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Homework #3

Econ 120B, Econometrics

1.  Suppose you are interested in the causal effect of taking a test prep course on a student’s SAT score.  (The SAT is a test which is often used by universities in the U.S. to help determine admission decisions.)  The following table shows the potential SAT scores for Amanda and Filipe if they had taken the course and if they had not taken the course.  Yi  denotes the actual SAT score for person i, Di  is an indicator for whether person i took the course, and the potential outcome notation is the same as used in the class notes.

Amanda

Felipe

Yi0

1180

1120

Yi1

1220

1230

Yi

1180

1230

D

0

1

a. What is the causal effect of taking the course on Amanda’s SAT score?  What is the causal effect of taking the course for Felipe?

b. What is the average treatment effect (ATE) for this population of two students?

c. What is the average effect of treatment on the treated (ATT) for this population of two students?

d.  Suppose you only have access to the variables Yi  and Di . What is the observed difference in SAT scores between Felipe and Amanda?  Decompose this difference into the true causal effect of taking the course for Felipe and the selection bias.

2.  Show that  is the solution to the following minimization problem: minm (Yi  − m)2

Once you do this, you will have shown that  is the “least squares” estimator of uY .  If you need help getting started, see the first set of lecture notes or Appendix 3.2 in Stock and Watson.  We   will return to the idea ofthe least squares estimator in the regression context later in the course.

3.  State the Central Limit Theorem.  Can the CLT be applied to a random sample from a discrete distribution?  To a random sample of size 10?

4.  Two hundred observationsfrom a random sample taken today are observed to have a sample mean of 6.5 and a sample variance of 3.8.

a) Test whether the population mean is different from 7 at the 5% significance level.  (note: this is a two-sided test)  Make sure to state the null and alternative hypotheses, the relevant test statistic, the distribution of the test statistic, the appropriate critical value, and what your conclusion is.

b) Test whether the population mean exceeds 7 at the 5% significance level.  (note: this is a one- sided test)  Make sure to state the null and alternative hypotheses, the relevant test statistic, the   distribution of the test statistic, the appropriate critical value, and what your conclusion is.

Five years later, you take a new random sample of 100 observations, and observe a sample mean of 7 and a sample variance of 3.6.

c) Test whether the population mean in the first sample equals the population mean in sample      taken five years later at the 5% significance level. You may assume that there are no overlapping observations in the two samples.  Carefully state the null and alternative hypotheses, the relevant test statistic, the distribution of the test statistic, the appropriate critical value, and what your        conclusion is.

5.  Stock and Watson, empirical exercise E3. 1, parts (a)-(f).

(Note: The E” in E3.1” indicates an empirical exercise which requires you to use Stata.)