  We analyse quantum properties of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric
gauge theories formulated in terms of ${\cal N}=1$ superfields and investigate
the conditions imposed on a renormalization prescription under which the
non-renormalization theorems are valid. For this purpose in these models we
calculate the two-loop contributions to the anomalous dimensions of all chiral
matter superfields and the three-loop contributions to the $\beta$-functions
for an arbitrary ${\cal N}=1$ supersymmetric subtraction scheme supplementing
the higher covariant derivative regularization. We demonstrate that, in
general, the results do not vanish due to the scheme dependence, which becomes
essential in the considered approximations. However, the two-loop anomalous
dimensions vanish if a subtraction scheme is compatible with the structure of
quantum corrections and does not break the relation between the Yukawa and
gauge couplings which follows from ${\cal N}=2$ supersymmetry. Nevertheless,
even under these conditions the three-loop contribution to the $\beta$-function
does not in general vanishes for ${\cal N}=2$ supersymmetric theories. To
obtain the purely one-loop $\beta$-function, one should also chose an NSVZ
renormalization prescription. The similar statements for the higher loop
contributions are proved in all orders.
