Regression modeling deals with explaining the movements in one variable by movements in one or more other variables. The classical linear model, or normal linear model, forms the basis of generalized linear modeling, and a thorough understanding is critical to an understanding of GLMs. Many of the regression concepts found in GLMs have their genesis in the normal linear model, and so are covered in this chapter. Response distributions encountered in the insurance world are typically strongly non-normal, with the result that the methodology covered in this chapter, while important background to understanding GLMs, is usually not directly applicable to insurance data.

History and terminology of linear modeling

There is a smooth line of development from Gauss' original idea of simple least squares to present day generalized linear modeling. This line of thought and development is surveyed in the current chapter.

1. (i) Simple linear modeling. The aim is to explain an observed variable y by a single other observed variable x. The variable y is called the response variable and x the explanatory variable. Alternative terminology used in the literature for y are dependent, outcome, or (in econometrics) endogenous variable. Alternative names for x are covariate, independent, predictor, driver, risk factor, exogenous variable, regressor or simply the “x” variable. When x is categorical it is also called a factor.

2. (ii) Multiple linear modeling. Here simple least squares is extended by supposing that x contains more than one explanatory variable, the combination of which serve to explain the response y.

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