In　this　paper,　we　study　the　empirical　likelihood　for　a　partially　linear　single-index　model　with　a　subset　of　covariates　and　response　missing　at　random.　By　using　the　bias　correction　and　the　imputation　method,　two　empirical　log-likelihood　ratios　are　proposed　such　that　any　of　two　ratios　is　asymptotically　chi-squared.　Two　maximum　empirical　likelihood　estimates　of　the　index　coefficients　and　the　estimator　of　link　function　are　constructed,　their　asymptotic　distributions　and　optimal　convergence　rate　are　obtained.　It　is　proved　that　our　methods　yield　asymptotically　equivalent　estimators　for　the　index　coefficients.　An　important　feature　of　our　methods　is　their　ability　to　handle　missing　response　and/or　partially　missing　covariates.　In　addition,　we　study　the　estimation　and　empirical　likelihood　for　two　special　cases　the　single-index　model　and　partially　linear　model　with　observations　are　missing　at　random.　A　simulation　study　indicates　that　the　proposed　methods　are　comparable　for　bias　and　standard　deviation,　as　well　as　in　terms　of　coverage　probabilities　and　average　areas　(lengths)　of　confidence　regions　(intervals).　The　proposed　methods　are　illustrated　by　an　example　of　real　data.　(C)　2019　Elsevier　B.V.　All　rights　reserved.