Discuss the fitted multiple regression model

Question

Assignment:

Question 1. Consider the population model y = β₀ + β₁x + u. Suppose that you have a sample of size n. The OLS estimator of β₁ is given by:

β1= _iΣⁿ=₁(x_i - x)y_i / _iΣⁿ= ₁(x_i -x)².

a) Clearly stating any assumptions that you make, show that βˆ 1 is an unbiased estimator of β₁.

b) Write down the formula for the variance of β^ˆ₁.

c) List and discuss three factors that affect the variance of β^ˆ₁.

Question 2. Consider the fitted multiple regression model wageˆ_i = βˆ₀ + β^ˆ₁ educ_i+ β^ˆ2exp_i , where wage is the hourly wage (in dollars), educ is years of schooling, and exp is year of experience in employment. Suppose that educi and exp_i are negatively correlated.

a) Do you expect β^ˆ₁ and β^ˆ₂ to be positive or negative? Why?

b) Interpret βˆ₂.

c) Suggest a reason for the negative correlation between experience and education.

d) What is the sign of δe1 in the following regression? Why?

e) Write down an equation that links βˆ₁ to β˜ ₁, where β˜₁ is defined by fitted simple regression wageˆ_i = β˜₀ + β˜₁ educ_i .

f) Using your answer to the previous question, explain whether β˜₁ will overestimate or underestimate βˆ₁.