2: Bivariate Regression in Stata ECO 321 DUE: Tuesday September 20 in class Instructions: You need to write a Stata do-file to answer questions (2) and (3). Your do-file must produce a log file that contains your Stata output. You must submit your log file along with your answers to the assignment. Assignments that do not contain log files will be marked down. You may work in groups (up to 4 people), but each student must submit his/her own assignment. If you work in a group and do not submit your own assignment, you will receive a zero for the assignment. Remember that late assignments will not be accepted. 1. Consider the OLS estimates of a bivariate regression: Yi = 0 + 1 Xi (SE0 ) (SE1 ) where 0 and 1 are the estimates of the coefficients and SE0 and SE1 are the standard errors of the estimates. a) What assumptions do we need for 1 and 0 to be unbiased? b) We want to to test the statistical significance of 1 . Write down the hypothesis test and the t-statistic. c) Why is the t-statistic for the significance test of k1 (i.e. H0 : k1 = 0, H1 : k1 6= 0) the same as the t-statistic for the significance test of 1 , where k is a positive constant? d) Write down the expression for a 95% confidence interval for 1 . e) Show that the 95% confidence interval for k1 can be found by multiplying the bounds of the 95% confidence interval of 1 by k (i.e. if CIL and CIH are the lower and upper bounds on the 95% confidence interval for 1 , then k CIL and k CIH are the bounds on the 95% confidence interval for k1 ). Again, assume that k is a positive constant. (Hint: You need to do more than 95% say CIk = k CI95% . You need to show why it is true.) 1 1 1 2. Smoking and birth outcomes. The data set bwght.dta contains two variables: the birth weight (in ounces) of a newborn and the number of cigarettes the mother smoked per day during pregnancy. a) Load the data set in Stata (via the use command). Construct a dummy variable (using the gen command) named anycig that equals one if the mother smoked at least one cigarette per day and zero otherwise. Use the tab command (or whatever command you like) to determine what share (percent) of moms smoked during pregnancy. b) Use the ttest command to test the null hypothesis that the average birth weight of babies born to smokers vs. non-smokers is the same. Report the means, t-statistic, and p-value. Can you reject the hypothesis at the 5% level? How about the 1% level? c) Rather than looking at the effect of smoking versus not smoking, we want to refine our analysis to look at the effect of how many cigarettes a pregnant woman smokes. Use the reg command to estimate by OLS the regression bwghti = 0 + 1 cigsi + ui . Report your regression results. What is the marginal effect on birthweight of smoking an additional cigarette? Is the effect statistically significanti.e. can you reject H0 : 1 = 0 and what is the t-statistic? d) Using your regression results in part (c), compute the expected birth weight of a child whose mother does not smoke, and of a child whose mother smokes a pack a day (assume 20 cigarettes per pack). Is the difference between these values significant? e) What is homoskedasticity, and is it likely to be a valid assumption in this example? f) Estimate the regression from part (c) without assuming homoskedasticity (i.e. use the reg command, but specify the option robust to account for heteroskedasticity in the error terms). Compare your standard errors to the regression with that assumes homoskedasticity of the standard errors. What is the difference? Do you suspect heteroskedasticity is a problem in this data? Is there a benefit or cost to using robust standard errors? g) Finally, run a regression of bwght on the dummy variable anycigs. How do you interpret 1 in this regression? How do you interpret 0 ? How do your results compare to what you found in part b? 3. Female labor supply. A question of interest to labor economists is what determines female labor supply, which increased throughout the 1970s and 1980s. CPS91.dta is a data set containing hours worked per week and husband weekly earnings for a sample of women. 2 a) Why are outliersextreme observationsa problem for OLS? b) Use the reg command to run a regression of hours worked on husband's earnings. Use robust standard errors and report your results. According to your regression, how many hours per week does a woman whose husband has zero earnings work? What happens to female labor supply when the husband's earnings increase by $100? c) Construct a 95% confidence interval for the effect of a $100 increase in husband earnings. What do you conclude about the effect of husband earnings on female labor supply? d) Use the command twoway scatter to plot the relationship between hours and husearns (it is helpful to specify the m(x) option to change the markers from big dots to small xs). Do you see an outlier in this data set? (Hint: there is one.) Do you think the outlying observation is a coding error or a true outlier (i.e. is there really an outlying couple corresponding to that observation)? Don't forget to attach your scatter plot to your assignment. e) Drop the outlier (and only the outlier) using the drop command and run the regression from part (b) again. What is the effect of a $100 increase in husband earnings? Is this effect significant? Does your conclusion about the relationship between husband earnings and female labor supply change? 3