1. EC220 - Introduction to econometrics (chapter 3)
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]
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2. F TESTS OF GOODNESS OF FIT
at least one
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. 1

3. F TESTS OF GOODNESS OF FIT
at least one
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. 2

4. F TESTS OF GOODNESS OF FIT
at least one
Of course we hope to reject it and conclude that the model does have some explanatory power. 3

5. F TESTS OF GOODNESS OF FIT
at least one
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. 4

6. F TESTS OF GOODNESS OF FIT
at least one
The alternative hypothesis is that at least one of these b coefficients is different from zero. 5

7. F TESTS OF GOODNESS OF FIT
at least one
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. 6

8. F TESTS OF GOODNESS OF FIT
at least one
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. 7

9. F TESTS OF GOODNESS OF FIT
at least one
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. 8

10. F TESTS OF GOODNESS OF FIT
at least one
It can be expressed in terms of R2 by dividing the numerator and denominator by TSS, the total sum of squares. 9

11. F TESTS OF GOODNESS OF FIT
at least one
ESS / TSS is the definition of R2. RSS / TSS is equal to (1 – R2). (See the last sequence in Chapter 2.) 10

12. F TESTS OF GOODNESS OF FIT
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. 11

13. F TESTS OF GOODNESS OF FIT
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. 12

14. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
Here is the regression output using Data Set 21. 13

15. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 14

16. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 15

17. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
The denominator is the residual sum of squares divided by the number of degrees of freedom remaining. 16

18. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
Hence the F statistic is All serious regression packages compute it for you as part of the diagnostics in the regression output. 17

19. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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 Hence we easily reject H0 at the 0.1% level. 18

20. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 19

21. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 20

22. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 21

23. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 22

24. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 23

25. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 24

26. F TESTS OF GOODNESS OF FIT
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. 25

27. F TESTS OF GOODNESS OF FIT
For example, in the original specification, Y may be written as a simple function of X2. In the second, we add X3 and X4. 26

28. F TESTS OF GOODNESS OF FITor
or both
and
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. 27

29. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
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. 28

30. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The ‘reduction in RSS’ is the reduction when the change is made, in this case, when the group of new variables is added. 29

31. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
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. 30

32. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
(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.) 31

33. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The ‘RSS remaining’ is the residual sum of squares after making the change. 32

34. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The ‘degrees of freedom remaining’ is the number of degrees of freedom remaining after making the change. 33

35. F TESTS OF GOODNESS OF FIT
. reg S ASVABC
Source | SS df MS Number of obs =
F( 1, 538) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
_cons |
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. 34

36. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 35

37. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The improvement in the fit on adding the parental variables is the reduction in the residual sum of squares. 36

38. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The cost is 2 degrees of freedom because 2 additional parameters have been estimated. 37

39. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The remaining unexplained is the residual sum of squares after adding SM and SF. 38

40. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The number of degrees of freedom remaining is n – k, that is, 540 – 4 = 536. 39

41. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The F statistic is 40

42. F TESTS OF GOODNESS OF FITor
or both
and
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The critical value of F(2,500) at the 0.1% level is 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. 41

43. F TESTS OF GOODNESS OF FIT
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. 42

44. F TESTS OF GOODNESS OF FIT
Suppose that in the original model Y is a function of X2 and X3, and that in the revised model X4 is added. 43

45. F TESTS OF GOODNESS OF FIT
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, b4. 44

46. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
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. 45

47. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM
Source | SS df MS Number of obs =
F( 2, 537) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
_cons |
Here is the regression of S on ASVABC and SM. We make a note of the residual sum of squares. 46

48. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
Now we add SF and again make a note of the residual sum of squares. 47

49. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The reduction in the residual sum of squares is the reduction on adding SF. 48

50. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The cost is just the single degree of freedom lost when estimating b4. 49

51. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The RSS remaining is the residual sum of squares after adding SF. 50

52. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The number of degrees of freedom remaining after adding SF is 540 – 4 = 536. 51

53. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
Hence the F statistic is 52

54. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The critical value of F at the 0.1% significance level with 500 degrees of freedom is The critical value with 536 degrees of freedom must be lower, so we reject H0 at the 0.1% level. 53

55. F TESTS OF GOODNESS OF FIT
F (cost in d.f., d.f. remaining) =
reduction in RSS
cost in d.f.
RSS remaining
degrees of freedom
remaining
The null hypothesis we are testing is exactly the same as for a two-sided t test on the coefficient of SF. 54

56. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
We will perform the t test. The t statistic is 3.48. 55

57. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
The critical value of t at the 0.1% level with 500 degrees of freedom is The critical value with 536 degrees of freedom must be lower. So we reject H0 again. 56

58. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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.) 57

59. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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.) 58

60. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
Hence the conclusions of the two tests must coincide. 59

61. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 60

62. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 61

63. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
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. 62

64. F TESTS OF GOODNESS OF FIT
. reg S ASVABC SM SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SM |
SF |
_cons |
If this is the case, the model is said to be suffering from the problem of multicollinearity discussed in the previous sequence. 63

65. Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 3.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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