2017, by a far-right white nationalist (Table \[tab:external\_events\]).

**Step 4:** As evaluating the effect of such attacks on various slices of social media is a causal question, we frame it as measuring the impact of an intervention (event) on a time series (temporal evolution of speech). Following techniques for causal inference on time series [@brodersen2015inferring], we estimate an event’s impact on various types of hate and counter-hate speech by comparing the behavior of corresponding time series after an event, with counterfactual predictions of this behavior had no event taken place (§\[sec:method\]).

The last sections present (§\[sec:analysis\]) and discuss (§\[sec:conclusion\]) our results.

Background & Prior Work {#sec:background}
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We are interested in the relation between online hate speech and events. To ground our study, we first review work defining hate and counter-hate speech. Given our focus on anti-Muslim rhetoric in the context of extremist violence, we outline previous works on hate speech after terror attacks, and studies of hateful narratives targeting Muslims. We also cover observational studies on social media, particularly those focusing on harmful speech online.

Hate Speech Online and Offline {#hate-context}
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[**Defining hate speech.** ]{} While hate speech is codified by law in many countries, the definitions vary 