  The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the
$\overline{\mbox{DR}}$ scheme is calculated starting from the three-loop result
recently obtained with the higher covariant derivative regularization. For this
purpose, for the theory regularized by higher derivatives we find a subtraction
scheme in which the Green functions coincide with the ones obtained with the
dimensional reduction and the modified minimal subtraction prescription for the
renormalization of the SQCD coupling constant and of the matter superfields.
Also we calculate the $D$-function in the $\overline{\mbox{DR}}$ scheme for all
renormalization constants (including the one for the electromagnetic coupling
constant which appears due to the SQCD corrections). It is shown that the
results do not satisfy the NSVZ-like equation relating the $D$-function to the
anomalous dimension of the matter superfields. However, the NSVZ-like scheme
can be constructed with the help of a properly tuned finite renormalization. It
is also demonstrated that the three-loop $D$-function defined in terms of the
bare couplings with the dimensional reduction does not satisfy the NSVZ-like
equation for an arbitrary renormalization prescription. We also investigate a
possibility to present the results in the form of the $\beta$-expansion and the
scheme dependence of this expansion.
