Why Predictor and Window Choices Matter
Synthetic control constructs a comparison for a treated unit from untreated donor units. The pre-intervention information used to describe the treated unit helps determine which donors can reproduce its observed history and characteristics. Choosing that information is a design decision: the goal is to represent features that are relevant to the treated unit’s untreated outcome trajectory, using measurements that treatment could not have changed.
A predictor list that is too sparse may leave important differences in baseline risk or outcome history unrepresented. A list that includes variables measured after treatment, or variables already responding to anticipation, can instead build treatment effects into the comparison. The choice of fitting period also matters: a long period can capture persistent patterns but may mix distinct regimes, while a short period can miss important trajectory features or overemphasize temporary noise.
Notation and the Role of Predictors
Suppose unit 1 receives an intervention after time \(T_0\), and units \(j=2,\ldots,J+1\) are potential donors. The pre-intervention period is \(t=1,\ldots,T_0\). Write \(Y_{it}\) for the outcome of unit \(i\) at time \(t\). Let \(X_{i,k}\) denote predictor \(k\) for unit \(i\), measured before the intervention. Some predictors are baseline characteristics, such as population size or age composition; others are summaries of past outcomes, such as an outcome value in a particular year or the mean across several years.
Predictors need not be causes of the outcome to be useful. A variable can be valuable because it predicts untreated outcomes, even if it is not itself a direct causal mechanism. Conversely, a variable that strongly predicts the observed outcome is not automatically suitable: its measurement must be genuinely pre-treatment, and its role in the causal timeline must be considered.
In standard synthetic control, selected predictor values for the treated unit are compared with the corresponding values for donors, and pre-intervention outcomes are used to assess whether a donor combination can reproduce the treated unit’s history. This tutorial concerns what information and time periods to include, not how to calculate the donor weights.
Selecting Outcomes and Covariates
Pre-intervention outcomes are often especially informative because they directly record the trajectory the synthetic comparison is meant to reproduce. A single baseline outcome may distinguish high-outcome from low-outcome units, but it may say little about trend, volatility, or a recent change in level. Multiple outcome lags, or summaries such as period averages, can represent these features more fully. Their usefulness depends on the application: a lag from a relevant period may be more informative than a large collection of nearly redundant observations.
Covariates can add information about why units have different outcome levels or trajectories. Examples include population, economic composition, healthcare capacity, or baseline disease burden. Prefer characteristics with a clear substantive relationship to untreated outcomes and reliable measurement before treatment. A predictor should not be included merely because it is available in the data.
Timing is essential. A covariate measured before formal implementation may already have responded to a public announcement or other anticipation of treatment. If the intervention could affect a variable, directly or through a pathway such as service use, behavior, or resource allocation, that variable is not a safe baseline predictor when measured after the effect could begin. Adjusting for a post-treatment mediator can remove part of the effect one intends to estimate, while conditioning on other treatment-affected variables can introduce additional bias.
This principle does not mean every pre-treatment variable must be included. A large set of noisy or highly redundant predictors can make the design harder to interpret, and the data may not support matching all of them well. State why each predictor is included, when it was measured, and how it is expected to relate to the untreated outcome.
Worked Example: Choosing Baseline Predictors for a Health Intervention
A region begins a new treatment-access program after 2022. Analysts have annual data on avoidable hospital admissions per 10,000 residents, population age composition, clinic capacity, and program enrollment. They want to choose predictors using information available through 2022. Assume the program was announced in 2022 and could have changed enrollment before implementation.
Outcome history. Admissions in 2019, 2021, and 2022 are potential outcome predictors, all measured before implementation. The 2019 value represents an earlier level, while 2021 and 2022 help describe more recent conditions. If 2022 admissions could already respond to announcement or anticipation, however, the 2022 value should not be treated as an unaffected baseline without further justification.
Baseline covariates. Age composition measured in 2021 is eligible if the program could not plausibly change it over that interval. Clinic capacity measured in 2021 may be useful if it reflects pre-existing service availability and was not altered in anticipation.
Program enrollment. Enrollment recorded in 2022 is not a safe predictor under the stated facts: the announcement could have changed it. It could therefore encode an early response to the intervention. Enrollment from a clearly earlier, unaffected year might be considered instead, if it has a defensible relationship to untreated admissions.
The resulting candidate set might use admissions in 2019 and 2021, age composition in 2021, and clinic capacity in 2021. The exclusion of 2022 admissions and enrollment follows from the timing concern, not from a rule that later calendar years are always unusable. A defensible decision depends on when effects could begin.
Choosing the Fitting Period
The fitting period should be defined using the intervention date and the history relevant to the estimand. A useful period has enough observations to represent the treated unit’s pre-intervention level and trajectory, but it should not automatically include every available year. Changes in outcome definition, data collection, population boundaries, or underlying policy regimes may make older observations less comparable to the period of interest.
A short period can make the comparison overly dependent on a few unusual years. It may fail to reveal a persistent trend or a recurring seasonal pattern. A long period can provide more information, but it can also give substantial influence to remote years that describe a different environment. It may combine multiple regimes whose relationship to the post-intervention untreated outcome is doubtful. The appropriate window is therefore a substantive choice, not simply the longest or shortest available interval.
Within a selected period, analysts may use individual outcome lags, averages over meaningful blocks, or both. Averages can reduce the influence of a one-year fluctuation, but they conceal variation within the block. Individual lags can preserve the shape of the trajectory, but many such predictors can be redundant. Report the years and construction of every outcome summary so the chosen representation is transparent.
