Parallel Trends as the Identifying Assumption

Why Parallel Trends Matters

Difference-in-Differences (DiD) compares the change in an outcome for a treated group with the change for a control group. The control group’s change is used to represent how the treated group’s outcome would have changed without treatment. That interpretation depends on a key assumption: parallel trends in the groups’ untreated potential outcomes.

The assumption concerns a comparison we cannot directly observe: after treatment begins, what would the treated group’s outcome have been if it had remained untreated? Parallel trends does not say that the treated and control groups have the same outcomes. It says that, over the time interval used for DiD, their mean outcomes would have changed by the same amount in the absence of treatment.

Definition: Let $G=1$ denote membership in the treated group and $G=0$ membership in the control group. Let $Y_t(0)$ be the outcome that would be observed at time $t$ if a unit were untreated. For a pre-treatment period $t=0$ and a post-treatment period $t=1$, parallel trends in untreated potential outcomes means $$ E[Y_1(0)-Y_0(0)\mid G=1] = E[Y_1(0)-Y_0(0)\mid G=0]. $$ The expected untreated outcome change is equal between groups over this specified interval.

What the Assumption Says—and Does Not Say

Parallel trends is an equality of changes, not of outcome levels. The treated group may begin with a higher or lower mean outcome than the control group. The assumption concerns whether their mean untreated outcomes would move by the same amount from the pre-treatment period to the post-treatment period. Equality of the two groups’ baseline means is neither the definition of parallel trends nor a substitute for it.

The assumption is about group means. It does not require each treated unit and each control unit to have the same individual outcome change. Nor does it require the groups to have identical outcome distributions. What must match is the average change in their untreated potential outcomes for the population and interval relevant to the analysis.

The period scope matters. In a two-period design, the assumption is about the change from the chosen pre-treatment period to the chosen post-treatment period. In a setting with several periods, assuming parallel trends for one interval does not automatically establish it for every other interval. A claim about a longer period, or about multiple adjacent periods, requires the relevant equalities for those periods.

From Parallel Trends to a Causal Contrast

To see why the assumption identifies a causal effect, define the average treatment effect on the treated (ATT) in the post-treatment period as $E[Y_1(1)-Y_1(0)\mid G=1]$, where $Y_1(1)$ is the outcome under treatment. Consider a design in which the treated group receives treatment only after period 0, and the control group remains untreated. Assume consistency: observed outcomes equal the potential outcomes under the treatment status actually received. Also assume no anticipation, so treatment does not affect the treated group’s period-0 outcome. Under these conditions, the treated group’s observed pre-treatment mean is its mean $Y_0(0)$, and the control group’s observed means are its means $Y_0(0)$ and $Y_1(0)$.

The observed DiD contrast is the treated group’s observed mean change minus the control group’s observed mean change. Add and subtract the treated group’s mean untreated change to express what this contrast contains:

\begin{aligned} \text{DiD} &= E[Y_1(1)\mid G=1]-E[Y_0(0)\mid G=1]\\ &\quad-\left(E[Y_1(0)\mid G=0]-E[Y_0(0)\mid G=0]\right)\\ &= E[Y_1(1)-Y_1(0)\mid G=1]\\ &\quad+\left(E[Y_1(0)-Y_0(0)\mid G=1] -E[Y_1(0)-Y_0(0)\mid G=0]\right). \end{aligned}

The first term on the final line is the ATT. The expression in parentheses is the difference between the treated and control groups’ mean untreated changes. Parallel trends sets that difference to zero. Therefore, under the stated treatment-timing and consistency conditions, the observed DiD contrast equals the ATT. Without parallel trends, the contrast combines the ATT with a difference in untreated trends; it need not be a causal effect.

Worked Example: Using the Assumption to Interpret DiD

Suppose a hypothetical program begins between periods 0 and 1. The treated group has mean observed outcomes of 50 units in period 0 and 64 units in period 1. The control group’s means are 48 and 54 units. Assume the control group remains untreated, the treated group has no anticipation, and parallel trends holds for the untreated outcomes over these two periods. Calculate DiD and state its interpretation.

The treated group’s observed change is $64-50=14$ units. The control group’s observed change is $54-48=6$ units. Thus, $$ \text{DiD}=(64-50)-(54-48)=14-6=8\text{ units}. $$

Under parallel trends, the treated group would have gained 6 units on average in the absence of the program, matching the control group’s untreated change. Its implied mean untreated outcome in period 1 is therefore $50+6=56$ units. The treated group’s observed mean is 64 units, so the estimated difference between its treated and untreated period-1 mean outcomes is $64-56=8$ units. The DiD contrast identifies an ATT of 8 outcome units under the assumptions; it is not justified by the arithmetic alone.

