Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: multiple restrictions and zero restrictions Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

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.

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

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

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

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

MULTIPLE 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,  5 in the model above. Taken in isolation, the appropriate test is of course the t test. 6

MULTIPLE 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

MULTIPLE RESTRICTIONS 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. or or bothand 8

MULTIPLE RESTRICTIONS or or bothand 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

MULTIPLE RESTRICTIONS Unrestricted model: Restricted model: Restrictions: RSS U RSS R 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

MULTIPLE RESTRICTIONS Unrestricted model: Restricted model: Restrictions: RSS U RSS R Hypotheses: Fitting the restricted model: at least one of the slope coefficients When we fit the restricted model, we find that the OLS estimator of  1 is the sample mean of Y (see Exercise 1.3). 11

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

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

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

MULTIPLE RESTRICTIONS Unrestricted model: Restricted model: Restrictions: RSS U RSS R Hypotheses: Fitting the restricted model: for all i at least one of the slope coefficients 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

MULTIPLE RESTRICTIONS 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. Unrestricted model: Restricted model: Restrictions: RSS U RSS R Hypotheses: at least one of the slope coefficients 16

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

MULTIPLE RESTRICTIONS 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 RSS U will in be smaller than RSS R and ESS U will be positive, instead of zero. Unrestricted model: Restricted model: Restrictions: RSS U RSS R Hypotheses: at least one of the slope coefficients 18

MULTIPLE RESTRICTIONS 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. Unrestricted model: Restricted model: Restrictions: RSS U RSS R Hypotheses: at least one of the slope coefficients 19

Copyright Christopher Dougherty 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 6.5 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 Individuals studying econometrics on their own and 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics