  Using the recent four-loop calculations of the QCD beta-function and the
three-loop matching coefficients we study the induced error in $\alpha_s(m_Z)$
obtained from $\alpha_s(m_tau)$ due to the evolution procedure. We show that,
when consistent matching and running is used at this order, these errors are
pushed below 0.0005 in $\alpha_s(m_Z)$.
