1. Testing Multiple Linear Restrictions: The F-test
Three situations call for this test
Testing exclusion restrictions.
Example: Does the genre of background music affect sales at Mitchell’s ice cream?
Testing the overall significance of a regression.
Testing more complicated general linear restrictions motivated by a model.

2. Testing Multiple Linear Restrictions: The F-test
Two ways to calculate F-statistic:
Using SSR:
Using R-sq:
Equivalent because 𝑆𝑆 𝑅 𝑢 =TSS 1− 𝑅 𝑢 2 and 𝑆𝑆 𝑅 𝑟 =TSS 1− 𝑅 𝑟 2
How to remember which one comes first in the numerator? Remember that the F-statistic is always >0, so the one that is larger comes first.
Note: When taking R-squared from Stata output to put it in this formula, do NOT square it!
𝐹= ( 𝑅 𝑢 2 − 𝑅 𝑟 2 )/𝑞 (1− 𝑅 𝑢 2 )/(𝑛−𝑘−1)

3. Testing Multiple Linear Restrictions: The F-test
Test of the overall significance of a regression
The test of overall significance is reported in most regression packages; the null hypothesis is usually overwhelmingly rejected
The null hypothesis states that the explanatory variables are not useful at all in explaining the dependent variable
Restricted model
(regression on constant)

4. More use of the F-test
Testing general linear restrictions with the F-test
Example: Test whether house price assessments are rational
The assessed housing value (before the house was sold)
Size of lot
(in feet)
Actual house price
Square footage
Number of bedrooms
In addition, other known factors should
not influence the price once the assessed value has been controlled for.
If house price assessments are rational, a 1% change in the
assessment should be associated with a 1% change in price.

5. More use of the F-test Unrestricted regression Restricted regression
Test statistic
The restricted model is actually a
regression of [y-x1] on a constant
cannot be rejected

6. More use of the F-test
Regression output for the unrestricted regression
The F-test works for general multiple linear hypotheses
For all tests and confidence intervals, validity of assumptions MLR.1 – MLR.6 has been assumed. Tests may be invalid otherwise.
When tested individually, there is also no evidence against the rationality of house price assessments

7. More examples
Exercises