Question 1: You have a regional country-year dataset over 10 years of 150 total observations, within this data you have 25 observations where the ruling party has lost power.
(a) Assuming a binomial distribution, what is the likelihood function for Π , the probability that a country's ruling party loses power over ten years?
(b) Calculate the MLE of Π.
(c) Calculate the standard error and the 95% con dence interval for this estimate. How did you know what test/sampling distribution to use?
(d) What is the value of the log-likelihood for this estimate?
(e) Say that for another region the estimated probability of a ruling party losing power over a ten-year span is 25%, using your region's data, what is the value of the log-likelihood function for a .25 value of Π?
(f) Using a Wald test, test whether your region's rate of over turning power is signi cantly di erent from the other region's.
Question 2: Each section of the SAT test is supposed to be distributed normally with a mean of 500 and a standard deviation of 100. Suppose 5 students in a class took the SAT math test. They received the following scores:
400, 450, 575, 600, and 625.
(a) Assuming a normal distribution, write out the likelihood function for estimating the mean μ and standard deviation σ of the average class score.
(b) Calculate the MLE estimate for μ and the standard deviation (σ) for this class.
(c) What is the value of the log-likelihood function at this estimate?
(d) What is the least-squares estimate for μ and σ in your data? Are there any di fferences? Why or why not?
(e) Using a likelihood-ratio test, test whether the MLE estimates from the class are signi cantly di erent from the national mean of 500 and national standard deviation of 100. (Hint : Start by plugging those values into your log-likelihood function. Remember you are testing two restrictions).
Question 3: (this is tough but give it a try) Use R (easiest) or Stata to simulate how often one makes a wrong small-sample inference in MLE vs. OLS in the following circumstance. Perform the following steps (be sure to attach your code):
A) Take 5 draws from a standard normal distribution (μ= 0, σ = 1)
B) Record the MLE and OLS estimate of (note: they have slightly di fferent estimates).
C) Calculate and record each estimate's squared error ((^σ - σ )2, where σ = 1).
D) Calculate and record each estimate's 95% con dence interval using z vs. t-scores.
E) Test and record whether ^μ is estimated as signi cantly different from 0.
F) Repeat for a total of 1000 simulations.
Answer the following:
Question a) What was your average estimate of σ in MLE and OLS for your simulations?
Question b) Taking the mean of the squared error, which estimator was closer to the true parameter (σ) on average?
Question c) How often did you make a spurious influence with MLE? How does this compare to the Type I error rate in OLS?
Question d) How would you evaluate the performance of the ML-estimator with 5 observations?