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We consider performing simulation experiments in the presence of covariates. Here, covariates refer to some input information other than system designs to the simulation model that can also affect the system performance. To make decisions, decision makers need to know the covariate values of the problem. Traditionally in simulation-based decision making, simulation samples are collected after the covariate values are known; in contrast, as a new framework, simulation with covariates starts the simulation before the covariate values are revealed, and collects samples on covariate values that might appear later. Then, when the covariate values are revealed, the collected simulation samples are directly used to predict the desired results. This framework significantly reduces the decision time compared to the traditional way of simulation. In this paper, we follow this framework and suppose there are a finite number of system designs. We adopt the metamodel of stochastic kriging (SK) and use it to predict the system performance of each design and the best design. The goal is to study how fast the prediction errors diminish with the number of covariate points sampled. This is a fundamental problem in simulation with covariates and helps quantify the relationship between the offline simulation efforts and the online prediction accuracy. Particularly, we adopt measures of the maximal integrated mean squared error (IMSE) and integrated probability of false selection (IPFS) for assessing errors of the system performance and the best design predictions. Then, we establish convergence rates for the two measures under mild conditions. Last, these convergence behaviors are illustrated numerically using test examples.