The variance of the OLS estimator when sigma2 is zero

Problem

If \(σ^2=0\) we have \(Var\left( b_2 \mid x \right)=0\) . Explain why this makes sense.

Solution

If \(σ^2=0\) then all error terms \(ε_i\) are zero. This means that \(y_i= β_1+β_2x_i\) and all observations will line up on a straight line. \(b_2\) , our estimate of \(β_2\) will become equal to \(β_2\) and there is no uncertainty in the estimate. \(Var\left( b_2 \mid x \right)\) should be (and will be) equal to zero.