Market Risk Analysis Volume I: Quantitative Methods in Finance

I.4

Introduction to Linear Regression

I.4.1 INTRODUCTION

A regression model is a statistical model of the influence of one (or more) random variables on another random variable. Usually we assume a simple linear relationship between the variables of the form

On the left of the above equality is the dependent variable, often denoted Y. On the right of the equality we have:

• k Independent Variables X₁, X₂, …, X_k. These are often called explanatory variables because they are put into the model to explain the behaviour of the dependent variable. Almost all models contain a constant term, which means that X₁ = 1.
• k Coefficients β₁, β₂, … β_k. These are not random variables and usually they are assumed to be constant. The coefficients are some of the model parameters that will be estimated using data on the dependent and independent variables.¹ Each coefficient measures the effect that a change in its independent variable will have upon Y. So if an estimated coefficient is insignificantly different from 0 then its explanatory variable can be excluded from the regression model.

The estimated model can be used to:

• predict or forecast values of the dependent variable using scenarios on the independent variables;
• test an economic or financial theory;
• estimate the quantities of financial assets to buy or sell when forming a diversified portfolio, a hedged portfolio or when implementing a ...