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Child-penalty estimates for one 2x2 comparison

Source: R/single_treatment_group_analysis.R
single_treatment_group_analysis.Rd

For one treatment group \(d\) (age at first birth), control group \(d'\) and target age \(a\), estimates the descriptive estimands of Leventer (2025): within-gender DID, the gender gaps under TD, the conventional estimator, the effect of parenthood on the gender earnings ratio (NTD and proportional TD), and the accounting of the gender gap. Returns 19 rows with standard errors clustered by person.

Usage

single_treatment_group_analysis(
  data,
  d,
  dp,
  a,
  pre = 1,
  Y_name = "Y",
  age_name = "age",
  D_name = "D",
  id_name = "id",
  female_name = "female"
)

Arguments

data

A data.frame (or data.table) in long format: one row per person and age, with the person id, age, age at first birth, a female indicator and earnings. The panel need not be balanced. Each person's female indicator and age at first birth must be constant; a missing age at first birth means the person never has a child. Rows with missing earnings are dropped (with a message). Earnings may be zero, but the ratio rows need positive cell means. data is not modified.

d

Whole number. Treatment group: age at first birth.

dp

Whole number. Control group \(d'\): an age at first birth that is not yet treated at the target age, dp > a (and dp > d). The paper uses the closest not-yet-treated group, dp = a + 1 (Section 1.3), as multiple_treatment_group_analysis does.

a

Whole number. Target age; event time is \(e = a - d\). With a >= d the rows are post-birth estimates; with a < d - pre they are pre-treatment validation (placebo) estimates (Sections 2.3, 4.2).

pre

Whole number \(\ge 1\) (default 1). The baseline age is \(d - pre\). The default, \(d - 1\), is the paper's choice; larger values move the baseline earlier in case earnings at \(d - 1\) reflect anticipation (Section 3.1).

Y_name, age_name, D_name, id_name, female_name

Names of the columns holding earnings, age, age at first birth, the person id (clustering unit) and the female indicator (1 / TRUE = female; also accepted: labels such as "female"/"male", "f"/"m", "woman"/"man"; any other label is an error).

Value

A data.frame with 19 rows and columns d, dp, a, event_time (\(a - d\)), control_offset (\(d' - d\)), estimand, method, est, se, ci_l, ci_h (est -/+ 1.96 se), t, p (two-sided, normal), and n_female_treat, n_female_control, n_male_treat, n_male_control (persons with \(D = d\) and \(D = d'\)). Attribute "if_scores": person-level sums of the influence functions, used by aggregate_estimands and bias_bounds.

Details

Let \(\mu(g, k, t)\) be the mean earnings of gender \(g \in \{f, m\}\) with age at first birth \(k\) at age \(t\), estimated by the cell mean, and \(b = d - pre\) the baseline age. The additive (DID) and multiplicative imputations of counterfactual earnings at age \(a\) are $$\delta_{APO}(g) = \mu(g,d,b) + \mu(g,d',a) - \mu(g,d',b), \qquad \delta^{\times}_{APO}(g) = \mu(g,d,b)\,\mu(g,d',a) / \mu(g,d',b).$$ The rows, with the assumption under which each identifies its target (all also need no anticipation):

methodestimandestimator; target
DID_Female, DID_MaleAPO\(\delta_{APO}(g)\); \(APO(g, \infty)\) under DID-PT
DID_Female, DID_MaleATE\(\mu(g,d,a) - \delta_{APO}(g)\); \(ATE(g)\) under DID-PT
DID_Female, DID_MalethetaATE row / APO row; \(\theta(g)\) under DID-PT
TDgap_ATEATE row f minus m; \(ATE(f) - ATE(m)\) under TD-PT
Conventionalgap_thetatheta row f minus m; \(\theta(f) - \theta(m)\) under DID-PT only (under NTD-PT it is biased, Theorem 1)
NTDrho_effect\(\mu(f,d,a)/\mu(m,d,a) - \delta_{APO}(f)/\delta_{APO}(m)\); \(\rho(d) - \rho(\infty)\) under NTD-PT (Theorem 2)
TD_E_LOGSrho_effectas NTD with \(\delta^{\times}_{APO}\); \(\rho(d) - \rho(\infty)\) under TD-E-LOGS (proposition in the online appendix on proportional DID and TD)
TD_PRgap_theta_pre\([\mu(g,d,a) - \delta^{\times}_{APO}(g)] / \mu(g,d,b)\), f minus m; \(ATE(g)/APO(g,d,\infty,b)\), f minus m, under TD-PR (same proposition)
NTD, TD_E_LOGSrho_obs\(\mu(f,d,a)/\mu(m,d,a)\); the observed gender earnings ratio \(\rho(d)\)
NTD, TD_E_LOGSrho_cf\(\delta_{APO}(f)/\delta_{APO}(m)\) (NTD) or with \(\delta^{\times}_{APO}\) (TD_E_LOGS); \(\rho(\infty)\) under NTD-PT (NTD) or TD-E-LOGS (TD_E_LOGS)
NTD, TD_E_LOGSshare_parenthood\((\rho_{cf} - \rho_{obs}) / (1 - \rho_{obs})\); parenthood's share of the observed gender gap (Section 4.2), under the same assumption as rho_cf
NTD, TD_E_LOGSshare_other\(1 -\) share_parenthood; the share of other factors

Gaps are female minus male. The conventional estimator is the 2x2 version of the normalized event-study child penalty; under NTD it equals the target gap times a gender-invariant bias factor (Theorem 1), see bias_bounds for the paper's bias-bounding exercise. The accounting rows and their standard errors are an extension of the paper's Section 4.2 accounting exercise.

Standard errors. Each row is a smooth function of the eight cell means; its influence function follows by the chain rule from those of the cell means, \(\psi_{\mu} = S (Y - \mu) / P(S)\) with \(S\) the cell indicator (the appendix on estimation and inference). The variance is \(n^{-2} \sum_c (\sum_{i \in c} \psi_i)^2 \, G / (G - 1)\), clustering by person (\(n\) person-age rows, \(G\) persons). vignette("estimators") writes out every estimator and influence function.

See also

multiple_treatment_group_analysis for the grid of comparisons the paper uses, aggregate_estimands, bias_bounds, vignette("estimators")

Examples

# \donttest{
sim <- simulate_data(n_individuals = 2000)
# Treatment group d = 25 one year after first birth (a = 26),
# closest not-yet-treated control group d' = a + 1 = 27
res <- single_treatment_group_analysis(sim, d = 25, dp = 27, a = 26)
res[res$method %in% c("Conventional", "NTD"), c("method", "estimand", "est", "se")]
#>          method         estimand        est         se
#> 8  Conventional        gap_theta -0.2410713 0.02185681
#> 9           NTD       rho_effect -0.2493495 0.02378134
#> 12          NTD          rho_obs  0.6782204 0.01755007
#> 13          NTD           rho_cf  0.9275699 0.02667726
#> 14          NTD share_parenthood  0.7749077 0.07769671
#> 15          NTD      share_other  0.2250923 0.07769671
# Pre-treatment validation (placebo) test: a < d - 1
val <- single_treatment_group_analysis(sim, d = 25, dp = 27, a = 22)
val[val$method %in% c("Conventional", "NTD") & val$estimand %in% c("gap_theta", "rho_effect"),
    c("method", "estimand", "est", "se")]
#>         method   estimand        est         se
#> 8 Conventional  gap_theta 0.04753514 0.03251608
#> 9          NTD rho_effect 0.04567309 0.03071295
# }