Semiparametric efficiency and minimax bounds for the ATE

Hahn's bound under unconfoundedness is $$V_{eff} = E\left[\frac{\sigma_1^2(X)}{e(X)} + \frac{\sigma_0^2(X)}{1 - e(X)} + (\tau(X) - \tau)^2\right],$$ a floor on the asymptotic variance of every regular asymptotically linear estimator, and \(c\sqrt{V_{eff}/n}\) is a floor on the worst-case welfare regret of every treatment rule.

Usage

morie_efficiency_bound_ate(
  y,
  D,
  X,
  family = c("gaussian", "binomial"),
  trim = 0.01,
  alpha = 0.05
)

Arguments

y

Outcome vector.

D

Binary treatment indicator.

X

Covariate matrix; an intercept is added.

family

"gaussian" or "binomial" for the arm regressions.

trim

Propensities are confined to [trim, 1 - trim].

alpha

Level for the interval on the ATE.

Value

A list with efficiency_bound, se_bound, estimate (the AIPW ATE), var_aipw, var_ipw, aipw_efficiency_ratio, ipw_efficiency_ratio, minimax_regret_bound, overlap_term, heterogeneity_term, tau_x, warnings.

Details

Three consequences follow from the shape of it, and the return value separates them so each can be checked rather than assumed.

The first term blows up as the propensity approaches zero or one. That is not a numerical artefact to be trimmed away: it is the bound stating that regions of covariate space with no comparison units carry no information about the contrast there.

The third term is the variance of the CONDITIONAL effect. It does not shrink with better nuisance estimation and does not vanish when the propensity is known. Effect heterogeneity has an irreducible price for anyone reporting an average.

Knowing the true propensity score does not lower the bound – Hahn's result is that it is the same either way, and that using an estimated score can give a smaller variance than plugging in the true one.

References

Hahn J (1998) Econometrica 66(2):315-331, doi:10.2307/2998560. Hirano K, Porter JR (2009) Econometrica 77(5):1683-1701, doi:10.3982/ECTA6630.

Examples

set.seed(1)
r <- morie_efficiency_bound_ate(y = rnorm(10), D = rbinom(10, 1, 0.5), X = rnorm(10)); TRUE
#> [1] TRUE