For the next two questions, consider a regression model of savings (sav) on income (inc): sav = ?0 + ?1 inc + u where u = inc^2 × e Suppose e is a random variable with the following properties: E(e|inc) = 0 Var(e|inc) = ?_e^2 Suppose assumptions SLR.1–SLR.3 are satisfied, and recall that in last week's problem set you evaluated whether assumptions SLR.4 and SLR.5 were satisfied. Question 1 For this regression model, we would like to know whether ordinary least squares (OLS) slope estimator will be unbiased. Based solely on this result (from Lecture 5): SLR.1 – SLR.4 ? OLS estimators of ?0 and ?1 are unbiased what can we conclude? Is the OLS slope estimator unbiased? Options: It will be unbiased It will be biased We cannot be sure whether it will be biased or unbiased
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1-SLR.3 are satisfied. We also know that if SLR.1 and SLR.4 are satisfied, then the OLS estimators of β0 and β1 are unbiased. However, we are not given any information about whether SLR.4 is satisfied or not. Since we don't have enough information about SLR.4, we Show more…
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We would like to obtain an estimator of β1 from the following regression model with only one independent regressor: yi = β0 + β1 x1i + ui. (2) However, there is another variable x2i, which is missing from the model and potentially correlated with x1i. That is, the true model would be: yi = β0 + β1 x1i + β2 x2i + vi, where vi is an observation error that satisfies E(vi | x1i, x2i) = 0. Show that the OLS estimator of β1 obtained from model (2) is biased. What is the bias equal to? When is this bias equal to 0, positive, or negative? Show all cases. What are the implications in terms of estimated coefficients in each of the cases?
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Question 2 (25 points) Briefly answer the following: (a) (7 points) In the context of classical linear regression model (CLRM), which assumptions are required in order to ensure that OLS estimator is unbiased and efficient? (b) (5 points) What is meant by the term heteroskedasticity? What are the statistical properties of the OLS estimator in the presence of heteroskedasticity (assuming all other CLRM assumptions are satisfied)? (c) (5 points) What is meant by the term serial correlation? What are the statistical properties of the OLS estimator in the presence of serial correlation (assuming all other CLRM assumptions are satisfied)? (d) (8 points) Recall that in the LSDV model, one panel unit needs to be selected as the "baseline". Suppose that you re-estimate an LSDV model using the same data twice, each time selecting different unit as baseline. You notice that while estimates of the dummy coefficients in the model change, estimates of the slope coefficients remain the same. Is this surprising? Why?
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