messages, propose a multidimensional taxonomy of online hate speech, and show that a causal inference approach contributes to understanding how online hate speech fluctuates. Among our findings, we observe that extremist violence attacks tend to lead to more messages directly advocating violence, demonstrating that concerns about a positive feedback loop between violence “offline” and hate speech online are, unfortunately, well-founded.

### Paper Outline and Methodology Overview.  {#paper-outline-and-methodology-overview. .unnumbered}

Outlined in Figure \[analysis-framework\], our approach consists of several steps:

**Step 1:** We create a longitudinal collection of hate speech messages in social media from Twitter and Reddit that covers a period of 19 months. This collection is based on a series of keywords that are obtained through an iterative expansion of known hate speech terms (§\[sec:data\]).

**Step 2:** We categorize the data along four dimensions:

the group each message refers to,

the attitude of speakers,

the severity of hateful expressions, particularly whether they advocate violence, and

the framing of content

(§\[sec:characterizing\]).

**Step 3:** We select 13 extremist attacks involving Arabs and Muslims as perpetrators or victims, like the Berlin Christmas market attack on Dec. 2016, perpetrated by a follower of jihadist group ISIL, or the Quebec City mosque shooting on Jan. 