1 Econ 326 - Chapter 6 Chapter 6 More Multiple Regression Model The F-test – Joint Hypothesis Tests Consider the linear regression equation: (1) i 4 i 4 3 i 3 2 i 2 1 i e x x x y + β + β + β + β = for i = 1, 2, . . . , N The t-statistic give a test of significance of an individual explanatory variable, given the other variables in the regression equation. An example of a joint hypothesis test is: 0 : H 4 0 = β = β 3 : H 1 at least one is not zero Why not use separate t-tests on each of the null hypotheses 0 : H 0 = β 3 and 0 : H 4 0 = β ? Typically 0 ) b , b cov( 4 3 ≠ . That is, the slope estimators may be correlated. Therefore, testing a series of single hypotheses is not equivalent to testing hypotheses jointly. An equation that assumes the null hypothesis is true and incorporates the restrictions 0 = β 3 and 0 4 = β is: (2) i 2 i 2 1 i v x y + β + β = ( i v is another random error) Model (1) is called the unrestricted model. Model (2) is called the restricted model.
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1 Econ 326 - Chapter 6
Chapter 6 More Multiple Regression Model
The F-test – Joint Hypothesis Tests
Consider the linear regression equation:
(1) i4i43i32i21i exxxy ++++ββββ++++ββββ++++ββββ++++ββββ==== for i = 1, 2, . . . , N
The t-statistic give a test of significance of an individual explanatory
variable, given the other variables in the regression equation.
An example of a joint hypothesis test is:
0:H 40 ====ββββ====ββββ3333
:H1 at least one is not zero
Why not use separate t-tests on each of the null hypotheses
0:H0 ====ββββ3333 and 0:H 40 ====ββββ ?
Typically 0)b,bcov( 43 ≠≠≠≠ .
That is, the slope estimators may be correlated.
Therefore, testing a series of single hypotheses is not equivalent to
testing hypotheses jointly.
An equation that assumes the null hypothesis is true and
incorporates the restrictions 0====ββββ3333 and 04 ====ββββ is:
(2) i2i21i vxy ++++ββββ++++ββββ==== ( iv is another random error)
Model (1) is called the unrestricted model.
Model (2) is called the restricted model.
2 Econ 326 - Chapter 6
The test method can proceed as follows.
STEP 1 Estimate the unrestricted model and get the sum of
squared residuals:
∑∑∑∑====
====N
1i
2iU eSSE
STEP 2 Estimate the restricted model and compute:
∑∑∑∑====
====N
1i
2iR vSSE
Note: RU SSESSE ≤≤≤≤
STEP 3 Construct the F-statistic:
)KN(SSE
J)SSESSE(F
U
UR
−−−−
−−−−====
where J is the number of restrictions.
In this example, J = 2 and K = 4.
The F-statistic is the ratio of two sum of squares.
The numerator degrees of freedom is J and
the denominator degrees of freedom is N – K.
3 Econ 326 - Chapter 6
The F-statistic can be compared with the F-distribution with
(J , N – K) degrees of freedom.
The F-distribution is defined for positive values and has a skewed
shape. The shape depends on the numerator and denominator
degrees of freedom.
Probability density function of the F-distribution
0 0.5 1 1.5 2 2.5 3 3.5 4
F(2,16)
F(3,16)
F(5,16)
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Statistical tables included in textbook appendixes report critical
values cF such that
05.0)FF(P c)2m,1m( ====>>>> or
01.0)FF(P c)2m,1m( ====>>>>
where m1 is the numerator degrees of freedom and m2 is the
denominator degrees of freedom.
By setting the significance level of the test at either 0.05 (5%) or 0.01
(1%), the decision rule is to reject the null hypothesis if the calculated
F-statistic exceeds the critical value cF .
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Example
Consider the joint hypothesis:
0:H 40 ====ββββ====ββββ3333
:H1 at least one is not zero
For an application with N=20, the calculated F-statistic is F = 7.4.
From the statistical tables, the 1% critical value from the
F-distribution with J=2 and N – K = 20 – 4 = 16 degrees of freedom is
cF = 6.23.
The F-statistic exceeds the critical value and therefore, there is
evidence to reject the null hypothesis.
This suggests that the p-value for the test must be less than 0.01.
The p-value is the probability:
)F(P)FF(Pp )16,2()2m,1m( 7.4>>>>====>>>>====
An exact p-value can be calculated with the Microsoft Excel function:
F.DIST.RT(7.4, 2, 16)
↑↑↑↑ right-tail
This gives the answer p = 0.005303.
6 Econ 326 - Chapter 6
With Stata, the test command can be used for joint hypothesis tests.
On the Stata results, the F-statistic is reported with an accompanying
p-value. The p-value can then be interpreted to make a decision.
Application of the F-test method with one restriction J = 1 is
equivalent to a t-test. In this special case the random variable
Ft ==== has a t-distribution with (N – K) degrees of freedom.
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Testing the Significance of the Model
To test the overall significance of the regression consider a joint test
that all slope coefficients are zero. Test:
0...:H K320 ====ββββ========ββββ====ββββ
:H1 at least one is not zero
In this case, the restricted model is:
i1i vy ++++ββββ====
Least squares estimation of the restricted model gives: yb1 ====
Therefore, ∑∑∑∑====
−−−−========N
1i
2iR )yy(SSTSSE
Recognize SSESSEU ==== .
The number of restrictions is K – 1.
The test statistic and accompanying p-value is:
)KN(SSE
)1K()SSESST(F
−−−−
−−−−−−−−====
)FF(Pp )KN,1K( >>>>==== −−−−−−−−
8 Econ 326 - Chapter 6
There is a relationship between the 2R and the F-test for the overall
significance of the regression.
Recall SST
SSESST
SST
SSE1R2 −−−−
====−−−−====
Express the F-statistic as:
SSE
)SSESST(
1K
KNF
−−−−⋅⋅⋅⋅
−−−−−−−−
====
Divide both numerator and denominator by SST to get:
2
2
R1
R
1K
KN
SST/SSE
SST/)SSESST(
1K
KNF
−−−−⋅⋅⋅⋅
−−−−−−−−
====
−−−−⋅⋅⋅⋅
−−−−−−−−
====
This shows that if 2R = 0 then F = 0.
As 2R increases the F-statistic also increases.
When 2R = 1, F is infinite.
The F-test for the overall significance of the regression can be viewed
as a test of significance of the 2R .
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The Use of Nonsample Information
It may be sensible to incorporate restrictions in the model estimation.
That is, better use of the information may give better estimates.
This is illustrated by continuing with the Cobb-Douglas production
function example that was introduced in the Chapter 5 lecture notes.