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 understood as the discrepancy between the post-event realisation of the response and the hypothetical realisation of that same outcome for the same treated units in the absence of the event. This theoretical discrepancy is clearly unobservable. To circumvent the implicit missing variable problem, DiD methods utilise the realisations of the response variable observed in comparable random samples of untreated units. The latter are samples of units drawn from the same population, but they are not exposed to the event under investigation. They function as the control or comparison group and serve as proxies for the non-existent untreated realisations of the responses in treated units during post-treatment periods. In summary, the DiD model posits that, in the absence of intervention and under specific conditions, treated units would exhibit behaviours that are indistinguishable from those of control or untreated units during the post-treatment periods. For the purpose of estimation, the method employs a combination of before–after and treatment–control group comparisons. The event that affects the response variables is referred to as “treatment.” However, it could also be referred to as “causal factor” to emphasise that, in the DiD approach, the objective is not to estimate a mere statistical association among variables. This review introduces the DiD techniques for researchers in economics, public policy, health research, management, environmental analysis, and other fields. It commences with the rudimentary methods employed to estimate the so-called Average Treatment Effect upon Treated (ATET) in a two-period and two-group case and subsequently addresses numerous issues that arise in a multi-unit and multi-period context. A particular focus is placed on the statistical assumptions necessary for a precise delineation of the identification process of the cause–effect relationship in the multi-period case. These assumptions include the parallel trend hypothesis, the no-anticipation assumption, and the SUTVA assumption. In the multi-period case, both the homogeneous and heterogeneous scenarios are taken into consideration. The homogeneous scenario refers to the situation in which the treated units are initially treated in the same periods. In contrast, the heterogeneous scenario involves the treatment of treated units in different periods. A portion of the presentation will be allocated to the developments associated with the DiD techniques that can be employed in the context of data clustering or spatio-temporal dependence. The present review includes a concise exposition of some policy-oriented papers that incorporate applications of DiD. The areas of focus encompass income taxation, migration, regulation, and environmental management.

This paper uses an Italian income tax treatment occurred in 2006/7 as a quasi-natural tax experiment and offers some fresh empirical evidence on how labour supply responds to exogenous income tax hikes. The specific features of the Italian tax measures, namely homogeneity and contemporaneity of the treatment across Italian regions, allow us to adopt the identification strategy of the two-way fixed effect (TWFE) panel data difference-in-differences (DID) model and to define the correct statistical framework of the study. We find that income tax hikes cause extensive negative adjustments of various response variables measuring the supply of labour services offered by treated taxpayers. DID estimated coefficients indicate that supply adjustments are significant, fast, and strong but not long-time lasting. Estimated time effects permit to evaluate the time impact of the tax measures and to conduct a dynamical policy evaluation of the tax hikes. We also show that treated families reduce in a similar manner their consumption with respect to families in the control groups and that analogous adjustment responses to income tax hikes characterise the growth of real per capita regional GDP. Altogether, estimates provide an overall view of the labour, consumption, and real growth effects of the income tax hikes. Results from DID estimations are further evaluated in comparison with the spatial–temporal patterns observed for every response variable in treated and untreated regions.

This paper analyzes the price behavior of Phase III (2013–2020) EU-ETS emission allowances of CO₂ by focusing on the dynamics of daily auction equilibrium prices and on the changes of the volatility of the underlying stochastic process. The paper derives the main testable statistical hypotheses (particularly that about the determinants of the conditional variance of prices) on the results derived in a model of optimal bidders’ behavior given the ETS auction rules. Tests are conducted by estimating various versions of GARCH models for both mean and variance equations of price return. Results show that the price dynamics is affected by factors including a measure of excess demand/offer and the number of winning bidders and that, contrary to expectations, reforms of the auction rules introduced at the end of Phase III explain a great part of the increased price volatility. The increased volatility is also positively associated with the bid spread and negatively associated with the number of bidders active in each auction round.

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.