Suppose that the population model determining y is y = 0 + 1 x 1

Suppose that the population model determining y is

y = β₀ + β₁x₁ + β₂x₂ + β₃x₃ + u,

and this model satisfies Assumptions MLR.1 through MLR.4. However, we estimate the model that omits x₃. Let β̃₀, β̃₁, and β̃₂ be the OLS estimators from the regression of y on x₁ and x₂. Show that the expected value of β̃₁ (given the values of the independent variables in the sample) is

where the r̂_i1 are the OLS residuals from the regression of x₁ on x₂. The formula for β̃₁ comes from equation (3.22). Plug y_i = β₀ + β₁x_i1 + β₂x_i2 + β₃x_i3 + u_i into this equation. After some algebra, take the expectation treating x_i3 and r̂_i1 as nonrandom.

Transcribed Image Text:

Ε(β ) -β, + β