Generalized Synthetic Control With Interactive Fixed Effects

Why Interactive Fixed Effects?

A conventional synthetic control constructs a comparison from observed untreated units. That comparison can be inadequate when treated and donor units respond differently to unobserved conditions over time. For example, a national policy change may affect regions differently because their underlying economic or demographic characteristics differ. Interactive fixed effects represent this kind of heterogeneity with latent unit characteristics whose effects vary over time.

Generalized synthetic control uses the panel of untreated observations to estimate those latent patterns and then predict the untreated outcomes missing for treated units. The approach expands the kinds of counterfactual patterns a synthetic control can represent. It does not, by itself, make a causal interpretation valid: the factor structure and the support in the observed panel must be credible.

Model and Target

Let \(i\) index units and \(t\) index time periods. Write \(D_{it}=1\) when unit \(i\) is exposed to treatment in period \(t\), and \(D_{it}=0\) otherwise. Let \(Y_{it}(0)\) be the outcome that unit \(i\) would have in period \(t\) without treatment, and \(Y_{it}(1)\) its outcome under treatment. The observed outcome is \(Y_{it}=Y_{it}(0)\) when \(D_{it}=0\), and \(Y_{it}=Y_{it}(1)\) when \(D_{it}=1\).

Definition: An interactive fixed-effects model specifies the untreated potential outcome as

Y_{it}(0)=X_{it}'\beta+\lambda_i'f_t+\varepsilon_{it}.

Here \(X_{it}\) is a vector of observed covariates, \(\beta\) is a common coefficient vector, \(\lambda_i\) is a vector of unit-specific latent factor loadings, \(f_t\) is a vector of time-specific latent factors, and \(\varepsilon_{it}\) is an idiosyncratic error. The factor dimension \(r\) is the length of \(\lambda_i\) and \(f_t\), so \(\lambda_i'f_t\) is a scalar.

The word interactive matters: the contribution of a latent factor depends both on a unit’s loading and on the factor’s value in that period. A model with one factor has contribution \(\lambda_i f_t\); units with different \(\lambda_i\) can therefore have different responses to the same movement in \(f_t\). Unit fixed effects and time fixed effects are special, more restrictive patterns, rather than a general substitute for this interaction.

The causal target can be stated for the treated observations under study. For example, an average treatment effect over those observations is estimated by comparing observed treated outcomes with predicted untreated outcomes:

\widehat{\tau} = \frac{1}{N_D} \sum_{(i,t):D_{it}=1} \left[Y_{it}-\widehat{Y}_{it}(0)\right], \qquad N_D=\sum_{i,t}D_{it}.

This expression is meaningful as a causal estimate only if \(\widehat{Y}_{it}(0)\) credibly estimates the outcome those treated units would have had without treatment. The model does not observe that counterfactual directly.

How Untreated Observations Supply the Counterfactual

For estimation, the treated outcomes cannot be used as observations of \(Y_{it}(0)\). Instead, the model is fitted to cells where \(D_{it}=0\), which can include never-treated units and untreated periods of units that later receive treatment. In simplified form, estimation chooses coefficients, loadings, and factors to minimize squared prediction errors in those untreated cells:

\min_{\beta,\{\lambda_i\},\{f_t\}} \sum_{(i,t):D_{it}=0} \left(Y_{it}-X_{it}'\beta-\lambda_i'f_t\right)^2,

subject to a chosen factor dimension and a normalization. The normalization is needed because the factorization is not unique: for any invertible \(r\times r\) matrix \(A\), replacing \(\lambda_i\) with \(A'\lambda_i\) and \(f_t\) with \(A^{-1}f_t\) leaves \(\lambda_i'f_t\) unchanged. A normalization fixes a convenient representation; it does not change the fitted counterfactuals.

Once the model is fitted, it predicts an untreated outcome in a treated cell by plugging in that unit’s covariates, its estimated loading, and the estimated factor for that period. Intuitively, untreated units help reveal the time factors, while untreated observations for a treated unit help reveal how strongly that unit loads on those factors. A prediction for a treated unit in a later period combines the two.

