编程代码 Problem Set 2 discrete random variable

Problem Set 2

编程代码 You may work in groups. If you do, please turn in one write-up per group. These problem sets should be turned in in a…

Instructions

You may work in groups. If you do, please turn in one write-up per group. These problem sets should be turned in in a presentable fashion. I am okay with handwritten submissions however the paper used needs to be paper not torn out of a spiral notebook or three-ring binder. I do not care what kind of paper you use but use the same type of paper for the entire assignment. I will ask that you print your programming code so it is probably best that you turn the assignment in on computer paper. Please staple the assignment before you turn it in, I will try to remember to bring a stapler but I will expect this to be done ahead of time. I am also okay with you typing the assignment and you emailing it to me or scanning the assignment.

IMPORTANT!!: I am not interested in whether the answer is correct of not. I want to see your thinking process and I want to see your work! A correct answer without the work present will receive no credit. If I can’t read or follow your work points will be deducted.

PART 1: Turn this part in. 编程代码

1. Writethe GLS estimator as a GMM
2. Generatethe following variables:

• X₁:1000 observations from N(2, 1)
• X₂:1000 observations from N(1, 2)
• ϵ:1000 observations from N(0, 1)
• η:1000 observations: even observations from N (0, 1) and odd observations from N (3, 2).
• y= 0.5X₁ + 0.8X₂ + ϵ
• z= 0.5X₁ + 0.8X₂ + η

Report the following (I do not need your code): 编程代码

(a) Estimate using OLS and test if β₁ = 0: y = β₀ + β₁X₁ + β₂X₂

(b) Estimateusing OLS and test if β₁ = 0: z = γ₀ + γ₁X₁ + γ₂X₂

(c) Estimate using OLS and test if β₁ = 0 (use White’s standard errors): y = β₀+β₁X₁+β₂X₂

(d) Estimate using OLS and test if β₁ = 0 (use White’s standard errors): z = γ₀+γ₁X₁+γ₂X₂

(e) Estimate using GLS and test if β₁ = 0: y = β₀ + β₁X₁ + β₂X₂

(f) Estimate using GLS and test if β₁ = 0: z = γ₀ + γ₁X₁ + γ₂X₂

Comment on the following: 编程代码

(a) Is the approach taken in (a) and (b) proper?

(b) Is the approach taken in (c) and (d) proper?

(c) Compare(d) and (f).

(d)Compare(a) and (e). Is it proper to take the approach taken in (e)?

3. The following distribution is called the Gamma distribution:

where x, α, and β > 0. Generate 1000 observations where α = 0.3 and β = 1.5. Do the following:

• Conducta method of moments (MM) estimation of α and β.
• Conducta GMM estimation of α and β.
• Explain the difference between GMM and MM both theoretically and empirically. I will not make you code the variances but which estimation would you trust more and why?
• Would you be able to easily estimate α and β using maximum likelihood estimation? Why or why not?

4. The following distribution is called the Poisson distribution: 编程代码

where θ > 0 and y 0 is a discrete random variable. Generate 1000 observations where θ = 2.5. Do the following:

• Estimate θ by method of moments.
• Estimate θ by maximum likelihood.
• Compare (a) and (b). Compare method of moment and maximum likelihood. Could you estimate θ by OLS? If so, how?

PART 2: Do not turn this part in. 编程代码

5. Write the IV estimator as a GMM estimator.

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