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Question answering (QA) has demonstrated impressive progress in answering questions from customized domains. Nevertheless, domain adaptation remains one of the most elusive challenges for QA systems, especially when QA systems are trained in a source domain but deployed in a different target domain. In this work, we investigate the potential benefits of question classification for QA domain adaptation. We propose a novel framework: Question Classification for Question Answering (QC4QA). Specifically, a question classifier is adopted to assign question classes to both the source and target data. Then, we perform joint training in a self-supervised fashion via pseudo-labeling. For optimization, inter-domain discrepancy between the source and target domain is reduced via maximum mean discrepancy (MMD) distance. We additionally minimize intra-class discrepancy among QA samples of the same question class for fine-grained adaptation performance. To the best of our knowledge, this is the first work in QA domain adaptation to leverage question classification with self-supervised adaptation. We demonstrate the effectiveness of the proposed QC4QA with consistent improvements against the state-of-the-art baselines on multiple datasets.