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Conditional Z-Estimation with DNN weights

Source: vignettes/conditional-z-estimation.Rmd
conditional-z-estimation.Rmd

Overview

The initial public release focuses on CATE and nonparametric regression. Start with Getting started for executable examples of those workflows. This article gives additional moment details, including experimental PLQR:

  • np_dnn() for localized conditional means
  • cate_dnn() for conditional average treatment effects
  • plqr_dnn() for experimental moment evaluation at supplied coefficients

Each wrapper supports:

  • matrix and data-frame interfaces
  • DNN method choice ("exact" or "asymp")
  • delete-1 jackknife SEs ("fast", "loo", "none")
  • modular kernel-order tuning via tuning = tuning_dnn_paper(...)
  • fitted-object diagnostics via diagnostics()

The wrappers implement fixed top-k DNN kernels for 1 <= k <= s. The accompanying manuscript derivations are stated for k = 1; larger fixed k is an implementation-supported generalization.

Local Mean (np_dnn) 

demo_np <- np_demo_data(n = 250, p = 2, design = "sinusoidal", seed = 1)

fit_np <- np_dnn(
  data = demo_np$data,
  x_cols = c("X1", "X2"),
  y_col = "Y",
  x0 = c(0, 0),
  method = "exact",
  k = 1L,
  tuning = tuning_default,
  se = TRUE,
  jackknife = "fast",
  return = c("W", "ess", "tuning")
)

fit_np$mu
#> [1] -0.02251305
fit_np$se
#> [1] 0.1782743
diagnostics(fit_np)
#> <dnn_diagnostics>
#>   targets: 1
#>   n_eff  : min=18.6582, median=18.6582, max=18.6582
#>   weights: max(top5 share)=0.4122, max(weight)=0.1
#>   dist   : available for 0/1 target(s)
#>   outcome: available for 1/1 target(s)

CATE (cate_dnn)

cate_dnn() supports either:

  • Full AIPW nuisance inputs (m0, m1, e)
  • Robinson-form nuisance inputs (m, e)
  • Direct pseudo_outcome 
demo_cate <- cate_demo_data(n = 300, p = 2, design = "hetero", seed = 1)
df <- demo_cate$data
df$m1_oracle <- df$m0_oracle + df$tau_oracle

fit_cate <- cate_dnn(
  data = df,
  x_cols = c("X1", "X2"),
  y_col = "Y",
  t_col = "D",
  m0_col = "m0_oracle",
  m1_col = "m1_oracle",
  e_col = "e_oracle",
  x0 = c(0, 0),
  method = "exact",
  k = 1L,
  tuning = tuning_default,
  se = TRUE,
  return = c("W", "psi", "nuisance", "tuning")
)

summary(fit_cate)
#> <summary.cate_result>
#>   targets: 1
#>          estimate std.error statistic p.value conf.low conf.high
#> target_1   0.5512    0.3181     1.733 0.08313 -0.07227     1.175
confint(fit_cate)
#>             conf.low conf.high
#> target_1 -0.07226519  1.174624
diagnostics(fit_cate)
#> <dnn_diagnostics>
#>   targets: 1
#>   n_eff  : min=20.098, median=20.098, max=20.098
#>   weights: max(top5 share)=0.3894, max(weight)=0.0933
#>   dist   : available for 0/1 target(s)
#>   overlap: available for 1/1 target(s)
predict(fit_cate, x0 = c(0.2, -0.1), interval = "confidence")
#>            fit       lwr      upr
#> [1,] 0.8336633 0.3016249 1.365702

Experimental PLQR moments and coefficient uncertainty (plqr_dnn)

All PLQR functions, including score and demo helpers, methods, and specialized plots, are experimental. Their APIs and inference interfaces may change. The current wrapper does not estimate a local coefficient root.

demo_plqr <- plqr_demo_data(n = 250, p = 2, p_d = 2, tau = 0.5, seed = 1)
# demo_plqr$data$density_oracle holds the true threshold density of the
# demo DGP (constant, 4 * tau * (1 - tau) / sigma).

