1. 6.4*The table gives the results of multiple and simple regressions of 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 869 households in the 1995 Consumer Expenditure Survey. The correlation coefficient for LGEXP and LGSIZE was 0.45. Explain the variations in the regression coefficients. 1 EXERCISE 6.4

2. 2 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

3. 3 The first column of the table gives the result of a multiple regression on LGEXP, the logarithm of total annual household expenditure, and LGSIZE, the logarithm of the number of people in the household (standard errors in parentheses). EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

4. 4 We will assume that this is the correct specification. The estimated elasticities are both significantly different from 0 at a very high significance level. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

5. 5 The second column gives the result of regressing LGFDHO on LGEXP only. We see that the coefficient of LGEXP is much larger than before. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

6. 6 The reason is that the coefficient is subject to omitted variable bias, and we can demonstrate that the bias is likely to be positive. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

7. 7 As a matter of common sense,  2 is certainly positive. The fact that the estimated coefficient in the multiple regression is positive and highly significant provides powerful supporting evidence. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

8. 8 The correlation between LGEXP and LGSIZE is also positive, as might be expected, and hence their covariance must be positive. (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

9. 9 The variance of LGEXP is positive. Hence all the components of the bias term are positive. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

10. 10 For similar reasons, the coefficient of LGSIZE is biased upwards when LGEXP is omitted. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

11. 11  2 is certainly positive. As with  3, the fact that the estimated coefficient in the multiple regression is positive and highly significant provides powerful supporting evidence. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

12. 12 We have already seen that Cov(LGEXP, LGSIZE) is positive, and Var(LGSIZE) is automatically positive, so the bias is positive. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

13. 13 Finally, note the values of R 2. In the simple regression on LGEXP only, R 2 is 0.31. However this exaggerates the explanatory power of LGEXP because it is acting partly as a proxy for the missing LGSIZE. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

14. 14 Similarly, in the simple regression on LGSIZE only, R 2 is 0.42. This exaggerates the explanatory power of LGSIZE because it is acting partly as a proxy for the missing LGEXP. EXERCISE 6.4 (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45

15. (1) (2) (3) LGEXP0.290.48–(0.02) LGSIZE0.49–0.63 (0.03)(0.02) constant4.723.177.50 (0.22)(0.24)(0.02) R 2 0.520.310.42 r LGEXP,LGSIZE = 0.45 15 The simple regressions might seem to suggest that LGEXP and LGSIZE jointly account for 0.31 + 0.42 = 0.73 of the variance of LGFDHO. However the multiple regression reveals that they account for only 0.52 of the variance. EXERCISE 6.4

16. Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 30.08.06