Twitter is a popular social media site on which people post millions of Tweets every day. As patients often share their experiences with drugs on Twitter, Tweets can also be considered as a rich alternative source of adverse drug reaction (ADR)-related information. This information can be useful for health authorities and drug manufacturing companies to monitor the post-marketing effectiveness of drugs. However, the automatic detection of ADRs in Tweets is challenging, as Tweets are informal and prone to grammatical errors. The existing approaches to automatically detecting ADRs do not consider the cause-effect relationships between a drug and an ADR. In this paper, we propose a novel shared causal attention network that exploits such cause-effect relationships to detect ADRs in Tweets. In our approach, we split a Tweet into the prefix, midfix, and postfix segments based on the position of the drug name in the Tweet and separately extract causal features from the segments. We then share these separate causal features with both word and parts-of-speech features, and apply the multi-head self-attention mechanism. We run extensive experiments on three publicly available benchmark datasets to illustrate the effectiveness of the proposed approach.