We consider in this paper the $d$-dimensional nonlinear autoregressive model of order~$1$ defined by: $$ X_{n+1}=F(X_n)+\sigma(X_n)\eps_{n+1}, \qquad n\geq 0, $$ where $(\eps_n)_{n\geq 1}$ is a sequence of independent identically distributed $\R^d$-valued random variables, and $F\: \R^d\to\R^d$ (resp\. $\sigma\: \R^d\to\R_+$) is a H\"{o}lder function. >From an $L_1$-method, we construct a uniformly consistent estimator $m_n$ of the function $F+\sigma\alpha_0$, where $\alpha_0\in \R^d$ is the unique $L_1$-median of $\eps_1$. Since $F(X_n)+\sigma(X_n)\alpha_0$ is the best predictor (for the $L_1$-norm) of $X_{n+1}$ we can get when $X_n$ is known, the estimator $m_n$ provides the statistical predictor $m_n(X_n)$.