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Transcript of Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: f tests in a...
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 3)Slideshow: f tests in a multiple regression model
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/129/
Available in LSE Learning Resources Online: May 2012
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
F TESTS OF GOODNESS OF FIT
1
This sequence describes two F tests of goodness of fit in a multiple regression model. The first relates to the goodness of fit of the equation as a whole.
uXXY kk ...221
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
2
We will consider the general case where there are k – 1 explanatory variables. For the F test of goodness of fit of the equation as a whole, the null hypothesis, in words, is that the model has no explanatory power at all.
uXXY kk ...221
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
3
Of course we hope to reject it and conclude that the model does have some explanatory power.
uXXY kk ...221
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
4
The model will have no explanatory power if it turns out that Y is unrelated to any of the explanatory variables. Mathematically, therefore, the null hypothesis is that all the coefficients 2, ..., k are zero.
uXXY kk ...221
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
uXXY kk ...221
5
The alternative hypothesis is that at least one of these coefficients is different from zero.
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
uXXY kk ...221
6
In the multiple regression model there is a difference between the roles of the F and t tests. The F test tests the joint explanatory power of the variables, while the t tests test their explanatory power individually.
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
uXXY kk ...221
7
In the simple regression model the F test was equivalent to the (two-sided) t test on the slope coefficient because the ‘group’ consisted of just one variable.
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
)()1()1(
)(
)1(
)()1(
),1(
2
2
knRkR
knTSSRSS
kTSSESS
knRSSkESS
knkF
uXXY kk ...221
8
The F statistic for the test was defined in the last sequence in Chapter 2. ESS is the explained sum of squares and RSS is the residual sum of squares.
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
)()1()1(
)(
)1(
)()1(
),1(
2
2
knRkR
knTSSRSS
kTSSESS
knRSSkESS
knkF
uXXY kk ...221
9
It can be expressed in terms of R2 by dividing the numerator and denominator by TSS, the total sum of squares.
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
10
)()1()1(
)(
)1(
)()1(
),1(
2
2
knRkR
knTSSRSS
kTSSESS
knRSSkESS
knkF
uXXY kk ...221
ESS / TSS is the definition of R2. RSS / TSS is equal to (1 – R2). (See the last sequence in Chapter 2.)
0 :
0...:
1
20
H
H k
at least one
F TESTS OF GOODNESS OF FIT
11
uSFSMASVABCS 4321
The educational attainment model will be used as an example. We will suppose that S depends on ASVABC, the ability score, and SM, and SF, the highest grade completed by the mother and father of the respondent, respectively.
F TESTS OF GOODNESS OF FIT
12
0: 4320 H
The null hypothesis for the F test of goodness of fit is that all three slope coefficients are equal to zero. The alternative hypothesis is that at least one of them is non-zero.
uSFSMASVABCS 4321
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
13
Here is the regression output using Data Set 21.
uSFSMASVABCS 4321 0: 4320 H
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
14
uSFSMASVABCS 4321 0: 4320 H
In this example, k – 1, the number of explanatory variables, is equal to 3 and n – k, the number of degrees of freedom, is equal to 536.
)/()1/(
),1(knRSS
kESSknkF
3.104
536/20243/1181
)536,3( F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
15
uSFSMASVABCS 4321 0: 4320 H
)/()1/(
),1(knRSS
kESSknkF
3.104
536/20243/1181
)536,3( F
The numerator of the F statistic is the explained sum of squares divided by k – 1. In the Stata output these numbers are given in the Model row.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
16
uSFSMASVABCS 4321 0: 4320 H
)/()1/(
),1(knRSS
kESSknkF
3.104
536/20243/1181
)536,3( F
The denominator is the residual sum of squares divided by the number of degrees of freedom remaining.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
17
uSFSMASVABCS 4321 0: 4320 H
)/()1/(
),1(knRSS
kESSknkF
3.104
536/20243/1181
)536,3( F
Hence the F statistic is 104.3. All serious regression packages compute it for you as part of the diagnostics in the regression output.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
18
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F
The critical value for F(3,536) is not given in the F tables, but we know it must be lower than F(3,500), which is given. At the 0.1% level, this is 5.51. Hence we easily reject H0 at the 0.1% level.
