Imputation and estimation under nonignorable nonresponse in household surveys with missing covariate information

In this article we develop and apply new methods for handling not missing at random (NMAR) nonresponse. We assume a model for the outcome variable under complete response and a model for the response probability, which is allowed to depend on the outcome and auxiliary variables. The two models define the model holding for the outcomes observed for the responding units, which can be tested. Our methods utilize information on the population totals of some or all of the auxiliary variables in the two models, but we do not require that the auxiliary variables are observed for the nonresponding units. We develop an algorithm for estimating the parameters governing the two models and show how to estimate the distributions of the missing covariates and the outcomes. The latter distributions are used for imputing the missing values of the nonresponding units and for estimating population means and the variances of the estimators. We consider several test statistics for testing the combined model fitted to the observed data, which enables validating the models used. The new developments are illustrated using a real data set collected as part of the Household Expenditure Survey carried out by the Israel Central Bureau of Statistics in 2005.