hws04 - Economics 140A Answers to Exercise 4 1. (Fall 1998...

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Economics 140A Answers to Exercise 4 1. (Fall 1998 Final Examination) Because the education regressor is an indicator variable for college graduation, E ( Y i j E i = 0) = & 0 and E ( Y i j E i = 1) = & 0 + 1 : The intercept is the average income of individuals who do not graduate from college. The coe¢ cient 1 captures the income gain (presumably) resulting from college graduation. That is, 1 is the di/erence in average incomes between individuals who have graduated from college and those who have not. b) The group averages for college graduates and non-college graduates con- tain the same information as the coe¢ cients from the model discussed in a). The two group averages and the two coe¢ cients line up exactly, as E Y G = & 0 + 1 and E Y NG = & 0 : We could estimate the regression coe¢ cients directly from the group averages, without recourse to OLS estimators. We simply equate Y NG = A 0 and Y G Y NG = A 1 : c) The second model explains individual income as a function of both edu- cation and sex. Both regressors are indicator variables, so the expected levels of income are E ( Y i j E i = 1 ;S i = 1) = ± 0 + ± 1 + ± 2 (male college graduates) E ( Y i j E i = 1 ;S i = 0) = ± 0 + ± 1 (female college graduates) E ( Y i j E i = 0 ;S i = 1) = ± 0 + ± 2 (male non-college graduates) E ( Y i j E i = 0 ;S i = 0) = ± 0 (female non-college graduates). The coe¢ cient ± 0 captures the average income of female non-college graduates. The coe¢ cient ± 1 captures the average, over males and females, of the change in income resulting from a college degree. As such it does not necessarily capture the gain for either females or males. The coe¢ cient ± 2 captures the average, over college graduates and non-graduates, of the di/erence between male and female incomes. As such the coe¢ cient does not necessarily capture the male-female di/erence for either college graduates or non-college graduates. d) We have four possible group averages, but only three coe¢ cients, so the two measures do not contain the same information. In fact, the four group averages contain more information than do the three coe¢ cients, because two of the coe¢ cients re±ected average changes across two groups. E Y F;NG = ± 0 ; 1
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so that our estimator of the intercept is in fact the average of one cell and B 0 = Y F;NG . Suppose that there are an equal number of individuals in each cell (that is half of the college graduates are male and half of the non-college graduates are male). Then 1 = 1 2 E Y F;G Y F;NG ² + ± Y M;G Y M;NG ²³ : Similarly 2 = 1 2 E Y M;G Y F;G ² + ± Y M;NG Y F;NG ²³ : Clearly one can learn the OLS coe¢ cient estimates from the cell average esti- mates, but one cannot infer the cell average estimates from the OLS coe¢ cient estimates. If there is no male-female di/erence in incomes for college gradu-
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hws04 - Economics 140A Answers to Exercise 4 1. (Fall 1998...

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