best_linear_projection {grf}

Estimate the best linear projection of a conditional average treatment effect.

Description

Let tau(Xi) = E[Y(1) - Y(0) | X = Xi] be the CATE, and Ai be a vector of user-provided covariates. This function provides a (doubly robust) fit to the linear model tau(Xi) ~ beta_0 + Ai * beta.

Usage

best_linear_projection(
  forest,
  A = NULL,
  subset = NULL,
  debiasing.weights = NULL,
  compliance.score = NULL,
  num.trees.for.weights = 500,
  vcov.type = "HC3",
  target.sample = c("all", "overlap")
)

Arguments

forest

The trained forest.

A

The covariates we want to project the CATE onto.

subset

Specifies subset of the training examples over which we estimate the ATE. WARNING: For valid statistical performance, the subset should be defined only using features Xi, not using the treatment Wi or the outcome Yi.

debiasing.weights

A vector of length n (or the subset length) of debiasing weights. If NULL (default) these are obtained via the appropriate doubly robust score construction, e.g., in the case of causal_forests with a binary treatment, they are obtained via inverse-propensity weighting.

compliance.score

Only used with instrumental forests. An estimate of the causal effect of Z on W, i.e., Delta(X) = E[W | X, Z = 1] - E[W | X, Z = 0], which can then be used to produce debiasing.weights. If not provided, this is estimated via an auxiliary causal forest.

num.trees.for.weights

In some cases (e.g., with causal forests with a continuous treatment), we need to train auxiliary forests to learn debiasing weights. This is the number of trees used for this task. Note: this argument is only used when debiasing.weights = NULL.

vcov.type

Optional covariance type for standard errors. The possible options are HC0, ..., HC3. The default is "HC3", which is recommended in small samples and corresponds to the "shortcut formula" for the jackknife (see MacKinnon & White for more discussion, and Cameron & Miller for a review). For large data sets with clusters, "HC0" or "HC1" are significantly faster to compute.

target.sample

Which sample to compute the BLP over. The default is "all". Option "overlap" uses weights equal to e(X)(1 - e(X)), where e(x) are estimates of the propensity score.

Details

Procedurally, we do so by regressing doubly robust scores derived from the forest against the Ai. Note the covariates Ai may consist of a subset of the Xi, or they may be distinct. The case of the null model tau(Xi) ~ beta_0 is equivalent to fitting an average treatment effect via AIPW.

In the event the treatment is continuous the inverse-propensity weight component of the double robust scores are replaced with a component based on a forest based estimate of Var[Wi | Xi = x]. These weights can also be passed manually by specifying debiasing.weights.

Value

An estimate of the best linear projection, along with coefficient standard errors.

References

Cameron, A. Colin, and Douglas L. Miller. "A practitioner's guide to cluster-robust inference." Journal of Human Resources 50, no. 2 (2015): 317-372.

Cui, Yifan, Michael R. Kosorok, Erik Sverdrup, Stefan Wager, and Ruoqing Zhu. "Estimating Heterogeneous Treatment Effects with Right-Censored Data via Causal Survival Forests." Journal of the Royal Statistical Society: Series B, 85(2), 2023.

MacKinnon, James G., and Halbert White. "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties." Journal of Econometrics 29.3 (1985): 305-325.

Semenova, Vira, and Victor Chernozhukov. "Debiased Machine Learning of Conditional Average Treatment Effects and Other Causal Functions". The Econometrics Journal 24.2 (2021).

Examples

n <- 800
p <- 5
X <- matrix(rnorm(n * p), n, p)
W <- rbinom(n, 1, 0.25 + 0.5 * (X[, 1] > 0))
Y <- pmax(X[, 1], 0) * W + X[, 2] + pmin(X[, 3], 0) + rnorm(n)
forest <- causal_forest(X, Y, W)
best_linear_projection(forest, X[,1:2])