The regression model to be analyzed is y = X 1 1 + X 2 2 + , where X 1 and X

The regression model to be analyzed is y = X₁β₁ + X₂β₂ + ε, where X₁ and X₂ have K₁ and K₂ columns, respectively. The restriction is β₂ = 0.

a. Using (5-23), prove that the restricted estimator is simply [b_1∗, 0], where b_1∗ is the least squares coefficient vector in the regression of y on X₁.

b. Prove that if the restriction is β₂ = β⁰₂ for a nonzero β⁰₂, then the restricted estimator of β₁ is b_1∗ = (X'₁X₁)⁻¹X'₁(y − X₂β⁰₂).