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## Statistics

problem 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) find out the MLE of Π.

(c) find out 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.

problem 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, prepare out the likelihood function for estimating the mean μ and standard deviation σ of the average class score.

(b) find out 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).

problem 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) find out and record each estimate's squared error ((^σ - σ )2, where σ = 1).

D) find out 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.

problem a) What was your average estimate of σ in MLE and OLS for your simulations?

problem b) Taking the mean of the squared error, which estimator was closer to the true parameter (σ) on average?

problem c) How often did you make a spurious influence with MLE? How does this compare to the Type I error rate in OLS?

problem d) How would you evaluate the performance of the ML-estimator with 5 observations?

Statistics and Probability, Statistics

• Category:- Statistics and Probability
• Reference No.:- M9613

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