  We show that the universal $\alpha$-attractor models of inflation can be
realized by including an auxiliary vector field $A_{\mu}$ for the Starobinsky
model with the Lagrangian $f(R)=R+R^2/(6M^2)$. If the same procedure is applied
to general modified $f(R)$ theories in which the Ricci scalar $R$ is replaced
by $R+A_{\mu} A^{\mu}+\beta \nabla_{\mu}A^{\mu}$ with constant $\beta$, we
obtain the Brans-Dicke theory with a scalar potential and the Brans-Dicke
parameter $\omega_{\rm BD}=\beta^2/4$. We also place observational constraints
on inflationary models based on auxiliary vector modified $f(R)$ theories from
the latest Planck measurements of the Cosmic Microwave Background (CMB)
anisotropies in both temperature and polarization. In the modified Starobinsky
model, we find that the parameter $\beta$ is constrained to be $\beta<25$
(68\,\%\,confidence level) from the bounds of the scalar spectral index and the
tensor-to-scalar ratio.
