Chapter Preview. This chapter introduces regression where the dependent variable is the time until an event, such as the time until death, the onset of a disease, or the default on a loan. Event times are often limited by sampling procedures and so ideas of censoring and truncation of data are summarized in this chapter. Event times are nonnegative and their distributions are described in terms of survival and hazard functions. Two types of hazard-based regression are considered, a fully parametric accelerated failure time model and a semiparametric proportional hazards models.

Introduction

In survival models, the dependent variable is the time until an event of interest. The classic example of an event is time until death (the complement of death being survival). Survival models are now widely applied in many scientific disciplines; other examples of events of interest include the onset of Alzheimer's disease (biomedical), time until bankruptcy (economics), and time until divorce (sociology).

Example: Time until Bankruptcy. Shumway (2001) examined the time to bankruptcy for 3,182 firms listed on Compustat Industrial File and the CRSP Daily Stock Return File for the New York Stock Exchange over the period 1962–92. Several explanatory financial variables were examined, including working capital to total assets, retained earnings to total assets, earnings before interest and taxes to total assets, market equity to total liabilities, sales to total assets, net income to total assets, total liabilities to total assets, and current assets to current liabilities. The dataset included 300 bankruptcies from 39,745 firm years.

Chapter Preview. This chapter extends the discussion of multiple linear regression by introducing statistical inference for handling several coefficients simultaneously. To motivate this extension, this chapter considers coefficients associated with categorical variables. These variables allow us to group observations into distinct categories. This chapter shows how to incorporate categorical variables into regression functions using binary variables, thus widening the scope of potential applications. Statistical inference for several coefficients allows analysts to make decisions about categorical variables and other important applications. Categorical explanatory variables also provide the basis for an ANOVA model, a special type of regression model that permits easier analysis and interpretation.

The Role of Binary Variables

Categorical variables provide labels for observations to denote membership in distinct groups, or categories. A binary variable is a special case of a categorical variable. To illustrate, a binary variable may tell us whether someone has health insurance. A categorical variable could tell us whether someone has

• Private group insurance (offered by employers and associations),

• Private individual health insurance (through insurance companies),

• Public insurance (e.g., Medicare or Medicaid) or

• No health insurance.

For categorical variables, there may or may not be an ordering of the groups. For health insurance, it is difficult to order these four categories and say which is larger. In contrast, for education, we might group individuals into “low,” “intermediate,” and “high” years of education.

Chapter Preview. Many datasets feature dependent variables that have a large proportion of zeros. This chapter introduces a standard econometric tool, known as a tobit model, for handling such data. The tobit model is based on observing a left-censored dependent variable, such as sales of a product or claim on a health-care policy, where it is known that the dependent variable cannot be less than zero. Although this standard tool can be useful, many actuarial datasets that feature a large proportion of zeros are better modeled in “two parts,” one part for frequency and one part for severity. This chapter introduces two-part models and provides extensions to an aggregate loss model, where a unit under study, such as an insurance policy, can result in more than one claim.

Introduction

Many actuarial datasets come in “two parts:”

1. One part for the frequency, indicating whether a claim has occurred or, more generally, the number of claims

2. One part for the severity, indicating the amount of a claim

In predicting or estimating claims distributions, we often associate the cost of claims with two components: the event of the claim and its amount, if the claim occurs. Actuaries term these the claims frequency and severity components, respectively. This is the traditional way of decomposing two-part data, where one can consider a zero as arising from a policy without a claim (Bowers et al., 1997, Chapter 2).