Step 1: Sort the four phenomena into two families.
Some optical effects happen because light bends while passing through air of changing density, refraction, while others happen because light gets scattered off tiny particles or molecules, scattering.
Step 2: Place phenomena (I), (II) and (III) in the bending family.
Advance sunrise and delayed sunset happen because sunlight bends through the denser lower atmosphere, twinkling of stars happens because starlight keeps bending unpredictably through turbulent air layers of changing density, and the wavering of objects seen through hot air happens because the hot, less dense air near a flame keeps refracting light unevenly. All three are examples of light bending due to changing refractive index.
Step 3: Place phenomenon (IV) in the scattering family instead.
The blue colour of the sky comes from sunlight bouncing off tiny air molecules, with blue light scattering far more than red, this is scattering, not bending of a ray path, so it does not belong with the other three.
Step 4: Combine the refraction based ones.
Only (I), (II) and (III) are due to atmospheric refraction.
\[ \boxed{\text{(I), (II) and (III)}} \]