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Use the data in CPS78 $\mathrm{CPS} 78_{-} 85$ for this exercise.(i) How do you interpret the coefficient on y85 in equation $(13.2) ?$ Does it have an interestinginterpretation? (Be careful here; you must account for the interaction terms y 85$\cdot$educ and $y 85$ female.)(ii) Holding other factors fixed, what is the estimated percent increase in nominal wage for a male with 12 years of education? Propose a regression to obtain a confidence interval for this estimate. [Hint: To get the confidence interval, replace y 85$\cdot$educ with $y 85 \cdot(e d u c-12) ;$ refer to Example $6.3 . ]$(iii) Reestimate equation $(13.2)$ but let all wages be measured in 1978 dollars. In particular, define the real wage as rwage $=$ wage for 1978 and as rwage $=$ wagel 1.65 for $1985 .$ Now, uselog(rwage) in place of log(wage) in estimating $(13.2) .$ Which coefficients differ from those inequation $(13.2) ?$(iv) Explain why the $R$ -squared from your regression in part (iii) is not the same as in equation$(13.2) .$ Hint: The residuals, and therefore the sum of squared residuals, from the two regressions are identical.)(v) Describe how union participation changed from 1978 to $1985 .$(vi) Starting with equation $(13.2),$ test whether the union wage differential changed over time. Thisshould be a simple $t$ test.)(vii) Do your findings in parts (v) and (vi) conflict? Explain.

(i) not useful (ii) see video (iii) coefficient of y85 differs (iv) $R^2=.356$ (v) fall in union membership (vi) no change in premium (vii) no

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Chapter 13

Pooling Cross Sections Across Time: Simple Panel Data Methods

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Pierce N.

November 15, 2021

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part one. The coefficient on the year dummy of 1985 is roughly the proportion it changed in wage for a male, and the person has zero years of education. This is not an interesting result because the US working population with any education is without any education is a small girl. Such people are in no way typical part two. What we want to estimate is fate or not equals Delta, not plus 12 Delta one. We could write. Delta not equals fate or not minus 12 Delta one. And we pluck that into equation 13.1 and rearrange We will get log of wage is regressed on beta. Not plus they dare not. Why 85 plus beta one education plus Delta One. Why 85 times education minus 12 and the rest is the same as before. The coefficient we need here is the estimate of fate or not. Seita, not that is 0.339 with a standard error of 0.34 This estimate implies the nominal increase in wage is about 33.9% and the 95% confidence interval is yeah, 33.9 plus and minus 1.96 the critical value times the standard error of 3.4, and we would get 27.2% as the lower bound and 40.6% as the upper bound for three. We find that only the coefficient on the year dummy of 1985 differs from Equation 13.2. The new coefficient is minus 0.383 with a standard error of 0.1 to 4. This shows that real wages have fallen over the seven year period, although less so for the more educated part four. The art square when lock of really, which is the dependent variable, is 40.356 as compared with 0.4 to 6 when the log of which is the dependent variable. If the sum of square residuals from the regressions are the same, but the R Square are not, then the total sum of squares must be different. This is the case for this part part 5 1978. About 30.6% of workers in the sample belong to a union. The number in 1985 is only 18%. Therefore, over the seven year period, there was a notable fall in union membership. Heart Sixth, When we add the interaction term of Ear 85 Union to the Equation, it's coefficients and standard Errol are about minus 0.4 and 0.6 one, respectively. This is very small, and the T statistic is almost zero. We can conclude that there has been no change in the union wage premium over time. March 7 Part four, part five and six do not contradict each other. The two parts implied that why the economic return to union membership has not changed, the fraction of people reaping those benefits has fallen just

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