  In this paper, we scrutinize a radiatively generated QCD $\theta$ parameter
at the two-loop level based on both full analytical loop functions with the
Fock-Schwinger gauge method and the effective field theory approach, using
simplified models. We observe that the radiatively generated $\theta$
parameters at the low energy scale precisely match between them. It provides
validity to perturbative loop calculations of the QCD $\theta$ parameter with
the Fock-Schwinger gauge method. Furthermore, it is also shown that the
ordinary Fujikawa method for the radiative $\theta$ parameter by using
$\bar\theta = - arg det M_q^{loop}$ does not cover all contributions in the
simplified models. But, we also find that when there is a scale hierarchy in
$CP$-violating sector, evaluation of the Fujikawa method is numerically
sufficient. As an application, we calculate the radiative $\theta$ parameter at
the two-loop level in a slightly extended Nelson-Barr model, where the
spontaneous $CP$ violation occurs to solve the strong $CP$ problem. It is found
a part of the radiative $\theta$ parameters cannot be described by the Fujikawa
method.
