Chapter Preview. This chapter considers regression in the case of only one explanatory variable. Despite this seeming simplicity, most of the deep ideas of regression can be developed in this framework. By limiting ourselves to the one variable case, we are able to express many calculations using simple algebra. This will allow us to develop our intuition about regression techniques by reinforcing it with simple demonstrations. Further, we can illustrate the relationships between two variables graphically because we are working in only two dimensions. Graphical tools prove important for developing a link between the data and a model.

Correlations and Least Squares

Regression is about relationships. Specifically, we will study how two variables, an x and a y, are related. We want to be able to answer questions such as, If we change the level of x, what will happen to the level of y? If we compare two subjects that appear similar except for the x measurement, how will their y measurements differ? Understanding relationships among variables is critical for quantitative management, particularly in actuarial science, where uncertainty is so prevalent.

It is helpful to work with a specific example to become familiar with key concepts. Analysis of lottery sales has not been part of traditional actuarial practice, but it is a growth area in which actuaries could contribute.

Example: Wisconsin Lottery Sales. State of Wisconsin lottery administrators are interested in assessing factors that affect lottery sales. Sales consists of online lottery tickets that are sold by selected retail establishments in Wisconsin.