01 May Question: Problem Set 1 EC420-Spring 2019 Rules: Answers are due in class on Thursday, January 25. You may …
Problem Set 1 EC420-Spring 2019 Rules: Answers are due in class on Thursday, January 25. You may work with classmates and/or ask classmates for help (in addition to asking the RA and instructor But each student must submit his or her own answers. And you may not simply copy someone another classmate’s work. Rather you must seek to understand and then answer in your own words. Q1. (Potential outcomes and causal effect) [10 points Consider the following table that shows the potential wages of Spencer and Shelby if they had gone to college and if they had not gone to college. D 1 if i did go to college, and 0 otherwise. Yi denotes wages of i, and the rest of the notation corresponds to the textbook and class notes (e.g., Yoi is the wage of i if they didn’t go to college). Spencer Shelby 20 40 Y 15 D.0 Yi 40 a. What is the causal effect of going to college on Spencer’s wages? 1 point) b. What is the causal effect of going to college on Shelby’s wages? 1 point] c. What is the average causal effect? 1 point d. Is there a difference between the potential wages of Spencer and Shelby when neither goes to college? If yes, what might be the potential causes of this difference? If no, why do you think there’s no difference? 3 points) e. What is the difference between the observed wages of Spencer and Shelby? Decompose this difference into the true causal effect for Shelby and the se- lection bias. 4 points Q2. (Randomized trials) [10 points Suppose a study of high school students in a developing country finds that owning a computer is associated with higher academic grades
a. Do you think this relationship is causal? If yes, why? If no, what else might be driving the correlation? 3 points b. How would you design an experiment to estimate the causal effect of owning a computer? 4 points) c. How would you empirically check that the “all else equal” condition holds in the experiment? 3 points) Q3. (Estimation and inference for the population mean of a normal distri bution) 10 points) Suppose that X is normally distributed, and that you are given the following random sample from its distribution: 5, 8, 7, 9, 3, 5, 10, 8, 7, 9, 6, 10, and 5. a. Estimate the mean and variance of the underlying normal distribution. 3 points) b. Test the null hypothesis that the true population mean is equal to three (against the one-sided alternative that it exceeds three) at the 95% confi- dence level. 3 points) c. Construct a 99% confidence interval for the true population mean using your estimates. 4 points Q4. (Sampling distributions and the central limit theorem) [10 points Sup pose you programmed a computer to do the following Step 1: Randomly choose an integer number from 1-5 (with equal proba- bility of choosing each value). Do this 147 times to get a sample of n=147 random numbers. Step 2: Using the sample in step 1, calculate μ and σ2 82 Step 3: Repeat steps 1-2 another 9,999 times to get a total of 10,000 differ- ent sainple means μ and variances 02 calculated in the same way. a. Suppose you created a histogram using your 10,000 values of μ. What does this histogram teu2 points b. what do you expect would be the mean of the 10.000 values of fi? what do you expect would be the variance? 2 points c. What can you say about the approximate shape of the histogram? 2 points d. Suppose you increased the sample size in step 1 above to n=289 observa- tions. How would this change (or not change) your answers to (a)-(? In (c), how does the histogram change (or not change)? 4 points
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