F TEST OF GOODNESS OF FIT FOR THE WHOLE EQUATION 1 This sequence describes two F tests of goodness...

F TEST OF GOODNESS OF FIT FOR THE WHOLE EQUATION

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

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.


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H

H k

at least one

3

Of course we hope to reject it and conclude that the model does have some explanatory power.


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H k

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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 b2, ..., bk are zero.


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5

The alternative hypothesis is that at least one of these b coefficients is different from zero.


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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.


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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.


uXXY kk ...221

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H

H k

at least one

)()1()1(

)(

)1(

)()1(

),1(

2

2

knRkR

knTSSRSS

kTSSESS

knRSSkESS

knkF

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H k

at least one

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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.

)()1()1(

)(

)1(

)()1(

),1(

2

2

knRkR

knTSSRSS

kTSSESS

knRSSkESS

knkF

uXXY kk ...221

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0...:

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H k

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It can be expressed in terms of R2 by dividing the numerator and denominator by TSS, the total sum of squares.


)()1()1(

)(

)1(

)()1(

),1(

2

2

knRkR

knTSSRSS

kTSSESS

knRSSkESS

knkF

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H k

at least one


ESS / TSS is the definition of R2. RSS / TSS is equal to (1 – R2). (See the last sequence in Chapter 2.)

11

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.


uSFSMASVABCS 4321

12

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

0 : ,0: 14320 HH at least one

13


Here is the regression output using Data Set 21.

uSFSMASVABCS 4321

0 : ,0: 14320 HH

. 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------------------------------------------------------------------------------

at least one

14


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.

uSFSMASVABCS 4321



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------------------------------------------------------------------------------

)/()1/(

,1knRSS

kESSknkF

3.104536/20243/1181

536,3 F

15


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.

uSFSMASVABCS 4321 uSFSMASVABCS 4321




3.104536/20243/1181

536,3 F )/()1/(

,1knRSS

kESSknkF

16

uSFSMASVABCS 4321


The denominator is the residual sum of squares divided by the number of degrees of freedom remaining.

uSFSMASVABCS 4321




3.104536/20243/1181

536,3 F )/()1/(

,1knRSS

kESSknkF

17


Hence the F statistic is 104.3. All serious regression packages compute it for you as part of the diagnostics in the regression output.





3.104536/20243/1181

536,3 F )/()1/(

,1knRSS

kESSknkF

18


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.





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536,3 F 51.5500,3crit,0.1% F

19


This result could have been anticipated because both ASVABC and SF have highly significant t statistics. So we knew in advance that both b2 and b4 were non-zero.





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536,3 F 51.5500,3crit,0.1% F

20


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.





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536,3 F 51.5500,3crit,0.1% F

21


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.





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536,3 F 51.5500,3crit,0.1% F

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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.





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536,3 F 51.5500,3crit,0.1% F

23


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.





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536,3 F 51.5500,3crit,0.1% F

24

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.






3.104536/20243/1181

536,3 F 51.5500,3crit,0.1% F

Copyright Christopher Dougherty 2012.

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F TEST OF GOODNESS OF FIT FOR THE WHOLE EQUATION 1 This sequence describes two F tests of goodness...

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