Introduction to Econometrics

Introduction to Econometrics

Subject: Economics    / General Economics
Question
UNIVERSITY OF SOUTHERN CALIFORNIA
DEPARTMENT OF ECONOMICS
ECON 318: Introduction to Econometrics,
Instructor: Yu-Wei Hsieh HOMEWORK 2
Problem 1. Consider two randomly selected corporate bonds: A and B.
Suppose that the probability that bond A defaults is 0.5. If bond A defaults,
the probability that bond B defaults is 0.75, whereas if bond A does not default
then the probability that bond B defaults is 0.25.
A) Write down a random variable representing bond A (ie., all possible
outcomes and the associated probabilities).
B) What is the probability that both bonds will default? Hint: think about
the tree representation for conditional probability.
C) What is the probability that bond B will default?
D) Let Y be the number of bonds that default (ie., Y=A+B). What is the
probability distribution of Y?
Hint: List events that correspond to each possible value of Y and then
use the conditional probabilities given in the problem to compute that
probabilities that Y will take for each of these values.
Problem 2. An insurance company is considering settling small claims by
mail rather than by the personal attention of agents, hoping thereby to reduce
the time of settling such claims. Under the old system claims were settled
with a mean time of 33.2 days. The company decides that the new system
is worth adopting if it will reduce the mean time to less than 30 days. The
company obtains a random sample of 900 small claims by the new system
¯ = 29.4 days and s = 8.2 days. Should the
and finds that for this sample X
company adopt a new plan? 1 A) Formulate this as a hypothesis testing problem: state the null and an
alternative hypothesis.
B) State the rejection rule of the test when the significance level of the test
is 0.05. State your conclusion, that is, would you recommend that the
company adopts a new system?
C) Compute p-value of the test and interpret it.
D) What would be your conclusion if significance level were 0.01 instead of
0.05?
Problem 3 (Conditional Expectation and Variance). Suppose the joint distribution of (X, Y ) is given by the following contingency (row represents x)
table[20 points]
(x, y) 2
4
6
1
2
3 0.3 0
0.1
0
0.2 0
0.1 0
0.3 A) Compute the marginal distributions of X and Y .
B) Are X and Y independent? Explain.
C) Find the conditional distribution of Y given X = 1
D) Compute E[Y |X = 1]
E) Compute E[Y |X = 2]
F) Compute E[XY |X = 2] 2