|University||National University of Singapore (NUS)|
1. Suppose the true regression model is 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝛽2𝑥2 + 𝑢. In addition, Gauss Markov assumptions are satisfied. A researcher accidentally excluded the 𝑥2 variable from the regression model and used a “wrong” regression model 𝑦 = 𝛽0 + 𝛽1𝑥1 + 𝑣.
(a) Let the 𝛽1‾ be the OLS estimators from the regression 𝑦 on 𝑥1 only. Derive the estimator 𝛽1‾.
(b) State the minimal assumptions that make 𝛽1‾ unbiased. You need to justify your answer.
(c) Suppose 𝑥1 and 𝑥2 are positively correlated, and 𝛽2 has a positive theoretical sign. Show that 𝛽1‾ is biased and on average it overestimates the true 𝛽1 population parameter.
2. The general formula for the sampling variance of the OLS estimator is Var (βˆ) = σ² / SSTj (1 – R²j).
(a) Show that when the model is a simple regression model, the sampling variance of the OLS estimator degenerates to Var (β1ˆ) = σ² / Σ(i=1, n) (xi – x‾)².
(b) Consider a regression model with 2 explanatory variables. What is the consequence of having two highly correlated explanatory variables? How would you solve the “problem”?
(c) Explain the Variance Inflation Factor (VIF). What is the significance of VIF?
3. A brochure inviting subscriptions for a new diet program states that the participants are expected to lose over 10.00 kg in three months. From the data of the three-month weight control program, we interviewed 200 participants about their weight loss. The sample mean and sample standard deviation are found to be 8.80 kg and 25.00 kg, respectively. Could the statement in the brochure be substantiated on the basis of these findings? Please test 𝐻0: 𝜇 = 10 against 𝐻1: 𝜇 < 10 at the α = 0.05 level. Draw the rejection region.