The assumption most appropriate for nonlinear relationships estimated on stratified cross-section data (e.g., the Current Population Surveys) is that of independent not identically distributed (i.n.i.d.) regressors, not fixed regressors. This study provides conditions which ensure the consistency and asymptotic normality of nonlenear weighted least squares estimators with i.n.i.d. regressors for both the known and estimated weights cases. A general statistic for testing hypotheses about the parameters is given, as well as a test for model misspecification. The usual conditional parameter covariance matrix estimator may be an inconsistent estimator of the unconditional covariance matrix, and a consistent estimator is provided.