Suppose you obtain alpha and varalpha describe how to

Question

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: Y_i = β₀ + β₁X_i + U_i

but the true model is Y_i = β₀ + β₁X_i + γZ_i + U_i Assuming that γ > 0, then β^{^}₁^OLS will be upward biased.

(c) Suppose your are studying the effect of Gender and Race on wages. You are working with the following variables

Y_i: Log wage

D_i^G: A gender dummy (0 if female, 1 if male)

D_i^R: A race dummy (1 if Black, 0 if White)

and the following model

Y_i = α₀ + α₁D_i^G + α₂D_i^R + U_i.

In this set up, you can interpret α0 as the average (log) wage for White females, i.e.,

E(Y_i|D_i^G = 0, D_i^R  = 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.