Faculty of Business & Economic Sciences
BED3332/ECO3022 LECTURE 6: THE F-TEST
DR QABHOBHO
Outline of the lecture
 This
lecture will look at the F-test in chapter 5
 This
is part hypothesis testing
 Pages
142 to 147 in the 7th edition
pages 165 to the top of 169 in the 6th edition
(appendix)
 From
Introduction
 In
the previous lecture we covered hypothesis testing
of individual slope coefficients
 In
this lecture we will look at conducting hypothesis
testing for multiple coefficients simultaneously
Introduction
 Hypothesis
testing involving multiple coefficients
simultaneously is done using the F-test
 We
will use both the manual method and the p-value
method to conduct the F-test
 You
will have to read from the F-distribution (Page
520 to 523 in the 7th edition)
Test of significance
 We
will go through the following:
1.
Joint hypothesis testing
2.
Testing the overall significance of a model
The F-test – Joint significance
A
null hypothesis with more than one parameter is
called a joint hypothesis
 Such
a hypothesis is specified as follows:
𝐻0 : 𝛽1 = 0, 𝛽2 = 0, 𝛽3 = 0
𝐻1 : 𝛽1 ≠ 0, 𝛽2 ≠ 0, 𝛽3 ≠ 0
The F-test – Joint significance
 It
is possible to test whether certain coefficients in a
regression model are equal to zero or not
simultaneously
 When
using such a test we are trying to figure out
whether these coefficients should be in the model or
not
The F-test – Joint significance
 These
are steps taken when conducting a joint
hypothesis test:
1.
Translate the null hypothesis into constraints that
will be placed on the model
2.
Estimate the constrained equation using OLS and
compare the fit of the constrained model with that
of the unconstrained model
The F-test – Joint significance
 If
the fits of the constrained and unconstrained
equations are not significantly different then the null
hypothesis should not be rejected
 If
the fit of the unconstrained equation is significantly
better then we reject the null hypothesis
The F-test – Joint significance
 The
fits of the equations is compared using the Fstatistic
(𝑅𝑆𝑆𝑀 − 𝑅𝑆𝑆)/𝑀
𝐹=
𝑅𝑆𝑆/(𝑁 − 𝐾 − 1)
K in this case is the number of parameters excluding
the intercept
The F-test – Joint significance
 The
decision rule to use for the F-test is:
Reject the null hypothesis if the F-statistic is
greater than the F-critical value from the Fdistribution
The F-distribution
 The
F-critical value is based on the following:
 The
chosen level of significance
 The numerator degrees of freedom
 The denominator degrees of freedom
 (NB)
You can use the F-statistic formula to figure out
the numerator and denominator degrees of freedom
Example
 Suppose
you estimate the following model:
𝑄𝑑 = 𝛽0 + 𝛽1 𝑃 + 𝛽2 𝑃𝑠 + 𝛽3 𝑃𝑐 + 𝛽4 𝐼𝑁𝐶 + 𝜀
 If
you believe that a model with no Ps and Pc would
do just as well as the one above
Example
 The
null hypothesis is as follows:
𝐻0 : 𝛽2 = 0, 𝛽3 = 0
𝐻1 : 𝛽2 ≠ 0, 𝛽3 ≠ 0
 How
will your constrained model look like?
Example
 Suppose
that:
𝑅𝑆𝑆𝑀 = 1896.391
 𝑅𝑆𝑆 = 1532.084
 𝑀 =?
 𝑁 = 28
 𝐾 =?

Example
 Test
the following hypothesis at 5% level of
significance
𝐻0 : 𝛽2 = 0, 𝛽3 = 0
𝐻1 : 𝛽2 ≠ 0, 𝛽3 ≠ 0
 What
is your conclusion?
The F-Test of Overall Significance
 The
test of overall significance is a formal hypothesis
test for of the overall fit of a model
 We
test whether all the slope coefficients are equal to
zero simultaneously or not
The F-Test of Overall Significance
 For
an equation with K independent variables the null
and alternative hypotheses are as follows:
𝐻0 : 𝛽1 = 𝛽2 = ⋯ = 𝛽𝐾 = 0
𝐻1 : 𝐻0 is not true
The F-Test of Overall Significance
 The
F-test of overall significance is specified as
follows:
𝐸𝑆𝑆/𝐾
𝐹=
𝑅𝑆𝑆/(𝑁 − 𝐾 − 1)
K
is the number of parameters excluding the
intercept or you could say the number of independent
variables
The F-Test of Overall Significance
 Is
the fit of the equation significantly better than that
provided by the mean alone?
 The
decision rule to use for the F-test is:
 Reject
the null hypothesis if the F-statistic is greater
than the F-critical value from the F-distribution
Example
 We
will use the Woody example introduced in chapter
three (Section 3.2)
 It
deals with finding the best location for a restaurant
 The
model is as follows:
𝑆𝐴𝐿𝐸𝑆 = 𝛽0 + 𝛽1 𝐶𝑂𝑀𝑃 + 𝛽2 𝑃𝑂𝑃 + 𝛽3 𝐼𝑁𝐶 + 𝜀
Example
 Suppose
we have the following:
N
= 33
 RSS = 6,133,300,000
 ESS = 9,928,900,000
 K =?
 Test
the overall significant at the 5% level
Another example
Dependent Variable: SALES
Method: Least Squares
Date: 08/27/16 Time: 23:26
Sample: 1 75
Included observations: 75
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
PRICE
ADVERT
ADVERT^2
109.7190
-7.640000
12.15124
-2.767963
6.799045
1.045939
3.556164
0.940624
16.13742
-7.304442
3.416950
-2.942688
0.0000
0.0000
0.0011
0.0044
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
0.508235
0.487456
4.645283
1532.084
-219.5540
24.45932
0.000000
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
77.37467
6.488537
5.961440
6.085039
6.010792
2.043061