Estimands and identification assumptions

Causal analysis starts by defining what quantity is being estimated and then stating the assumptions under which observed data identify that quantity. The estimator comes after those two decisions.

This repository uses the potential-outcomes framework as its common notation.

Potential outcomes

For unit (i), let

[ Y_i(1) ]

denote the outcome that would be observed under treatment and

[ Y_i(0) ]

the outcome that would be observed without treatment.

Only one potential outcome is observed for each unit:

[ Y_i = D_iY_i(1) + (1-D_i)Y_i(0), ]

where (D_i in {0,1}) is the realised treatment.

The individual treatment effect is

[ au_i = Y_i(1)-Y_i(0), ]

but it is generally not identifiable because both potential outcomes are never observed simultaneously for the same unit.

Core estimands

Average treatment effect

The population average treatment effect is

[ operatorname{ATE} = mathbb{E}[Y(1)-Y(0)]. ]

It answers: what would the average outcome difference be if the population were treated rather than untreated?

Average treatment effect on the treated

The ATT is

[ operatorname{ATT} = mathbb{E}[Y(1)-Y(0)mid D=1]. ]

It answers: what was the average causal effect for units that actually received treatment?

ATT and ATE coincide only under additional conditions. Treatment-effect heterogeneity and selective assignment can make them materially different.

Conditional average treatment effect

For covariates (X=x),

[ operatorname{CATE}(x) = mathbb{E}[Y(1)-Y(0)mid X=x]. ]

CATE describes systematic effect heterogeneity. It is useful when the policy question concerns where or for whom treatment works rather than only the population mean.

Local average treatment effect

With instrument (Z), treatment (D), and monotonic treatment response to the instrument, the local average treatment effect is

[ operatorname{LATE} = mathbb{E}[Y(1)-Y(0)mid ext{compliers}]. ]

LATE is not generally the population ATE. It applies to units whose treatment status changes because of the instrument.

Dynamic and event-time effects

For panel and time-series settings, treatment effects may depend on event time (k). A generic event-time ATT is

[ operatorname{ATT}k = mathbb{E}[Y(0)mid t-G_i=k, D_i=1], ]}(1)-Y_{it

where (G_i) is the treatment-adoption period.

A cumulative effect over a horizon (H) can be written as

[ Delta^{(H)} = sum_{h=0}^{H} mathbb{E}[Y_{i,t+h}(1)-Y_{i,t+h}(0)]. ]

The canonical simulator also exposes a realised policy-path effect because lagged and persistent exposures need not map one-to-one to a binary treatment indicator.

---

Identification assumptions

An estimand is a mathematical target. Identification assumptions determine whether that target can be learned from observed data.

Consistency

If unit (i) receives treatment (D_i=d), the observed outcome equals the potential outcome under that treatment:

[ Y_i = Y_i(d). ]

This requires a sufficiently well-defined intervention. Ambiguous versions of "treatment" can break consistency.

SUTVA and interference

The stable unit treatment value assumption combines two ideas:

1. no multiple hidden versions of treatment;
2. one unit's outcome does not depend on another unit's treatment.

For regional pricing or promotion policies, the second condition can be questionable if customers cross regions, competitors react, or treatment spills over geographically.

The repository therefore distinguishes generic SUTVA from the more explicit assumption of no interference across units.

Conditional exchangeability

For observed covariates (X),

[ (Y(1),Y(0)) perp D mid X. ]

Conditional on (X), treatment assignment must contain no residual information about the potential outcomes.

This is the central identifying assumption behind adjustment, matching, inverse-probability weighting, and many doubly robust estimators.

It is not testable from observed data alone.

Positivity

For covariate values with positive population density,

[ 0 < P(D=1mid X=x) < 1. ]

Both treatment states must be possible at relevant values of (X). Severe lack of overlap creates extrapolation even when exchangeability holds.

Parallel trends

Difference-in-Differences replaces level exchangeability with a trend restriction. In the absence of treatment, treated and comparison groups must have followed the same expected change:

[ mathbb{E}[Y_t(0)-Y_{t-1}(0)mid D=1] = mathbb{E}[Y_t(0)-Y_{t-1}(0)mid D=0]. ]

Observed pre-treatment trends can provide evidence about plausibility but cannot prove post-treatment parallel trends.

