Why Baseline Outcome Levels Need Not Match

Why Starting at Different Levels Is Not the Problem

A treated group and a comparison group may differ substantially before an intervention. One group might have lower average earnings, higher average symptom severity, or a different baseline outcome for other reasons. Difference-in-differences (DiD) does not require those initial averages to be equal. Its identifying requirement concerns something different: how the groups’ outcomes would have changed over time in the absence of treatment.

This distinction is easiest to see by separating levels from changes. A level is an outcome mean at a particular time. A change is the difference between outcome means across times. DiD compares the groups’ changes, so a baseline gap that remains constant under a shared untreated trend cancels from the contrast. A difference in their untreated trends, however, does not cancel and can be mistaken for a treatment effect.

Definition: Let $G=1$ denote membership in the group exposed to treatment after the intervention date, and $G=0$ denote membership in the comparison group. Let $Y_t$ be the outcome at time $t$, where $t=0$ is before treatment and $t=1$ is after treatment. The two-period DiD contrast is $$ \bigl(E[Y_1\mid G=1]-E[Y_0\mid G=1]\bigr) - \bigl(E[Y_1\mid G=0]-E[Y_0\mid G=0]\bigr). $$ It is the treated group’s observed mean change minus the comparison group’s observed mean change.

The contrast subtracts changes, not baseline levels. This arithmetic does not by itself establish a causal effect: its causal interpretation depends on whether the comparison group’s change represents the treated group’s change in the absence of treatment. In particular, a baseline difference is allowed, but a difference in those counterfactual changes can distort the estimate.

Separate Baseline Levels From Untreated Changes

Use potential outcomes to state the distinction precisely. For each group, $Y_t(0)$ denotes the outcome that would be observed at time $t$ without treatment, and $Y_1(1)$ denotes the post-treatment outcome under treatment. The treated group’s average treatment effect at the post period is $E[Y_1(1)-Y_1(0)\mid G=1]$, often called the average treatment effect on the treated.

Assume treatment begins between times 0 and 1, the pre-treatment outcome is unaffected by anticipation, the comparison group remains untreated, and group membership and outcome definitions are comparable across periods. Under these conditions, the pre-period observed outcome is $Y_0(0)$ in both groups, the comparison group’s post-period outcome is $Y_1(0)$, and the treated group’s post-period outcome is $Y_1(1)$. The relevant parallel-trends condition is equality of the groups’ average changes in their untreated potential outcomes:

E[Y_1(0)-Y_0(0)\mid G=1] = E[Y_1(0)-Y_0(0)\mid G=0].

This condition says nothing about whether the groups’ average untreated outcomes at time 0 are equal. They may start at different levels and still have equal average untreated changes. Conversely, equal starting levels do not imply equal untreated changes.

Worked Example: A Large Baseline Gap With Comparable Changes

Suppose, in a hypothetical program evaluation, the outcome is average weekly work hours. Before the program, the treated group averages 40 hours and the comparison group averages 70 hours. Afterward, the treated group averages 53 hours and the comparison group averages 75 hours. Assume the groups would each have gained 5 hours on average over this period without the program, so the parallel-trends condition is appropriate for this example.

The treated group’s observed change is $53-40=13$ hours. The comparison group’s observed change is $75-70=5$ hours. Therefore,

\widehat{\mathrm{DiD}} =(53-40)-(75-70) =13-5 =8\text{ hours per week}.

The baseline gap was $40-70=-30$ hours, but it does not appear in the change contrast. Under the stated assumptions, the comparison group’s 5-hour increase represents the treated group’s average untreated increase. The treated group’s observed increase exceeds that counterfactual increase by 8 hours, so the DiD estimate is an average treatment effect of 8 additional work hours per week for the treated group.

The interpretation depends on the assumed comparability of untreated changes, not on the observed baseline gap being small. The example stipulates that condition to show the arithmetic; in an actual study, the post-treatment untreated outcome for the treated group cannot be observed directly.

Why the Baseline Difference Cancels

The cancellation can be shown by writing the four means in terms of untreated outcomes and the treatment effect. For compactness, define the treated group’s average treatment effect at time 1 as $\tau=E[Y_1(1)-Y_1(0)\mid G=1]$. Also define each group’s average untreated change as $\delta_g^0=E[Y_1(0)-Y_0(0)\mid G=g]$.

The observed DiD contrast equals the treatment effect plus the difference in untreated changes. To see this, add and subtract the treated group’s average post-period untreated outcome:

\begin{aligned} \mathrm{DiD} &=\{E[Y_1(1)\mid G=1]-E[Y_0(0)\mid G=1]\}\\ &\quad-\{E[Y_1(0)\mid G=0]-E[Y_0(0)\mid G=0]\}\\ &=\{E[Y_1(1)-Y_1(0)\mid G=1]\}\\ &\quad+\{E[Y_1(0)-Y_0(0)\mid G=1] -E[Y_1(0)-Y_0(0)\mid G=0]\}\\ &=\tau+(\delta_1^0-\delta_0^0). \end{aligned}

The first term is the average treatment effect on the treated. The second is the difference between the groups’ average untreated changes. Under parallel trends, $\delta_1^0=\delta_0^0$, so the second term is zero and the DiD contrast equals $\tau$. Notice what is absent from this expression: there is no requirement that $E[Y_0(0)\mid G=1]$ equal $E[Y_0(0)\mid G=0]$. Baseline levels do not appear as a condition for identification.

