Potential-Outcomes Derivation of the DiD Contrast

Why Write DiD in Potential-Outcomes Terms?

A two-period difference-in-differences (DiD) calculation compares outcome changes in a treated group with outcome changes in a control group. The arithmetic can be computed from observed outcomes, but a causal interpretation requires asking a different question: what would the treated group’s post-period outcome have been without treatment? Potential outcomes make that missing comparison explicit.

This tutorial derives the two-period DiD contrast using potential outcomes. The derivation clarifies which outcomes are observed in each group and period, which remain counterfactual, and what the observed contrast equals before any identifying assumption is added.

Groups, Periods, and Potential Outcomes

Let \(D_i=1\) indicate that unit \(i\) belongs to the group exposed to treatment in the post period, and let \(D_i=0\) indicate membership in the untreated control group. Index the baseline period by \(t=0\) and the post period by \(t=1\). For each unit and period, define \(Y_{it}(1)\) as the outcome that would be observed if the unit were treated at time \(t\), and \(Y_{it}(0)\) as the outcome that would be observed if it were untreated at time \(t\).

The two-period setup assumes that neither group is treated at baseline, the treated group receives treatment in the post period, and the control group remains untreated. We also assume no anticipation: future treatment does not change baseline outcomes. Under consistency, the observed outcome is the potential outcome corresponding to the treatment status actually received. The setup also presumes no interference between units, so one unit’s treatment does not change another unit’s potential outcome.

Thus, treated-group units have \(A_{i0}=0\) and \(A_{i1}=1\); control-group units have \(A_{i0}=A_{i1}=0\). The distinction between a potential outcome and an observed outcome matters: for any unit-period, only the potential outcome matching its actual treatment status is observed.

The table describes availability, not whether a potential outcome is conceptually meaningful. For example, \(Y_{i1}(0)\) for a treated-group unit is well-defined as the outcome that unit would have had in the post period without treatment, but it is not observed for that unit. Likewise, \(Y_{i1}(1)\) for a control-group unit is not observed. The DiD calculation uses observed outcomes only; its causal interpretation depends on how these missing potential outcomes relate to observed ones.

The Population DiD Contrast

Write \(E[Y_t(a)\mid D=d]\) for the mean potential outcome at time \(t\), under treatment status \(a\), among units in group \(d\). The population DiD contrast is the treated group’s observed mean change minus the control group’s observed mean change. By the treatment timing and consistency conditions, its observed-outcome form is:

This contrast is formed from observed cells: treated-group outcomes under no treatment at baseline and treatment after baseline, and control-group outcomes under no treatment in both periods. It is an algebraic contrast of observed means. To understand its causal content, add and subtract the treated group’s unobserved post-period mean under no treatment, \(E[Y_1(0)\mid D=1]\).

The first term is the average treatment effect on the treated in the post period: the treated group’s mean post-period outcome under treatment minus its mean post-period outcome under no treatment. The bracketed term is the difference between the groups’ mean changes in outcomes under no treatment. This second term compares two untreated potential-outcome trends, one of which is counterfactual for the treated group.

What the Decomposition Does—and Does Not—Tell Us

The decomposition identifies the key gap between a descriptive contrast and a causal effect. The DiD contrast equals the treated group’s average post-period effect when the bracketed untreated-trend difference is zero. Whether that equality is credible is an identifying question, not an algebraic consequence of computing the contrast. The next step in studying DiD is to examine the assumption that makes this equality plausible; here, the important point is that the counterfactual trend appears explicitly in the formula.

The contrast is not generally the difference in treatment effects between groups. For this two-period design, the control group is never treated, so its observed post-period outcomes do not reveal a treatment effect for controls. The first term in the decomposition concerns the treated group. Nor is the DiD contrast automatically the treated group’s change in outcomes: it subtracts the control group’s observed change, and this subtraction equals the causal effect only under an additional condition on untreated trends.

From Population Means to a Sample Calculation

In data, population means are replaced by sample means for the relevant groups and periods. Let \(\bar{Y}_{d,t}\) denote the observed sample mean in group \(d\) at time \(t\). The sample DiD estimate is:

This sample statistic estimates the corresponding population contrast, subject to the study’s sampling and design. It does not make the missing treated-group post-period outcomes under no treatment observable. The same potential-outcomes decomposition applies to the population target; in a sample, its components are not all directly calculable because the counterfactual mean remains missing.

Common Mistake: Treating the Contrast as the Effect by Definition

A frequent error is to write “the treated group increased by 12 points, controls increased by 4 points, so treatment caused an 8-point increase,” as though subtracting changes automatically removes every non-treatment difference. The subtraction removes the control group’s observed change. It removes the treated group’s untreated change only when the treated and control groups would have had the same mean untreated change. The potential-outcomes decomposition shows the remaining term explicitly.

A careful report therefore distinguishes the observed DiD contrast from the causal effect it may identify. State the contrast in the outcome’s units, identify the causal estimand of interest, and explain why the unobserved treated-group outcome trend without treatment can be related to the control group’s trend. Do not describe the counterfactual as an observed fact.