A result on the sign of restricted least-squares estimates

When a variable is dropped from a least-squares regression equation, there can be no change in sign of any coefficient that is more significant than the coefficient of the omitted variable. More generally, a constrained least squares estimate of a parameter β_j must lie in the interval $\text{(}\text{β}\text{?}{}_{\mathrm{j}}\text{?V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}\text{|t|,}\text{β}\text{?}{}_{\mathrm{j}}\text{+ V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}\text{|t|)}$ where $\text{β}\text{?}{}_{\mathrm{j}}$ is the unconstrained estimate of $\text{β}{}_{\mathrm{j}}\text{, V}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{jj}}$ is the standard error of $\text{β}\text{?}{}_{\mathrm{j}}$ and t is the t-value for testing the (univariate) restriction.