Can anyone help me on solving these questions how to approach:

Module 1 Problem Set

R Exercise. There are 10 parts to the following question.

- Create a dataset (data.frame) with 100 observations, and each of the following:

(a) A variable, t, containing numbers from 1 to 100 (sequentially). What is the mean of t? Use one place after the decimal point.

(b) A variable we will call α, which has value 3 for all 100 observations. What is the vari- ance of α? Report your answer as an integer value.

(c) Another variable, εt , a random normal variable with mean 0 and standard deviation 1. Then create xt, a random uniform variable over [0,1]. What is E [xt]? Use one place after the decimal point.

(d) What is the variance of xt? Hint: the variance of a random uniform variable over [a,b] 12

is equal to 12 (b − a) . Enter as a ratio of two integers.

(e) An outcome variable yt defined as yt = α+βxt +εt where β = 2. What is E[yt]?

Report your answer as an integer value.

(f) Estimate β. Suppose the standard OLS 95% confidence interval of β is (1.61, 3.04) (Note that your confidence interval will depend on the exact draws of εt, and so it might be slightly different).

True or False: We can reject the hypothesis that β = 1 with 95% confidence. i. True

ii. False

(g) Create vt as a random normal variable with mean 0 and standard deviation 1. Create a new variable εt (replace the old variable εt), a random normal variable with mean 0 and standard deviation 1. Generate qt as qt = xt + 2x3t + vt.

What is E [qt]? Hint: If w is a random variable uniformly distributed over [0,1], then E[wn] = 1/(n + 1). Report your answer as an integer value.

(h) Generate outcome variable zt defined as zt = α+βxt +γqt +εt, where β = 2, γ = 3. What is E [zt]? Report your answer as an integer value.

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(i) Estimate β from the (misspecified) model:

zt = α + βxt + ut

Assume that the standard OLS 95% confidence interval is (5.832,10.014). Which point estimate derives this confidence interval? Use three places after the decimal point.

(j) Now suppose that the point estimate is 7.8 and the standard error is 1.04. In this instance, what is the upper bound of the standard OLS 95% confidence interval (recall the critical value for a standard OLS 95% confidence interval is 1.96)? Use two places after the decimal point

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Thanks to all in advance.

It’s urgent.please help