Question 1- Short questions
Please answer True or False. Answer without explanation will not receive credits. Full credit requires a formal proof.
(a) The inclusion of an irrelevant variable will always reduce precision (False or True?)
(b) Assume the model: Yi = β0 + β1Xi + Ui
but the true model is Yi = β0 + β1Xi + γZi + Ui Assuming that γ > 0, then β^1OLS will be upward biased.
(c) Suppose your are studying the effect of Gender and Race on wages. You are working with the following variables
Yi: Log wage
DiG: A gender dummy (0 if female, 1 if male)
DiR: A race dummy (1 if Black, 0 if White)
and the following model
Yi = α0 + α1DiG + α2DiR + Ui.
In this set up, you can interpret α0 as the average (log) wage for White females, i.e.,
E(Yi|DiG = 0, DiR = 0) = α0.
(False or True?)
Question 2 - In the context of the linear model:
Y = α + βX + U
(a) Suppose you obtain α^ and Var^(α^). Describe how to obtain the confidence interval at the 90% confidence level for α.
(b) Suppose you obtain β^ and Var^(β^). Describe the procedure to test H0: β = 1 against the alternative hypothesis H1: β ≠ 1.
Question 3 - Generate a data of 1000 observations using the following structure:
Y = -0.3 - 0.2X1 + 0.3X2 + U
where X1 ∼ N(1, 4), X2 = 0.3 × X1 +V , U ∼ N(0, 1) and V ∼ N(0, 1).
(a) Does the model satisfy the GAUSS-MARKOV theorem? Would your estimates be BLUE?
(b) Drop V, U from your model. Present the OLS results of a regression of Y on X1. Interpret your findings. Are they biased? Why?
(c) Add X2 to your regression. Interpret your results. Did your R2 increase? Why? Present a formal proof.
(d) Suppose that now X2 = V and V ∼ N(2, 1). Run the regression Y on X1 and X2 and compare your results with those from a regression of Y on X1. Are you getting biased results? Why?
(e) Suppose now you got information on a third variable X3. Construct X3 ∼ Uniform(-1, 4). Add X3 to your original regression model (i.e., where X2 = 0.5 × X1 + V, U ∼ N(0, 1)).
Interpret the new R2. Explain the changes on your coefficients and t-tests.