Econometrics Assignment -

PART A - SHORT ANSWERS

Question 1 - In the context of the simple statistical model

1. Probability model: {f(x; θ), θ ∈ Θ ⊂ R, x ∈ R}.  

2. Sampling model: X = (X₁, X₂, . . . , X_n) is a random sample, carefully define what it means for θ^{^} to be an estimator of θ, and for this estimator to be consistent.

Question 2 - Let (Ω, F, P) be a probability space. What is the domain and range of P? Provide the axioms P must satisfy in order for it to be a probability set function.

Question 3 - Consider the following definite integral

_-1∫¹ e^{-x^2} dx.

Explain how to construct an approximate 95% confidence interval for the value of this integral using the method of Monte Carlo Integration. Justify the steps of your answer.

PART B - ANSWER ALL QUESTIONS

REMINDER: When performing statistical tests, always state the null and alternative hypotheses, the test statistic and its distribution under the null hypothesis, rejection rule, level of significance and the conclusion of the test.

Question 4 - Consider the following one-parameter discrete uniform simple statistical model

1. Probability model: {f(x; θ) = 1/(θ+1), θ ∈ Z₊, x = 0, 1, 2, . . . , θ}.

2. Sampling model: X = (X₁, X₂, . . . , X_n) is a random sample.

(i) Write down the likelihood and log-likelihood functions. Show all of your derivations.

(ii) Compute the maximum likelihood estimator of θ. Justify your answer. Hint: you won't be able to use differential calculus to solve this problem.

(iii) Is this probability model a regular probability model? Justify your answer.

(iv) Let θ^{^} denote the MLE of θ. Suppose that the true values of, the parameter is θ₀. Compute a closed-form expression for the sampling distribution of n(θ^{^}₁ - θ₀). Hint: The marginal CDF is given by F(x; θ₀) = 1+x/1+θ, for x = 0, 1, . . . , θ₀.

(v) Compute the asymptotic distribution of n(θ^{^} - θ₀). Justify your answer. Hint: The limit as n → +∞ of (1 + x/n)ⁿ, for arbitrary real x, is e^x.

Question 5 - Consider the simple one parameter Normal model

1. Probability model: {f(x; θ) = 1/√(2π) exp {-(x- θ)²/2}, θ ∈ R, x ∈ R}.

2. Sampling model: {X₁, X₂, . . . , X_n) is a random sample.

Let 0 < a < 1 be any given real number. Define

where X^-_n, denotes the sample mean estimator.

(i) Is T_n an estimator of θ? Justify your answer.

(ii) Prove that {|X^-_n|< n ^-1/4} ∈ B(Rⁿ) has the following large sample property:

as n → +∞.

(iii) Using the result in part (ii), prove that T_n is a consistent estimator of θ.

(iv) Using the result in part (ii), derive the asymptotic distribution of √n(T_n - θ) as it depends on a and θ. Remember to justify your answer.

(v) Does the asymptotic variance of √n(T_n - θ) achieve the asymptotic Cramer Rao Lower Bound? Explain your answer.