  Suppose $n\geq 3$ and let $B$ be the open unit ball in $\mathbb{R}^n$. Let
$\varphi: B\to B$ be a $C^2$ map whose Jacobian does not change sign, and let
$\psi$ be a $C^2$ function on $B$. We characterize bounded weighted composition
operators $W_{\varphi,\psi}$ acting on harmonic Hardy spaces $h^p(B)$. In
addition, we compute the operator norm of $W_{\varphi,\psi}$ on $h^p(B)$ when
$\varphi$ is a M\"obius transformation of $B$.
