(Anti)commutators at different times

Why does the commutator of two operators evaluated at different times vanish? Take for instance a fermonic field $psi_sigma (vec{x},t)$, which satisfies the well known anti-commutation relations at equal times

begin{equation}
[psi_sigma (vec{x},t),psi^dagger_{sigma’} (vec{x}’,t)]_+ = delta^{(3)}(vec{x}-vec{x}’) , delta_{sigma sigma’}
end{equation}
where $sigma$ is a flavour index.

Is that correct to state that the commutator at different times vanishes? In other words
begin{equation}
[psi_sigma (vec{x},t),psi_{sigma} (vec{x},t’)]_- = 0quad ?
end{equation}