Econometric test prep, please help!

Question

A. Show that under Gauss-Markov assumptions, for a multiple linear regression model:

𝑦 = 𝛽₀ + 𝛽₁𝑥₁ + ⋯ + 𝛽_k𝑥_k + 𝜀

OLS estimators for β₀, β₁, …, β_k, 𝛽̂₀. 𝛽̂₁, ⋯ , 𝛽̂_k are BLUE.

B. Show that omitting variables has the possibility to cause omitted variable bias. According to your analysis, what is an omitted variable? What determines the sign and magnitude of the omitted variable bias? Using mathematical proof and economic examples to illustrate your idea.

Answer 1

1. OVB refers to the case where you don't include a "relevant" variable in your regression, for example

y = β₀ + β₁x₁ + β₂x₂ + ε

1. This implicitly throws that variable into the error term:

y = β₀ + β₁x₁ + η, where η = β₂x₂ + ε

1. For a simple linear regression, it can be shown that

β₁_ols = β₁ + Cov(x₁,η) = β₁ + Cov(x₁, β₂x₂ + ε)

1. If β₂ ≠ 0 and Cov(x₁,x₂) ≠ 0 then we will run into trouble since:

β₁_ols = β₁ + β_{2 *} Cov(x₁, x₂)

1. The standard example is when we take y = log(wages), x₁ = education and x₂ = IQ. We expect people with higher IQ to earn more money (β₂>0) but they are also more likely to get education Cov(x₁, x₂) > 0.