For a cluster-robust t-statistic under cluster heterogeneity we establish that the cluster-robust t-statistic has a gaussian asymptotic null distribution and develop the effective number of clusters, which scales down the actual number of clusters, as a guide to the behavior of the test statistic. The implications for hypothesis testing in applied work are that the number of clusters, rather than the number of observations, should be reported as the sample size, and the effective number of clusters should be reported to guide inference. If the effective number of clusters is large, testing based on critical values from a normal distribution is appropriate.

In 2001 a large and sustained supply shock halted a heroin epidemic in Australia. We use drug offenses to identify individual opioid users and examine how the shock affected their mortality risks and criminal activity over the next eight years. Initially, gains from fewer overdoses are offset by drug substitution and more crime, including homicides. Most adverse effects dissipate over time, whereas persistent mortality reductions save the lives of around one in 48 individuals in our sample. Our results demonstrate that reducing the supply of illicit opioids can lead to meaningful longer-term improvements, even when the short-term effects are ambiguous.