REFERENCE
Notation & conventions
Symbol conventions used consistently across all chapters. Where a chapter deviates from the defaults, it notes the substitution at the top of the page.
VARIABLESConventions used throughout all chapters.
YOutcomeThe variable whose causal determinants we study. Also written Yᵢₜ in panel settings.
D or TTreatmentBinary by default: 1 = treated, 0 = control. Continuous or multi-valued extensions noted per chapter.
XCovariatesPre-treatment observed characteristics. Used to control for confounding under CIA.
ZInstrumentVariable that shifts D but has no direct path to Y. Required for IV identification.
UUnobserved confounderCommon cause of D and Y not in the data. The source of endogeneity.
iUnit indexIndividual, firm, county, state, or any cross-sectional unit.
tTime indexYear, quarter, or month. Relevant for panel and DiD designs.
gTreatment cohortFirst period in which group g is treated. Used in staggered DiD (Callaway–Sant'Anna).
POTENTIAL OUTCOMESRubin causal model — the formal language for individual causal effects.
Yᵢ(1)Potential outcome under treatmentThe outcome unit i would realize if assigned to treatment. Counterfactual for untreated units.
Yᵢ(0)Potential outcome under controlThe outcome unit i would realize if assigned to control. Counterfactual for treated units.
τᵢ = Yᵢ(1) − Yᵢ(0)Individual treatment effectUnobservable for any unit — only one potential outcome is ever realized. The fundamental problem of causal inference.
Yᵢ = DᵢYᵢ(1) + (1−Dᵢ)Yᵢ(0)Switching equationLinks the observed outcome to potential outcomes via treatment status.
ESTIMANDSWhat quantity a method identifies. Always clarify which estimand you are targeting.
ATEAverage treatment effectE[Y(1) − Y(0)]. Average over the full population. Requires overlap for all units.
ATTATE on the treatedE[Y(1) − Y(0) | D = 1]. DiD, matching, and IPW typically target ATT.
ATUATE on the untreatedE[Y(1) − Y(0) | D = 0]. Rarely the primary target but relevant for policy extrapolation.
LATELocal ATEE[Y(1) − Y(0) | complier]. IV estimand — identified for units whose treatment status is moved by Z.
CATEConditional ATEE[Y(1) − Y(0) | X = x]. Effect for a subgroup defined by covariates. Target of causal ML methods.
ATT(g,t)Group-time ATTATT for cohort g at calendar time t. Callaway–Sant'Anna building block for staggered DiD.
REGRESSION & FIXED EFFECTSStandard notation for linear models and panel estimators.
αᵢUnit fixed effectTime-invariant unobserved heterogeneity absorbed by within-unit demeaning.
λₜTime fixed effectPeriod-specific intercept capturing common shocks across all units.
εᵢₜIdiosyncratic errorResidual variation after fixed effects. TWFE assumes E[εᵢₜ | D, α, λ] = 0.
β̂ₒₗₛOLS estimator(XᵀX)⁻¹Xᵀy. Consistent only when regressors are uncorrelated with the error.
β̂_RF / β̂_FSReduced form / first stageRF: effect of Z on Y. FS: effect of Z on D. IV estimator = RF ÷ FS.
FFirst-stage F-statisticTests instrument relevance. Rule of thumb: F > 10 (Stock–Yogo). Below 10 use Anderson-Rubin or LIML.
ρ (rho)BandwidthRDD: window around the cutoff used for local polynomial estimation. MSE-optimal via rdrobust.
OPERATORS & SYMBOLSMathematical shorthand used in proofs, assumptions, and estimator expressions.
E[·]ExpectationPopulation mean of a random variable.
E[· | X]Conditional expectationMean of a variable given covariates X. The CEF is the best linear predictor in L² loss.
Var(·)VarianceE[(X − E[X])²].
Cov(·,·)CovarianceE[(X − E[X])(Y − E[Y])].
P(·)ProbabilityProbability of an event. P(D = 1 | X) is the propensity score.
⊥⊥IndependenceX ⊥⊥ Y means X and Y are statistically independent.
⊥⊥ |Conditional independenceD ⊥⊥ U | X: D is independent of U after conditioning on X. The CIA in formal terms.
→ₚConvergence in probabilityEstimator converges to the true parameter as n → ∞. Consistency.
→ᵈConvergence in distributionUsed in CLT statements: √n(β̂ − β) →ᵈ N(0, V).
n⁻¹/⁴Nuisance rateRequired convergence rate for cross-fitted ML nuisance models in DML/AIPW. Slower than parametric n⁻¹/².
≡Defined asDefinitional equality — not a claim that both sides are empirically equal.
Notation follows the conventions of Angrist & Pischke (2009), Imbens & Rubin (2015), and Chernozhukov et al. (2018). Where sources conflict, the more common applied-econometrics convention is used.