51.5)500,3(crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
19
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
This result could have been anticipated because both ASVABC and SF have highly significant t statistics. So we knew in advance that both 2 and 4 were non-zero.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
20
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
It is unusual for the F statistic not to be significant if some of the t statistics are significant. In principle it could happen though. Suppose that you ran a regression with 40 explanatory variables, none being a true determinant of the dependent variable.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
21
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
Then the F statistic should be low enough for H0 not to be rejected. However, if you are performing t tests on the slope coefficients at the 5% level, with a 5% chance of a Type I error, on average 2 of the 40 variables could be expected to have ‘significant’ coefficients.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
22
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
The opposite can easily happen, though. Suppose you have a multiple regression model which is correctly specified and the R2 is high. You would expect to have a highly significant F statistic.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
23
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
However, if the explanatory variables are highly correlated and the model is subject to severe multicollinearity, the standard errors of the slope coefficients could all be so large that none of the t statistics is significant.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
24
uSFSMASVABCS 4321 0: 4320 H
3.104536/20243/1181
)536,3( F51.5)500,3(crit,0.1% F
In this situation you would know that your model is a good one, but you are not in a position to pinpoint the contributions made by the explanatory variables individually.
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
25
We now come to the other F test of goodness of fit. This is a test of the joint explanatory power of a group of variables when they are added to a regression model.
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
26
For example, in the original specification, Y may be written as a simple function of X2. In the second, we add X3 and X4.
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
27
The null hypothesis for the F test is that neither X3 nor X4 belongs in the model. The alternative hypothesis is that at least one of them does, perhaps both.
0 :
0:
31
430
H
H
04 04 3or or both and
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
28
For this F test, and for several others which we will encounter, it is useful to think of the F statistic as having the structure indicated above.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
29
The ‘reduction in RSS’ is the reduction when the change is made, in this case, when the group of new variables is added.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
30
The ‘cost in d.f.’ is the reduction in the number of degrees of freedom remaining after making the change. In the present case it is equal to the number of new variables added, because that number of new parameters are estimated.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
31
(Remember that the number of degrees of freedom in a regression equation is the number of observations, less the number of parameters estimated. In this example, it would fall from n – 2 to n – 4 when X3 and X4 are added.)
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
32
The ‘RSS remaining’ is the residual sum of squares after making the change.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
33
uXXXY 4433221
uXY 221 1RSS
2RSS
The ‘degrees of freedom remaining’ is the number of degrees of freedom remaining after making the change.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
. reg S ASVABC
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 274.19 Model | 1081.97059 1 1081.97059 Prob > F = 0.0000 Residual | 2123.01275 538 3.94612035 R-squared = 0.3376-------------+------------------------------ Adj R-squared = 0.3364 Total | 3204.98333 539 5.94616574 Root MSE = 1.9865
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .148084 .0089431 16.56 0.000 .1305165 .1656516 _cons | 6.066225 .4672261 12.98 0.000 5.148413 6.984036------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
34
We will illustrate the test with an educational attainment example. Here is S regressed on ASVABC using Data Set 21. We make a note of the residual sum of squares.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
35
Now we have added the highest grade completed by each parent. Does parental education have a significant impact? Well, we can see that a t test would show that SF has a highly significant coefficient, but we will perform the F test anyway. We make a note of RSS.
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
36
The improvement in the fit on adding the parental variables is the reduction in the residual sum of squares.
16.13536/6.2023
2/)6.20230.2123()4540(2)(
)4540,2(2
21
RSS
RSSRSSF
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
37
The cost is 2 degrees of freedom because 2 additional parameters have been estimated.
16.13536/6.2023
2/)6.20230.2123()4540(2)(
)4540,2(2
21
RSS
RSSRSSF
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
38
The remaining unexplained is the residual sum of squares after adding SM and SF.
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2/)6.20230.2123()4540(2)(
)4540,2(2
21
RSS
RSSRSSF
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
39
The number of degrees of freedom remaining is n – k, that is, 540 – 4 = 536.
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2/)6.20230.2123()4540(2)(
)4540,2(2
21
RSS
RSSRSSF
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
16.13536/6.2023
2/)6.20230.2123()4540(2)(
)4540,2(2
21
RSS
RSSRSSF
40
The F statistic is 13.16.
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
uXXXY 4433221
uXY 221 1RSS
2RSS
41
The critical value of F(2,500) at the 0.1% level is 7.00. The critical value of F(2,536) must be lower, so we reject H0 and conclude that the parental education variables do have significant joint explanatory power.
00.7)500,2(crit,0.1% F
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)4540,2(2
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RSS
RSSRSSF
0 :
0:
31
430
H
H
04 04 3or or both and
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
1RSS
2RSS
uXXY 33221
uXXXY 4433221
42
This sequence will conclude by showing that t tests are equivalent to marginal F tests when the additional group of variables consists of just one variable.
F TESTS OF GOODNESS OF FIT
1RSS
2RSS
uXXY 33221
uXXXY 4433221
43
Suppose that in the original model Y is a function of X2 and X3, and that in the revised model X4 is added.