A short derivation makes the role of support precise. Suppose \(\beta\) and the factor vectors are known for a treated unit’s untreated periods, and let \(z_t=Y_{it}-X_{it}'\beta\) in those periods. Stack the corresponding factor row vectors into a matrix \(F_{\mathrm{pre}}\) and the residual outcomes into a vector \(z\). The model implies \(z=F_{\mathrm{pre}}\lambda_i+\varepsilon\). If \(F_{\mathrm{pre}}\) has full column rank \(r\), least squares gives

\widehat{\lambda}_i = (F_{\mathrm{pre}}'F_{\mathrm{pre}})^{-1} F_{\mathrm{pre}}'z.

The inverse exists precisely when the available factor variation distinguishes all \(r\) loading dimensions. The predicted untreated outcome in another period \(t\) is then \(X_{it}'\widehat{\beta}+\widehat{\lambda}_i'\widehat{f}_t\). In practice, the factors and coefficients are estimated too, so their estimation uncertainty also matters.

Worked Example: A Rank-One Counterfactual

Consider a hypothetical panel with no observed covariates and one latent factor. Suppose untreated donor observations are:

Unit	Period 1	Period 2	Period 3
Donor A	10	12	15
Donor B	20	24	30

For illustration, take the factor normalization \(\lambda_A=1\). The model \(Y_{it}(0)=\lambda_i f_t\) then gives \(f_1=10\), \(f_2=12\), and \(f_3=15\) from Donor A. Donor B is consistent with this factor sequence because its outcomes are \(2f_t\) in every period.

A treated unit T has untreated outcomes 15 and 18 in periods 1 and 2. Its loading is \(15/10=1.5\) using period 1 and \(18/12=1.5\) using period 2. Thus the factor model predicts its untreated outcome in period 3 as

\widehat{Y}_{T3}(0)=\widehat{\lambda}_T\widehat{f}_3 =1.5(15)=22.5.

If the hypothetical observed treated outcome in period 3 is 28.0, the estimated treatment effect for that cell is \(28.0-22.5=5.50\) outcome units. This exact-fit calculation illustrates the prediction logic; real panels have estimation error and may not fit a rank-one model exactly.

Worked Prediction With Covariates and Noisy Pre-Treatment Outcomes

Worked Example: Estimating a Loading by Least Squares

Suppose the factor sequence has been estimated from untreated observations as \(f_1=1\), \(f_2=2\), and \(f_3=3\). For simplicity, suppose the covariate coefficient is known to be \(\beta=2\), and the treated unit has \(X_{it}=4\) in all three periods. Its observed untreated outcomes in periods 1 and 2 are 11.2 and 13.8. Its observed outcome in period 3, when treated, is 19.0. Estimate its loading and counterfactual under the one-factor model.

First remove the covariate contribution, \(X_{it}\beta=4(2)=8\). The two untreated residual outcomes are \(11.2-8=3.2\) and \(13.8-8=5.8\). The least-squares loading minimizes \((3.2-\lambda)^2+(5.8-2\lambda)^2\), so

\widehat{\lambda} = \frac{1(3.2)+2(5.8)}{1^2+2^2} = \frac{14.8}{5} = 2.96.

The period-3 counterfactual is the covariate contribution plus the estimated factor contribution:

\widehat{Y}_{i3}(0)=4(2)+2.96(3)=8+8.88=16.88.

The estimated treatment effect for period 3 is \(19.0-16.88=2.12\) outcome units. This calculation treats the coefficient and factors as given to isolate the loading step; an applied analysis estimates them from untreated panel data and should account for that additional uncertainty.

Restrictions Needed for Identification

A low residual sum of squares is not enough to establish that a counterfactual is identified. The prediction relies on restrictions connecting the observed untreated cells to the unobserved treated counterfactuals. The main restrictions are:

A stable, adequate factor structure. Over the analysis period, the untreated outcome must be represented well by the specified covariates and a fixed-dimensional set of common factors with unit-specific loadings. The loadings and the meaning of the factors cannot change because a unit is treated.
Enough untreated support. Untreated units and periods must provide information about the factors needed for prediction. For a treated unit, its untreated observations must contain enough independent factor variation to identify its loading. Covariate patterns used for prediction must also be supported by the observations used to estimate the model.
No treatment effects in the estimation cells. Cells treated in the observed data are excluded from fitting the untreated-outcome model. There must also be no anticipation: a period classified as untreated cannot already have an outcome changed by expected treatment.
Residuals do not create a treatment-aligned counterfactual shift. Conditional on the modeled covariates and latent factors, the remaining errors must not contain a systematic shock that both predicts treatment and changes treated units’ untreated outcomes. In particular, an unmodeled shock specific to treated units at the time of treatment can be mistaken for a treatment effect.

