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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 8

Linear Regression Analysis

This chapter provides a basic introduction to linear regression models. At the end of the chapter, we will explore two risk management applications, factor analysis and stress testing.

LINEAR REGRESSION (ONE REGRESSOR)

One of the most popular models in statistics is the linear regression model. Given two constants, α and β, and a random error term, , in its simplest form the model posits a relationship between two variables, X and Y:

(8.1) Numbered Display Equation

As specified, X is known as the regressor or independent variable. Similarly, Y is known as the regressand or dependent variable. As dependent implies, traditionally we think of X as causing Y. This relationship is not necessary, and in practice, especially in finance, this cause-and-effect relationship is either ambiguous or entirely absent. In finance, it is often the case that both X and Y are being driven by a common underlying factor.

The linear regression relationship is often represented graphically as a plot of Y against X, as shown in Figure 8.1. The solid line in the chart represents the deterministic portion of the linear regression equation, Y = α + βX. For any particular point, the distance above or below the line is the error, , for that point.

FIGURE 8.1 Linear Regression Example

Because there ...

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