This is an algebraic result, not a guarantee that parallel trends holds in a particular application. If the treated group would have changed differently even without treatment, then $\delta_1^0-\delta_0^0$ remains in the contrast. DiD then combines the treatment effect with that untreated-trend difference.

Worked Example: Equal Untreated Changes Despite Unequal Baselines

Consider a hypothetical study of a policy and average monthly spending, measured in dollars. Before the policy, the treated group averages $1,200 and the comparison group averages $900. Suppose both groups’ average spending would have fallen by $60 without the policy. The comparison group is observed to fall to $840; the treated group is observed to fall to $1,220 because the policy raises its spending relative to its untreated outcome.

The treated group’s observed change is $1{,}220-1{,}200=20$ dollars. The comparison group’s change is $840-900=-60$ dollars. Thus,

\widehat{\mathrm{DiD}} =(1{,}220-1{,}200)-(840-900) =20-(-60) =80\text{ dollars per month}.

To check the interpretation, the treated group’s untreated post-period mean would be $1{,}200-60=1{,}140$ dollars under the scenario’s assumptions. The treatment effect is therefore $1{,}220-1{,}140=80$ dollars per month. The baseline means differ by $300, but both untreated changes are \(-$60, so the level difference does not bias the DiD contrast.

The treated group’s observed spending rises by $20 even though its untreated spending would have fallen by $60. A positive observed change alone is not the treatment effect: the relevant comparison is the treated group’s change relative to the change expected without treatment.

Equal Baselines Do Not Guarantee Parallel Trends

The reverse distinction is just as important. A matched baseline is only one equality at one time. It does not determine what the groups’ outcomes would have done afterward without treatment. If their untreated changes differ, the DiD contrast includes that discrepancy even when their starting means are identical.

Worked Example: Equal Starting Means but Different Untreated Changes

In a hypothetical evaluation, both groups begin with a mean outcome of 50 points. Without treatment, suppose the treated group’s mean would rise by 9 points, while the comparison group’s mean would rise by 3 points. The comparison group is observed to increase from 50 to 53. The treated group is observed to increase from 50 to 63, and its assumed treatment effect is 4 points. Thus its untreated post-period mean would be $50+9=59$, and treatment raises that mean to 63.

The observed changes are $63-50=13$ points for the treated group and $53-50=3$ points for the comparison group. The DiD estimate is

\widehat{\mathrm{DiD}} =(63-50)-(53-50) =13-3 =10\text{ points}.

But the treatment effect in the stated scenario is $63-59=4$ points. The remaining 6 points are the difference in untreated changes: $9-3=6$. The decomposition confirms the result:

\mathrm{DiD} =\tau+(\delta_1^0-\delta_0^0) =4+(9-3) =10\text{ points}.

Here equal baseline levels do not rescue the estimate. The DiD contrast overstates the treatment effect by 6 points because the treated group would have improved more even without treatment.

What the Condition Does—and Does Not—Require

Parallel trends concerns equality of average untreated changes for the two groups, on the outcome scale used in the analysis. It does not require equal baseline means, identical outcomes for every individual, or identical observed changes after treatment. The treated group’s post-period observed change includes treatment, so it is not expected to match the comparison group’s change when treatment has an effect.

The condition is also about a counterfactual: the treated group’s average change had it remained untreated. That outcome is not observed for the treated group after treatment. A baseline table can describe level differences, but by itself it cannot establish whether the missing untreated change would have matched the comparison group’s change. The causal interpretation therefore depends on the design and substantive justification for using the comparison group, rather than on baseline equality.

The outcome scale matters to the meaning of “comparable changes.” In the additive DiD contrast presented here, parallel trends means equal changes in average outcome units. For example, a common increase of 5 hours is an additive trend; a common percentage increase need not correspond to equal changes in hours when baseline levels differ. The chosen scale determines which changes are being compared.

Common Mistake: Treating Baseline Balance as the Assumption

Wrong reasoning: “The baseline means differ, so DiD cannot be valid.” Correction: Different baseline levels are allowed. The relevant condition is equality of the groups’ average untreated changes.
Wrong reasoning: “The baseline means are equal, so parallel trends must hold.” Correction: Equality at one time says nothing by itself about how untreated outcomes would change over time.
Wrong reasoning: “The treated group changed by 13 units, so the treatment effect is 13 units.” Correction: The treated group might have changed without treatment. DiD subtracts the comparison group’s change to account for the counterfactual trend, subject to the identifying assumption.
Wrong reasoning: “Parallel trends means both groups’ observed outcomes must move together after treatment.” Correction: It concerns untreated potential-outcome changes. A treatment effect can make observed post-treatment changes differ.

Key takeaway: DiD permits different baseline outcome levels because it compares changes. Its causal interpretation requires the treated and comparison groups to have the same average change in their untreated outcomes; equal starting means neither are required nor sufficient.

Check Your Understanding

Use the distinction between baseline levels and untreated changes to answer these questions.

A treated group starts at 30 units and a comparison group at 55 units. What does this baseline difference, by itself, imply about whether DiD can identify an effect?
In a hypothetical setting, the treated group’s observed outcome rises by 12 units and the comparison group’s outcome rises by 7 units. Calculate the DiD contrast and state what additional assumption is needed for a causal interpretation.
Why does equality of the groups’ baseline means not establish parallel trends?
If the treated group’s untreated outcome would rise by 8 units and the comparison group’s by 5 units, what term contaminates the DiD contrast, and by how much?
Does parallel trends require each treated individual to have the same untreated change as each comparison individual? Explain what it requires instead.