Overview

This appendix runs two robustness/validation components as a standalone file (it loads data and applies the manual distance overrides):

  1. Donut RD robustness (true cutoff only): excludes observations within 2.5 km of the cutoff on both sides (keeps |X| >= 2.5) and re-estimates RD at the true cutoff c = 0 for all outcomes, using fixed bandwidth h = 5 km.

  2. Placebo cutoffs (fixed bandwidth): estimates RD effects at artificial cutoffs every 5 km up to ±30 km, including 0, using fixed bandwidth h = 5 km.

Both are run for: - All Italy (full merged sample, with region FE via covs= in rdrobust) - Lombardia (subset, no region FE)

Inputs (edit paths here)

# DATA (CSV)
DATA_PATH <- "/Users/murugesana/Dropbox/Research/history_behavior/Experimental_Institutions/R/data/RDD_MERGED_CSVS/All_italy_df_merged.csv"

# MANUAL DISTANCE OVERRIDES (XLSX)
MANUAL_DISTANCE_PATH <- "/Users/murugesana/Dropbox/Research/history_behavior/Experimental_Institutions/R/data/RDD_MERGED_CSVS/TreatedControl_Distance_AsCrowFlies_Manual.xlsx"

# Column names
RUNNING <- "distance_treated_positive_x"  # signed distance in km (cutoff at 0)
REGION  <- "COD_REG"                     # region id (for All-Italy FE)

# Lombardia region code (as in your main RMD)
REGION_CODE_LOMBARDIA <- 3

# Outcomes (full list used in main RMD; edit if needed)
OUTCOMES <- c(
  "gb_intensity","gb_reg_rate","evasione","services_level", "Admin_Tax_Emp",
  "edu_serv_lvl","edu_muni_school_area_per1000", "edu_emp_per1000",
  "PublS_CyclePath_per","pol_mun_road","Pillar2_pol",
  "marr_civil","marr_rel","incomepc","income",
  "expend_level","expenditure"
)

# Placebo outcomes: default = OUTCOMES, or specify a subset
PLACEBO_OUTCOMES <- OUTCOMES

# Donut and placebo settings
DONUT_KM <- 2.5

PLACEBO_STEP_KM <- 5
PLACEBO_MAX_KM  <- 30

# Fixed bandwidths
H_DONUT <- 40
H_PLACEBO <- 5

placebo_cutoffs <- seq(-PLACEBO_MAX_KM, PLACEBO_MAX_KM, by = PLACEBO_STEP_KM)

Load data and apply manual distance overrides (standalone)

df0 <- readr::read_csv(DATA_PATH, show_col_types = FALSE)

# Key column for municipalities
KEYCOL <- if ("istat" %in% names(df0)) "istat" else if ("ISTAT" %in% names(df0)) "ISTAT" else NA_character_
if (is.na(KEYCOL)) stop("No istat/ISTAT column found in df0.")

stopifnot(RUNNING %in% names(df0), REGION %in% names(df0))
stopifnot("COMUNE" %in% names(df0), "Treated" %in% names(df0))

# Region FE factor + compatibility alias
df0 <- df0 %>%
  mutate(
    COD_REG = as.integer(.data[[REGION]]),
    .region_fe = factor(.data[[REGION]])
  )

if (!("ireg" %in% names(df0))) df0 <- df0 %>% mutate(ireg = as.integer(.data[[REGION]]))

# Manual distance corrections
manual0 <- readxl::read_excel(MANUAL_DISTANCE_PATH)

needed_manual <- c("COD_REG", "COMUNE", "Treated", "distance_updated")
stopifnot(all(needed_manual %in% names(manual0)))

manual <- manual0 %>%
  mutate(
    COD_REG = as.integer(COD_REG),
    Treated = as.integer(Treated),
    COMUNE_clean = stringr::str_squish(stringr::str_to_upper(as.character(COMUNE))),
    distance_updated = as.numeric(distance_updated)
  ) %>%
  filter(!is.na(distance_updated)) %>%
  select(COD_REG, Treated, COMUNE_clean, distance_updated)

df0 <- df0 %>%
  mutate(
    Treated = as.integer(Treated),
    COMUNE_clean = stringr::str_squish(stringr::str_to_upper(as.character(COMUNE))),
    .running_original = .data[[RUNNING]]
  ) %>%
  left_join(manual, by = c("COD_REG", "Treated", "COMUNE_clean")) %>%
  mutate(
    "{RUNNING}" := dplyr::if_else(!is.na(distance_updated), distance_updated, .data[[RUNNING]]),
    .running_overwritten = !is.na(distance_updated)
  )

# Exclude Tronzano Lago Maggiore, province code 12 (as in main RMD)
if ("COD_PROV" %in% names(df0)) {
  df0 <- df0 %>%
    filter(!(COMUNE_clean == "TRONZANO LAGO MAGGIORE" & as.integer(.data[["COD_PROV"]]) == 12))
}

cat("Manual overrides applied: overwritten rows = ", sum(df0$.running_overwritten, na.rm = TRUE), "\n", sep = "")
## Manual overrides applied: overwritten rows = 68
cat("All-Italy rows after override+drop: ", nrow(df0), "\n", sep = "")
## All-Italy rows after override+drop: 6984
# Subsamples
df_all <- df0
df_lom <- df0 %>% filter(as.integer(.data[[REGION]]) == REGION_CODE_LOMBARDIA)

cat("All-Italy N:", nrow(df_all), "\n")
## All-Italy N: 6984
cat("Lombardia N:", nrow(df_lom), "\n")
## Lombardia N: 2812
# Basic checks for requested outcomes (skip missing)
OUTCOMES_PRESENT <- OUTCOMES[OUTCOMES %in% names(df_all)]
if (length(OUTCOMES_PRESENT) == 0) stop("None of OUTCOMES are present in df0. Check column names.")
MISSING_OUTCOMES <- setdiff(OUTCOMES, OUTCOMES_PRESENT)
if (length(MISSING_OUTCOMES) > 0) {
  cat("Note: missing outcomes will be skipped:\n")
  print(MISSING_OUTCOMES)
}

PLACEBO_OUTCOMES_PRESENT <- PLACEBO_OUTCOMES[PLACEBO_OUTCOMES %in% names(df_all)]
if (length(PLACEBO_OUTCOMES_PRESENT) == 0) stop("None of PLACEBO_OUTCOMES are present in df0. Check column names.")

