  The Starobinsky model of inflation, consistent with Planck 2015, has a
peculiar form of the action, which contains the leading Einstein term $R$, the
$R^2$ term with a huge coefficient, and negligible higher-order terms. We
propose an explanation of this form based on compactification of extra
dimensions. Once tuning of order $10^{-4}$ is accepted to suppress the linear
term $R$, we no longer have to suppress higher-oder terms, which give
nontrivial corrections to the Starobinsky model. We show our predictions of the
spectral index, its runnings, and the tensor-to-scalar ratio. Finally, we
discuss a possibility that quantum gravity may appear at the scale $\Lambda
\gtrsim 5 \times 10^{15}$ GeV.
