Suppose that a researcher, using wage data on 250 randomly selected male workers and 280 female workers, estimates the OLS regression
Wage = 12.52+2.12× Male,R^2 = 0.06,SER = 4.2, (.23) (0.36)
where Wage is measured in dollars per hour and Male is a binary variable ,that is equal to 1 if the person is a male and 0 if the person is a female. Define the wage gender gap as the difference in mean earnings between men and women.
a. What is the estimated gender gap?
b. Is the estimated gender gap significantly different from zero? (Compute the p-value for testing the null hypothesis that there is no gender gap.)
c. Construct a 95% confidence interval for the gender gap.
d. In the sample, what is the mean wage of women? Of men?
e. Another researcher uses these same data but regresses Wages on Female, a variable that is equal to 1 if the person is female and 0 if the person a male. What are the regression estimates
calculated from this regression?
WAGE =_______________ +_______________ ×Female,R^2 =_______________,SER=_______________.
(2) In the1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to "regular" and "small" classes, and given standardized tests at the end of the year. (Regular classes contained approximately 24 students, and small classes contained approximately 15 students.) Suppose that, in the population, the standardized tests have a mean score of 925 points and a standard deviation of 75 points. Let Small Class denote a binary variable equal to 1 if the student is assigned to a small class and equal to 0 otherwise. A regression of TestScore on Small Class yields
TestScore = 918.0+13.9× SmallClass,R^2 = 0.01,SER = 74.6. (1.6) (2.5)
a. Do small classes improve test scores? By how much? Is the effect large? Explain.
b. Is the estimated effect of class size on test scores statistically significant? Carry out a test at the 5% level.
c. Construct a 99% confidence interval for the effect of SmallClass on test score.
(3) Use the dataset provided on eLearning called CollegeDistancewest.dta This dataset contains a random sample of high school seniors. The variable "ed" is the years of completed schooling. The variable "dist" is the distance to the nearest college measured in tens of miles
(so dist=20 means the high school student lived 200 miles from the nearest college). The variable "female" is a variable that equals 1 if the student is female and equals 0 if the student is male.
a. After reading the data into Stata (using the "use" command), summarize the two key variables using the command "summarize ed dist female". In words, describe the basic summary statistics of these three variables.
b. Run a regression of years of completed education (ED) on distance to the nearest college (Dist) and carry out the following exercises. (Use the command regress ed dist) Is the estimated regression slope coefficient statistically significant? That is, can you reject the
null hypothesis HO: (Beta1)= 0 versus a two-sided alternative at the 10%, 5% or 1% significance level? What is the p-value associated with coefficient's t-statistic?
c. Construct a 95% confidence interval for the slope coefficient. Show the calculations behind the results that Stata gives you.
d. Run the regression using data only on females and repeat (b). (Use the command regressed dist if female==1)
e. Run the regression using data only on males and repeat (b). (You should be able to figure out the command...)
f. Is the effect of distance on completed years of education different for men than for women? (Hint: See Exercise 5.15 on page 172 in the textbook)
(4) Data were collected from a random sample of 220 home sales from a community in 2003. Let Price denote the selling price (in $1000), BDR denote the number of bedrooms, Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size (in square feet), Age denote the age of the house (in years), and Poor denote a binary variable that is equal to 1 if the condition of the house is reported as "poor." An estimated regression yields
Price = 119.2+0.485BDR +23.4Bath +0.156Hsize +0.002Lsize + 0.090Age ?48.8Poor,R^2 = 0.72,SER = 41.5.
a. Suppose that a homeowner converts part of an existing family room in her house into a new bathroom. What is the expected increase in the value of the house?
b. Suppose that a homeowner adds a new bathroom to her house, which increases the size of the house by 100 square feet. What is the expected increase in the value of the house?
c. What is the loss in value if a homeowner lets his house run down so that its condition becomes "poor"?
(5) A researcher plans to study the causal effect of police on crime using data from a random sample of U.S. counties. He plans to regress the county's crime rate on the (per capita) size of the county's police force.
a. Explain why this regression is likely to suffer from omitted variable bias. Which variables would you add to the regression to control for important omitted variables?
b. Use your answer to (a) and the expression for omitted variable bias given in Equation (6.1) of the textbook to determine whether the regression will likely over- or underestimate the effect of police on the crime rate. (Do you think that (Beta1hat) 1 > (Beta1) or (Beta1hat)
1 < (Beta1) ?