Assignment Questions:
1. Suppose the true regression model is ? = ?0 + ?1?1 + ?2?2 + ?. In addition, Gauss Markov assumptions are satisfied. A researcher accidentally excluded the ?2 variable from the regression model and used a “wrong” regression model ? = ?0 + ?1?1 + ?.
(a) Let the ?1‾ be the OLS estimators from the regression ? on ?1 only. Derive the estimator ?1‾.
(b) State the minimal assumptions that make ?1‾ unbiased. You need to justify your answer.
(c) Suppose ?1 and ?2 are positively correlated, and ?2 has a positive theoretical sign. Show that ?1‾ is biased and on average it overestimates the true ?1 population parameter.
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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 ?0: ? = 10 against ?1: ? < 10 at the α = 0.05 level. Draw the rejection region.
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