Unformatted text preview: able xj has no partial e¤ect on the dVar y .
The hypothesis Ho : βj = 0 all j 1 is an “extreme” exclusion
restriction since we test the overall signi…cance of the regression:
if we fail to reject Ho then there is no evidence that any of the eVars
help explain y (STATA outputs this test by default when we use the
command regress)
Melissa Tartari (Yale) Econometrics 36 / 41 Testing General Hypothesis: Remarks II For examples of applications of the command test look at the
STATA tutorial within the tutorial for the command regress or type
help test (also check out the very convenient command testparm). Melissa Tartari (Yale) Econometrics 37 / 41 Testing General Hypothesis: The F Test Here we provide an intuitive overview of the F test. Melissa Tartari (Yale) Econometrics 38 / 41 Testing General Hypothesis: The F Test Here we provide an intuitive overview of the F test.
Suppose we want to test the joint hypothesis Ho involving q exclusion
restrictions (on q slope coe¢ cients); Melissa Tartari (Yale) Econometrics 38 / 41 Testing General Hypothesis: The F Test Here we provide an intuitive overview of the F test.
Suppose we want to test the joint hypothesis Ho involving q exclusion
restrictions (on q slope coe¢ cients);
What would you start looking at? Melissa Tartari (Yale) Econometrics 38 / 41 Testing General Hypothesis: The F Test Here we provide an intuitive overview of the F test.
Suppose we want to test the joint hypothesis Ho involving q exclusion
restrictions (on q slope coe¢ cients);
What would you start looking at? The sum of squared residuals SSR would seem to provide a
convenient basis for testing such hypothesis in the sense that how
much SSR increases when we exclude the q explanatory variables tells
us something about the truth in Ho . We know that SSR increases
when we leave out x 0 s from the model so what really matters is
whether such increase is large enough relative to the SSR in
the model with all of the variables to warrant rejecting Ho . Melissa Tartari (Yale) Econometrics 38 / 41 Testing General Hypothesis: The F Test
The F statistics presented above does exactly this: it combines such
information in such a way that the we know its distribution under Ho !
Indeed, we can rewrite it as follow:
F= (SSRr SSRu ) /q
SSRu /n K 1 (4.37) where: Melissa Tartari (Yale) Econometrics 39 / 41 Testing General Hypothesis: The F Test
The F statistics presented above does exactly this: it combines such
information in such a way that the we know its distribution under Ho !
Indeed, we can rewrite it as follow:
F= (SSRr SSRu ) /q
SSRu /n K 1 (4.37) where:
SSRr = SSR of the restricted model i.e. the model estimated with
the restrictions imposed, Melissa Tartari (Yale) Econometrics 39 / 41 Testing General Hypothesis: The F Test
The F statistics presented above does exactly this: it combines such
information in such a way that the we know its distribution under Ho !
Indeed, we can rewrite it as follow:
F= (SSRr SSRu ) /q
SSRu /n K 1 (4.37) where:
SSRr = SSR of the restricted model i.e. the model estimated with
the restrictions imposed,
SSRu = SSR of the unrestricted model i.e. the model estimated
without the restrictions imposed, Melissa Tartari (Yale) Econometrics 39 / 41 Testing General Hypothesis: The F Test
The F statistics presented above does exactly this: it combines such
information in such a way that the we know its distribution under Ho !
Indeed, we can rewrite it as follow:
F= (SSRr SSRu ) /q
SSRu /n K 1 (4.37) where:
SSRr = SSR of the restricted model i.e. the model estimated with
the restrictions imposed,
SSRu = SSR of the unrestricted model i.e. the model estimated
without the restrictions imposed,
q number of restrictions Melissa Tartari (Yale) Econometrics 39 / 41 Testing General Hypothesis: The F Test
The F statistics presented above does exactly this: it combines such
information in such a way that the we know its distribution under Ho !
Indeed, we can rewrite it as follow:
F= (SSRr SSRu ) /q
SSRu /n K 1 (4.37) where:
SSRr = SSR of the restricted model i.e. the model estimated with
the restrictions imposed,
SSRu = SSR of the unrestricted model i.e. the model estimated
without the restrictions imposed,
q number of restrictions
the statistics F (a RV) is distributed as a F (q , n K 1) Melissa Tartari (Yale) Econometrics 39 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics? Melissa Tartari (Yale) Econometrics 40 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics?
We knew that we could use the t statistics to test
Ho : βj = 0tβ =
ˆ
1 Melissa Tartari (Yale) ˆ
β
h j i jx
ˆ
se βj jx Econometrics tn K1 40 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics?
We knew that we could use the t statistics to test
Ho : βj = 0tβ =
ˆ
1 ˆ
β
h j i jx
ˆ
se βj jx tn K1 Now we have learnt that there is another way to test the same
hypothesis. We call this other statistics Fβ (it has distribution
ˆ
j F (1, n K Melissa Tartari (Yale) 1)). Econometrics 40 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics?
We knew that we could use the t statistics to test
Ho : βj = 0tβ =
ˆ
1 ˆ
β
h j i jx
ˆ
se βj jx tn K1 Now we have learnt that there is another way to test the same
hypothesis. We call this other statistics Fβ (it has distribution
ˆ
j F (1, n K 1)). Questions: Melissa Tartari (Yale) Econometrics 40 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics?
We knew that we could use the t statistics to test
Ho : βj = 0tβ =
ˆ
1 ˆ
β
h j i jx
ˆ
se βj jx tn K1 Now we have learnt that there is another way to test the same
hypothesis. We call this other statistics Fβ (it has distribution
ˆ
j F (1, n K 1)). Questions:
How are the two related? Melissa Tartari (Yale) Econometrics 40 / 41 Testing General Hypothesis: the F & the t
What is the relationship between F and t Statistics?
We knew that we could use the t statistics to test
Ho : βj = 0tβ =
ˆ
1 ˆ
β
h j i jx
ˆ
se βj jx tn K1 Now we have learnt that there is another way to test the same
hypothesis. We call this other statistics Fβ (it has distribution
ˆ
j F (1, n K 1)). Questions:
How are the two related?
Does inference change depending on which one we use? Melissa Tartari (Yale) Econometrics 40 / 41 Testing General Hypothesis: the F & the t They are related as follows and inference does not change,
q
tβ = Fβ
ˆ
ˆ
j Melissa Tartari (Yale) Econometrics j 41 / 41 Testing General Hypothesis: the F & the t They are related as follows and inference does not change,
q
tβ = Fβ
ˆ
ˆ
j j The t statistics is more ‡exible than the F for testing a single
hypothesis because it can be used to test against a onesided H1 . Melissa Tartari (Yale) Econometrics 41 / 41...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.
 Fall '10
 DonaldBrown
 Econometrics

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