<em>Vis Inertiae</em> and Statistical Inference: A Review of Difference-in-Differences Methods Employed in Economics and Other Subjects

Difference-in-Differences (DiD) is a useful statistical technique employed by researchers to estimate the effects of exogenous events on the outcome of some response variables in random samples of treated units (i.e. units exposed to the event) ideally drawn from an infinite population. The term effect should be intended as the difference between the actual post-event realization of the response and the (non-existing and therefore unobservable) hypothetical realization of that same response for the same treated units, were the event absent. To circumvent the implicit missing variables problem, DiD methods use the realizations of the response variable observed in comparable random samples of untreated units. The latter are samples of units drawn from the same infinite population, but they are not exposed to the event. They serve as control or comparison groups. They provide the “substitutes” for the non-existing untreated realizations of the responses in treated units during post-treatment periods. In short, DiD assumes that without treatment, and under certain circumstances, treated units would behave exactly as the control or untreated units during post treatment periods. Then for the estimation purposes, the method adopts a combination of before-after and treatment-control group comparisons. The event that affects the response variables was termed “treatment”, but it could be equally termed “causal factor” to emphasise that with DiD we are not estimating a mere statistical association among variates. With DiD we cultivate the ambition of evaluating whether a precise causative link between causes and effects –defined according to a model based on a proper identification of the relationship among variables– is actually consistent with the data, and estimate how intensive and statistically robust the causal-effect link actually is. DiD analysis has been widely employed in economics, public policy, health research, management, environment analysis, and other fields. There is a discussion about the true “fatherhood” of the method and, not surprisingly, there are clear pioneering antecedents of DiD applications outside economics. Examples include medicine (the study of the causes of London’ worst cholera epidemics of 1849 with 14,137 victims) and agriculture (studies of changes of soil productivity enhanced by new cultivation techniques in Africans’ neighbour areas in the 1980s conducted by revolutionary governments after the victory of their anti-colonialist movements in the second half of the 1970s). A recognised common methodological basis is R. A. Fisher’s analysis of variance (ANOVA). This Review is an introduction to the DiD techniques. It starts from the very basic methods used to estimate the so-called Average Treatment Effect upon Treated (ATET) in a 2–period and 2–group case and proceeds by covering many of the issues that emerge in a multi-unit and multi-period context. Particular attention will be devoted to the statistical assumptions needed for a correct definition of the identification process of the causal-effect relationship in the multi-period case, namely to the parallel trend hypothesis, to the no anticipation assumption, and to the SUTVA assumption. In the multi-period case, both the Homogeneous case (when treated units start being treated in the same periods) and the Heterogeneous case (when treated units start being treated in different periods) will be considered. Some space will be devoted to the developments associated to the DiD techniques employable in the presence of data clustering or spatial-temporal dependence. The Review includes brief presentations of some policy-oriented applications of DiD. Areas covered are income taxation, migration, regulation and environment management.