ECON 4211
Problem Set #1
Spring 2025
Due February 13th by 8pm
1. [25 points] Remind yourself of the terminology we developed in Chapter 1 for causal questions. Suppose we are interested in the causal effect of having health insurance on an individual’s health status.
(a) [5 points] We run a phone survey where we ask 5,000 respondents about their current insurance and health conditions. The data we collect is an example of a .
(b) [5 points] The US government has Census data onevery elderly American’s current insurance and health status. This is an example of data for the .
(c) [5 points] Suppose we take our phone survey data and calculate the difference in health between individuals who do and do not have insurance. This difference is an example of an _______.
(d) [5 points] The difference in health between all Americans who do and don’t have insurance is an example of an _______. The effect of insurance on health is an example of a _______.
(e) [5 points] When the two objects in (d) coincide,we have an example of . Give one reason why the two objects in (d) might not coincide.
2. [35 points] Let Y = a + X3 /b where a and b are some constants with b > 0, and where X ∼ N(0, 1).
(a) [2 points] State the definition of the cumulative density function of Y , which we’llcall Fa,b (y).
(b) [5 points] Express Fa,b (y) in terms of the CDF of the standard normal distribution Φ(·). Hint: can you re-write the inequality Y ≤ y as an inequality involving X?
(c) [3 points] Express E[Y] in terms of E[X3], then use the fact that E[X3] = 0 when X ∼ N(0, 1) to derive E[Y].
(d) [4 points] Express Cov(Y, X) in terms of E[X4], then use the fact that E[X4] = 3 when X ∼ N(0, 1) to derive Cov(Y, X).
(e) [2 points] Suppose E[Y] = 0 and Cov(Y, X) = 0.3. What can you conclude about a and b?
(f) [6 points] Given your answers to (b) and (e), what is the probability that a draw of Y is bigger than zero? What is the probability that a draw of Y falls between −0.1 and 0.1?
(g) [3 points] Let W = a + X3 /b + Z where Z is mean-zero and independent of X . How does the
distribution of E[W | X] (recall this is a random variable) compare to the distribution of Y?
3. [40 points] In this problem, we will talk about how to estimate the variance of a random variable X ,
σX(2), with an iid sample of n observations. Recall that the definition of variance is
σX(2) = E[(Xi − µX )2]
Note that to use this estimator, you need to know µX . However, we rarely (or never) know the true value of µX . This implies one has to use some sort of replacement for µX . A seemingly natural option
is to use as an estimator for µX .
(a) [7 points] Show that is an unbiased estimator. Is it consistent? What law would you use to prove that?
(b) [7˜ points] Suppose that somebody gave you the advice of using the following estimator: and said that this estimator is more efficient. Is this claim true?
(c) [7 points] Suppose you decided to use the formula below to estimate sample variance:
which can be conveniently rewritten as
Show that
(d) [7 points] Show that
Hint: think about the variance of the sample mean (computed in class).
(e) [6 points] Using (d),calculate E[˜(σ)X(2)].
(f) [6 points] Is the estimator ˜(σ)X(2) unbiased? If not, calculate the bias of this estimator and propose an unbiased estimator.