Cohort-Specific Time Effects and Unequal Exposure

Why Cohort-Specific Time Effects Matter

A panel can contain shocks shared by all units in a calendar period and changes that differ across cohorts. For example, a policy change or macroeconomic event may coincide with a common shift, while units in different cohorts respond differently. A model with only common time effects can miss that heterogeneity; a model with cohort-by-time interactions can describe it.

Unequal observation windows complicate the comparison. Cohorts may enter the panel in different periods, and the units observed in a cohort-period cell may change over time. A difference between raw cell means can then reflect changing unit composition as well as time effects. The key questions are which comparisons the model defines and whether the observed pattern of repeated observations can identify them.

Model and Interpretation

Definition: Let $y_{it}$ be the outcome for unit $i$ in calendar period $t$, and let $c(i)$ denote its fixed cohort. A cohort-specific time-effects model is $$ y_{it}=\alpha_i+\lambda_t+\sum_{c\ne c_0}\sum_{s\ne t_0}\gamma_{cs}\,\mathbf{1}\{c(i)=c\}\mathbf{1}\{t=s\} +x_{it}'\beta+\varepsilon_{it}, $$ where $\alpha_i$ is a unit effect, $\lambda_t$ is a common calendar-time effect, $c_0$ and $t_0$ are reference cohort and period, and $x_{it}$ contains any included covariates. The displayed interaction notation means a separate coefficient for each non-reference cohort and non-reference period: $\gamma_{ct}$ multiplies the indicator that unit $i$ belongs to cohort $c$ and observation $it$ is in period $t$.

The reference coding sets the interaction to zero for cohort $c_0$ and period $t_0$. Under this coding, $\lambda_t-\lambda_{t_0}$ is the modeled time change for the reference cohort, holding covariates fixed. The coefficient $\gamma_{ct}$ is the additional change for cohort $c$ in period $t$, relative to the reference cohort and relative to the reference period. It is a differential time effect, not the cohort’s total change and not automatically a causal effect.

The model assumes that the specified common time component and cohort-by-time terms, together with unit effects and covariates, adequately represent conditional mean differences. Interpreting a coefficient causally requires further assumptions, including that no omitted cohort-specific time-varying factors account for the differential change. A model specification alone does not establish those assumptions.

What the Interaction Contrast Compares

With two cohorts $A$ and $B$, and two periods $0$ and $1$, take $A$ and period $0$ as references. For predictions at fixed covariates, after accounting for unit effects, the model implies that the change for $A$ is $\lambda_1-\lambda_0$, while the change for $B$ is $\lambda_1-\lambda_0+\gamma_{B1}$. Subtracting these adjusted changes gives $\gamma_{B1}$. This is the familiar difference-in-differences contrast, but its identity concerns model-adjusted predictions or comparable within-unit changes, not arbitrary raw cell means.

One way to see why unit effects matter is to subtract each unit’s time average from its observations. For a unit observed in multiple periods, this removes its fixed $\alpha_i$. For a two-period comparison, the same cancellation appears directly by differencing within that unit:

y_{i1}-y_{i0} =(\lambda_1-\lambda_0)+\gamma_{c(i),1} +(x_{i1}-x_{i0})'\beta+(\varepsilon_{i1}-\varepsilon_{i0}),

where $\gamma_{A1}=0$ for the reference cohort. If covariates are held fixed, or their changes are adjusted for, the cohort contrast in mean within-unit changes estimates the interaction under the model. If different units supply the two period means, their unit effects need not cancel; the raw-mean contrast need not equal the interaction.

Worked Example: A Cohort Contrast With Repeated Units

Suppose two cohorts are observed in periods 0 and 1, with each listed unit observed in both periods. The outcome values are hypothetical, and there are no changing covariates in this example.

Cohort	Unit	Outcome, period 0	Outcome, period 1	Within-unit change
A	A1	10	12	2
A	A2	20	23	3
B	B1	100	105	5
B	B2	110	116	6

The mean within-unit change for cohort $A$ is $(2+3)/2=2.5$ outcome units. For cohort $B$, it is $(5+6)/2=5.5$ outcome units. Their difference is $$ (5.5)-(2.5)=3.0\ \text{outcome units}. $$

In the unit-effects model with $A$ as reference, the estimated common time change is 2.5 outcome units for this two-period comparison, and the estimated differential change for $B$ is 3.0 outcome units. This calculation uses within-unit changes, so the fixed level differences between units do not enter. It describes a differential change; a causal interpretation would require additional assumptions.

Unequal Exposure: Which Observations Identify the Effect?

An unbalanced panel is not necessarily a problem. It is common for some units to have longer observation windows than others. What matters for an interaction is whether the observed rows provide variation in the interaction after unit effects and the other included regressors have been accounted for. In linear algebra terms, a parameter contrast must be estimable from the design matrix after projecting out unit indicators and other included terms. A cell count by itself does not establish that condition.

For intuition, in a two-period fixed-effects model a unit observed in both periods contributes a within-unit change. A unit observed in only one period has no within-unit time change: subtracting its own observed value from itself leaves zero. Thus a single observation may help describe a raw cell mean, but by itself it cannot separate that unit’s fixed effect from the time or cohort-by-time component. The pattern of repeat observations across cohorts and periods matters, not just the number of observations in each cell.

Worked Example: Unequal Cell Counts With an Identified Contrast

Consider the following hypothetical panel. All three $A$ units are observed in both periods. Two $B$ units are observed in both periods, while a third $B$ unit is observed only in period 1. Covariates are absent or held fixed.