fit_plqr <- plqr_dnn(
  data = demo_plqr$data,
  x_cols = c("X1", "X2"),
  D_cols = c("D1", "D2"),
  g_col = "g_oracle",
  mD_cols = c("mD1", "mD2"),
  y_col = "Y",
  density_col = "density_oracle",
  theta_eval = demo_plqr$truth$theta_true,
  tau = 0.5,
  x0 = c(0, 0),
  method = "exact",
  k = 1L,
  tuning = tuning_default,
  se = TRUE,
  return = c("W", "score", "ess", "tuning")
)

summary(fit_plqr)
confint(fit_plqr)
predict(fit_plqr, x0 = c(0.25, -0.25), interval = "confidence")
fit_plqr$moment
diagnostics(fit_plqr)

PLQR nuisances and inference

plqr_dnn() localizes the moment

𝔼[(τ−𝟏{Y≤D⊤θ+g0(U)})(D−mD(U))|X≈x0] \mathbb{E}\!\left[ \left(\tau - \mathbf{1}\{Y \le D^\top \theta + g_0(U)\}\right) \left(D - m_D(U)\right) \;\middle|\; X \approx x_0 \right]

at the supplied coefficient value theta_eval. Here g is the partially-linear model’s nonparametric shape nuisance (paper notation g0(U)g_0(U)), not itself a quantile — the sum D′θ0+g0D'\theta_0 + g_0 is the conditional quantile. mD residualizes the design (m0(U)=E[D∣U]m_0(U) = E[D \mid U]). Both are supplied by the user; no nuisance model is fit internally, and no root solving is performed: the result’s theta repeats the supplied theta_eval (meta$theta_source = "supplied").

Coefficient-level standard errors and confidence intervals require supplying density/density_col (an estimate of the local conditional density of Y at the fitted quantile). This is not an optional convenience: every valid coefficient route needs the density-backed sandwich bread. Two coefficient-meat routes are available through theta_meat: the analytic score covariance (default) and a full delete-1 jackknife covariance of the localized vector moment (theta_meat = "jackknife", stored in moment_vcov). A fit without a density input can still report moment_se (a componentwise jackknife SE on the moment, useful as a diagnostic) but confint()/predict(se.fit = TRUE) will error, by design, until refit with a density input — a componentwise moment_se alone is never coefficient inference. Coefficient-level results live in theta_se / theta_vcov; the routes are never mixed in one field.

Diagnostics and Reporting

The three wrappers return enough retained state for diagnostics() to compute fit-level summaries without rerunning the estimator.

d_np <- diagnostics(fit_np)
d_cate <- diagnostics(fit_cate)
d_plqr <- diagnostics(fit_plqr)

d_np[, c("target_id", "n_eff", "weighted_mean_y")]
d_cate[, c("target_id", "treated_mass", "control_mass", "treated_share")]
d_plqr[, c("target_id", "gram_min_eig", "gram_condition_number")]

General diagnostics summarize weight concentration, realized distance radii when dx was retained, and tuning metadata. Estimator-specific diagnostics add NP outcome summaries, CATE overlap summaries, and PLQR residual-regressor Gram conditioning.

Notes on Standard Errors

np_dnn() and cate_dnn() always use delete-1 jackknife SEs when se = TRUE.

  • "fast" reuses full-sample ranks and applies rank updates per deletion.
  • "loo" recomputes leave-one-out weights directly (reference-equivalent).
  • "none" skips SE computation.

s < n is required when jackknife SEs are requested.

Experimental plqr_dnn() differs: with density/density_col, se = TRUE reports a density-backed coefficient sandwich. Its meat uses either the analytic score covariance (default) or the vector-moment jackknife covariance with theta_meat = "jackknife". Without density, moment_se is a moment jackknife diagnostic and does not support coefficient intervals. See “PLQR nuisances and inference” above.