1. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
The F test approach to testing a restriction may be extended to cover the case where we wish to test whether several restrictions are valid simultaneously. 1

2. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
Suppose that there are p restrictions. Let RSSU be RSS for the fully unrestricted model and RSSR be RSS for the model where all p restrictions have been imposed. The test statistic is then as shown. 2

3. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
The numerator is the reduction in RSS comparing the fully restricted model with the unrestricted model, divided by the number of degrees of freedom lost when the restrictions are relaxed. 3

4. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
The denominator is the RSS for the unrestricted model, divided by the number of degrees of freedom remaining when that model is fitted. k is the number of parameters in the unrestricted model. 4

5. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
The t test approach can be used, as before, to test individual restrictions in isolation. 5

6. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
You will often encounter references to zero restrictions. This just means that a particular parameter is hypothesized to be equal to zero, for example, b5 in the model above. Taken in isolation, the appropriate test is of course the t test. 6

7. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
It can be considered to be a special case of the t test of a restriction discussed above where there is no need for reparameterization. The test statistic is just the t statistic for the parameter in question. 7

8. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
or both
and
Likewise the testing of multiple zero restrictions can be thought of as a special case of the testing of multiple restrictions. The example shown is for a model where there are two zero restrictions. 8

9. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
or both
and
The F test of the joint explanatory power of a group of explanatory variables discussed in Section 3.5 in the text can be thought of in this way. 9

10. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Even the F statistic for the equation as a whole can be treated as a special case. Here the unrestricted and restricted models are as shown. 10

11. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Fitting restricted model:
When we fit the restricted model, we find that the OLS estimator of b1 is the sample mean of Y (see Exercise 1.3). 11

12. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Fitting restricted model:
for all i
Hence the fitted value of Y in all observations is equal to the sample mean of Y. 12

13. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Fitting restricted model:
for all i
Now we know that for any OLS regression, TSS = ESS + RSS. 13

14. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Fitting restricted model:
for all i
Hence TSS = RSS for the restricted regression. 14

15. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Fitting restricted model:
for all i
Obviously, if there are no explanatory variables, none of the variation in Y is explained by the model and so RSS is equal to TSS. 15

16. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Here is the F statistic for the comparison of the unrestricted model with all of the X variables and the restricted model with only the intercept. 16

17. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
We have just seen that RSS from the restricted version is equal to TSS. 17

18. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Now we refer to the decomposition of TSS in the case of the unrestricted regression. This is similar to the decomposition for the restricted model, with the difference that RSSU will in be smaller than RSSR and ESSU will be positive, instead of zero. 18

19. MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS
RSSU
Unrestricted model:
RSSR
Restricted model:
Restrictions:
Hypotheses:
at least one of the slope coefficients
Given the decomposition for the unrestricted version, we can rewrite the F statistic as shown. This is the expression for the F statistic for the equation as a whole that was given in Section 3.5. 19

20. Copyright Christopher Dougherty 2012.
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