Helper functions

# Region FE matrix for rdrobust(covs=...) (All-Italy)
make_region_covs <- function(data, region_var) {
  data <- data %>% mutate(.region_fe = factor(.data[[region_var]]))
  model.matrix(~ .region_fe - 1, data = data)
}

# Fixed-h rdrobust (used for donut and placebo)
run_rd_fixed_h <- function(data, y, x, covs_mat = NULL, c0, h_fixed, p = 1, kernel = "tri", min_each_side = 30) {
  yv <- data[[y]]
  xv <- data[[x]]
  ok <- is.finite(yv) & is.finite(xv)
  yv <- yv[ok]; xv <- xv[ok]
  covs_use <- if (!is.null(covs_mat)) covs_mat[ok, , drop = FALSE] else NULL

  # support check (overall, not within h) — avoids hard failures
  nL <- sum(xv < c0)
  nR <- sum(xv >= c0)
  if (nL < min_each_side || nR < min_each_side) {
    return(tibble(outcome = y, cutoff = c0, n = length(yv), h = h_fixed,
                  tau_bc = NA_real_, se_rb = NA_real_, p_rb = NA_real_))
  }

  fit <- tryCatch(
    rdrobust(yv, xv, c = c0, p = p, h = h_fixed, kernel = kernel, covs = covs_use),
    error = function(e) NULL
  )

  if (is.null(fit)) {
    return(tibble(outcome = y, cutoff = c0, n = length(yv), h = h_fixed,
                  tau_bc = NA_real_, se_rb = NA_real_, p_rb = NA_real_))
  }

  tibble(
    outcome = y,
    cutoff  = c0,
    n       = length(yv),
    h       = h_fixed,
    tau_bc  = as.numeric(fit$coef[2]),
    se_rb   = as.numeric(fit$se[2]),
    p_rb    = as.numeric(fit$pv[2])
  )
}

Appendix B1. Donut RD robustness (exclude |X| < 2.5 km), true cutoff only (fixed h = 40)

cat("\n\n## Appendix B1. Donut RD robustness: exclude |X| < ", DONUT_KM, " km (true cutoff only), fixed h = ", H_DONUT, " km\n\n", sep = "")
## 
## 
## ## Appendix B1. Donut RD robustness: exclude |X| < 2.5 km (true cutoff only), fixed h = 40 km
# Apply donut
df_all_donut <- df_all %>% filter(abs(.data[[RUNNING]]) >= DONUT_KM)
df_lom_donut <- df_lom %>% filter(abs(.data[[RUNNING]]) >= DONUT_KM)

# Region FE covariates (All-Italy only)
covs_all <- make_region_covs(df_all_donut, REGION)

# Run donut RD for all outcomes present (fixed h = 40)
donut_all_tbl <- purrr::map_dfr(OUTCOMES_PRESENT, ~run_rd_fixed_h(df_all_donut, y = .x, x = RUNNING, covs_mat = covs_all,
                                                                 c0 = 0, h_fixed = H_DONUT))
donut_lom_tbl <- purrr::map_dfr(OUTCOMES_PRESENT, ~run_rd_fixed_h(df_lom_donut, y = .x, x = RUNNING, covs_mat = NULL,
                                                                 c0 = 0, h_fixed = H_DONUT))

donut_out <- bind_rows(
  donut_all_tbl %>% mutate(sample = "All Italy (donut)"),
  donut_lom_tbl %>% mutate(sample = "Lombardia (donut)")
) %>%
  select(sample, outcome, n, h, tau_bc, se_rb, p_rb) %>%
  arrange(sample, outcome)

knitr::kable(donut_out, digits = 4)
sample outcome n h tau_bc se_rb p_rb
All Italy (donut) Admin_Tax_Emp 1834 40 1.041000e-01 3.090000e-02 0.0008
All Italy (donut) Pillar2_pol 6884 40 -1.024900e+00 6.668000e-01 0.1243
All Italy (donut) PublS_CyclePath_per 6242 40 4.698400e+00 7.099000e-01 0.0000
All Italy (donut) edu_emp_per1000 3388 40 -9.000000e-04 2.160000e-02 0.9662
All Italy (donut) edu_muni_school_area_per1000 6306 40 2.648000e+00 5.083000e-01 0.0000
All Italy (donut) edu_serv_lvl 5960 40 6.317000e-01 1.908000e-01 0.0009
All Italy (donut) evasione 6884 40 -2.103200e+00 4.669000e-01 0.0000
All Italy (donut) expend_level 6314 40 2.852000e-01 1.501000e-01 0.0574
All Italy (donut) expenditure 6884 40 -7.212221e+05 1.722566e+05 0.0000
All Italy (donut) gb_intensity 6924 40 -1.440000e-02 1.300000e-03 0.0000
All Italy (donut) gb_reg_rate 6924 40 6.350000e-02 1.260000e-02 0.0000
All Italy (donut) income 6314 40 -9.261885e+07 1.622913e+07 0.0000
All Italy (donut) incomepc 6314 40 5.789608e+02 1.791153e+02 0.0012
All Italy (donut) marr_civil 6314 40 -3.377600e+00 1.487400e+00 0.0232
All Italy (donut) marr_rel 6314 40 -2.393500e+00 6.746000e-01 0.0004
All Italy (donut) pol_mun_road 6314 40 6.651600e+00 4.615200e+00 0.1495
All Italy (donut) services_level 6314 40 1.556200e+00 1.704000e-01 0.0000
Lombardia (donut) Admin_Tax_Emp 774 40 -8.250000e-02 4.780000e-02 0.0846
Lombardia (donut) Pillar2_pol 2746 40 -1.298400e+00 1.073200e+00 0.2264
Lombardia (donut) PublS_CyclePath_per 2722 40 -1.892000e-01 1.460700e+00 0.8969
Lombardia (donut) edu_emp_per1000 1358 40 -8.610000e-02 2.870000e-02 0.0027
Lombardia (donut) edu_muni_school_area_per1000 2746 40 2.200600e+00 9.551000e-01 0.0212
Lombardia (donut) edu_serv_lvl 2668 40 1.608000e-01 2.688000e-01 0.5496
Lombardia (donut) evasione 2746 40 -6.344000e-01 8.589000e-01 0.4602
Lombardia (donut) expend_level 2746 40 6.625000e-01 2.652000e-01 0.0125
Lombardia (donut) expenditure 2746 40 -1.432557e+06 1.524924e+05 0.0000
Lombardia (donut) gb_intensity 2766 40 4.000000e-04 2.000000e-03 0.8584
Lombardia (donut) gb_reg_rate 2766 40 9.250000e-02 1.350000e-02 0.0000
Lombardia (donut) income 2746 40 -1.765973e+08 1.775563e+07 0.0000
Lombardia (donut) incomepc 2746 40 1.098302e+03 2.440615e+02 0.0000
Lombardia (donut) marr_civil 2746 40 -1.230470e+01 1.878900e+00 0.0000
Lombardia (donut) marr_rel 2746 40 -5.375700e+00 1.009700e+00 0.0000
Lombardia (donut) pol_mun_road 2746 40 -3.730910e+01 6.763900e+00 0.0000
Lombardia (donut) services_level 2746 40 1.770300e+00 2.964000e-01 0.0000

Appendix B2. Placebo cutoffs (every 5 km to ±30 km), fixed h = 5

cat("\n\n## Appendix B2. Placebo cutoffs: every ", PLACEBO_STEP_KM, " km to ±", PLACEBO_MAX_KM,
    " (including 0), fixed h = ", H_DONUT, " km\n\n", sep = "")
## 
## 
## ## Appendix B2. Placebo cutoffs: every 5 km to ±30 (including 0), fixed h = 40 km
# Covariates for All-Italy placebo (region FE)
covs_all_placebo <- make_region_covs(df_all, REGION)

