1. 1 REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION Linear restrictions can also be tested using a t test. This involves the reparameterization of a regression model and we will start with this.

2. 2 Suppose that you have a regression model shown and that the regression model assumptions are valid. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

3. 3 We fit the model in the usual way. This enables us to obtain estimates of the parameters and their standard errors. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

4. 4 Suppose, however, that we are interested in a linear combination, , of the parameters, where the j are weights. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

5. 5 Obviously, the can obtain a point estimate of  as the corresponding linear combination of the estimates of the individual parameters. If the regression model assumptions are valid, it can easily be shown that it is unbiased and that it is the most efficient estimator of . REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

6. 6 However you do not have information on its standard error and hence you are not able to construct confidence intervals for  or to perform t tests. There are three ways that you might use to obtain such information. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

7. 7 (1) Some regression applications have a special command that produces it. For example, Stata has the lincom command. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

8. 8 (2) Given the appropriate command, most regression applications will produce the variance-covariance matrix for the estimates of the parameters. This is the complete list of the estimates of their variances and covariances, for convenience arranged in matrix form. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

9. 9 The standard errors in the ordinary regression output are the square roots of the variances. The estimate of the variance of the estimate of  is given by the expression shown, where s subscript b p b j is the estimate of the covariance between b p and b j. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

10. 10 This method is cumbersome and avoided when possible. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

11. 11 (3) The third method is to reparameterize the model, manipulating it so that  and its standard error are estimated directly as part of the regression output. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

12. 12 To do this, we rewrite the expression for  so that one of the  parameters is expressed in terms of  and the other  parameters. This will be illustrated with two simple examples, the general case being left as an additional exercise (see the study guide on the website). REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

13. 13 Suppose the regression model is as shown and suppose we are interested in the sum of  2 and  3. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

14. 14 We rewrite the relationship as shown, expressing one of the  parameters in terms of  and the other  parameter. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

15. 15 Substituting in the original model, we reparameterize it as shown. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

16. 16 Thus if we define a new variable Z = X 2 – X 3, and regress Y on Z and X 3, the coefficient of Z will be an estimate of  2 and that of X 3 will be an estimate of . REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

17. 17 The estimate of  will be exactly the same as that obtained by summing the estimates of  2 and  3 in the original model. The difference is that we obtain its standard error directly from the regression results. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

18. 18 The estimate of  2 and its standard error will be the same as those obtained with the original model. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

19. 19 Suppose instead that we were interested in the difference between  2 and  3. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

20. 20 We rewrite this as shown and substitute in the original model. Thus if we define a new variable Z = X 2 + X 3 and regress Y on Z and X 3, the coefficient of Z will be an estimate of  2 and that of X 3 will be an estimate of . REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

21. 21 The estimate of  will be exactly the same as that obtained by taking the difference of the estimates of  2 and  3 in the original model and we obtain its standard error directly from the regression results. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

22. 22 The estimate of  2 and its standard error will be the same as those obtained with the original model. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

23. 23 An obvious application of reparameterization is its use in the testing of linear restrictions. Suppose that your hypothetical restriction is as shown, where  is a scalar. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

24. 24 Define  as shown and reparameterize.  will become the coefficient of one of the variables in the model, and a t test of H 0 :  = 0 is effectively a t test of the restriction. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

25. 25 As an illustration. we will use the example discussed in the previous sequence. The model relates years of schooling, S, to the cognitive ability score ASVABC and years of schooling of the mother and the father, SM and SF. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

26. 26 It was hypothesized that mother’s education and father’s education are equally important for educational attainment, implying the restriction  4 =  3. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

27. 27 The restriction may be rewritten as  4 –  3 = 0. Define  as shown. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

28. 28 Hence express one of the  parameters in terms of  and the other  parameter, and substitute in the original model. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

29. 29 SP is defined as the sum of SM and SF. A t test on the coefficient of SF is a test of the restriction. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

30. . reg S ASVABC SP 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 SP |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.0584401.0617051 0.95 0.344 -.0627734.1796536 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------ 30 Here is the corresponding regression. We see that the coefficient of SF is not significantly different from zero, indicating that the restriction is not rejected. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION t = 0.95

31. 31 It can be shown mathematically that the F test of a restriction and the corresponding t test are equivalent. The F statistic is the square of the t statistic and the critical value of F is the square of the critical value of t. (We saw in the previous sequence that the F statistic is 0.90.) REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION. reg S ASVABC SP 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 SP |.0492424.0390901 1.26 0.208 -.027546.1260309 SF |.0584401.0617051 0.95 0.344 -.0627734.1796536 _cons | 5.370631.4882155 11.00 0.000 4.41158 6.329681 ------------------------------------------------------------------------------ t = 0.95 F = 0.90 = t 2

32. 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 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 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. 2012.11.09