F TESTS OF GOODNESS OF FIT
1RSS
2RSS
0 :
0:
41
40
H
H
uXXY 33221
uXXXY 4433221
44
The null hypothesis for the F test of the explanatory power of the additional ‘group’ is that all the new slope coefficients are equal to zero. There is of course only one new slope coefficient, 4.
F TESTS OF GOODNESS OF FIT
45
1RSS
2RSS
The F test has the usual structure. We will illustrate it with an educational attainment model where S depends on ASVABC and SM in the original model and on SF as well in the revised model.
0 :
0:
41
40
H
H
uXXY 33221
uXXXY 4433221
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
. reg S ASVABC SM
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 147.36 Model | 1135.67473 2 567.837363 Prob > F = 0.0000 Residual | 2069.30861 537 3.85346109 R-squared = 0.3543-------------+------------------------------ Adj R-squared = 0.3519 Total | 3204.98333 539 5.94616574 Root MSE = 1.963
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1328069 .0097389 13.64 0.000 .1136758 .151938 SM | .1235071 .0330837 3.73 0.000 .0585178 .1884963 _cons | 5.420733 .4930224 10.99 0.000 4.452244 6.389222------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
46
Here is the regression of S on ASVABC and SM. We make a note of the residual sum of squares.
F TESTS OF GOODNESS OF FIT
47
Now we add SF and again make a note of the residual sum of squares.
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221
uXXY 33221 1RSS
2RSS
48
The reduction in the residual sum of squares is the reduction on adding SF.
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)4540,1(2
21
RSS
RSSRSSF
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
49
The cost is just the single degree of freedom lost when estimating 4.
uXXY 33221
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)4540,1(2
21
RSS
RSSRSSF
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
50
The RSS remaining is the residual sum of squares after adding SF.
uXXY 33221
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)4540,1(2
21
RSS
RSSRSSF
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
51
The number of degrees of freedom remaining after adding SF is 540 – 4 = 536.
uXXY 33221
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)4540,1(2
21
RSS
RSSRSSF
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
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RSS
RSSRSSF
uXXY 33221
52
Hence the F statistic is 12.10.
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
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RSS
RSSRSSF
uXXY 33221
53
96.10)500,1( crit,0.1% F
The critical value of F at the 0.1% significance level with 500 degrees of freedom is 10.96. The critical value with 536 degrees of freedom must be lower, so we reject H0 at the 0.1% level.
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
0 :
0:
41
40
H
H
uXXXY 4433221 1RSS
2RSS
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)4540,1(2
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RSS
RSSRSSF
uXXY 33221
54
The null hypothesis we are testing is exactly the same as for a two-sided t test on the coefficient of SF.
96.10)500,1( crit,0.1% F
F (cost in d.f., d.f. remaining) =reduction in RSS cost in d.f.
RSS remaining degrees of freedomremaining
F TESTS OF GOODNESS OF FIT
55
We will perform the t test. The t statistic is 3.48.
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
F TESTS OF GOODNESS OF FIT
56
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
The critical value of t at the 0.1% level with 500 degrees of freedom is 3.31. The critical value with 536 degrees of freedom must be lower. So we reject H0 again.
31.3crit,0.1% t
F TESTS OF GOODNESS OF FIT
57
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% tIt can be shown that the F statistic for the F test of the explanatory power of a ‘group’ of one variable must be equal to the square of the t statistic for that variable. (The difference in the last digit is due to rounding error.)
11.1248.3 2
F TESTS OF GOODNESS OF FIT
58
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 96.1031.3 2 It can also be shown that the critical value of F must be equal to the square of the critical value of t. (The critical values shown are for 500 degrees of freedom, but this must also be true for 536 degrees of freedom.)
F TESTS OF GOODNESS OF FIT
59
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 Hence the conclusions of the two tests must coincide.
96.1031.3 2
F TESTS OF GOODNESS OF FIT
60
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 96.1031.3 2 This result means that the t test of the coefficient of a variable is a test of its marginal explanatory power, after all the other variables have been included in the equation.
F TESTS OF GOODNESS OF FIT
61
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 96.1031.3 2 If the variable is correlated with one or more of the other variables, its marginal explanatory power may be quite low, even if it genuinely belongs in the model.
F TESTS OF GOODNESS OF FIT
62
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 96.1031.3 2 If all the variables are correlated, it is possible for all of them to have low marginal explanatory power and for none of the t tests to be significant, even though the F test for their joint explanatory power is highly significant.
F TESTS OF GOODNESS OF FIT
63
96.10crit,0.1% F
. reg S ASVABC SM SF
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943
------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------
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F
31.3crit,0.1% t11.1248.3 2 96.1031.3 2 If this is the case, the model is said to be suffering from the problem of multicollinearity discussed in the previous sequence.
Copyright Christopher Dougherty 2011.
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11.07.25