MF-852 Financial Econometrics

1
MF-852 Financial Econometrics

• Lecture 8
• Introduction to Multiple Regression
• Roy J. Epstein
• Fall 2003

2
Topics

• Formulation and Estimation of a Multiple
  Regression
• Interpretation of the Regression Coefficients
• Omitted Variables
• Collinearity
• Advanced Hypothesis Testing

3
Multiple Regression

• Used when 2 or more independent variables explain
  the dependent variable
• Yi ?0 ?1X1i ?2X2i
• ?kXki ei
• or Yi Xi? ei

4
The Error Term

• Same assumptions as before
• E(ei) 0
• var(ei) ?2
• cov(X,e) 0
• cov(ei, ej) 0

5
The Error Term

• Same assumptions as before
• E(ei) 0
• var(ei) ?2
• cov(X,e) 0
• cov(ei, ej) 0

6
The Estimated Coefficients

• Measure the marginal effect of an independent
  variable, controlling for the other effects.
• I.e., effect of Xi all else equal
• Can be sensitive to what other variables are
  included in the regression.

7
Omitted Variables

• Suppose true model is
• Yi ?0 ?1X1i ?2X2i ei
• But you leave out X2. (by ignorance or lack of
  data) Does it matter?

8
Analysis of Omitted Variables

• Error term now includes e and X2
• Yi ?0 ?1X1i ui
• ?0 ?1X1i ?2X2i ei
• Two cases
• X2 correlated with X1. biased picks up
  effect of X2 and attributes it to X1.
• X2 uncorrelated with X1. No bias.

9
Case Study MIT Lawsuit
10
(No Transcript)
11
Collinearity

• Let Yi ?0 ?1X1i ?2X2i ei
• Suppose X1 and X2 highly correlated.
• What difference does it make?
• Hard to estimate ?1 and ?2.
• No bias, but large standard errors.

12
CollinearityDiagnosis

• Neither X1 or X2 has a significant t statistic
  BUT
• X1 is significant when X2 is left out of the
  regression and vice versa.
• Test joint significance with F test.

13
Exact Collinearity

• Let Yi ?0 ?1X1i ?2X2i ei
• Suppose X2 is exact linear function of X1
• E.g., X2 a bX1
• Then cannot estimate model at all!
• Can also occur with 3 or more Xs.

14
Exact CollinearityExample

• Regression to explain calories as function of fat
  content of foods
• X1 is fat in ounces per portion
• X2 is fat in same food in grams
• Then X2i 28.35 X1i
• Cant estimate Yi ?0 ?1X1i ?2X2i ei
• Intuition no independent information in X2.

15
Tests of Restrictions

• Suppose H0 ?2 2?1 in
• Yi ?0 ?1X1i ?2X2i ei
• Test H0 with reformulated model that embeds
  restriction
• Yi ?0 ?1(X1i 2X2i) ?2X2i ei
• Under H0, ?2 0
• Can test with usual t statistic

16
Test your Understanding!

• What is difference between exact collinearity,
  e.g.,
• X2i 2X1i
• And a coefficient restriction, e.g.,
• H0 ?2 2?1 ?
• Relate the concepts to the model.