For this question I used data from a survey conducted by the department of education in 1980. A description of the data can be found here. We want to test the effect of an extra year of education on wages. We estimate a regression:

wagei= 0 + 1educi + i

where wagei is the individual hourly wage, and educi is the individual years of education. This regression yields the following parameter estimates:

wagei=9.253 + 0.01791 educi

(0.1518) (0.0109)

where standard errors for each parameter are given below in parenthesis.

0a. Conduct the appropriate statistical test to determine whether or not education has a statistically significant impact on wages. Write out all steps and be clear with your conclusion.

0b. Now write out the formula for the standard error of 1. Is the standard error increasing or decreasing in the sample size n? Next write out the formula for the test statistic you calculated in 0a.. Is the test statistic increasing or decreasing in the sample size? Lastly, use your answers to this question to determine whether or not probability that you reject the null hypothesis is increasing or decreasing in the size. Hint: You don't have to write out this probability explicitly. Just explain the intuition behind what the test-statistic is telling you and how this helps you answer the question.

Oc. For the above regression, we have n = 4739 observations and k = 2 parameters. This means that we have 4737 degrees of freedom. For a = 5 significance level, this gives us a t-critical value of

t0.025,4737 = 1.96. Use this information combined with the information above to construct a 95% confidence interval for the parameter 1. Write out the steps you took to get to the lower and upper bounds. Provide a careful interpretation of what this confidence interval tells you.

0d. Now suppose we think we omitted an important variable: gender. State the two conditions this variable must meet (in the context of this example) for it to cause omitted variables bias. Would increasing the sample size (working with "big data") alleviate the issues caused by omitting gender from this regression?

Luckily, our data contains information on whether or not individuals in our data are male or female. We now include two indicators in our regression. One for male, and one for female -- and drop the intercept. We have the following coefficient estimates and standard errors.

wagei = 9.29589 malei + 9.22308 femalei + 0.01771 educi

(0.15351) (0.15264) (0.0109)

0e. You don't need to calculate the next test (I have not given you enough information to do so), but write

out how you would use this model to statistically test the null hypothesis that wages for males and females2 / 8 are different from each other. Be very precise. Write out each step.