Bounds on the omitted variable bias for causal machine learning

Computes confidence bounds on the target parameter of interest accounting for omitted variable biases.

dml_bounds(model, cf.y, cf.d, rho2 = 1)

confidence_bounds(...)

# S3 method for class 'numeric'
confidence_bounds(
  theta.s,
  S2,
  se.theta.s,
  se.S2,
  cov.theta.S2,
  cf.y,
  cf.d,
  rho2 = 1,
  combine.method = "median",
  level = 0.95,
  ...
)

# S3 method for class 'dml'
confidence_bounds(
  model,
  cf.y,
  cf.d,
  rho2 = 1,
  level = 0.95,
  combine.method = "median",
  ...
)

# S3 method for class 'dml.bounds'
confidence_bounds(
  model,
  cf.y = NULL,
  cf.d = NULL,
  rho2 = NULL,
  level = 0.95,
  combine.method = "median",
  return = c("lwr", "upr"),
  ...
)

Arguments

model: an object of class dml or dml_bounds.
cf.y: (nonparametric) partial R2 of the omitted variables with the outcome. Must be a number between (0, 1).
cf.d: how much variation latent variables create in the Riesz Representer of the target parameters. Must be a number between (0, 1). When the target of interest is the ATE in a partially linear model, this corresponds to the partial R2 of omitted variables with the treatment. When the target of interest is the ATE in a non-parametric model with a binary treatment, this corresponds to the gains in precision (i.e, 1/variance) when predicting who is assigned to treatment.
rho2: degree of adversity. Default is rho2 = 1, which assumes the maximum degree of adversity of confounding.
...: arguments passed to other methods.
theta.s: short estimate of the target parameter (used as the first argument for the numeric method).
S2: estimated variance product (sigma2 * nu2).
se.theta.s: standard error of the short estimate.
se.S2: standard error of S2.
cov.theta.S2: covariance of theta.s and S2.
combine.method: method to combine the results of each repetition. Options are mean and median. Default is median.
level: confidence level. Default is 0.95.
return: character vector specifying which bounds to return. Options are "lwr" and "upr".

Value

For dml_bounds: an object of class dml.bounds. For confidence_bounds: a matrix or numeric vector of confidence bounds.

References

Chernozhukov, V., Cinelli, C., Newey, W., Sharma, A., and Syrgkanis, V. (2026). "Long Story Short: Omitted Variable Bias in Causal Machine Learning." Review of Economics and Statistics. doi:10.1162/REST.a.1705

Wang, J., Sant'Anna, P. H. C., Chernozhukov, V., and Cinelli, C. (2026). "Omitted Variable Bias in Difference-in-Differences Designs." Working Paper. (Conditional ATT/ATU formulation.)

Examples

data("pension")
set.seed(3); idx <- sample(nrow(pension), 2000)
y <- pension$net_tfa[idx]
d <- pension$e401[idx]
x <- model.matrix(~ -1 + age + inc + educ + fsize + marr + twoearn + pira + hown,
                  data = pension[idx, ])
fit <- dml(y, d, x, model = "plm", cf.folds = 2, cf.seed = 3, verbose = FALSE)

# bounds on the ATE under postulated confounding strength
dml_bounds(fit, cf.y = 0.03, cf.d = 0.03)
#> 
#> Debiased Machine Learning: Bounds on Omitted Variable Bias
#> 
#> Sensitivity Parameters
#>   cf.y = 0.03
#>   cf.d = 0.03
#>   rho2 = 1 
#> 
#> 
#> Bounds on Average Treatment Effect
#> 
#>                Estimate Std. Error t value P(>|t|)    
#> Short Estimate    12966       3354     3.9   1e-04 ***
#> |Bias| Bound       3611        611     5.9   3e-09 ***
#> Lower Bound        9356       3212     2.9   0.004 ** 
#> Upper Bound       16577       3596     4.6   4e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Note: DML estimates combined using the median method.
confidence_bounds(fit, cf.y = 0.03, cf.d = 0.03, level = 0.95)
#>         lwr     upr
#> ate  4073.0 22491.7
#> 
#> Confidence level: point = 95%; region = 90%.
#> Sensitivity parameters: cf.y = 0.03; cf.d = 0.03; rho2 = 1.