A period should not be shortened solely because the treated unit’s earlier history is difficult to match. Doing so can make apparent agreement look better by omitting inconvenient evidence. If there is a substantive reason to exclude a period—such as a documented measurement change or a different policy regime—describe it and examine whether the design conclusion depends on that choice. Later diagnostics can assess the quality of pre-intervention fit; the window itself should first have a clear, treatment-independent rationale.
Worked Example: Comparing Candidate Fitting Windows
A hypothetical region is treated after 2020, and its annual outcome and a simple reference trajectory are available for 2012–2020. The reference values below are only a planning summary used to compare candidate windows; they are not proposed synthetic control weights. Define the discrepancy in each year as treated outcome minus reference outcome.
| Year | Treated outcome | Reference outcome | Discrepancy |
|---|---|---|---|
| 2017 | 50 | 49 | 1 |
| 2018 | 52 | 51 | 1 |
| 2019 | 54 | 53 | 1 |
| 2020 | 55 | 54 | 1 |
For 2017–2020, the mean squared discrepancy is
Suppose 2012–2016 contain an older measurement regime, while the outcome definition is stable from 2017 onward. Then 2017–2020 is a defensible candidate fitting window because it is both pre-intervention and measured under the relevant regime. The calculation summarizes agreement over that candidate period; it does not establish that the window is causally valid or that the eventual synthetic control will fit equally well.
If the earlier years were instead measured comparably and represented an important part of the untreated trajectory, omitting them merely because they were less similar would be hard to defend. The decision turns on the measurement and substantive history, not on choosing whichever period yields the smallest discrepancy.
Why Prognostic Predictors Can Help
The basic rationale for using predictors is that units with similar relevant pre-treatment characteristics may have more similar untreated outcomes. This is a design rationale, not a guarantee: predictive variables cannot by themselves establish that the treated unit’s untreated trajectory would have followed the donor combination after treatment.
A simple squared-error identity illustrates why a predictor of untreated outcomes can matter. Let \(Y\) denote an untreated outcome at a target time and \(X\) a predictor. For any prediction \(a(X)\), if the conditional mean exists, then
To see this, write \(Y-a(X)=(Y-E[Y\mid X])+(E[Y\mid X]-a(X))\), square, and take the conditional expectation. The cross-product has conditional expectation zero because \(E[Y-E[Y\mid X]\mid X]=0\). A predictor that captures systematic differences in \(E[Y\mid X]\) can therefore help distinguish units whose untreated outcomes are expected to differ. Synthetic control is not simply this prediction exercise, and matching on a predictor does not ensure the necessary counterfactual assumptions. The identity explains the predictive motivation, not causal identification.
Worked Example: Using a Pre-Outcome Average
Two hypothetical regions have annual pre-intervention admission rates, per 10,000 residents. Region A has rates 40, 50, and 60 in 2017–2019; Region B has rates 50, 50, and 50. Their three-year means are
Matching only the three-year average would treat these histories as equal in level, but it would not represent their different trajectories: A is rising, while B is stable. If trend is substantively relevant, individual pre-period outcomes or a clearly defined trend summary should supplement or replace the average. If short-term variation is mostly measurement noise and the average level is the primary concern, the average may be a useful predictor. The choice follows the outcome process of interest; neither summary is universally superior.
The unit for each rate and each average is admissions per 10,000 residents. The calculation uses only pre-intervention outcomes, so it does not use the effect being estimated to choose the predictor.
A Practical Selection Sequence
Identify implementation and any plausible anticipation date. Treat measurements potentially affected by either as ineligible baseline predictors unless their unaffected status is justified.
Decide whether level, trend, recent conditions, or persistent baseline risk matters for the post-intervention outcome in this application.
Select pre-treatment outcome values or summaries and covariates with a substantive predictive rationale. Record their measurement dates, units, and definitions.
Use years with comparable measurement and relevant conditions. Explain why the start and end dates represent the history needed for the counterfactual comparison.
When reasonable alternatives exist, document them for sensitivity assessment rather than silently selecting a window or predictor set because it gives a preferred result.
Common Mistake: Treating Every Pre-Implementation Measurement as Baseline
A frequent error is to assume that a variable is safe because it was recorded before formal implementation. For example, an analyst might include service enrollment measured after a public announcement, arguing that treatment had not yet begun. But if the announcement changed enrollment, that measurement is already treatment-affected in the relevant causal sense. The correction is to define the earliest plausible response date, use measurements from before that date when possible, and explain any exception.
A related error is to add many outcome lags without considering what they represent. More years do not automatically mean better prediction: the additional observations may come from a different measurement regime or may duplicate nearly identical information. Choose a period and summaries that represent the untreated history of interest, and give a substantive reason for the choices.
Check Your Understanding
Use the timing and outcome rationale in each answer, not only whether a variable was recorded before formal implementation.
- A policy is announced in March and begins in October. Could a variable measured in July automatically be treated as an unaffected baseline predictor? Explain.
- A three-year outcome average is identical for two regions, but one region’s outcome is rising and the other’s is falling. What information does the average omit, and what alternative summaries might represent it?
- Give one reason a long fitting period could be undesirable even when every year precedes treatment.
- Why might a pre-treatment covariate be useful even if it is not a direct cause of the outcome?
- An analyst removes early years only because the treated unit is difficult to match in those years. What is the design concern, and what would make exclusion more defensible?