I. DERIVING OLS ESTIMATORS AND THEIR PROPERTIES
Suppose the population regression function can be written as: yi = β0 + β1xi1 + β2xi2 + ui, where E[ui|xi1, xi2] = 0. The sample equivalent to this restriction implies 1/n i=1∑n(u^i) = 0.
(1) Use 1/ni=1∑n(u^i) = 0 to demonstrate that the OLS estimator for β0 can be written as:
Β^0 = y- - β^1x-1 - β^2x-2, where y- = 1/n i=1∑nyi, x-1 = 1/n i=1∑n xi1, and x-2 = 1/n i=1∑nxi2.
(2) Suppose you omit x2 and estimate the univariate model: y~i = β~0 + β~1xi,1 + vi. The relationship between the univariate and multivariate estimators for β1 can be expressed as β~1 = β^1 + β^2δ1, where xi,2 = δ0 + δ1xi,1 + ri,2. Derive the bias of the univariate estimator for β1.
(3) Use your result from (2) to characterize the two special cases in which the univariate estimator is unbiased. Define each case and then explain it in words.
(4) Use the fact that β^1 = β1 + (i=k∑nr^i,1ui/i=1∑nr^i,12) in the multivariate model, where xi,1 = γ^0 + γ^2xi,2 + r^i,1, to demonstrate that Var(β^1) = σ2/i=1∑nr^i,12 if the error term satisfies the homoskedasticity assumption.
II. DISCUSSION QUESTIONS
(5) Because the OLS estimator is a linear model, it cannot realistically depict nonlinear relationships between economic variables. Please explain why you agree or disagree with this statement, using an example to support your answer.
(6) Consider the model y = β0 + β1x1 + β2x2 + β3x3 + u, where x1 and x2 are highly correlated. Discuss how this correlation would affect the OLS estimators for β0, β1, β2 and β3. How could the analyst mitigate the effects of this correlation?
(7) Explain the difference between the concepts of bias and consistency. As part of your answer, state the conditions under which the OLS estimator can be guaranteed to be consistent.
(8) As the sample size grows, it becomes less important to assume that the error term of the population regression function is normally distributed. Please explain why you agree or disagree with this statement.
III. INTERPRETING REGRESSION RESULTS
The following model is designed to estimate the return to education.
ln(wage) = β0 + β1 ln(education) + β2 experience + β3IQ + u,
where wage is measured in dollars per week, education and experience are measured in years, and IQ is the worker's score on a standardized exam designed to measure IQ. Coefficients and standard errors from a sample of 31 randomly selected workers are as follows:
ln(wage) = 2.96 + 0.94ln(education) + 0.23experience + 0.04IQ + u
(0.28) (0.70) (0.46) (0.12)
N=31, R2 = 0.42, SSR=110.08
(9) Interpret the coefficient on IQ
(10) Interpret the coefficient on ln(education).
(11) Suppose you want to conduct a statistical test of whether IQ has a positive effect on wages. State the critical value for a hypothesis test conducted at the 5% level of significance. State whether you would reject or fail to reject the null hypothesis.
(12) Dropping education and experience from the model gives:
Ln(wage) = 3.08 + 1.03ln(IQ) + u
(0.61) (0.12)
N=31, R2 = 0.37, SSR=114.87
Set up a test to determine whether education and experience are jointly significant in the original model at the 1% level. State the null hypothesis, the alternative hypothesis, and the critical value above which you would reject the null hypothesis.
(13) Suppose the model errors are heteroskedastic. Would this affect your interpretation of the t-test from problem 11? Explain your reasoning.