Cohort	Unit	Outcome, period 0	Outcome, period 1
A	A1	10	11
A	A2	20	22
A	A3	30	33
B	B1	100	104
B	B2	110	116
B	B3	—	150

The $A$ within-unit changes are $1,2,3$, giving a mean change of $(1+2+3)/3=2$ outcome units. The observed $B$ within-unit changes are $4,6$, giving a mean change of $(4+6)/2=5$ outcome units. The fixed-effects difference in changes is $5-2=3$ outcome units. Here both cohorts have repeated units spanning the two periods, so the comparison is supported by within-unit variation. The single period-1 observation for $B3$ supplies no within-unit change.

The raw means give a different answer. Cohort $A$’s means are $20$ in period 0 and $22$ in period 1, for a change of $2$. Cohort $B$’s means are $105$ in period 0 and $(104+116+150)/3=123.333$ in period 1, for a change of $18.333$. The difference between these raw changes is $18.333-2=16.333$, not 3. The discrepancy arises because the period-1 raw mean for $B$ includes a unit with no period-0 observation and a potentially different unit effect. The adjusted, within-unit calculation and the raw cell-mean calculation answer different questions.

Worked Example: When a Cohort Interaction Is Not Identified

Now suppose cohort $A$ has units observed in both periods, but every observed $B$ unit appears only in period 1. Use $A$ as the reference cohort and period 0 as the reference period. The $B$-by-period-1 interaction indicator equals 1 for each observed $B$ row and 0 elsewhere.

For each $B$ unit $i$, its only observed outcome has the form $$ y_{i1}=\alpha_i+\lambda_1+\gamma_{B1}+\varepsilon_{i1}. $$

There is no period-0 observation for that unit to subtract. Its one observed outcome cannot distinguish its unit effect $\alpha_i$ from the cohort-period component $\lambda_1+\gamma_{B1}$. After removing unit effects, the interaction indicator for these singleton observations has no within-unit variation. Therefore, the $B$-specific differential time effect is not identified by this observation pattern, even if the raw $B$-period-1 cell contains many units. Repeating the same period-1 observation across more singleton units does not create the missing within-unit time comparison.

The common time effect may still be informed by repeated $A$ units, but that does not identify a separate $B$ response. A model can report a coefficient only if its parameterization or extra restrictions resolve the aliasing; such a coefficient would then rely on those added restrictions, not on the observed within-unit variation alone.

Reading Results and Checking Support

With more than two periods or cohorts, each interaction coefficient is relative to the specified reference cohort and period. For example, $\gamma_{c2}$ compares cohort $c$’s adjusted change from the reference period to period 2 with the corresponding adjusted change for the reference cohort. It is not a comparison of cohort $c$’s level in period 2 with another cohort’s level, because fixed cohort level differences are absorbed by unit effects.

Before interpreting estimates, inspect the observation pattern at the unit level. Useful checks include:

Which units are observed in more than one calendar period?
Does each relevant cohort have within-unit observations spanning the periods needed for its contrast?
Are the cohorts’ observation windows sufficiently connected to separate a cohort-specific change from common period effects?
Do covariates change over time, and are their changes included or otherwise accounted for?
Are estimated interactions driven by a small set of repeat units or by an assumed restriction rather than direct panel variation?

A cohort defined by entry date or exposure history can have a restricted window by construction. In that setting, a cohort-by-calendar-period interaction describes differential change across the calendar periods that are actually supported by the panel design. It should not be silently reinterpreted as an effect of exposure duration. Calendar time and time since entry are distinct indices; separating their roles requires observations that provide the relevant variation.

Common Mistake and Practical Importance

A common mistake is to calculate differences between raw cohort-period means and label the result a fixed-effects interaction. The reasoning is: “The means changed by these amounts, so their difference is the cohort-specific time effect.” That conclusion is not generally valid. In an unbalanced panel, the units contributing to a cell mean may differ across periods, and the averages of their unit effects can change. The correction is to use the model’s adjusted predictions or a valid within-unit contrast, and then verify that the observation pattern identifies the contrast after unit effects are removed.

This distinction matters because a model with cohort-time interactions can otherwise suggest more information than the panel contains. Unequal numbers of observations are not automatically fatal, and equal cell counts do not guarantee identification. The decisive issue is the support for the required comparisons: which units are repeated, in which periods, and in which cohorts. For inferential claims, standard errors must also reflect the dependence structure of repeated observations, commonly by accounting for within-unit dependence.

Key takeaway: Common time effects describe a shared calendar component under the model; cohort-by-time interactions describe differential adjusted changes relative to a reference cohort. In unbalanced panels, interpret neither from raw cell means alone: check within-unit support and estimability after accounting for unit effects.

Check Your Understanding

Answer each question using the model and identification ideas developed above.

With cohort $A$ and period 0 as references, what does $\gamma_{B2}$ compare?
Why can the difference in raw cohort-period mean changes fail to equal a fixed-effects interaction?
A unit is observed only in period 1. What happens to its observation after removing its own unit mean, and what does that imply for within-unit time identification?
In a two-period panel, cohort $A$’s mean within-unit change is 1.5 and cohort $B$’s is 4.0. Under the stated reference coding, what is the estimated differential change for $B$?
Why do many single-period observations in a cohort not necessarily identify that cohort’s differential time response?