The authors of this paper (http://andrewleigh.org/pdf/GunBuyback_Panel.pdf) appear to be essentially regressing the change in the death rate to the change in guns from a gun buy back in Australia, at the state level: $\Delta$death rate involving guns= $\Delta$guns + $\epsilon$
Specifically, they state on page 16 of the pdf:
In this paper, we ask whether firearm deaths dropped proportionately more in states where relatively more firearms were bought back. If the gun buyback itself was effective in reducing firearm-related deaths, then this would imply that states where more firearms were removed from the population should have seen a greater reduction in firearm death rates than the Australian average
On page 17 (of the pdf) it states the following:
For the purposes of our empirical strategy, what matters is that differences in buyback rates were not correlated with other factors that might have affected gun deaths. In particular, we are concerned about two potential confounders. First, if differences in buyback rates were driven by pre-existing gun ownership rates and if the relationship between gun ownership and gun deaths is non-linear, this could lead to a spurious correlation. However, although a non-linear relationship is theoretically plausible, we have been unable to locate any studies supporting such a theory.
I think the concern that the differences in buy back rates were driven by pre-existing gun ownership is because if this was the case (and given that we assume that pre-existing gun ownership could also impact the change in deaths) we would have confounding. Is this correct?
But I don't understand the concern that the relationship is non-linear?
Can anyone explain the econometric issues?
I was also interested in page 17 where this is stated. I wanted to link what I read in a paper with the textbook econometrics. Is this an example of the authors arguing that if they find drivers of why the gun buy back rates differ between states, that is not the death rates, that this is evidence that there is no reverse causation?
Second, our empirical strategy relies on the assumption that the state- level gun buyback rate is exogenous with respect to firearms death rates. It is thus important to consider the various factors that might explain why the buyback rate varied across states.
The final model used is on page 26 of the pdf. I am a little confused by it but what I think they are doing is the following (I note there is data and stata code HERE but I dont know how to read stata): I believe the change in guns (gun buy back) here is the difference in average guns pre and post 1997 and thus is a constant value for a given state, repeated for each records for a state in the data (each state as multiple records, one for each year).