Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: f test of a linear restriction Original citation: Dougherty, C. (2012)
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Christopher Dougherty
EC220 - Introduction to econometrics (chapter 6)Slideshow: f test of a linear restriction
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]
This version available at: http://learningresources.lse.ac.uk/132/
Available in LSE Learning Resources Online: May 2012
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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
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
F TEST OF A LINEAR RESTRICTION
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
F TEST OF A LINEAR RESTRICTION
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
F TEST OF A LINEAR RESTRICTION
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
F TEST OF A LINEAR RESTRICTION
. g SP=SM+SF
. reg S ASVABC SP
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675-------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429
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.
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.
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 TEST OF A LINEAR RESTRICTION
. 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
. reg S ASVABC SP
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675-------------+------------------------------ 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 TEST OF A LINEAR RESTRICTION
. 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
. reg S ASVABC SP
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675-------------+------------------------------ 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 TEST OF A LINEAR RESTRICTION
. 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
. reg S ASVABC SP
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675-------------+------------------------------ 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.
F TEST OF A LINEAR RESTRICTION
uSFSMASVABCS 4321
34
uSPASVABC
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SFSMSP
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14
The null hypothesis is that the restriction is valid, and the alternative one is that it is invalid.
F TEST OF A LINEAR RESTRICTION
uSFSMASVABCS 4321
34
uSPASVABC
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61.202300.2027)/(
1/)(),1(
knRSS
RSSRSSknF
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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).
F TEST OF A LINEAR RESTRICTION
uSFSMASVABCS 4321
34
uSPASVABC
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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.
F TEST OF A LINEAR RESTRICTION
uSFSMASVABCS 4321
34
uSPASVABC
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61.202300.2027)/(
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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.
F TEST OF A LINEAR RESTRICTION
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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