1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: exercise 6.13 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/132/http://learningresources.lse.ac.uk/132/ 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. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

2. 6.13*The first regression shows the result of regressing LGFDHO, the logarithm of annual household expenditure on food eaten at home, on LGEXP, the logarithm of total annual household expenditure, and LGSIZE, the logarithm of the number of persons in the household, using a sample of 868 households in the 1995 Consumer Expenditure Survey. In the second regression, LGFDHOPC, the logarithm of food expenditure per capita (FDHO/SIZE), is regressed on LGEXPPC, the logarithm of total expenditure per capita (EXP/SIZE). In the third regression LGFDHOPC is regressed on LGEXPPC and LGSIZE. 1 EXERCISE 6.13

3. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 2 EXERCISE 6.13

4. . reg LGFDHOPC LGEXPPC Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 313.04 Model | 51.4364364 1 51.4364364 Prob > F = 0.0000 Residual | 142.293973 866.164311747 R-squared = 0.2655 ---------+------------------------------ Adj R-squared = 0.2647 Total | 193.73041 867.223449146 Root MSE =.40535 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.376283.0212674 17.693 0.000.3345414.4180246 _cons | 3.700667.1978925 18.700 0.000 3.312262 4.089072 ------------------------------------------------------------------------------ 3 EXERCISE 6.13

5. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 4 1.Explain why the second model is a restricted version of the first, stating the restriction. 2.Perform an F test of the restriction. 3.Perform a t test of the restriction. 4.Summarize your conclusions from the analysis of the regression results. EXERCISE 6.13

6. 5 The first regression is a straightforward logarithmic regression of expenditure on food consumed at home on total household expenditure and size of household. EXERCISE 6.13

7. 6 The second regression is a simple regression of LGFDHOPC, defined as log FDHO/SIZE, on LGEXPPC, defined as log EXP/SIZE. EXERCISE 6.13

8. 7 The logarithmic ratios have been split.

9. 8 EXERCISE 6.13 The LGSIZE terms have been brought together.

10. 9 EXERCISE 6.13 Comparing this equation with that for the first regression, we see that the second specification is a restricted version of the first with the restriction  3 = 1 –  2.

11. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 10 Before performing a test of the restriction, we should check whether the estimates of the elasticities in the unrestricted version appear to satisfy it. EXERCISE 6.13

12. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP |.2866813.0226824 12.639 0.000.2421622.3312003 LGSIZE |.4854698.0255476 19.003 0.000.4353272.5356124 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 11 b 3 is 0.49. 1 – b 2 is 0.71. The discrepancy is rather large, compared with the standard errors of the estimates. We should expect the restriction to be rejected. EXERCISE 6.13

13. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388. reg LGFDHOPC LGEXPPC Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 313.04 Model | 51.4364364 1 51.4364364 Prob > F = 0.0000 Residual | 142.293973 866.164311747 R-squared = 0.2655 ---------+------------------------------ Adj R-squared = 0.2647 Total | 193.73041 867.223449146 Root MSE =.40535 12 We see that the residual sum of squares increases from 130.2 to 142.3 when we impose the restriction. EXERCISE 6.13

14. . reg LGFDHO LGEXP LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 460.92 Model | 138.776549 2 69.3882747 Prob > F = 0.0000 Residual | 130.219231 865.150542464 R-squared = 0.5159 ---------+------------------------------ Adj R-squared = 0.5148 Total | 268.995781 867.310260416 Root MSE =.388. reg LGFDHOPC LGEXPPC Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 313.04 Model | 51.4364364 1 51.4364364 Prob > F = 0.0000 Residual | 142.293973 866.164311747 R-squared = 0.2655 ---------+------------------------------ Adj R-squared = 0.2647 Total | 193.73041 867.223449146 Root MSE =.40535 13 The F statistic is far above the critical value of F(1,750) at the 0.1% level. The critical value of F(1,865) must be lower than that for F(1,750). Therefore we reject the null hypothesis and conclude that the restriction is invalid. EXERCISE 6.13

15. 14 We will also use the t test approach to test the restriction. First we rewrite the restriction so that the right side of the definition is zero. EXERCISE 6.13

16. 15 We define  to be equal to the left side. The restriction is now  = 0. EXERCISE 6.13

17. 16 EXERCISE 6.13 We express one of the  coefficients in terms of  and the other  coefficient.

18. 17 EXERCISE 6.13 We substitute for this  in the regression model.

19. 18 EXERCISE 6.13 We bring the  2 components together and take the (+1)log SIZE to the left side of the equation.

20. 19 EXERCISE 6.13 This allows us to rewrite the model with the dependent variable the logarithm of expenditure on food per capita and the explanatory variables the logarithms of total household expenditure per capita and household size.

21. 20 EXERCISE 6.13 Having reparameterized the model in this way, we can test the restriction with a simple t test on the coefficient of LGSIZE.

22. 21 EXERCISE 6.13 If the coefficient of LGSIZE is significantly different from zero, we need the term and should stay with the unrestricted model. If it is not, the term could be dropped, giving us the restricted model as the appropriate specification.

23. 22 EXERCISE 6.13 Note that the null hypothesis for the t test is that the restriction is valid. This ties in with the reasoning in the previous slide. If the restriction is valid, we do not need the LGSIZE term and the restricted version is the appropriate specification.

24. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 23 Here is the corresponding regression result. We find that the coefficient has a very high (negative) t statistic. The null hypothesis is rejected and again we conclude that the restriction is invalid. EXERCISE 6.13

25. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 24 The F test and the t test approaches are of course equivalent. The F statistic, 80.4, is the square of the t statistic and the critical value of F is the square of the critical value of t. EXERCISE 6.13

26. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 25 Should we have anticipated this outcome? The restricted version effectively controls for the size of the household. Why should the size variable have a separate effect? EXERCISE 6.13

27. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 26 One possibility is that there are economies of scale in feeding a larger household, or perhaps less wastage. EXERCISE 6.13

28. . reg LGFDHOPC LGEXPPC LGSIZE Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 2, 865) = 210.94 Model | 63.5111811 2 31.7555905 Prob > F = 0.0000 Residual | 130.219229 865.150542461 R-squared = 0.3278 ---------+------------------------------ Adj R-squared = 0.3263 Total | 193.73041 867.223449146 Root MSE =.388 ------------------------------------------------------------------------------ LGFDHOPC | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXPPC |.2866813.0226824 12.639 0.000.2421622.3312004 LGSIZE | -.2278489.0254412 -8.956 0.000 -.2777826 -.1779152 _cons | 4.720269.2209996 21.359 0.000 4.286511 5.154027 ------------------------------------------------------------------------------ 27 Another is that there may be a compositional effect, large households tending to have more children, who eat less. Perhaps we should be controlling by some notion of the number of equivalent adults, rather than the unadjusted number of people in the household. EXERCISE 6.13

29. Copyright Christopher Dougherty 2000–2007. This slideshow may be freely copied for personal use. 07.12.07