# Estimate placebo curves (All-Italy + Lombardia) at fixed h = 40
placebo_all <- purrr::map_dfr(PLACEBO_OUTCOMES_PRESENT, function(y) {
  purrr::map_dfr(placebo_cutoffs, function(c0) {
    run_rd_fixed_h(df_all, y = y, x = RUNNING, covs_mat = covs_all_placebo,
                   c0 = c0, h_fixed = H_DONUT)
  })
}) %>% mutate(sample = "All Italy")

placebo_lom <- purrr::map_dfr(PLACEBO_OUTCOMES_PRESENT, function(y) {
  purrr::map_dfr(placebo_cutoffs, function(c0) {
    run_rd_fixed_h(df_lom, y = y, x = RUNNING, covs_mat = NULL,
                   c0 = c0, h_fixed = H_DONUT)
  })
}) %>% mutate(sample = "Lombardia")

placebo_df <- bind_rows(placebo_all, placebo_lom) %>%
  mutate(
    ci_low  = tau_bc - 1.96 * se_rb,
    ci_high = tau_bc + 1.96 * se_rb
  )

# Table (all estimates)
knitr::kable(
  placebo_df %>% select(sample, outcome, cutoff, h, tau_bc, se_rb, p_rb) %>% arrange(sample, outcome, cutoff),
  digits = 4
)
sample outcome cutoff h tau_bc se_rb p_rb
All Italy Admin_Tax_Emp -30 40 2.460000e-01 7.670000e-02 0.0013
All Italy Admin_Tax_Emp -25 40 1.685000e-01 6.120000e-02 0.0059
All Italy Admin_Tax_Emp -20 40 -4.930000e-02 7.390000e-02 0.5046
All Italy Admin_Tax_Emp -15 40 -2.851000e-01 7.170000e-02 0.0001
All Italy Admin_Tax_Emp -10 40 -6.480000e-02 4.340000e-02 0.1357
All Italy Admin_Tax_Emp -5 40 1.030000e-01 3.010000e-02 0.0006
All Italy Admin_Tax_Emp 0 40 9.260000e-02 3.070000e-02 0.0026
All Italy Admin_Tax_Emp 5 40 -2.228000e-01 2.930000e-02 0.0000
All Italy Admin_Tax_Emp 10 40 -8.940000e-02 4.210000e-02 0.0338
All Italy Admin_Tax_Emp 15 40 8.270000e-02 7.770000e-02 0.2869
All Italy Admin_Tax_Emp 20 40 5.071000e-01 1.490000e-01 0.0007
All Italy Admin_Tax_Emp 25 40 -4.432000e-01 1.173000e-01 0.0002
All Italy Admin_Tax_Emp 30 40 -9.400000e-02 8.210000e-02 0.2521
All Italy Pillar2_pol -30 40 4.209000e-01 7.762000e-01 0.5876
All Italy Pillar2_pol -25 40 7.103000e-01 7.747000e-01 0.3592
All Italy Pillar2_pol -20 40 -1.624900e+00 7.339000e-01 0.0268
All Italy Pillar2_pol -15 40 -1.239900e+00 6.724000e-01 0.0652
All Italy Pillar2_pol -10 40 9.345000e-01 6.334000e-01 0.1401
All Italy Pillar2_pol -5 40 -7.877000e-01 6.141000e-01 0.1996
All Italy Pillar2_pol 0 40 -6.059000e-01 6.324000e-01 0.3380
All Italy Pillar2_pol 5 40 9.044000e-01 6.325000e-01 0.1528
All Italy Pillar2_pol 10 40 9.535000e-01 6.493000e-01 0.1420
All Italy Pillar2_pol 15 40 1.357100e+00 6.806000e-01 0.0461
All Italy Pillar2_pol 20 40 -4.367100e+00 7.064000e-01 0.0000
All Italy Pillar2_pol 25 40 8.581000e-01 7.469000e-01 0.2506
All Italy Pillar2_pol 30 40 5.295000e-01 8.089000e-01 0.5128
All Italy PublS_CyclePath_per -30 40 -1.895600e+00 7.017000e-01 0.0069
All Italy PublS_CyclePath_per -25 40 2.203300e+00 6.553000e-01 0.0008
All Italy PublS_CyclePath_per -20 40 -1.015800e+00 7.565000e-01 0.1794
All Italy PublS_CyclePath_per -15 40 8.725000e-01 8.700000e-01 0.3159
All Italy PublS_CyclePath_per -10 40 -5.300500e+00 9.690000e-01 0.0000
All Italy PublS_CyclePath_per -5 40 1.068400e+00 7.270000e-01 0.1416
All Italy PublS_CyclePath_per 0 40 5.444700e+00 7.359000e-01 0.0000
All Italy PublS_CyclePath_per 5 40 1.133500e+00 7.636000e-01 0.1377
All Italy PublS_CyclePath_per 10 40 1.260200e+00 7.228000e-01 0.0813
All Italy PublS_CyclePath_per 15 40 -3.249800e+00 7.768000e-01 0.0000
All Italy PublS_CyclePath_per 20 40 -4.358000e-01 8.097000e-01 0.5904
All Italy PublS_CyclePath_per 25 40 -1.036200e+00 7.752000e-01 0.1814
All Italy PublS_CyclePath_per 30 40 -5.670000e-01 8.276000e-01 0.4932
All Italy edu_emp_per1000 -30 40 -1.404000e-01 6.220000e-02 0.0240
All Italy edu_emp_per1000 -25 40 9.010000e-02 2.380000e-02 0.0002
All Italy edu_emp_per1000 -20 40 -1.112000e-01 2.810000e-02 0.0001
All Italy edu_emp_per1000 -15 40 -4.800000e-02 2.310000e-02 0.0380
All Italy edu_emp_per1000 -10 40 -1.300000e-03 2.180000e-02 0.9529
All Italy edu_emp_per1000 -5 40 -1.720000e-02 2.200000e-02 0.4333
All Italy edu_emp_per1000 0 40 -1.090000e-02 2.020000e-02 0.5893
All Italy edu_emp_per1000 5 40 6.180000e-02 1.970000e-02 0.0017
All Italy edu_emp_per1000 10 40 -3.900000e-03 2.320000e-02 0.8653
All Italy edu_emp_per1000 15 40 -4.520000e-02 2.670000e-02 0.0911
All Italy edu_emp_per1000 20 40 2.290000e-02 3.140000e-02 0.4663
All Italy edu_emp_per1000 25 40 1.110000e-02 2.710000e-02 0.6833
All Italy edu_emp_per1000 30 40 -2.570000e-02 1.940000e-02 0.1849
All Italy edu_muni_school_area_per1000 -30 40 1.949900e+00 7.508000e-01 0.0094
All Italy edu_muni_school_area_per1000 -25 40 1.793100e+00 7.055000e-01 0.0110
All Italy edu_muni_school_area_per1000 -20 40 -1.617100e+00 7.156000e-01 0.0238
All Italy edu_muni_school_area_per1000 -15 40 -5.808400e+00 6.268000e-01 0.0000
All Italy edu_muni_school_area_per1000 -10 40 -1.341000e-01 5.405000e-01 0.8041
