  In this paper, we study some relation between the cocenter $\bar H(G)$ of the
Hecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedean
local field and the cocenter $\bar H(M)$ of its Levi subgroups $M$.
  Given any Newton component of $\bar H(G)$, we construct the induction map
$\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We show
that this map is surjective. This leads to the Bernstein-Lusztig type
presentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2}
on the affine Hecke algebras. We also show that the map $\bar i$ we constructed
is adjoint to the Jacquet functor and in characteristic $0$, the map $\bar i$
is an isomorphism.
