Consider the regression model \((Y_i=b(X_i)+ \sigma \, u_i, i=1,\ldots , n)\) with error process \(u_i=\sqrt{\rho }\varepsilon _0+ \sqrt{1-\rho }\varepsilon _i\) where \((\varepsilon _i, i=0, \ldots , n)\) are independent centered random variables (r.v.) with unit variance, \(\sigma >0\) and \((X_1, \ldots , X_n)\) are independent and identically distributed r.v. independent of \((\varepsilon _i, i=0, \ldots , n)\) . Thus there is a common noise \(\varepsilon _0\) to all \(u_i\) ’s. We study the nonparametric estimation of the regression function b on a subset A of \({{\mathbb {R}}}\) from the observations \((X_i, Y_i,,i=1, \ldots , n)\) using a projection method on sieves. The standard least-squares contrast fails to provide consistent estimators. Therefore, we introduce a least-squared contrast taking into account the covariance matrix of the noise. By minimizing the contrast over a finite dimensional space \(S_m\) , we obtain a projection estimator and study its risk. The estimators are implemented on simulated data and show excellent performances.