The factors can absorb important unobserved confounding when that confounding has the shared interactive structure in the model. They cannot absorb every possible unobserved change. For example, if treated units experience a new, unit-specific event at the same time as treatment, and that event is not represented by observed covariates or the common factors, the predicted untreated outcome may be wrong even when pre-treatment fit looks excellent.

Worked Example: When Pre-Treatment Variation Is Not Enough

Consider a two-factor model with no covariates. A treated unit has two untreated observations. In both periods, the factor vector is \((1,1)'\), and the residual outcome is 5. The pre-treatment equations are therefore

\lambda_1+\lambda_2=5, \qquad \lambda_1+\lambda_2=5.

The factor matrix has two identical rows, so its rank is 1 rather than 2. The observations identify only the sum of the loadings, not the two loadings separately. For example, both \((\lambda_1,\lambda_2)=(2,3)\) and \((4,1)\) fit the untreated observations exactly.

Suppose the future factor vector is \((1,0)'\). The first loading pair predicts an untreated outcome of 2, while the second predicts 4. Because both pairs fit the observed untreated data, the counterfactual is not uniquely determined. By contrast, if the future factor vector were \((1,1)'\), both pairs would predict 5. The first case lacks the factor support needed for this particular prediction; a good pre-treatment fit alone does not resolve that failure.

Estimation Choices and Interpretation

The factor dimension \(r\) is a substantive and statistical choice. Too few factors may leave systematic untreated-outcome patterns in the residuals; too many may fit noise and produce unstable loadings, especially when a treated unit has few untreated observations. Researchers commonly compare candidate dimensions using held-out untreated predictions or other model checks. Such checks assess predictive performance on observed untreated cells; they do not prove that the same model correctly predicts the missing treated counterfactuals.

Likewise, factor loadings themselves are not generally interpretable as uniquely defined economic quantities because different normalizations can represent the same fitted values. The central output is the implied untreated prediction and, consequently, the treated-minus-predicted contrast. A fitted counterfactual should be reported with its uncertainty and with enough information about the factor specification and support to make its dependence on modeling assumptions transparent.

The causal interpretation is conditional: it is justified only to the extent that the untreated-outcome model, support conditions, and treatment-exogeneity restrictions are credible. Excellent fit before treatment is useful evidence, but it is not a guarantee that the model extrapolates correctly after treatment.

Common Mistake

Wrong reasoning: “The model matches treated units’ pre-treatment outcomes, so its post-treatment predictions must be valid.”
Correction: Pre-treatment fit shows agreement in observed untreated cells. It does not establish that the factor structure remains adequate in the missing counterfactual cells, that there is sufficient factor support, or that no treatment-aligned residual shock is omitted. Those are identifying restrictions, not consequences of a low pre-treatment error.

Key takeaway: Generalized synthetic control predicts missing untreated outcomes by combining time-varying latent factors, unit-specific loadings, and observed covariates estimated from untreated panel cells. Its causal credibility depends on a stable factor structure, sufficient untreated support, and the absence of unmodeled treatment-aligned changes in untreated potential outcomes.

Check Your Understanding

Answer each question using the model and restrictions developed above.

In the interactive fixed-effects model, what does the product \(\lambda_i'f_t\) allow to vary across units and time?
Why are treated observations not used as observations of \(Y_{it}(0)\) when fitting the untreated-outcome model?
What rank condition allows a treated unit’s loading to be estimated from its untreated periods?
In the two-factor example, why can a future factor vector \((1,0)'\) yield different counterfactual predictions even though the observed untreated outcomes fit exactly?
Give one reason why a good pre-treatment fit does not by itself establish a valid causal estimate.