The best information about the benefits of long-term treatment is obtained from a long-term placebo-controlled trial. However, once efficacy has been demonstrated in relatively brief trials, it may not be possible to conduct long-term placebo-controlled trials, for ethical or practical reasons. This paper presents a method for estimating long-term effects of a treatment from a placebo-controlled trial in which some participants originally randomized to active-treatment volunteer to continue on treatment during an extension study, but follow-up of participants originally assigned to placebo ends with the trial, or they are crossed over to active treatment during the extension. We propose using data from the trial to project the outcomes for a 'virtual twin' for each active-treatment volunteer under the counterfactual placebo condition, and using bootstrap methods for inference. The proposed method is validated using simulation, and applied to data from the Fracture Intervention Trial and its extension, FLEX.