All Italy edu_muni_school_area_per1000 -5 40 1.266700e+00 4.775000e-01 0.0080
All Italy edu_muni_school_area_per1000 0 40 1.352600e+00 4.779000e-01 0.0047
All Italy edu_muni_school_area_per1000 5 40 -3.271000e-01 4.665000e-01 0.4831
All Italy edu_muni_school_area_per1000 10 40 1.429900e+00 4.653000e-01 0.0021
All Italy edu_muni_school_area_per1000 15 40 -3.428000e-01 4.545000e-01 0.4507
All Italy edu_muni_school_area_per1000 20 40 -3.350800e+00 4.576000e-01 0.0000
All Italy edu_muni_school_area_per1000 25 40 -8.927000e-01 4.564000e-01 0.0505
All Italy edu_muni_school_area_per1000 30 40 -7.190000e-01 4.605000e-01 0.1185
All Italy edu_serv_lvl -30 40 -6.323000e-01 2.144000e-01 0.0032
All Italy edu_serv_lvl -25 40 -1.415000e-01 2.092000e-01 0.4988
All Italy edu_serv_lvl -20 40 6.180000e-02 2.109000e-01 0.7697
All Italy edu_serv_lvl -15 40 -5.290000e-02 2.114000e-01 0.8022
All Italy edu_serv_lvl -10 40 -7.439000e-01 2.046000e-01 0.0003
All Italy edu_serv_lvl -5 40 -1.709000e-01 1.814000e-01 0.3461
All Italy edu_serv_lvl 0 40 3.057000e-01 1.812000e-01 0.0916
All Italy edu_serv_lvl 5 40 2.579000e-01 1.821000e-01 0.1567
All Italy edu_serv_lvl 10 40 9.543000e-01 1.884000e-01 0.0000
All Italy edu_serv_lvl 15 40 -6.708000e-01 1.906000e-01 0.0004
All Italy edu_serv_lvl 20 40 -7.580000e-01 1.981000e-01 0.0001
All Italy edu_serv_lvl 25 40 -2.597000e-01 1.959000e-01 0.1851
All Italy edu_serv_lvl 30 40 -1.051000e-01 2.020000e-01 0.6028
All Italy evasione -30 40 -3.221000e+00 8.045000e-01 0.0001
All Italy evasione -25 40 -3.086700e+00 7.602000e-01 0.0000
All Italy evasione -20 40 6.290000e-01 6.353000e-01 0.3221
All Italy evasione -15 40 3.950600e+00 5.307000e-01 0.0000
All Italy evasione -10 40 2.473600e+00 4.884000e-01 0.0000
All Italy evasione -5 40 -4.803000e-01 4.480000e-01 0.2837
All Italy evasione 0 40 -1.151800e+00 4.435000e-01 0.0094
All Italy evasione 5 40 -4.651000e-01 3.982000e-01 0.2428
All Italy evasione 10 40 -9.741000e-01 3.996000e-01 0.0148
All Italy evasione 15 40 -1.213000e-01 4.138000e-01 0.7694
All Italy evasione 20 40 2.719700e+00 4.376000e-01 0.0000
All Italy evasione 25 40 -1.139500e+00 4.651000e-01 0.0143
All Italy evasione 30 40 -4.034000e-01 4.885000e-01 0.4089
All Italy expend_level -30 40 -2.197000e-01 1.656000e-01 0.1846
All Italy expend_level -25 40 2.282000e-01 1.701000e-01 0.1799
All Italy expend_level -20 40 3.445000e-01 1.680000e-01 0.0403
All Italy expend_level -15 40 -3.175000e-01 1.741000e-01 0.0682
All Italy expend_level -10 40 -2.331000e-01 1.562000e-01 0.1356
All Italy expend_level -5 40 -2.268000e-01 1.396000e-01 0.1043
All Italy expend_level 0 40 2.311000e-01 1.433000e-01 0.1067
All Italy expend_level 5 40 -6.380000e-02 1.441000e-01 0.6578
All Italy expend_level 10 40 1.815000e-01 1.475000e-01 0.2186
All Italy expend_level 15 40 1.521000e-01 1.706000e-01 0.3726
All Italy expend_level 20 40 -5.206000e-01 1.725000e-01 0.0025
All Italy expend_level 25 40 -2.584000e-01 1.660000e-01 0.1195
All Italy expend_level 30 40 1.282600e+00 1.647000e-01 0.0000
All Italy expenditure -30 40 -2.029096e+06 5.413200e+05 0.0002
All Italy expenditure -25 40 4.598891e+05 4.530146e+05 0.3100
All Italy expenditure -20 40 -1.250840e+06 3.980632e+05 0.0017
All Italy expenditure -15 40 5.415566e+05 3.014317e+05 0.0724
All Italy expenditure -10 40 -2.702768e+04 3.033418e+05 0.9290
All Italy expenditure -5 40 6.384092e+04 2.354373e+05 0.7863
All Italy expenditure 0 40 -5.864452e+05 1.605011e+05 0.0003
All Italy expenditure 5 40 -1.330702e+05 1.745823e+05 0.4459
All Italy expenditure 10 40 4.513561e+05 5.300651e+05 0.3945
All Italy expenditure 15 40 1.654436e+06 1.099033e+06 0.1322
All Italy expenditure 20 40 4.679464e+06 1.606868e+06 0.0036
All Italy expenditure 25 40 -2.352458e+06 1.155721e+06 0.0418
All Italy expenditure 30 40 -6.841257e+05 6.672265e+05 0.3052
All Italy gb_intensity -30 40 -6.500000e-03 1.000000e-03 0.0000
All Italy gb_intensity -25 40 -2.600000e-03 1.100000e-03 0.0227
All Italy gb_intensity -20 40 4.600000e-03 1.300000e-03 0.0007
All Italy gb_intensity -15 40 6.900000e-03 1.400000e-03 0.0000
All Italy gb_intensity -10 40 -1.900000e-03 1.400000e-03 0.1619
All Italy gb_intensity -5 40 -3.900000e-03 1.200000e-03 0.0010
All Italy gb_intensity 0 40 -5.000000e-03 1.200000e-03 0.0000
All Italy gb_intensity 5 40 -3.000000e-04 1.300000e-03 0.8374
All Italy gb_intensity 10 40 7.600000e-03 1.600000e-03 0.0000
All Italy gb_intensity 15 40 3.900000e-03 1.600000e-03 0.0175
All Italy gb_intensity 20 40 -1.080000e-02 1.500000e-03 0.0000
All Italy gb_intensity 25 40 -4.300000e-03 1.200000e-03 0.0005
All Italy gb_intensity 30 40 2.900000e-03 1.300000e-03 0.0229
All Italy gb_reg_rate -30 40 4.580000e-02 1.650000e-02 0.0055
All Italy gb_reg_rate -25 40 -2.700000e-03 1.610000e-02 0.8664
All Italy gb_reg_rate -20 40 1.660000e-02 1.480000e-02 0.2624
