Bias-corrected conventional estimates for assumed fathers' effects

The bias-bounding exercise of Leventer (2025, Section 3.3). Under NTD the conventional estimator (the gender gap in normalized DID) is biased for the gender gap in normalized effects \(\theta(f) - \theta(m)\) when parallel trends fails (Theorem 1). If fathers' normalized effect \(\theta(m)\) were known, the gap would be identified (Proposition 2). This function computes the bias-corrected gap for each assumed value of \(\theta(m)\), for every post-treatment cell of results (Proposition 2 applies after treatment, \(d' > a \ge d\)); comparing it with the conventional estimate bounds the bias.

Usage

bias_bounds(results, theta_m = c(-0.1, 0.1))

Arguments

results

The data.frame returned by multiple_treatment_group_analysis, unmodified.

theta_m

Numeric vector of assumed fathers' normalized effects (default c(-0.1, 0.1), the paper's range).

Value

A data.frame with one row per cell and value of theta_m: d, dp, a, event_time, theta_m, est (the bias-corrected gap), se, ci_l, ci_h.

Details

For a post-treatment cell \((d, d', a)\), with \(\hat\Delta\) the NTD rho_effect estimate (the effect of parenthood on the gender earnings ratio) and \(\hat\delta_{APO}(g)\) the DID-imputed counterfactual mean of gender \(g\) (rows APO, DID_Female / DID_Male), the bias-corrected gap is (Proposition 2, in the form of the appendix on estimation and inference) $$\hat\Delta \, \frac{\hat\delta_{APO}(m)}{\hat\delta_{APO}(f)} \, (1 + \theta(m)).$$ It is linear in \(\theta(m)\), so the values at the ends of an interval bound it over the whole interval. Standard errors come from the delta method on the stored person-level influence functions, clustered by person.

See also

multiple_treatment_group_analysis

Examples

# \donttest{
sim <- simulate_data(n_individuals = 2000)
res <- multiple_treatment_group_analysis(sim, treatment_groups = 24:26,
  periods_post = 2, periods_pre = NULL, verbose = FALSE)
#> 1 cells could not be estimated (see attr(result, "skipped")), e.g.: No women with age at first birth 26 observed at age 28.
bb <- bias_bounds(res)
# Conventional estimate and the bias-corrected band, d = 25
subset(res, method == "Conventional" & d == 25, c(event_time, est))
#>     event_time        est
#> 65           0 -0.3594221
#> 84           1 -0.2410713
#> 103          2 -0.2355921
subset(bb, d == 25, c(event_time, theta_m, est, se))
#>    event_time theta_m        est         se
#> 7           0    -0.1 -0.3267637 0.01833033
#> 8           0     0.1 -0.3993778 0.02240374
#> 9           1    -0.1 -0.2419381 0.01831832
#> 10          1     0.1 -0.2957022 0.02238905
#> 11          2    -0.1 -0.2329846 0.01721372
#> 12          2     0.1 -0.2847590 0.02103900
# }