1. Show your work and justify all steps. You will lose points if your reasoning is incomplete.
Problem 1. OLS in the population.
(a) Consider f (xk) = b0 +b1xk, the Ordinary Least Squares univariate affine regression function of yk on a constant and xk in the population. Explain in words the meaning of “Least Squares”. (2 points)
(b) Suppose someone suggests that you should minimize the sum of the prediction error, n k=1
(yk − b0 − b1xk). Explain why this is a good or bad idea? What are the ramifications for our estimates of the yk values? (2 points)
(c) Explain in words why it is a good idea to include a constant in a univariate regression. Give a real world example in your explanation. (2 points)
(d) Write down the OLS parameters, b0 and b1, for the population regression specification,yk = b0 + b1xk + ek. (2 point)
(e) Are the OLS parameters you wrote down above random variables? Explain. (2 points)
Problem 2. Suppose you have a sample of n observations from the population where Yi represents the dependent variable and Xi represents the independent variable.
(a) Write down expressions for the estimators from a regression of Yi on a constant and Xi in the sample. That is, the regression specification, Yi = b0 + b1Xi + ei. (2 points)
(b) Assume that limn→∞ βˆ1 = β1. Now use the law of large numbers to show that limn→∞ βˆ0 =β0. Remember to show each step and provide explanations. (5 points)
(c) As in the lecture notes, let eˆi = (Yi − βˆ0 − βˆ1Xi), representing the error we make when we use the sample OLS regression to predict Yi. Show that the sample covariance between Xi and eˆi is zero. Explain why this would be the case. Recall that the sample covariance between two variables, Wi and Zi, is 1 Σn (Wi − W¯ )(Zi − Z¯), where W¯= 1 ΣNW i and Z¯ = 1 ΣN Zi Remember to show each step and provide explanations. (5 points)
Problem 3. Suppose that you are tasked with estimating the primary determinants of annual salaries in the USA. You initially obtain data (a sample) of n observations which contains the following variables for each individual in your sample: salaryi, genderi, educationi and parentSalaryi. Here salaryi is specified in dollars per year, genderi is an indicator variable equal to 1 if female and 0 for non-female, agei is specified in years, educationi is specified in years, and parentSalaryi is the average of the individuals parents salaries specified in dollars per year.
(a) Consider the regression specification wagesi = β0 + β1genderi + ei. You use some statistical software and estimate that coefficients.(2 points)
βˆ0 = 55,000 and βˆ1 = −11, 000.
Interpret both estimated
(b) Now consider instead the regression specification ln(wagesi) = β0+β1genderi+β2educationi+ ei. You use some statistical software and estimate that βˆ2 = 0.053. Interpret the estimated value of β2.(2 points)
(c) Now consider instead the regression specification ln(wagesi) = β0+β1genderi+β2educationi+ β3ln(parentSalaryi)ei. You use some statistical software and estimate that βˆ3 = 0.8. In- terpret the estimated value of β3.(2 points)
(d) Suppose you obtain more variables. Specifically, you now have n observations and j inde- pendent variables, where n = j − 1. Suppose you run an OLS regression of Yi on your j − 1 covariates and a constant. What is the R2? Explain why you get this result and whether you have good explanatory power of the Yi values in your sample and outside your sample. (2 points)
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