  Weak gravitational lensing is one of the most powerful tools for cosmology,
while subject to challenges in quantifying subtle systematic biases. The Point
Spread Function (PSF) can cause biases in weak lensing shear inference when the
PSF model does not match the true PSF that is convolved with the galaxy light
profile. Although the effect of PSF size and shape errors - i.e., errors in
second moments - is well studied, weak lensing systematics associated with
errors in higher moments of the PSF model require further investigation. The
goal of our study is to estimate their potential impact for LSST weak lensing
analysis. We go beyond second moments of the PSF by using image simulations to
relate multiplicative bias in shear to errors in the higher moments of the PSF
model. We find that the current level of errors in higher moments of the PSF
model in data from the Hyper Suprime-Cam (HSC) survey can induce a $\sim 0.05 $
per cent shear bias, making this effect unimportant for ongoing surveys but
relevant at the precision of upcoming surveys such as LSST.
