Causal Factor Model (CFM)

Causal Factor Model (CFM)#

When to Use This Method#

You have a single treated unit and a pool of untreated control units observed over many periods, an intervention switches on at a known date, and the units move together through shared, unobserved forces — an aggregate business cycle, an industry-wide demand shift, a regional shock — but not in parallel. Each unit responds to those common forces with its own sensitivity. This is the regime where difference-in-differences (which assumes the treated and control units share a common time effect, the parallel-trends assumption) and plain synthetic control (which pins the counterfactual to a convex combination of controls) are hard to justify.

The causal factor model of Bai and Wang ([BaiWang2026]) targets that regime with a different idea about what the intervention does. It writes each unit’s outcome as a small number of latent common factors \(\mathbf{f}_t\) (the shared forces) weighted by unit-specific factor loadings \(\boldsymbol{\lambda}_i\) (how strongly that unit feels them), and it lets the intervention change the treated unit’s loadings. The treatment effect is then a structural break in how the treated unit responds to the common shocks, and the target is the systematic causal effect

\[\tau^*_{it} = [\boldsymbol{\lambda}_i(1) - \boldsymbol{\lambda}_i(0)]^\top \mathbf{f}_t + [a_i(1) - a_i(0)],\]

the difference between the treated unit’s post- and pre-intervention systematic components. Here \(\boldsymbol{\lambda}_i(0)\) and \(\boldsymbol{\lambda}_i(1)\) are the loadings before and after the intervention, and \(a_i(0), a_i(1)\) are unit-specific intercepts.

Why the systematic target matters#

The methods you already know model only the untreated potential outcome \(Y_{it}(0)\), form a fitted counterfactual \(\widehat Y_{it}(0)\), and report the difference \(Y_{it}(1) - \widehat Y_{it}(0)\). Bai and Wang show this carries a one-sided idiosyncratic error: the observed treated outcome contains its own noise term \(\varepsilon_{it}(1)\) that the imputed counterfactual cannot cancel, and that term does not vanish for a fixed unit and date. When the idiosyncratic variation is large relative to the predictable part — common in panel applications with low explanatory power — the naive gap is an unstable object for inference.

CFM instead models both potential outcomes inside the same factor structure and re-estimates the treated unit’s loading after the intervention, so the reported effect is the change in the systematic component directly. The individual effect \(\tau_{it} = \tau^*_{it} + [\varepsilon_{it}(1) - \varepsilon_{it}(0)]\) is not point-identified for a fixed unit and date, because the idiosyncratic difference is unobserved; the systematic part \(\tau^*_{it}\) is. This is the essential contrast with Factor Model Approach (FMA), which is a single-equation imputation estimator on the same factor scaffolding. On data with no loading break the two coincide; they diverge exactly when the intervention changes the treated unit’s exposure to the common shocks.

Reach for CFM when#

  • the treated and control units co-move through latent common factors but not in parallel (heterogeneous trends);

  • the pre-period is long enough to estimate the treated unit’s factor loading;

  • you want a formal confidence band on the effect path without asserting parallel trends or a convex donor-weight fit;

  • you specifically suspect the intervention changed the treated unit’s response to shared shocks — a loading break — rather than adding a fixed increment.

Do not use CFM when#

  • you need the individual (not systematic) effect and the idiosyncratic variance is large — that object is not identified here;

  • the pre-period is too short to estimate loadings on the chosen number of factors;

  • the intervention plausibly changed the common factor process itself and the treated group is large — that is the potential-factors regime (Bai and Wang Section 4.2, Proposition 2), which this estimator does not yet cover.

Notation#

Let \(i = 1\) denote the treated unit and \(i = 2, \dots, n\) the controls, observed for \(t = 1, \dots, T\). The intervention occurs at \(T_0 + 1\), so periods \(t \le T_0\) are pre-treatment and \(t > T_0\) are post-treatment. Each outcome follows a factor model

\[y_{it} = a_i + \boldsymbol{\lambda}_i^\top \mathbf{f}_t + \varepsilon_{it},\]

with \(\mathbf{f}_t\) an \(r \times 1\) vector of latent common factors, \(\boldsymbol{\lambda}_i\) the unit’s loadings, \(a_i\) a unit intercept, and \(\varepsilon_{it}\) idiosyncratic noise. For the treated unit the loadings and intercept take the value \((a_1(0), \boldsymbol{\lambda}_1(0))\) before the intervention and \((a_1(1), \boldsymbol{\lambda}_1(1))\) after. The estimand is the systematic causal effect \(\tau^*_{1t}\) for \(t > T_0\), and its post-period average, the systematic ATT \(\bar\tau^* = (T - T_0)^{-1} \sum_{t > T_0} \tau^*_{1t}\). The intercept shift \(\kappa = a_1(1) - a_1(0)\) collects the constant part of the break, including any constant shift in the factor process.