All Italy gb_reg_rate -15 40 -3.040000e-02 1.220000e-02 0.0128
All Italy gb_reg_rate -10 40 -3.170000e-02 1.080000e-02 0.0035
All Italy gb_reg_rate -5 40 -3.640000e-02 1.070000e-02 0.0007
All Italy gb_reg_rate 0 40 3.780000e-02 1.160000e-02 0.0011
All Italy gb_reg_rate 5 40 6.310000e-02 1.260000e-02 0.0000
All Italy gb_reg_rate 10 40 -3.440000e-02 1.190000e-02 0.0039
All Italy gb_reg_rate 15 40 -1.360000e-02 1.220000e-02 0.2658
All Italy gb_reg_rate 20 40 -4.220000e-02 1.360000e-02 0.0019
All Italy gb_reg_rate 25 40 5.000000e-02 1.570000e-02 0.0014
All Italy gb_reg_rate 30 40 6.440000e-02 1.720000e-02 0.0002
All Italy income -30 40 -1.167725e+08 3.478761e+07 0.0008
All Italy income -25 40 3.155573e+07 3.122797e+07 0.3123
All Italy income -20 40 -1.018863e+08 2.990158e+07 0.0007
All Italy income -15 40 5.341101e+07 2.676626e+07 0.0460
All Italy income -10 40 -5.478871e+06 2.974583e+07 0.8539
All Italy income -5 40 -1.229272e+07 2.454716e+07 0.6165
All Italy income 0 40 -9.190051e+07 1.508990e+07 0.0000
All Italy income 5 40 -2.311387e+07 1.779191e+07 0.1939
All Italy income 10 40 4.958805e+07 5.762339e+07 0.3895
All Italy income 15 40 2.073770e+08 1.214646e+08 0.0878
All Italy income 20 40 5.228651e+08 1.778249e+08 0.0033
All Italy income 25 40 -2.777767e+08 1.277668e+08 0.0297
All Italy income 30 40 -9.216039e+07 7.291486e+07 0.2062
All Italy incomepc -30 40 -2.339395e+02 1.878854e+02 0.2131
All Italy incomepc -25 40 -1.358311e+02 1.858468e+02 0.4649
All Italy incomepc -20 40 -2.374455e+02 1.916437e+02 0.2153
All Italy incomepc -15 40 -5.625573e+02 1.857296e+02 0.0025
All Italy incomepc -10 40 -7.842552e+02 1.910082e+02 0.0000
All Italy incomepc -5 40 7.823980e+01 1.681200e+02 0.6417
All Italy incomepc 0 40 6.508961e+02 1.767874e+02 0.0002
All Italy incomepc 5 40 1.499011e+02 1.970435e+02 0.4468
All Italy incomepc 10 40 1.123670e+03 2.446427e+02 0.0000
All Italy incomepc 15 40 -1.510012e+03 2.586102e+02 0.0000
All Italy incomepc 20 40 -1.792299e+03 2.706013e+02 0.0000
All Italy incomepc 25 40 -1.153234e+02 2.708157e+02 0.6702
All Italy incomepc 30 40 1.782282e+03 2.568818e+02 0.0000
All Italy marr_civil -30 40 -1.106720e+01 3.375300e+00 0.0010
All Italy marr_civil -25 40 3.525500e+00 3.125500e+00 0.2593
All Italy marr_civil -20 40 -1.236650e+01 3.080200e+00 0.0001
All Italy marr_civil -15 40 1.643100e+00 2.415400e+00 0.4963
All Italy marr_civil -10 40 1.607800e+00 2.412700e+00 0.5052
All Italy marr_civil -5 40 2.308500e+00 1.920200e+00 0.2293
All Italy marr_civil 0 40 -3.811000e+00 1.394500e+00 0.0063
All Italy marr_civil 5 40 -3.069800e+00 1.539000e+00 0.0461
All Italy marr_civil 10 40 4.208700e+00 4.126300e+00 0.3077
All Italy marr_civil 15 40 1.475590e+01 8.392100e+00 0.0787
All Italy marr_civil 20 40 3.504160e+01 1.221770e+01 0.0041
All Italy marr_civil 25 40 -1.658050e+01 8.802400e+00 0.0596
All Italy marr_civil 30 40 -8.369100e+00 5.117800e+00 0.1020
All Italy marr_rel -30 40 -3.965700e+00 1.083300e+00 0.0003
All Italy marr_rel -25 40 1.835300e+00 1.069700e+00 0.0862
All Italy marr_rel -20 40 -3.584200e+00 1.094800e+00 0.0011
All Italy marr_rel -15 40 4.955000e-01 8.994000e-01 0.5817
All Italy marr_rel -10 40 -1.565000e-01 8.946000e-01 0.8612
All Italy marr_rel -5 40 5.056000e-01 7.504000e-01 0.5005
All Italy marr_rel 0 40 -2.932300e+00 6.360000e-01 0.0000
All Italy marr_rel 5 40 6.470000e-02 6.844000e-01 0.9246
All Italy marr_rel 10 40 2.402600e+00 1.405100e+00 0.0873
All Italy marr_rel 15 40 3.734700e+00 2.641100e+00 0.1573
All Italy marr_rel 20 40 9.181700e+00 3.781700e+00 0.0152
All Italy marr_rel 25 40 -5.158700e+00 2.745400e+00 0.0602
All Italy marr_rel 30 40 -3.019700e+00 1.674000e+00 0.0712
All Italy pol_mun_road -30 40 1.598400e+00 9.020400e+00 0.8593
All Italy pol_mun_road -25 40 1.287330e+01 9.057500e+00 0.1552
All Italy pol_mun_road -20 40 -2.942370e+01 9.340600e+00 0.0016
All Italy pol_mun_road -15 40 -4.203600e+00 7.380900e+00 0.5690
All Italy pol_mun_road -10 40 -5.322300e+00 7.096000e+00 0.4532
All Italy pol_mun_road -5 40 2.104500e+00 5.071300e+00 0.6782
All Italy pol_mun_road 0 40 -2.273600e+00 4.258600e+00 0.5934
All Italy pol_mun_road 5 40 1.018950e+01 4.001100e+00 0.0109
All Italy pol_mun_road 10 40 2.650100e+00 6.295000e+00 0.6738
All Italy pol_mun_road 15 40 4.975000e+00 9.805800e+00 0.6119
All Italy pol_mun_road 20 40 1.824650e+01 1.350470e+01 0.1767
All Italy pol_mun_road 25 40 -1.833220e+01 9.898500e+00 0.0640
All Italy pol_mun_road 30 40 -5.557000e+00 6.516800e+00 0.3938
All Italy services_level -30 40 -2.148000e-01 2.209000e-01 0.3310
All Italy services_level -25 40 5.324000e-01 2.072000e-01 0.0102
All Italy services_level -20 40 -3.922000e-01 2.029000e-01 0.0532
All Italy services_level -15 40 -1.454300e+00 1.969000e-01 0.0000
