Multiple Regression

Topic 4. Multiple regression

Aims
• Explain the meaning of partial regression coefficient and calculate and interpret multiple regression models • Derive and interpret the multiple coefficient of determination R2and explain its relationship with the the adjusted R2 • Apply interval estimation and tests of significance to individual partial regression coefficients d d l ff • Test the significance of the whole model (F-test)

Introduction
• The basic multiple regression model is a simple extension of the bivariate equation. • By adding extra independent variables, we are creating a multiple-dimensioned space, where the model fit is a some appropriate space. , p , • For instance, if there are two independent variables, we are fitting the points to a ‘plane in space’. trick. • Visualizing this in more dimensions is a good trick

Model specification – scalar version
• The basic linear model: • Yi = ß0 + ß1 X1i+ ß2X2i+ ß3X3i +….+ ßkXki +ui …. u • If bivariate regression can be described as a line on a plane, multiple regression represents a k-dimensional object in a k+1 d dimensional space. l

Matrix version
• We can use a different type of mathematical g structure to describe the regression model Frequently called Matrix or Linear Algebra • The multiple regression model may be easily represented in matrix terms. • Y=X B +u • Where theY X B and u are all matrices theY, X,

MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE EARNINGS = 1 + 2S + 3EXP + u

pure S effect

1
EARNINGS EXP S

1 + 2S

The next term on the right side of the equation gives the effect of variations in S. A one year increase in S The next term on the right side of the equation gives the effect of variations in S. A one year increase in S causes EARNINGS to increase by 2 dollars, holding EXP constant.
5

MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE EARNINGS = 1 + 2S + 3EXP + u

1 + 3EXP

pure EXP effect

1
EARNINGS EXP S