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Observational Studies

9. Internal Validity

Example 5

Domestic violence exacts a well-documented and costly toll on victims and their families. In the United States "on the average, 3.5 people are killed by intimate partners every day, and many others are injured" (Vigdor and Mercy, 2006:313). An important question, therefore, is whether laws restricting access to firearms for individuals with a history of domestic violence can reduce intimate partner homicides (IPH). Vigdor and Mercy try answer this question using states as the observational units.

"The states that have laws limiting access to guns by abusers passed their legislation at different times. We exploit this time variation by effectively comparing IPH rates before and after passage of the law in states that enacted these laws with those in states that did not pass such a law. Although we cannot be certain that we are isolating the impact of the laws, the time variation in the effective date of the laws reduces the likelihood that we are capturing the effect of an omitted shock affecting all IPH rates."

Fully appreciating states can vary on the other factors that can affect IPH, they use a negative binomial regression model that includes a large number of covariates. They find that restraining- order laws help to keep perpetrators and victims apart and reduce IPH. Other kinds of interventions, such as gun confiscation laws, had no demonstrable impact.

Each observational unit is assumed to have two potential outcomes. There is an outcome if the unit is exposed to the intervention. There is another outcome if the unit is exposed to the alternative. These outcomes can vary across units and are hypothetical. Using the Vigdor and Mercy case study, a given state has a potential number of IPHs if a law is passed restricting the access of batterers to firearms, and a potential number of IPHs if there is no such law.

Suppose we let Yi(1) denote the hypothetical IPH count if state i enacts the relevant legislation, and Yi(0) denote the hypothetical IPH count if state i does not enact that legislation. The causal effect of the legislation can be defined as [Yi(1)–Yi(0)], although occasionally [Yi(1)/Yi(0)] is used instead.

It is impossible to observe both Yi(1) and Yi(0). Either a given state passes the relevant legislation or it does not. Suppose we let Wi = 1 if state i enacts the legislation, and Wi = 0 if state i does not enact the legislation.

Then the observed outcome for a given state is:

(2) Yi = (1 - Wi) ŸYi(0) +Wi ŸYi(1)

Yi and Wi can be observed. But there is no way to map what can be observed back to the definition of a causal effect for a given state. Either the legislation passes or it does not. Consequently, we shift to group comparisons. In this example, attention is directed to the average IPH count of the states that passed the relevant legislation compared to average IPH count of the states that did not pass the relevant legislation. The difference between the two is an estimate of the average treatment effect (ATE).