All Italy services_level -10 40 -5.231000e-01 1.727000e-01 0.0024
All Italy services_level -5 40 4.858000e-01 1.594000e-01 0.0023
All Italy services_level 0 40 1.235700e+00 1.604000e-01 0.0000
All Italy services_level 5 40 -7.478000e-01 1.568000e-01 0.0000
All Italy services_level 10 40 -4.870000e-01 1.475000e-01 0.0010
All Italy services_level 15 40 1.943000e-01 1.513000e-01 0.1990
All Italy services_level 20 40 5.369000e-01 1.672000e-01 0.0013
All Italy services_level 25 40 -2.693000e-01 1.783000e-01 0.1310
All Italy services_level 30 40 -1.400000e-03 1.888000e-01 0.9940
Lombardia Admin_Tax_Emp -30 40 NA NA NA
Lombardia Admin_Tax_Emp -25 40 NA NA NA
Lombardia Admin_Tax_Emp -20 40 NA NA NA
Lombardia Admin_Tax_Emp -15 40 -9.660000e-02 8.720000e-02 0.2683
Lombardia Admin_Tax_Emp -10 40 5.060000e-02 7.460000e-02 0.4971
Lombardia Admin_Tax_Emp -5 40 5.730000e-02 4.960000e-02 0.2479
Lombardia Admin_Tax_Emp 0 40 -5.290000e-02 4.530000e-02 0.2427
Lombardia Admin_Tax_Emp 5 40 -3.008000e-01 3.700000e-02 0.0000
Lombardia Admin_Tax_Emp 10 40 -7.920000e-02 4.360000e-02 0.0692
Lombardia Admin_Tax_Emp 15 40 1.167000e-01 7.800000e-02 0.1345
Lombardia Admin_Tax_Emp 20 40 5.171000e-01 1.491000e-01 0.0005
Lombardia Admin_Tax_Emp 25 40 -4.718000e-01 1.343000e-01 0.0004
Lombardia Admin_Tax_Emp 30 40 -8.040000e-02 8.470000e-02 0.3424
Lombardia Pillar2_pol -30 40 NA NA NA
Lombardia Pillar2_pol -25 40 NA NA NA
Lombardia Pillar2_pol -20 40 -3.246000e+00 2.670900e+00 0.2243
Lombardia Pillar2_pol -15 40 -3.357700e+00 1.408300e+00 0.0171
Lombardia Pillar2_pol -10 40 -1.255400e+00 1.233800e+00 0.3089
Lombardia Pillar2_pol -5 40 -2.223200e+00 1.099100e+00 0.0431
Lombardia Pillar2_pol 0 40 -1.374300e+00 1.039500e+00 0.1862
Lombardia Pillar2_pol 5 40 9.294000e-01 7.230000e-01 0.1986
Lombardia Pillar2_pol 10 40 5.252000e-01 6.883000e-01 0.4455
Lombardia Pillar2_pol 15 40 1.273100e+00 7.117000e-01 0.0736
Lombardia Pillar2_pol 20 40 -4.325900e+00 7.418000e-01 0.0000
Lombardia Pillar2_pol 25 40 1.300600e+00 7.790000e-01 0.0950
Lombardia Pillar2_pol 30 40 8.873000e-01 8.246000e-01 0.2819
Lombardia PublS_CyclePath_per -30 40 NA NA NA
Lombardia PublS_CyclePath_per -25 40 NA NA NA
Lombardia PublS_CyclePath_per -20 40 -7.596000e-01 8.539000e-01 0.3737
Lombardia PublS_CyclePath_per -15 40 9.110000e-02 5.062000e-01 0.8571
Lombardia PublS_CyclePath_per -10 40 -2.681000e+00 7.638000e-01 0.0004
Lombardia PublS_CyclePath_per -5 40 3.469600e+00 8.920000e-01 0.0001
Lombardia PublS_CyclePath_per 0 40 7.766000e-01 1.447300e+00 0.5915
Lombardia PublS_CyclePath_per 5 40 9.180000e-01 9.311000e-01 0.3242
Lombardia PublS_CyclePath_per 10 40 9.129000e-01 7.703000e-01 0.2360
Lombardia PublS_CyclePath_per 15 40 -2.614900e+00 8.076000e-01 0.0012
Lombardia PublS_CyclePath_per 20 40 4.626000e-01 8.373000e-01 0.5806
Lombardia PublS_CyclePath_per 25 40 -4.273000e-01 8.132000e-01 0.5993
Lombardia PublS_CyclePath_per 30 40 -2.459000e-01 8.453000e-01 0.7711
Lombardia edu_emp_per1000 -30 40 NA NA NA
Lombardia edu_emp_per1000 -25 40 NA NA NA
Lombardia edu_emp_per1000 -20 40 NA NA NA
Lombardia edu_emp_per1000 -15 40 -1.770000e-02 8.700000e-03 0.0435
Lombardia edu_emp_per1000 -10 40 -8.300000e-03 1.280000e-02 0.5174
Lombardia edu_emp_per1000 -5 40 -4.420000e-02 2.220000e-02 0.0471
Lombardia edu_emp_per1000 0 40 -8.810000e-02 2.680000e-02 0.0010
Lombardia edu_emp_per1000 5 40 7.020000e-02 2.020000e-02 0.0005
Lombardia edu_emp_per1000 10 40 -6.200000e-03 2.450000e-02 0.8010
Lombardia edu_emp_per1000 15 40 -3.960000e-02 2.750000e-02 0.1505
Lombardia edu_emp_per1000 20 40 2.430000e-02 3.180000e-02 0.4450
Lombardia edu_emp_per1000 25 40 1.560000e-02 2.810000e-02 0.5787
Lombardia edu_emp_per1000 30 40 -1.670000e-02 1.970000e-02 0.3955
Lombardia edu_muni_school_area_per1000 -30 40 NA NA NA
Lombardia edu_muni_school_area_per1000 -25 40 NA NA NA
Lombardia edu_muni_school_area_per1000 -20 40 4.991500e+00 2.403000e+00 0.0378
Lombardia edu_muni_school_area_per1000 -15 40 -6.805300e+00 1.603900e+00 0.0000
Lombardia edu_muni_school_area_per1000 -10 40 -4.224900e+00 1.058000e+00 0.0001
Lombardia edu_muni_school_area_per1000 -5 40 3.137100e+00 7.490000e-01 0.0000
Lombardia edu_muni_school_area_per1000 0 40 1.869200e+00 9.372000e-01 0.0461
Lombardia edu_muni_school_area_per1000 5 40 -8.850000e-01 5.351000e-01 0.0982
Lombardia edu_muni_school_area_per1000 10 40 1.151300e+00 5.038000e-01 0.0223
Lombardia edu_muni_school_area_per1000 15 40 -2.593000e-01 4.836000e-01 0.5919
Lombardia edu_muni_school_area_per1000 20 40 -3.213700e+00 4.776000e-01 0.0000
Lombardia edu_muni_school_area_per1000 25 40 -4.852000e-01 4.779000e-01 0.3099
Lombardia edu_muni_school_area_per1000 30 40 -2.893000e-01 4.721000e-01 0.5400
Lombardia edu_serv_lvl -30 40 NA NA NA
Lombardia edu_serv_lvl -25 40 NA NA NA