Worked Examples: Levels and Time Scope

Worked Example: Different Baseline Means, Equal Untreated Changes

In a hypothetical evaluation, the treated group’s mean outcome is 70 units before treatment, and the control group’s mean is 45 units. Suppose their untreated mean outcomes would each increase by 4 units over the study interval. After treatment, the observed treated-group mean is 80 units, while the control-group mean is 49 units. Calculate the DiD and explain what the baseline difference tells us.

The treated group’s observed change is $80-70=10$ units. The control group’s observed change is $49-45=4$ units. The contrast is $$ \text{DiD}=(80-70)-(49-45)=10-4=6\text{ units}. $$

The assumed untreated change for the treated group is 4 units, so its mean untreated outcome in period 1 would be $70+4=74$ units. Comparing the observed treated mean of 80 units with that counterfactual mean gives $80-74=6$ units. The groups’ baseline means differ by $70-45=25$ units, but this difference does not contradict equal untreated changes. Under the assumptions stated, the 6-unit contrast identifies the ATT.

Worked Example: A Matching Pre-Trend Does Not Establish the Post-Period Assumption

Consider hypothetical group means over three years. Treatment begins after year 2020. The treated group’s observed means are 100 in 2019, 104 in 2020, and 116 in 2021; the control group’s means are 80, 84, and 90. The treated group is untreated through 2020 and treated in 2021. What do these data show about the scope of parallel trends?

From 2019 to 2020, the observed treated-group change is $104-100=4$ units, and the control-group change is $84-80=4$ units. The observed pre-treatment changes match over that interval.

For the 2020-to-2021 interval, the control group’s change is $90-84=6$ units. The treated group’s observed change is $116-104=12$ units, but that change combines its untreated trend and the treatment effect. It cannot be treated as the treated group’s untreated change. The data alone do not reveal whether its mean untreated change would have been 6 units or some other amount. Thus the matching 2019-to-2020 changes do not establish parallel trends for the 2020-to-2021 interval.

If the treated group’s untreated mean would have risen by 6 units, its period-1 counterfactual mean would be $104+6=110$ units, giving an effect of $116-110=6$ units. If its untreated mean would instead have risen by 12 units, its counterfactual mean would be $104+12=116$ units, giving an effect of 0 units. Both calculations use the same observed means; they differ because the unobserved treated-group trend is not determined by the observed pre-treatment equality. Identifying the effect requires an assumption about that specific post-period untreated trend.

Substantive Meaning and Assessment

Parallel trends is a substantive claim about the process generating outcomes, not a pattern guaranteed by the DiD calculation. It is plausible when, absent treatment, the factors driving outcome changes would have affected the treated and control groups similarly over the stated interval. It may be less plausible if a separate event, policy, or changing population affects one group’s untreated outcomes differently during that interval.

Observed outcomes in the post-treatment period cannot directly reveal the treated group’s untreated potential outcomes. Consequently, the identifying assumption cannot generally be verified from the treated group’s post-treatment data. Similar pre-treatment changes can provide relevant context, but they do not prove that the untreated changes would remain equal after treatment begins. A careful analysis should state the groups, outcome, interval, and population for which parallel trends is being assumed, and explain why the assumption is substantively credible.

Specify the contrast.
Name the treated and control groups, outcome, and pre- and post-treatment periods used in the DiD calculation.

State the counterfactual assumption.
Write that the groups would have had equal mean untreated outcome changes over that exact interval.

Assess its substantive basis.
Consider whether the groups would plausibly have faced similar forces affecting the outcome in the absence of treatment.

Limit the conclusion.
Interpret the DiD as an ATT only when the identifying assumptions, including parallel trends, are credible for the population and interval at issue.

Common Mistake

Wrong reasoning: “The treated and control groups had similar outcome levels before treatment, so parallel trends holds.” Similar baseline levels do not establish anything by themselves about the groups’ untreated changes. The reverse error is also common: concluding that parallel trends is impossible because baseline levels differ.

Correction: Compare changes, not just levels, and state the period those changes concern. Parallel trends requires equality of the groups’ mean untreated changes over the interval used for causal interpretation. Even similar observed pre-treatment changes are supporting evidence rather than proof of the unobserved post-treatment counterfactual trend.

Key takeaway: Parallel trends is an identifying assumption about equal mean changes in untreated potential outcomes for the treated and control groups over a specified interval. It is not an assumption of equal outcome levels, and its causal role is to make the control group’s untreated change a valid counterfactual change for the treated group.

Check Your Understanding

Answer each question by identifying the groups, outcome changes, or period at issue.

State parallel trends for a treated and control group observed before and after treatment, using untreated potential outcomes.
Can parallel trends hold when the groups’ baseline mean outcomes differ? Explain what must be equal instead.
Under parallel trends, the control group’s mean outcome rises by 3 units. What does the assumption imply about the treated group’s mean untreated change over the same interval?
Why does a matching change in observed outcomes during a pre-treatment interval not prove parallel trends for the post-treatment interval?
In a two-period DiD design, which interval does the parallel-trends assumption concern, and why should the interval be stated explicitly?