No anticipation

Treatment must not affect outcomes before the treatment date:

[ Y_{it}(1)=Y_{it}(0) quad ext{for } t<G_i. ]

Announcements, behavioural preparation, inventory changes, or early policy leakage can violate this assumption.

Instrument relevance

An instrument (Z) must change treatment exposure:

[ operatorname{Cov}(Z,D) eq 0. ]

Weak relevance leads to unstable IV estimates and poor finite-sample behaviour.

Exclusion restriction

The instrument may affect the outcome only through treatment:

[ Z ightarrow D ightarrow Y, ]

with no direct path (Z ightarrow Y) and no alternative causal channel.

This assumption is substantive and usually cannot be verified statistically.

Instrument independence

The instrument must be independent of unobserved causes of the outcome, either unconditionally or conditional on an explicit set of covariates.

Relevance alone is not enough to make an instrument valid.

Monotonicity

For binary instruments, monotonicity excludes "defiers":

[ D_i(1) geq D_i(0) ]

for every unit. Together with the standard IV assumptions, this gives the LATE interpretation.

---

Panel and time-series assumptions

Longitudinal data introduce additional failure modes.

Stable untreated evolution

Counterfactual methods require enough structural stability that the untreated process observed before intervention remains informative about the untreated process after intervention.

Structural breaks unrelated to treatment can invalidate this extrapolation.

No time-varying unmeasured confounding

When treatment changes over time, past outcomes or covariates may affect both future treatment and future outcomes.

Static adjustment is insufficient when there are time-varying confounders that are themselves affected by prior treatment.

Serial dependence

Autocorrelation does not automatically destroy identification, but it affects uncertainty estimation and can invalidate standard errors that assume independent observations.

Seasonality and common shocks

Seasonal patterns and common time shocks must either be modelled, differenced out, or absorbed through comparison units/time effects. Otherwise they can be mistaken for treatment effects.

Treatment carry-over

If treatment effects persist after treatment ends, the relevant exposure is a treatment history rather than only current treatment.

The estimand must then be defined with respect to that exposure path.

---

Method-to-assumption map

The table below is a design contract for future implementation. It states the primary target and the minimum assumptions that must be discussed before a causal interpretation is made.

Method family	Primary estimand	Core identifying assumptions
Covariate adjustment / outcome regression	ATE or ATT	consistency, conditional exchangeability, positivity
Matching	ATT or ATE	consistency, conditional exchangeability, positivity
IPW	ATE or ATT	consistency, conditional exchangeability, positivity, correct treatment model
Doubly robust estimation	ATE or ATT	consistency, exchangeability, positivity, at least one nuisance model correctly specified
Difference-in-Differences	ATT	consistency, parallel trends, no anticipation, stable composition/no interference as required
Event study	event-time ATT	parallel trends, no anticipation, correct treatment-timing comparison
Fixed effects panel regression	model-dependent	strict/sequential exogeneity appropriate to the specification, correct error dependence treatment
Instrumental variables / 2SLS	LATE under heterogeneous effects	relevance, exclusion, instrument independence, monotonicity
Interrupted time series	policy-path effect	stable untreated evolution, no simultaneous intervention/confounding shock, adequate time structure
Synthetic control	ATT / policy-path counterfactual	valid donor pool, stable pre/post relationship absent treatment, no spillovers
Distributed-lag models	dynamic/cumulative effect	appropriate temporal specification, treatment-history identification, no omitted time-varying confounding

The table is intentionally conservative. Method-specific documentation may add stronger assumptions.

---

Code-level methodology contracts

The package exposes Estimand, IdentificationAssumption, and MethodologySpec from causal_econometrics.methodology.

Future estimator modules should declare a methodology contract close to the implementation. For example:

DID_SPEC = MethodologySpec(
    estimand=Estimand.ATT,
    assumptions=(
        IdentificationAssumption.CONSISTENCY,
        IdentificationAssumption.PARALLEL_TRENDS,
        IdentificationAssumption.NO_ANTICIPATION,
    ),
)

The metadata is not a substitute for methodological documentation. Its purpose is to make the estimand and identification requirements discoverable in code and difficult to omit accidentally.

Rule for future analyses

Every estimator or case-study result that is given a causal interpretation must state:

1. the estimand;
2. the identifying assumptions;
3. which assumptions are empirically diagnosable;
4. which assumptions are substantive and not testable from observed data;
5. the expected direction or nature of failure when assumptions are violated.