Linear Regression with Multiple Regressors

1
Chapter 5

• Linear Regression with Multiple Regressors

2
Omitted Variable Bias

• The population regression model of Chapter 4
  allowed for only one explanatory or cause
  variable Yi?0?1Xiui
• The error term ui was interpreted to represent
  all variables other than X that affect Y.
• In this model, we assumed that Xi and ui are
  uncorrelated. This assumption is necessary for
  OLS to yield unbiased estimates of the regression
  parameters.

3
Example

• Suppose we are interested in the effect of gender
  on wages (the regression studied in problem 4.2).
  
• Wagei12.682.79Maleiûi
• Is the slope coefficient an unbiased estimate of
  the gender wage gap?
• This only happens if all variables that affect
  wages that were left out of the equation are
  uncorrelated with gender? Is this believable?

4
Example (Continued)

• Variables, other than gender, that affect wages.
  Are these variables correlated with gender?
• Years of schooling
• Years of work experience
• Race
• Industry of employment
• Occupation

5
Possible Solution

• Include additional explanatory or cause
  variables in the regression
• Estimate Yi?0?1X1i?2X2iui
• This equation has two explanatory variables. A
  simple extension would be to include more.
• The idea here is to include all explanatory
  variables that affect Y (wages).
• If other variables are correlated with gender and
  also affect wages, then the must be accounted for
  to get an unbiased estimate of the gender wage gap

6
Multiple Regression

• In theory, OLS with more than one X variable
  works about the same as in the bivariate case.
• Algebra to get OLS estimates is a little more
  complicated, but this is unimportant
• Assumptions of the model are the same except that
  we need one more
• Explanatory variables are not perfectly
  multicollineareach explanatory variable entered
  must provide new information

7
Interpretation of Coefficients

• In the model Yi?0?1X1i?2X2iui, the question
  arises as to how to interpret the coefficients
• ?0 still the constant or Y-intercept term
• ?1 is interpreted as the effect of X1 on Y,
  holding X2 constant. Or, it is the effect of X1
  on Y, controlling for the effect of X2
• Mathematically, slope coefficients have the
  interpretation of partial derivatives.