Lombardia edu_serv_lvl -20 40 7.153000e-01 3.704000e-01 0.0535
Lombardia edu_serv_lvl -15 40 2.473000e-01 3.647000e-01 0.4977
Lombardia edu_serv_lvl -10 40 -1.894600e+00 3.759000e-01 0.0000
Lombardia edu_serv_lvl -5 40 -6.373000e-01 2.576000e-01 0.0134
Lombardia edu_serv_lvl 0 40 1.158000e-01 2.597000e-01 0.6557
Lombardia edu_serv_lvl 5 40 2.497000e-01 2.042000e-01 0.2214
Lombardia edu_serv_lvl 10 40 9.569000e-01 2.010000e-01 0.0000
Lombardia edu_serv_lvl 15 40 -5.918000e-01 1.986000e-01 0.0029
Lombardia edu_serv_lvl 20 40 -6.061000e-01 2.067000e-01 0.0034
Lombardia edu_serv_lvl 25 40 -1.439000e-01 2.060000e-01 0.4849
Lombardia edu_serv_lvl 30 40 9.300000e-03 2.059000e-01 0.9639
Lombardia evasione -30 40 NA NA NA
Lombardia evasione -25 40 NA NA NA
Lombardia evasione -20 40 2.363400e+00 2.488000e+00 0.3422
Lombardia evasione -15 40 3.661100e+00 1.015300e+00 0.0003
Lombardia evasione -10 40 4.205100e+00 1.004500e+00 0.0000
Lombardia evasione -5 40 6.170000e-02 8.200000e-01 0.9400
Lombardia evasione 0 40 -6.620000e-02 8.308000e-01 0.9365
Lombardia evasione 5 40 -2.324000e-01 4.583000e-01 0.6121
Lombardia evasione 10 40 -7.813000e-01 4.247000e-01 0.0658
Lombardia evasione 15 40 -1.689000e-01 4.323000e-01 0.6960
Lombardia evasione 20 40 2.539900e+00 4.578000e-01 0.0000
Lombardia evasione 25 40 -1.599200e+00 4.868000e-01 0.0010
Lombardia evasione 30 40 -8.179000e-01 4.983000e-01 0.1007
Lombardia expend_level -30 40 NA NA NA
Lombardia expend_level -25 40 NA NA NA
Lombardia expend_level -20 40 1.199800e+00 3.831000e-01 0.0017
Lombardia expend_level -15 40 5.659000e-01 3.722000e-01 0.1284
Lombardia expend_level -10 40 -7.661000e-01 3.456000e-01 0.0266
Lombardia expend_level -5 40 -8.630000e-01 2.556000e-01 0.0007
Lombardia expend_level 0 40 7.570000e-01 2.581000e-01 0.0034
Lombardia expend_level 5 40 -5.080000e-02 1.580000e-01 0.7479
Lombardia expend_level 10 40 2.741000e-01 1.553000e-01 0.0777
Lombardia expend_level 15 40 1.982000e-01 1.772000e-01 0.2634
Lombardia expend_level 20 40 -5.079000e-01 1.860000e-01 0.0063
Lombardia expend_level 25 40 -2.061000e-01 1.745000e-01 0.2377
Lombardia expend_level 30 40 1.367000e+00 1.675000e-01 0.0000
Lombardia expenditure -30 40 NA NA NA
Lombardia expenditure -25 40 NA NA NA
Lombardia expenditure -20 40 -1.575181e+05 1.838620e+05 0.3916
Lombardia expenditure -15 40 4.580354e+05 2.527029e+05 0.0699
Lombardia expenditure -10 40 1.135609e+04 3.531286e+05 0.9743
Lombardia expenditure -5 40 -6.621972e+05 2.928470e+05 0.0237
Lombardia expenditure 0 40 -1.294804e+06 1.442799e+05 0.0000
Lombardia expenditure 5 40 -1.665005e+05 1.627168e+05 0.3062
Lombardia expenditure 10 40 1.096587e+05 5.230459e+05 0.8339
Lombardia expenditure 15 40 1.905644e+06 1.097258e+06 0.0824
Lombardia expenditure 20 40 4.847774e+06 1.606246e+06 0.0025
Lombardia expenditure 25 40 -2.320052e+06 1.276253e+06 0.0691
Lombardia expenditure 30 40 -4.960393e+05 6.695548e+05 0.4588
Lombardia gb_intensity -30 40 NA NA NA
Lombardia gb_intensity -25 40 NA NA NA
Lombardia gb_intensity -20 40 8.600000e-03 3.900000e-03 0.0288
Lombardia gb_intensity -15 40 1.790000e-02 3.600000e-03 0.0000
Lombardia gb_intensity -10 40 -1.730000e-02 3.600000e-03 0.0000
Lombardia gb_intensity -5 40 -1.500000e-03 2.200000e-03 0.4773
Lombardia gb_intensity 0 40 3.500000e-03 2.000000e-03 0.0776
Lombardia gb_intensity 5 40 8.000000e-04 1.500000e-03 0.6259
Lombardia gb_intensity 10 40 9.200000e-03 1.700000e-03 0.0000
Lombardia gb_intensity 15 40 2.600000e-03 1.800000e-03 0.1429
Lombardia gb_intensity 20 40 -1.210000e-02 1.600000e-03 0.0000
Lombardia gb_intensity 25 40 -4.500000e-03 1.300000e-03 0.0004
Lombardia gb_intensity 30 40 3.300000e-03 1.300000e-03 0.0116
Lombardia gb_reg_rate -30 40 NA NA NA
Lombardia gb_reg_rate -25 40 NA NA NA
Lombardia gb_reg_rate -20 40 3.300000e-03 2.250000e-02 0.8820
Lombardia gb_reg_rate -15 40 -2.050000e-02 1.460000e-02 0.1615
Lombardia gb_reg_rate -10 40 -4.880000e-02 1.310000e-02 0.0002
Lombardia gb_reg_rate -5 40 -4.040000e-02 1.300000e-02 0.0018
Lombardia gb_reg_rate 0 40 8.020000e-02 1.250000e-02 0.0000
Lombardia gb_reg_rate 5 40 7.630000e-02 1.440000e-02 0.0000
Lombardia gb_reg_rate 10 40 -6.020000e-02 1.280000e-02 0.0000
Lombardia gb_reg_rate 15 40 -1.480000e-02 1.250000e-02 0.2354
Lombardia gb_reg_rate 20 40 -3.460000e-02 1.400000e-02 0.0137
Lombardia gb_reg_rate 25 40 6.320000e-02 1.620000e-02 0.0001
Lombardia gb_reg_rate 30 40 7.220000e-02 1.770000e-02 0.0000
Lombardia income -30 40 NA NA NA
Lombardia income -25 40 NA NA NA
Lombardia income -20 40 -2.572292e+07 3.265561e+07 0.4309
Lombardia income -15 40 5.778366e+07 2.821561e+07 0.0406
Lombardia income -10 40 1.551929e+07 3.932325e+07 0.6931
Lombardia income -5 40 -6.996220e+07 3.262107e+07 0.0320
Lombardia income 0 40 -1.597944e+08 1.689269e+07 0.0000
