Imputation estimator under parallel trends for the bad control

Implements the two-step imputation estimator and its influence function under parallel trends for the bad control itself, rather than Covariate Unconfoundedness: ass:bad-control-parallel-trends and cor:att-under-bad-control-parallel-trends-and-linearity in app:bad-control-parallel-trends of the supplementary appendix, and dev/bad_control_parallel_trends_influence_function.md for the influence function derivation. Step 1 regresses the bad control's own change, rather than its post-period level, on (W, Z) among the comparison group; Step 2 (the outcome regression) is unchanged from imputation_unconfoundedness. Called by imputation_attgt when bad_control_identification_strategy = "did".

Usage

imputation_did(
  wide_data,
  pre_data,
  post_data,
  D,
  n,
  x_names,
  dx_names,
  bc_cov_names,
  bc_dcov_names,
  comparison_idx,
  bad_control_formula
)

Arguments

wide_data

one row per unit, as constructed by imputation_attgt

pre_data, post_data

the pre/post-period long-format subsets of the original gt_data, as constructed by imputation_attgt

D

treatment indicator vector, aligned to wide_data

n

number of units (nrow(wide_data))

x_names, dx_names

names of the general exogenous covariate columns (level and change, respectively) already merged into wide_data

bc_cov_names, bc_dcov_names

names of the bad-control auxiliary covariate columns (level and change, respectively, W in the paper) already merged into wide_data

comparison_idx

row indices of wide_data in the comparison (untreated) group

bad_control_formula

One-sided formula naming the bad control variable, or NULL for no bad control at all