We consider estimation of finite population totals in the presence of nonresponse, assuming nonresponses arise randomly within response classes. We compare two regression estimators: one based on the adjusted for nonresponse probability weights, the other based on unadjusted weights. We show that when the auxiliary variables used for nonresponse adjustment are included in the estimators, they differ only slightly. In this case, the nonresponse adjustment step can be omitted from the estimation process without loss of generality (from Result 5 of Deville and Särndal (1992), it follows the same remains correct for a wide class of calibration estimators). We consider a multivariate analog of a "regression through the origin" estimator and show the "adjusted" and "unadjusted" estimators coincide in this case. We also consider calibration estimators under restrictions on weights. We show that if there exists even one set of weights satisfying the calibration equations and restrictions, the benchmark estimator does not depend on the restrictions.