1. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES 1 We now come to more general F tests 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.

2. 2 For example, in the original specification, Y may be written as a simple function of X 2. In the second, we add X 3 and X 4. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

3. 3 The null hypothesis is that neither X 3 nor X 4 belongs in the model. The alternative hypothesis is that at least one of them does, perhaps both. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES or or bothand

4. 4 When new variables are added to the model, RSS cannot rise. In general, it will fall. If the new variables are irrelevant, it will fall only by a random amount. The test evaluates whether the fall in RSS is greater than would be expected on a pure chance basis. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES or or bothand

5. 5 The appropriate test is an F test. For this test, and for several others which we will encounter, it is useful to think of the F statistic as having the structure indicated above. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

6. 6 The ‘reduction in RSS’ is the reduction when the change is made, in this case, when the group of new variables is added. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

7. 7 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. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

8. 8 (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 X 3 and X 4 are added.) F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

9. 9 The ‘RSS remaining’ is the residual sum of squares after making the change. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

10. 10 The ‘degrees of freedom remaining’ is the number of degrees of freedom remaining after making the change. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining or or bothand

11. . 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 ------------------------------------------------------------------------------ 11 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. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

12. . 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 ------------------------------------------------------------------------------ 12 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES

13. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 13 F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES The improvement in the fit on adding the parental variables is the reduction in the residual sum of squares. or or bothand

14. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 14 F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES The cost is 2 degrees of freedom because 2 additional parameters have been estimated. or or bothand

15. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 15 F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES The remaining unexplained is the residual sum of squares after adding SM and SF. or or bothand

16. or or bothand F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 16 F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES The number of degrees of freedom remaining is n – k, that is, 540 – 4 = 536.

17. 17 F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES The F statistic is 13.16. or or bothand F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

18. 18 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 H 0 and conclude that the parental education variables do have significant joint explanatory power. or or bothand F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

19. 19 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES

20. 20 Suppose that in the original model Y is a function of X 2 and X 3, and that in the revised model X 4 is added. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

21. 21 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES

22. 22 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. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

23. 23 Here is the regression of S on ASVABC and SM. We make a note of the residual sum of squares. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

24. 24 Now we add SF and again make a note of the residual sum of squares. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

25. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 25 The reduction in the residual sum of squares is the reduction on adding SF. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

26. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 26 The cost is just the single degree of freedom lost when estimating  4. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

27. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining 27 The RSS remaining is the residual sum of squares after adding SF. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES degrees of freedom remaining

28. F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining 28 The number of degrees of freedom remaining after adding SF is 540 – 4 = 536. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES

29. 29 Hence the F statistic is 12.10. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

30. 30 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 H 0 at the 0.1% level. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

31. 31 The null hypothesis we are testing is exactly the same as for a two-sided t test on the coefficient of SF. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES F (cost in d.f., d.f. remaining) = reduction in RSScost in d.f. RSS remaining degrees of freedom remaining

32. 32 We will perform the t test. The t statistic is 3.48. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

33. 33 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 H 0 again. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

34. 34 It 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.) F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

35. 35 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

36. 36 Hence the conclusions of the two tests must coincide. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

37. 37 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

38. 38 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

39. 39 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 RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

40. 40 If this is the case, the model is said to be suffering from the problem of multicollinearity discussed in the previous sequence. F TESTS RELATING TO GROUPS OF EXPLANATORY VARIABLES. 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 ------------------------------------------------------------------------------

41. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 3.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.10.29