Assumptions#

  1. Factor structure. Both potential outcomes admit the same factor model with a fixed number of common factors \(r\); the factors are common across treated and control units.

    Remark. The controls identify the factors, so their outcomes must be driven by the same latent forces as the treated unit. This is what replaces parallel trends: the units need not move together, only respond to the same shocks.

  2. The intervention does not change the factor process. The policy alters the treated unit’s loadings (and possibly slope coefficients on covariates) but leaves the common factors \(\mathbf{f}_t\) themselves unchanged.

    Remark. This is the Proposition 1 benchmark: appropriate for a targeted or partial-equilibrium intervention whose feedback to the aggregate trends is negligible. A constant shift in the factor process is permitted, because with unit intercepts it is absorbed into the post-treatment intercept (the constant-shift specification of Section 4.2), and is what \(\kappa\) measures.

  3. Long panel on both sides. The number of controls, the pre-period length, and the post-period length all grow; the treated unit may be unique (\(n_0 = 1\)).

    Remark. The factors are learned from the controls, so a wide control pool sharpens them; the treated loadings are identified from the time-series variation within each regime, so both the pre- and post-periods must be long enough to estimate an \(r\)-vector.

  4. Weak dependence of the idiosyncratic errors. The \(\varepsilon_{it}\) are weakly cross-sectionally and serially dependent with finite variance, so cross-sectional and time averages of the noise vanish asymptotically.

    Remark. This is why the systematic effect is identified while the individual effect is not: averaging over units or time kills the idiosyncratic term, but for a fixed \((i, t)\) it survives.

Inference and diagnostics#

The standard error of \(\widehat\tau^*_{1t}\) has two asymptotically-uncorrelated pieces (Bai and Wang appendix A.2), and the total variance is their sum:

  • a treated-regression component \(V^{\text{reg}}\), from estimating the pre- and post-treatment loadings and intercept. It is a block-additive heteroskedasticity-robust (HC1) sandwich: the pre- and post-block scores are asymptotically independent, so \(\mathrm{Var}(\widehat\tau^*_{1t}) = \mathbf{c}_t^\top \widehat{\mathrm{Var}}(\widehat{\boldsymbol\theta}_1)\, \mathbf{c}_t + \mathbf{c}_t^\top \widehat{\mathrm{Var}}(\widehat{\boldsymbol\theta}_0)\, \mathbf{c}_t\) with \(\mathbf{c}_t = (1, \mathbf{f}_t^\top)^\top\);

  • a factor-estimation component \(V^f\), from learning the common factors off the controls (appendix A.20). It enters through \([\boldsymbol\lambda_1(1) - \boldsymbol\lambda_1(0)]^\top \mathrm{Var}(\widehat{\mathbf f}_t) [\boldsymbol\lambda_1(1) - \boldsymbol\lambda_1(0)]\) and can be turned off with factor_variance=False to report the regression component alone.

The per-period (1 - alpha) interval is \(\widehat\tau^*_{1t} \pm z_{\alpha/2}\,\widehat{\mathrm{SE}}_t\), and the ATT interval aggregates the same components over the post-period.

Two diagnostics accompany the fit. The intercept-shift test reports \(\widehat\kappa\) and its t-statistic (a post-treatment level break, or a constant shift in the factor process); a Chow F-statistic tests parameter stability of the treated-unit factor regression at the intervention date. A large Chow statistic supports allowing the treated loadings to break.

Number of factors#

The factor count is chosen from the controls by the Ahn and Horenstein (2013) eigenvalue-ratio (factor_selection="er") or growth-ratio ("gr") criteria — maximisers of successive eigenvalue ratios of the control covariance — or by the Bai and Ng (2002) information criterion ("bai_ng"). Pass n_factors to fix it directly; a one-versus-two factor robustness pass is the paper’s habit.

