Prove whether the ordinary Least-Squares estimators for the remaining regressors in the final model have a smaller variance than the respective ordinary least-squares estimators when using the sum of squares to test whether x₁ or x₂ should be excluded and fitting the final regression model.

The original model y = B₀ + B₁x₁ + B₂x₂ + , has a response variable y which is continuous, and regressor vector X = (x₁, x₂)^Twith a mean vector of (_x1, _x2)^Tand a positive definite variance-covariance matrix _X. The random errors _i are conditional on X_i, and are independent and normally distributed with mean zero and variance ² which does not depend on X.

Answer with mathematical proof.