I. One debate in the economics of education literature centers on whether or not attending a charter school improves educational outcomes. This problem examines the causal effect of charter school attendance on 8th grade standardized math test scores.

i. Let the Bernoulli random variable CSi represent the charter school status for an individual 8th grader, where CSi = 1 if an individual is enrolled in a charter school and CSi = 0 if an individual is not enrolled in a charter school. Suppose that 10% of individuals in this age group are enrolled in s charter school. What is the expected value and variance of CSi?

ii. Suppose we use the sample average, CS for a random sample of 1,000 students to estimate the expected value of CSi. What is the expected value and variance of the estimator, CS?

iii. Let the random variable MTi represent the math test score for an individual 8th grader. Ex plain mathematically and in words, what is the causal effect of charter school enrollment on the math test score for an individual? What is the average causal effect for a random sample of size n? What are the two counterfactuals in this case?

iv. Suppose we compare the average math test score for individuals who are enrolled in charter schools to that of individuals who are not enrolled in charter schools. We find that on average, 8th graders enrolled in charter schools score 7 percentage points higher than their non-charter school peers. Explain mathematically and in words, why might this average difference be a biased estimate for the causal effect of interest? Your answer should include a mathematical expression for the comparison of averages and the potential selection bias in this case. Provide some economic reasoning for whether you believe the bias would likely be positive or negative.

v. Each year Portland Public Schools holds a lottery for it's charter school enrollments. Suppose we compare math test scores for individuals who were selected by the lottery and attended charter schools to that of individuals who applied but were not selected by the lottery and did not attend charter schools. Explain mathematically and in words, why might this comparison more accurately estimate the average causal effect of interest? Your answer should address your mathematical expression of the selection bias from part (iv).

II. The Portland metropolitan area includes 354 Census tracts, which are roughly the size of neighborhoods. Suppose that you are working for Portland metro on a project to estimate the effect of access to bicycle lanes on bicycle commuting rates across the city. Suppose the following model describes this relationship at the census tract level and that MLRM 1 - 4 are satised. (100 points) ln(bike share) = 50 + Bnane share) + zdistance + u The variable bike share is the share of census tract residents who bike commute to downtown, lane share is the share of census tract roads that have bicycle lanes, and distance is the census tract distance (in miles) from city hall, in the center of downtown. i. Your supervisor doesn't want to measure each tract's distance to downtown, and recommends that you instead simply regress ln(bike share) on marine share): ln(bike share) = 60 + Bllnane share) + v, v = Bgdistance + u, where E[u|a:1,:r2] = 0 and Var[u|a:1,m2] = 03. If the share of roads with bicycle lanes is higher for neighborhoods close to downtown than for parts of the city further away, show your supervisor, mathematically, that using this model to estimate 31 will lead to a biased estimator. Specically, solve for the expected value of the SLRM estimator for ,81, conditional on lane share and distance, and indicate the bias term. For efciency, use :31, m2 and y to denote the model variables. Hint: We can write the SLRM estimator, 571, as (40 points) 2(3),: Law ~ .=1 [31 = 31 + i n . Z (x,- - 9'3? i=1 ii. Explain, mathematically and in words, whether you expect the omitted variables bias to be positive or negative in this case. Your answer should include the equation for the omitted variables bias term. (40 points) iii. Now use a diagram relating explained variation to total variation to explain to your supervisor the intuition of the partialiing out interpretation for the MLRM OLS estimator for ,81. What portion of your diagram represents the source of the bias for the SLRM OLS estimator for 51? (20 points)