CFM — Causal Inference Using Factor Models (Bai & Wang 2026)#

Estimator:: Causal Factor Model (CFM) — mlsynth.CFM
Source:: Bai, J., & Wang, P. (2026), “Causal Inference Using Factor Models” [BaiWang2026].
Replication type:: Path A — the paper’s two empirical applications (California Prop 99 and German reunification) reproduced on the authors’ data.
Status:: Verified — factor counts, the structural-break diagnostics, and the intercept-shift tests reproduce the paper’s reported numbers.

Validation strategy#

Bai and Wang re-analyze the two canonical synthetic-control panels — Abadie, Diamond and Hainmueller’s California tobacco-control (Proposition 99) and German reunification data — with the causal factor model. The paper does not release code, so the target is the set of numbers it reports: the number of factors chosen by the Ahn-Horenstein criteria, the Chow structural-break statistic at the intervention date, the Quandt likelihood-ratio break date, and the intercept-shift t-statistics. These are sensitive to the extracted factor, so matching them confirms the factor extraction and the treated regressions, not just an endpoint.

Both datasets ship with mlsynth: basedata/smoking_data.csv (California plus 38 control states, 1970-2000) and basedata/german_reunification.csv (West Germany plus 16 control countries, 1960-2003).

Path A — California Prop 99#

import pandas as pd
from mlsynth import CFM
from mlsynth.utils.cfm_helpers.setup import prepare_cfm_inputs
from mlsynth.utils.cfm_helpers.factors import extract_cfm_factors
from mlsynth.utils.cfm_helpers.pipeline import chow_break_statistic

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", "n_factors": 1,
           "display_graphs": False}).fit()
res.att, res.metadata["kappa_t"], res.metadata["chow_fstat"]

The intervention starts in 1989, so \(T_0 = 1988\) with a 1970-1988 pre-period and a 1989-2000 post-period. The reproduced quantities:

Quantity	CFM (mlsynth)	Bai & Wang
ER factor count	1	1
GR factor count	1	1
Chow F, break at 1989 (1 factor)	16.84	16.84
QLR sup-F break date (15% trim)	1984	1984
intercept-shift \(t(\kappa)\), 1 factor	1.38	1.38
intercept-shift \(t(\kappa)\), 2 factors	0.10	0.10

The one-factor systematic effect path has a post-period mean of about \(-20.7\) packs and tracks the synthetic-control estimate (\(\approx -19.5\)) at correlation \(0.80\), consistent with the paper’s Figure 5. The small intercept-shift t-statistics indicate little evidence of a post-treatment level break, so the effect operates through the loading change.

Path A — German reunification#

df = pd.read_csv("basedata/german_reunification.csv")
# paper convention: 1990 marked; treated periods 1991-2003 (T0 = 1990)
df["treat"] = ((df.country == "West Germany") & (df.year >= 1991)).astype(int)

res = CFM({"df": df, "outcome": "gdp", "treat": "treat",
           "unitid": "country", "time": "year", "n_factors": 1,
           "display_graphs": False}).fit()

Quantity	CFM (mlsynth)	Bai & Wang
ER / GR factor count	1 / 1	1 / 1
Chow F, break at 1991 (1 factor)	634.5	634.5
QLR sup-F break date (15% trim)	1993	1993
intercept-shift \(t(\kappa)\), 1 factor	11.77	11.81

The one-factor path tracks synthetic control at correlation \(0.98\). Unlike California, the intercept-shift test is strongly significant, matching the paper’s finding of a post-treatment level shift for West Germany (either a direct intercept change or a constant shift in the factor process, which the specification cannot separate).

Seam notes#

Two details were pinned while reproducing these numbers, and both are held by unit tests:

The intercept-shift t-statistic reproduces only under the paper’s block-additive heteroskedasticity-robust construction — \(\mathrm{Var}(\widehat\kappa) = \mathrm{Var}(\widehat a_1(0)) + \mathrm{Var}(\widehat a_1(1))\) from separate pre- and post-regressions, each with an HC1 sandwich (appendix A.14-A.19). A naive pooled-OLS t-statistic is badly off on the heteroskedastic German panel (about 21 against the reported 11.8); the robust block form recovers 11.77.
The German treat flag follows the paper’s convention: 1990 is the marked reunification year but the treated periods run 1991-2003, so \(T_0 = 1990\) and the Chow break is tested at 1991 (matching the reported F of 634.5).

Not reproduced here#

The per-period confidence bands’ factor-estimation component \(V^f\) (appendix A.20) is implemented but not separately checked against a paper number; its acceptance target is the Monte Carlo coverage of Section 6, left as a future simulation benchmark.
The potential-factors regime (Proposition 2, large treated cross section) is out of scope for this estimator.

The durable check lives in benchmarks/cases/cfm.py.

CFM — Causal Inference Using Factor Models (Bai & Wang 2026)

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