Inference of a causal connection between the treatment and the outcome.
Underlying problem:
Variables:
Concepts:
Treatment effect on a unit is the (hypothetical) difference between the unit's outcome in treated and untreated states. Average treatment effect (ATE) is the population mean of treatment effect. Average treatment effect on the treated (ATET) is the treated sub-population mean of treatment effect.
Selection bias arises when the treatment variable is correlated with the error in the outcome equation. Selection on observables refers to correlation of omitted observable variables with treatment assignment; selection on unobservables refers to correlation of unobserved factors with both treatment assignment and outcome determination. Average selection bias is the difference between program participants and nonparticipants in the base state.
Conditional independence assumption: conditional on observable variables, the outcomes are independent of treatment.
$$y_0, y_1 ∐ D| \mathbf{x}$$
Implications:
Ignorability assumption: conditional on observable variables, the outcome of the untreated is independent of treatment.
$$y_0 ∐ D| \mathbf{x}$$
Implications:
Conditional mean independence assumption
$$\mathbb{E}[y_0 | D, \mathbf{x}] = \mathbb{E}[y_0 | \mathbf{x}]$$
Matching (Overlap) assumption: for each value of x there are both treated and nontreated cases.
$$0 < P[D = 1|\mathbf{x}] < 1$$
Stable unit treatment value assumption (SUTVA): the treatment effect on a particular unit does not affect other units.
$$\begin{matrix} y_1 = \mu_1 (\mathbf{x}) + u_1, \text{ if } D = 1 \\ y_0 = \mu_0 (\mathbf{x}) + u_0, \text{ if } D = 0 \\ \mathbb{E}[u_1|\mathbf{x}] = \mathbb{E}[u_0|\mathbf{x}] = 0 \end{matrix}$$
Concepts:
If the assignment to the treatment and control is random, ATET can be estimated as a simple average of the differential due to treatment.
A major ATET estimator is the propensity score matching estimator. {Rosenbaum and Rubin, 1983}
Other estimators includes:
Exact matching: each treated unit is matched with untreated units that have the same observable variables $\mathbf{x}$, applicable when $\mathbf{x}$ take value over a discrete finite set.
Inexact matching: a treated unit is matched with untreated units in a neighborhood of observable variables $\mathbf{x}$. Typically $\mathbf{x}$ is mapped into a lower dimensional measure, e.g. propensity score $p(x)$.
The average outcome for the untreated matched group identifies the mean counterfactual outcome for the treated group in the absence of the treatment.
Assumptions:
ATET is identifiable under assumption 2.
Propensity score is the probability of treatment conditioning on observable variables.
$$p(\mathbf{x}) = P[D = 1|\mathbf{X} = \mathbf{x}]$$
Implications of conditional independence assumption:
Propensity score estimator:
Although logit and probit regressions are usually used to estimate propensity scores, a better statistical fit for the propensity score is more likely to result from a flexible parametric or nonparametric model.
Matching estimator estimates ATET at specific values of $\mathbf{x}$ without functional form assumptions.
General formula:
$$\text{ATET}^M = \frac{1}{N_T} \sum_{D_i=1} \left( y_{1,i} − \sum_{j \in A_i(\mathbf{x})} w(i,j) y_{0,j} \right)$$
Notations:
Matching methods:
Matching without replacement means that any observation in the comparison group is matched to no more than one treated observation, that which is the closest match. Matching with replacement means that there can be multiple matches.
Balancing condition: $D ∐ \mathbf{x} | p(\mathbf{x})$