Chapter 6 f Test of a Linear Restriction (EC220)

Christopher Dougherty

EC220 - Introduction to econometrics

(chapter 6) Slideshow: f test of a linear restriction

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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]

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1

F TEST OF A LINEAR RESTRICTION

In the last sequence it was argued that educational attainment might be related to cognitive

ability and family background, with mother's and father's educational attainment proxying

for the latter.

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

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

2

However, when we run the regression using Data Set 21, we find the coefficient of mother's

education is not significant.




-------------+------------------------------ F( 3, 536) = 104.30

Model | 1181.36981 3 393.789935 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3651

Total | 3204.98333 539 5.94616574 Root MSE = 1.943

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


-------------+----------------------------------------------------------------

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

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

3

As was noted in one of the sequences for Chapter 3, this might be due to multicollinearity,

because mother's education and father's education are correlated.

. cor SM SF

(obs=540)

| SM SF

--------+------------------

SM| 1.0000

SF| 0.6241 1.0000


4

In the discussion of multicollinearity, several measures for alleviating the problem were

suggested, among them the use of an appropriate theoretical restriction.

uSFSMASVABCS 4321


5

In particular, in the case of the present model, it was suggested that the impact of parental

education might be the same for both parents, that is, that 3 and 4 might be equal.

uSFSMASVABCS 4321

34


6

If this is the case, the model may be rewritten as shown. We now have a total parental

education variable, SP, instead of separate variables for mother’s and father’s education,

and the multicollinearity caused by the correlation between the latter has been eliminated.

uSFSMASVABCS 4321

34

uSPASVABC

uSFSMASVABCS

321

321 )(

SFSMSP


. g SP=SM+SF

. reg S ASVABC SP


-------------+------------------------------ F( 2, 537) = 156.04

Model | 1177.98338 2 588.991689 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3652

Total | 3204.98333 539 5.94616574 Root MSE = 1.9429

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


-------------+----------------------------------------------------------------

ASVABC | .1253106 .0098434 12.73 0.000 .1059743 .1446469

SP | .0828368 .0164247 5.04 0.000 .0505722 .1151014

_cons | 5.29617 .4817972 10.99 0.000 4.349731 6.242608

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

7

Here is the regression with SP replacing SM and SF.



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


-------------+----------------------------------------------------------------

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

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

. reg S ASVABC SP

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


-------------+----------------------------------------------------------------

ASVABC | .1253106 .0098434 12.73 0.000 .1059743 .1446469

SP | .0828368 .0164247 5.04 0.000 .0505722 .1151014

_cons | 5.29617 .4817972 10.99 0.000 4.349731 6.242608

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

8

A comparison of the regressions reveals that the standard error of the coefficient of SP is

much smaller than those of SM and SF, and consequently its t statistic is higher. Its

coefficient is a compromise between those of SM and SF, as might be expected.



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


-------------+----------------------------------------------------------------

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

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

. reg S ASVABC SP

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


-------------+----------------------------------------------------------------

ASVABC | .1253106 .0098434 12.73 0.000 .1059743 .1446469

SP | .0828368 .0164247 5.04 0.000 .0505722 .1151014

_cons | 5.29617 .4817972 10.99 0.000 4.349731 6.242608

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

9

However, the use of a restriction will lead to a gain in efficiency only if the restriction is

valid. If it is not valid, its use will lead to biased coefficients and invalid standard errors and

tests.



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


-------------+----------------------------------------------------------------

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

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

. reg S ASVABC SP

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


-------------+----------------------------------------------------------------

ASVABC | .1253106 .0098434 12.73 0.000 .1059743 .1446469

SP | .0828368 .0164247 5.04 0.000 .0505722 .1151014

_cons | 5.29617 .4817972 10.99 0.000 4.349731 6.242608

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

10

Do the coefficients of SM and SF in the unrestricted regression look as if they satisfy the

restriction? Not really, in this case. The coefficient of SM is much smaller than that of SF,

but then it should be noted that the standard errors are quite large.




-------------+------------------------------ F( 3, 536) = 104.30

Model | 1181.36981 3 393.789935 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3651

Total | 3204.98333 539 5.94616574 Root MSE = 1.943

. reg S ASVABC SP


-------------+------------------------------ F( 2, 537) = 156.04

Model | 1177.98338 2 588.991689 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3652

Total | 3204.98333 539 5.94616574 Root MSE = 1.9429

11

We will now perform a proper test. The imposition of a restriction makes it more difficult for

the regression model to fit the data because there is one fewer parameter to adjust. There

will therefore be an increase in RSS (and a decrease in R2) when it is imposed.




-------------+------------------------------ F( 3, 536) = 104.30

Model | 1181.36981 3 393.789935 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3651

Total | 3204.98333 539 5.94616574 Root MSE = 1.943

. reg S ASVABC SP


-------------+------------------------------ F( 2, 537) = 156.04

Model | 1177.98338 2 588.991689 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3652

Total | 3204.98333 539 5.94616574 Root MSE = 1.9429

12

If the restriction is valid, the deterioration in the fit should be a small, random amount.

However, if the restriction is invalid, the distortion caused by its imposition will lead to a

significant deterioration in the fit.




-------------+------------------------------ F( 3, 536) = 104.30

Model | 1181.36981 3 393.789935 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3651

Total | 3204.98333 539 5.94616574 Root MSE = 1.943

. reg S ASVABC SP


-------------+------------------------------ F( 2, 537) = 156.04

Model | 1177.98338 2 588.991689 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3652

Total | 3204.98333 539 5.94616574 Root MSE = 1.9429

13

In the present case, we can see that the increase in RSS is very small, and hence we are

unlikely to reject the restriction.


uSFSMASVABCS 4321

34

uSPASVABC

uSFSMASVABCS

321

321 )(

SFSMSP

341340 :,: HH

14

The null hypothesis is that the restriction is valid, and the alternative one is that it is invalid.


uSFSMASVABCS 4321

34

uSPASVABC

uSFSMASVABCS

321

321 )(

SFSMSP

90.0536/61.2023

61.202300.2027

)/(

1/)(),1(

knRSS

RSSRSSknF

U

UR

341340 :,: HH

15

The test statistic is a member of the family of F tests where the numerator is the

improvement in the fit on relaxing the restriction, divided by the cost of relaxing it (one

degree of freedom, because one additional parameter has to be estimated).


uSFSMASVABCS 4321

34

uSPASVABC

uSFSMASVABCS

321

321 )(

SFSMSP

90.0536/61.2023

61.202300.2027

)/(

1/)(),1(

knRSS

RSSRSSknF

U

UR

341340 :,: HH

16

The denominator of the test statistic is RSS after making the improvement (that is, RSS for

the unrestricted model), divided by n – k, the number of degrees of freedom remaining. k is

the number of parameters in the unrestricted model.


uSFSMASVABCS 4321

34

uSPASVABC

uSFSMASVABCS

321

321 )(

SFSMSP

90.0536/61.2023

61.202300.2027

)/(

1/)(),1(

knRSS

RSSRSSknF

U

UR

341340 :,: HH

17

The F statistic is 0.90. An F statistic below 1 is never significant (look at the F table), so we

do not reject H0. The restriction appears to be valid. At least, it is not rejected by the data.


Copyright Christopher Dougherty 2011.

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The content of this slideshow comes from Section 6.5 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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Chapter 6 f Test of a Linear Restriction (EC220)

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