Description

(a) Write the expression for Ehβˆ1i in the estimated regression of y on x.
(b) What is the interpretation for βˆ1?
(c) Suppose that, instead of using x in the regression, we use z, defined as . What is the expression for now? How do we interpret its magnitude?
h i h ˆ i.
(d) Show that E βe1 = σ E β1
2. Omitted variable bias: Suppose you are given data on the variables y, x1, and x2. Suppose that y =
, where Cov( ) = Cov( ) = 0 and Var( .
(a) Find Cov(y,). (Hint: Plug in the equation for y and use the rules for covariances, in particular the fact that Cov(U + W,Z) = Cov(U,Z) + Cov(W,Z) for any random variables U,W,Z.)
(b) Suppose we attempt to estimate . Write an expression for ˜ in terms of and x2.
(c) What is Cov( )? Under what condition will x1 not be correlated with the error term ˜ in this case?
3. Suppose you wish to estimate a linear regression of Y on X1 and X2. Write a model describing each of the following scenarios, and fill in actual numbers which match the patterns described.
(a) Write a model where Y increases at an increasing rate with X1 and a constant rate with X2.
(b) Write a model where Y increases with X2, but the increase is larger at smaller values of X1.
(c) Write a model where Y increases at a constant rate with X1 and X2, but X2 only enters the model if X1 is greater than 5.
(d) Write a model where the coefficients on both X1 and X2 can be interpreted as elasticities.
1
(a) from the Data folder on Blackboard.
(b) Load the dataset berkeley.dta. How many departments are represented in the dataset? You can use the tabulate or distinct commands to determine this.
(c) Generate an indicator variable which equals 1 if the applicant is female and zero otherwise.
(d) What share of applicants were women?
(e) Generate an indicator variable which equals 1 if the applicant was admitted.
(f) Find out what share of women are admitted by computing the sample mean.
(g) Regress the admission status indicator on the female indicator and interpret the estimated coefficient; are women accepted at a lower or higher rate than men?
(h) Generate indicator variables for each department and re-estimate the above regression separately for each individual department. How does the admissions rate for women vs. men compare by department? (i) Estimate the overall acceptance rate for each department.
(j) Estimate the share of applicants that are female for each department. How can this help explain the previous results?
5. In this exercise, you will create and analyze data on U.S. cities pertaining to education, income, and income inequality. We will rank cities from most to least educated, examine the correlation between city size and income, and see how these relate to inequality.
(b) Now, decide on the level of geography you would like to use. Since we’re interested in “cities,” tractor block-level data are unnecessary. Choose either Census place-level or CBSA-level data.
(c) Select the variable best representing the education data you need – it’s generally sensible to focus on those above a certain age (say, 25).
(d) Select the measure of income you prefer – median household or per capita would probably be best. Also, select the Gini index of income inequality. Download the data in fixed-width format.
(e) The income inequality measure will be in a separate dataset from the rest of your data. You can use the merge command to join these two data sets.
(f) Convert the education data into a usable variable for measuring “most educated.” You must decide on how you will measure “most educated” – fraction with a BA or higher? Graduate degree? Something else?
(g) You can additionally decide if you want to restrict your analysis to larger cities. Why might you want (or not want) to impose such a restriction?
(h) Once you’ve generated a measure of education level for the city, use the sort command on this variable to put the list in order. Comment on the pattern you observe; what types of cities seem to be the most/least educated? (You do NOT need to print out the whole list for me.)
(i) Now, you’ll examine the relationship between city size and your measure of residents’ incomes. Regress income on city population. What relationship do you observe?
(j) Repeat the previous regression, but replace income with the Gini index. Do larger cities exhibit more or less inequality?
(k) Now, regress the Gini index on income. Are richer cities associated with more or less inequality?
(l) Include a table in your writeup that reports the results of the previous three regressions (using the estout package).