Example#

import pandas as pd
from mlsynth import CFM

# California Prop 99: cigarette sales, treated from 1989.
df = pd.read_csv("basedata/smoking_data.csv")
df["treat"] = ((df.state == "California") & (df.year >= 1989)).astype(int)

res = CFM({
    "df": df, "outcome": "cigsale", "treat": "treat",
    "unitid": "state", "time": "year",
    "factor_selection": "er",       # ER and GR both pick 1 factor here
    "display_graphs": False,
}).fit()

res.att                              # mean systematic reduction in pack sales
res.att_ci                           # asymptotic 95% CI for the ATT
res.design.tau                       # per-period systematic effect path
res.inference_detail.ci_lower_t      # per-period CI band
res.metadata["kappa_t"]              # intercept-shift test t-statistic
res.metadata["chow_fstat"]           # structural-break diagnostic

The reported gap is the systematic causal effect \(\tau^*\) — not observed - counterfactual — because that is the estimand; counterfactual is the systematic untreated path \(a_1(0) + \boldsymbol\lambda_1(0)^\top \mathbf{f}_t\).

Verification#

CFM reproduces Bai and Wang’s two empirical applications — California Prop 99 and German reunification — on the authors’ data. See CFM — Causal Inference Using Factor Models (Bai & Wang 2026) for the cell-by-cell comparison (both applications select a single factor by ER and GR; the Chow break statistic and the intercept-shift t-statistics match the paper), and the durable case benchmarks/cases/cfm.py.

Core API#

class mlsynth.CFM(config: CFMConfig | dict)#

Causal Factor Model (Bai & Wang 2026) estimator.

Parameters:

config (CFMConfig or dict) – Configuration object. See mlsynth.config_models.CFMConfig.

Returns:

CFMResults – Frozen container with the systematic-effect design, asymptotic inference, the systematic causal-effect path, and the ATT.

fit() → CFMResults#

Run the CFM pipeline end to end.

class mlsynth.config_models.CFMConfig(*, df: ~pandas.DataFrame, outcome: str, treat: str, unitid: str, time: str, display_graphs: bool = True, save: bool | str = False, counterfactual_color: ~typing.List[str] = <factory>, treated_color: str = 'black', plot: ~mlsynth.config_models.PlotConfig = <factory>, factor_selection: ~typing.Literal['er', 'gr', 'bai_ng'] = 'er', n_factors: ~typing.Annotated[int | None, ~annotated_types.Ge(ge=1)] = None, max_factors: ~typing.Annotated[int, ~annotated_types.Ge(ge=1)] = 10, factor_variance: bool = True, alpha: ~typing.Annotated[float, ~annotated_types.Gt(gt=0.0), ~annotated_types.Lt(lt=1.0)] = 0.05)#

Configuration for the Causal Factor Model (CFM) estimator.

Implements Bai & Wang (2026), “Causal Inference Using Factor Models”. CFM models both potential outcomes within a single factor structure and lets the treated unit’s factor loadings break at the intervention date, targeting the systematic causal effect for a single treated unit. This is the Proposition 1 / constant-shift regime: the intervention changes treated units’ exposure to the common shocks (loadings) but not the factor process itself.

Parameters:

  • factor_selection ({“er”, “gr”, “bai_ng”}) – How to choose the number of common factors. "er" and "gr" are the Ahn-Horenstein (2013) eigenvalue-ratio and growth-ratio estimators (the paper’s primary criteria); "bai_ng" uses the Bai-Ng (2002) information criterion. Ignored when n_factors is supplied.

  • n_factors (int or None) – Override the data-driven factor count. None triggers factor_selection.

  • max_factors (int) – Upper bound passed to the factor-selection routine.

  • factor_variance (bool) – Whether to add the factor-estimation variance component V_f (Bai & Wang appendix A.2.2) to the standard errors. False reports the treated-regression component V_reg only.

  • alpha (float) – Two-sided significance level for CIs.

model_config: ClassVar[ConfigDict] = {'arbitrary_types_allowed': True, 'extra': 'forbid'}#

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

References#

[BaiWang2026]

Bai, J., & Wang, P. (2026). “Causal Inference Using Factor Models.” Working paper.

Ahn, S. C., & Horenstein, A. R. (2013). “Eigenvalue Ratio Test for the Number of Factors.” Econometrica 81(3):1203-1227.

Bai, J. (2009). “Panel Data Models with Interactive Fixed Effects.” Econometrica 77(4):1229-1279.

Bai, J., & Ng, S. (2002). “Determining the Number of Factors in Approximate Factor Models.” Econometrica 70(1):191-221.