Lombardia income 5 40 -2.945151e+07 1.756364e+07 0.0936
Lombardia income 10 40 2.064159e+07 5.746486e+07 0.7194
Lombardia income 15 40 2.266632e+08 1.213928e+08 0.0619
Lombardia income 20 40 5.374229e+08 1.778245e+08 0.0025
Lombardia income 25 40 -2.729822e+08 1.411699e+08 0.0531
Lombardia income 30 40 -6.860914e+07 7.315652e+07 0.3483
Lombardia incomepc -30 40 NA NA NA
Lombardia incomepc -25 40 NA NA NA
Lombardia incomepc -20 40 -2.063234e+03 4.241674e+02 0.0000
Lombardia incomepc -15 40 -5.312796e+02 2.245492e+02 0.0180
Lombardia incomepc -10 40 -1.132407e+03 2.687176e+02 0.0000
Lombardia incomepc -5 40 1.388767e+02 2.550851e+02 0.5861
Lombardia incomepc 0 40 1.288767e+03 2.531422e+02 0.0000
Lombardia incomepc 5 40 1.750214e+02 2.259083e+02 0.4385
Lombardia incomepc 10 40 8.637562e+02 2.677281e+02 0.0013
Lombardia incomepc 15 40 -1.613312e+03 2.736305e+02 0.0000
Lombardia incomepc 20 40 -1.649290e+03 2.834907e+02 0.0000
Lombardia incomepc 25 40 1.585662e+02 2.846325e+02 0.5775
Lombardia incomepc 30 40 1.989750e+03 2.624977e+02 0.0000
Lombardia marr_civil -30 40 NA NA NA
Lombardia marr_civil -25 40 NA NA NA
Lombardia marr_civil -20 40 -1.110500e+00 2.864000e+00 0.6982
Lombardia marr_civil -15 40 1.486200e+00 2.167000e+00 0.4928
Lombardia marr_civil -10 40 3.339700e+00 2.783500e+00 0.2302
Lombardia marr_civil -5 40 1.398200e+00 2.390200e+00 0.5586
Lombardia marr_civil 0 40 -1.011730e+01 1.811100e+00 0.0000
Lombardia marr_civil 5 40 -3.856400e+00 1.514000e+00 0.0109
Lombardia marr_civil 10 40 2.508500e+00 4.083800e+00 0.5390
Lombardia marr_civil 15 40 1.694290e+01 8.386200e+00 0.0433
Lombardia marr_civil 20 40 3.625290e+01 1.221840e+01 0.0030
Lombardia marr_civil 25 40 -1.616830e+01 9.714400e+00 0.0960
Lombardia marr_civil 30 40 -7.156900e+00 5.141700e+00 0.1639
Lombardia marr_rel -30 40 NA NA NA
Lombardia marr_rel -25 40 NA NA NA
Lombardia marr_rel -20 40 -1.723900e+00 1.318200e+00 0.1909
Lombardia marr_rel -15 40 -2.610000e-02 8.839000e-01 0.9764
Lombardia marr_rel -10 40 -1.340000e-02 9.534000e-01 0.9888
Lombardia marr_rel -5 40 6.924000e-01 8.570000e-01 0.4191
Lombardia marr_rel 0 40 -5.052900e+00 9.668000e-01 0.0000
Lombardia marr_rel 5 40 -5.660000e-02 6.712000e-01 0.9328
Lombardia marr_rel 10 40 9.098000e-01 1.396100e+00 0.5146
Lombardia marr_rel 15 40 4.152600e+00 2.643500e+00 0.1162
Lombardia marr_rel 20 40 9.676000e+00 3.785400e+00 0.0106
Lombardia marr_rel 25 40 -4.743200e+00 3.023100e+00 0.1167
Lombardia marr_rel 30 40 -2.403400e+00 1.681600e+00 0.1529
Lombardia pol_mun_road -30 40 NA NA NA
Lombardia pol_mun_road -25 40 NA NA NA
Lombardia pol_mun_road -20 40 -3.134010e+01 1.316160e+01 0.0173
Lombardia pol_mun_road -15 40 -5.728600e+00 4.252000e+00 0.1779
Lombardia pol_mun_road -10 40 1.771900e+00 4.775900e+00 0.7106
Lombardia pol_mun_road -5 40 7.621700e+00 4.733200e+00 0.1073
Lombardia pol_mun_road 0 40 -3.782250e+01 6.504100e+00 0.0000
Lombardia pol_mun_road 5 40 5.715700e+00 3.598000e+00 0.1122
Lombardia pol_mun_road 10 40 -1.268400e+00 5.877100e+00 0.8291
Lombardia pol_mun_road 15 40 1.348130e+01 9.698300e+00 0.1645
Lombardia pol_mun_road 20 40 2.383720e+01 1.345960e+01 0.0766
Lombardia pol_mun_road 25 40 -1.550490e+01 1.081930e+01 0.1518
Lombardia pol_mun_road 30 40 -4.782600e+00 6.554700e+00 0.4656
Lombardia services_level -30 40 NA NA NA
Lombardia services_level -25 40 NA NA NA
Lombardia services_level -20 40 1.307100e+00 3.560000e-01 0.0002
Lombardia services_level -15 40 -1.268400e+00 4.657000e-01 0.0065
Lombardia services_level -10 40 -7.098000e-01 3.886000e-01 0.0678
Lombardia services_level -5 40 4.170000e-01 3.016000e-01 0.1668
Lombardia services_level 0 40 1.082300e+00 2.984000e-01 0.0003
Lombardia services_level 5 40 -9.769000e-01 1.807000e-01 0.0000
Lombardia services_level 10 40 -3.754000e-01 1.613000e-01 0.0199
Lombardia services_level 15 40 4.812000e-01 1.583000e-01 0.0024
Lombardia services_level 20 40 6.732000e-01 1.753000e-01 0.0001
Lombardia services_level 25 40 -2.006000e-01 1.867000e-01 0.2828
Lombardia services_level 30 40 4.600000e-02 1.925000e-01 0.8113 
# One graph per outcome, facet by sample
for (y in unique(placebo_df$outcome)) {
  dfy <- placebo_df %>% filter(outcome == y, is.finite(tau_bc), is.finite(se_rb))
  p <- ggplot(dfy, aes(x = cutoff, y = tau_bc)) +
    geom_hline(yintercept = 0) +
    geom_vline(xintercept = 0, linetype = "dashed") +
    geom_errorbar(aes(ymin = ci_low, ymax = ci_high), width = 0) +
    geom_point(size = 1.6) +
    facet_wrap(~ sample, ncol = 1, scales = "free_y") +
    labs(
      x = "Artificial cutoff (km; true cutoff at 0)",
      y = "RD estimate (bias-corrected Ï„; robust SE)",
      title = paste0("Placebo cutoff RD estimates: ", y, " (fixed h = ", H_PLACEBO, " km)")
    ) +
    theme_minimal(base_size = 12)
  print(p)
}