1. Consider the following regression model:
Yi = β1 + β2X2i + ... + βkXki + ei, i = 1, 2, ..., n
ei ∼ i.i.d. N(0, σ2)
Assume all the classical assumptions hold.
a) Provide an unbiased estimator of σ2.
b) What is the distribution of 1/σ2 i=1∑n ei2?
c) What is the distribution of 1/σ2 i=1∑n e^i2? (Here e^i = Yi - β^1 - β^2X2i - ... - β^kXki.)
d) What is the distribution of B = (β^j - βj/√(var^( β^j)))? Here, var^(β^j) is an estimate for var(β^j). (j = 1, 2, .., or k)
2. Consider the following regression equation:
Yi = β2X2i + β3X3i + ei, i = 1, 2, .., n,
a) Write down each of the classical assumptions specifically.
b) Suppose that one of the assumptions is violated, and we have X2i = 2 + 4X3i. Show that β2 and β3 cannot be separately estimated.
c) Provide a sample regression equation, and explain how you might estimate the coefficients of the model by applying the ordinary least squares (OLS) criterion. Set up the problem, and then derive the first-order conditions.
d) Suppose that Y is blood pressure, X2i is the amount of cigarette consumption, and X3i is the amount of physical exercise. Someone estimated the following sample regression equation, by omitting the X3i variable:
Yi = β^2X2i + u^i
d-i) Provide the OLS estimator β^2. Then derive the expected value of β^2. Is β^2 biased?
d-ii) Does β^2 from the above regression overstate the true value of β2 or understate the true value of β2? Provide a logic behind your answer.