still most common premium ratemaking techniques in insurance industry. The earliest works in credibility theory date back to the beginning of the 20th century, when Mowbray (1914) and Whitney (1918) laid the foundation for limited fluctuation credibility theory.

Chapter Preview. In this chapter we approach many of the topics of the previous chapters, but from a Bayesian viewpoint. Initially we cover the foundations of Bayesian inference. We then describe the Bayesian linear and generalized regression models. We concentrate on the regression models with zero-one and count response and illustrate the models with real datasets. We also cover hierarchical prior specifications in the context of mixed models. We finish with a description of a semi-parametric linear regression model with a nonparametric specification of the error term. We also illustrate its advantage with respect to the fully parametric setting using a real dataset.

Introduction

The use of Bayesian concepts and techniques in actuarial science dates back to Whitney (1918) who laid the foundations for what is now called empirical Bayes credibility. He noted that the solution of the problem “depends upon the use of inverse probabilities.” This is the term used by T. Bayes in his original paper (e.g., Bellhouse 2004). However, Ove Lundberg was apparently the first one to realize the importance of Bayesian procedures (Lundberg 1940). In addition, Bailey (1950) put forth a clear and strong argument in favor of using Bayesian methods in actuarial science. To date, the Bayesian methodology has been used in various areas within actuarial science; see, for example, Klugman (1992), Makov (2001), Makov, Smith, and Liu (1996), and Scollnik (2001).

Preview of the Chapter. We start with a discussion of model families for multilevel data outside the Gaussian framework. We continue with generalized linear mixed models (GLMMs), which enable generalized linear modeling with multilevel data. The chapter includes highlights of estimation techniques for GLMMs in the frequentist as well as Bayesian context. We continue with a discussion of nonlinear mixed models (NLMMs). The chapter concludes with an extensive case study using a selection of R packages for GLMMs.

Introduction

Chapter 8 (Section 3) motivates predictive modeling in actuarial science (and in many other statistical disciplines) when data structures go beyond the cross-sectional design. Mixed (or multilevel) models are statistical models suitable for the analysis of data structured in nested (i.e., hierarchical) or non-nested (i.e., cross-classified, next to each other, instead of hierachically nested) clusters or levels. Whereas the focus in Chapter 8 is on linear mixed models, we now extend the idea of mixed modeling to outcomes with a distribution from the exponential family (as in Chapter 5 on generalized linear models (GLMs)) and to mixed models that generalize the concept of linear predictors. The first extension leads to the family of generalized linear mixed models (GLMMs), and the latter creates nonlinear mixed models (NLMMs). The use of mixed models for predictive modeling with multilevel data is motivated extensively in Chapter 8.

Chapter Preview. We give a general discussion of linear mixed models and continue by illustrating specific actuarial applications of this type of model. Technical details on linear mixed models follow: model assumptions, specifications, estimation techniques, and methods of inference. We include three worked-out examples with the R lme4 package and use ggplot2 for the graphs. Full code is available on the book's website.

Mixed Models in Actuarial Science

8.1.1 What Are Linear Mixed Models?

A First Example of a Linear Mixed Model. As explained in Chapter 7, a panel dataset follows a group of subjects (e.g., policyholders in an insurance portfolio) over time. We therefore denote variables (e.g., yit, xit) in a panel dataset with double subscripts, indicating the subject (say, i) and the time period (say, t). As motivated in Section 1.2 of Chapter 7, the analysis of panel data has several advantages. Panel data allow one to study the effect of certain covariates on the response of interest (as in usual regression models for cross–sectional data), while accounting appropriately for the dynamics in these relations. For actuarial ratemaking the availability of panel data is of particular interest in a posteriori ratemaking. An a posteriori tariff predicts the current year loss for a particular policyholder, using (among other factors) the dependence between the current year's loss and losses reported by this policyholder in previous years.

Chapter Preview. This chapter presents regression models where the dependent variable is categorical, whereas covariates can either be categorical or continuous. In the first part binary dependent variable models are presented, and the second part is aimed at covering general categorical dependent variable models, where the dependent variable has more than two outcomes. This chapter is illustrated with datasets, inspired by real-life situations. It also provides the corresponding R programs for estimation, which are based on R packages glm and mlogit. The same output can be obtained when using SAS or similar software programs for estimating the models presented in this chapter.

Coding Categorical Variables

Categorical variables measure qualitative traits; in other words, they evaluate concepts that can be expressed in words. Table 3.1 presents examples of variables that are measured in a categorical scale and are often found in insurance companies databases. These variables are also called risk factors when they denote characteristics that are associated with losses.

Categorical variables must have mutually exclusive outcomes. The number of categories is the number of possible response levels. For example, if we focus on insurance policies, we can have a variable such as TYPE OF POLICY CHOSEN with as many categories as the number of possible choices for the contracts offered to the customer.