Method Comparison Demonstrations

Exogeneity: training is uncorrelated with error term

1.23

[1.11, 1.34]

In this simulated DGP, likely biased upward: firms that adopt training tend to already be higher-performing, though the magnitude of bias depends on the strength of selection.

No time-varying confounders

0.18

[-0.05, 0.41]

Controls for firm-level time-invariant factors. Smaller estimate is consistent with selection bias in OLS, though other explanations are possible.

Parallel trends: absent training, treated and control firms would have trended similarly

0.18

[0.01, 0.36]

An event study can provide evidence about whether pre-trends are present, though a non-significant pre-trend test does not guarantee parallel trends holds (Roth, 2022). Pre-treatment dynamics should be stable for the design to be credible.

Conditional independence + correct outcome functional form

0.29

[0.20, 0.37]

Preferred when the conditional expectation E[Y|D,X] is well-approximated by a linear function; under correct specification it is efficient. The linear specification may be misspecified if the relationship is nonlinear in X.

Selection on observables (CIA); overlap/common support; correctly specified propensity score

0.33

[0.24, 0.43]

Preferred when the analyst is unwilling to impose a linear outcome model but is willing to model the treatment assignment process. Wider CI (738 matched pairs) reflects information loss from using only matched units; targets the ATT on the matched sample rather than the ATE.

CIA holds; consistent if either outcome or PS model is correct

0.28

[0.19, 0.37]

Consistent if EITHER the outcome model OR the propensity score model is correctly specified (robust to misspecification of one of the two nuisance models, but not both). Preferred when the analyst wants insurance against misspecification in a single nuisance model; still requires CIA and overlap. Performance in finite samples depends on the quality of both nuisance estimates.

Random assignment only

0.04

[-0.04, 0.13]

Estimates the effect of being offered treatment (the offer-of-treatment estimand), not the effect of taking treatment. Under random assignment with full follow-up it is unbiased for the ITT estimand; with imperfect compliance it is attenuated toward zero relative to the LATE on compliers. Differential attrition can introduce bias even with random assignment.

Random assignment + exclusion restriction + monotonicity

0.10

[-0.11, 0.32]

Identifies the Local Average Treatment Effect for compliers only — not the ATE; the LATE equals the ATE only if treatment effects are homogeneous across compliance types. Larger in magnitude than ITT here because the Wald estimator scales up by 1/first-stage to adjust for ~60% non-compliance.

Monotonicity of sample selection

[-0.29, 0.38]

Bounds, not point estimate

Accounts for 15 percentage-point differential attrition (Lee, 2009). Width of the bounds reflects the cost of partial identification when sample selection is non-random; preferred when point identification via assumed selection mechanisms is not defensible.