The identification problem in case of data with missing values is challenging and currently not fully understood. For example, there areno general nonconservative identifiability results, nor provably correct data efficient methods. In this paper, we consider a special caseof periodically missing output samples, where all but one output sample per period may be missing. The novel idea is to use a liftingoperation that converts the original problem with missing data into an equivalent standard identification problem. The key step is theinverse transformation from the lifted to the original system, which requires computation of a matrix root. The well-posedness of theinverse transformation depends on the eigenvalues of the system. Under an assumption on the eigenvalues, which is not verifiable fromthe data, and a persistency of excitation-type assumption on the data, the method based on lifting recovers the data-generating system.(Preprint submitted to Automatica)