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This paper studies theCox model with time-varying coefficients for cause-specific hazard functions when the causes of failure are subject to missingness. Inverse probability weighted and augmented inverse probability weighted estimators are investigated. The latter is considered as a two-stage estimator by directly utilizing the inverse probability weighted estimator and through modeling available auxiliary variables to improve efficiency. The asymptotic properties of the two estimators are investigated. Hypothesis testing procedures are developed to test the null hypotheses that the covariate effects are zero and that the covariate effects are constant. We conduct simulation studies to examine the finite sample properties of the proposed estimation and hypothesis testing procedures under various settings of the auxiliary variables and the percentages of the failure causes that are missing. These simulation results demonstrate that the augmented inverse probability weighted estimators are more efficient than the inverse probability weighted estimators and that the proposed testing procedures have the expected satisfactory results in sizes and powers. The proposed methods are illustrated using the Mashi clinical trial data for investigating the effect of randomization to formula-feeding versus breastfeeding plus extended infant zidovudine prophylaxis on death due to mother-to-child HIV transmission in Botswana.