Full Empirical Analysis — Classical R Workflow

This skill is the canonical 8-step pipeline an applied economist runs on every empirical paper, written in the modern tidyverse + econometrics R ecosystem — dplyr/tidyr/haven for data, fixest as the panel/IV/DID workhorse, did/bacondecomp/HonestDiD for modern DID, rdrobust/rddensity for RD, Synth/gsynth/synthdid for synthetic control, MatchIt/WeightIt/cobalt/ebal for matching, grf/DoubleML for ML causal, mediation for causal mediation, marginaleffects for post-estimation, modelsummary/kableExtra/gt for publication tables, ggplot2/iplot/binsreg for figures.

Companion skills: this is the R sibling of 00-StatsPAI_skill (Python DSL), 00.1-Full-empirical-analysis-skill (explicit Python), and 00.2-Full-empirical-analysis-skill_Stata (Stata .do). All four implement the same 8 steps, in their respective ecosystems.

Philosophy

1. Tidyverse + fixest, the modern R idioms. feols(... | unit + year, cluster = ~unit), not Frankenstein-y lm(y ~ x + factor(unit) + factor(year)).
2. Reproducible scripts / Quarto. Every example below is paste-runnable. renv for package locking; Quarto (.qmd) for combined narrative + code + tables/figures.
3. 8 steps, first-class. R users historically over-invest in Step 5; this skill treats Steps 1–4 and 6–8 as core.
4. Rich outputs. Every step yields at least one table or figure — tex/docx/png/pdf.
5. Progressive disclosure. SKILL.md gives the canonical call per step; references/ holds variant-specific depth.

SkillOpt-style execution gate

Use this long playbook as a seed skill, not as a script to exhaustively apply. SkillOpt discipline: treat each local R/Quarto change as a candidate patch that must beat a selection check and survive a held-out check before it becomes reusable boilerplate. Before writing or revising an R script/Quarto workflow, compress the user's request into a task-local best_skill card:

best_skill: <mode + design + artifact target>
train_signal: <current failure, user goal, or missing evidence>
selection_split: <focal dataset/spec/output used to judge the candidate>
heldout_gate: <checks the patch must pass beyond the focal example>
accepted_patterns: <rules to reuse after validation>
rejected_patterns: <failed shortcuts not to retry without new evidence>
patch_scope: <one estimator/sample/export/robustness change>
reject_if: <conditions that force rollback to the last passing spec>

1. Route card: record the mode (econ, epi, or ml-causal), estimand, identification design, focal outcome/treatment, R package family, and required artifacts.
2. Bounded edit: change one decision at a time (sample rule, estimator, clustering, export format, or robustness check). Prefer the smallest patch that can pass validation.
3. Selection split discipline: treat the user's immediate failure or requested artifact as the selection split. Reserve at least one alternate outcome, sample window, estimator family, or export target as the held-out gate.
4. Held-out gate: define checks before running code: row counts, distinct() key uniqueness, treatment support, missingness thresholds, expected table/figure files, and one non-focal robustness/specification that the change must not break.
5. Reject buffer: if a candidate spec fails the gate, log the failure, R/Quarto diff, and gate output in analysis_log.md; revert to the last passing spec and do not retry the same unchecked pattern.
6. Slow/meta update: at the end of the task, write down accepted_patterns and rejected_patterns from the trajectory. Do not widen the canonical project template from a single passing run.
7. Promote only after validation: only turn a one-off fix into reusable project boilerplate after it passes the current data and at least one alternate outcome/sample/specification.

Three domain modes (default = AER econ; alternates = epi & ML-causal)

The default playbook above is AER-style applied econometrics — the AEA convention: written-out estimating equation, identifying assumption, design horse-race, full robustness gauntlet. The skill also ships two parallel sub-pipelines for the other two big causal-inference traditions, each reusing the same Steps 1–4 (cleaning / construction / Table 1 / diagnostics) and Step 8 (tables/figures) — only Step 5 (estimator) and Step 6/7 swap packages:

How to invoke a non-default mode (Claude / agent picks this up from the user's wording):

The three modes share the same Step 1–4 cleaning / Table 1 / diagnostics scaffolding, the same Step 8 export stack, and the same DAG-first identification logic — switching modes only changes which Step-5 estimator family you reach for, not the surrounding paper structure. If you only want descriptive stats / Table 1 / a balance check, the AER gtsummary::tbl_summary / modelsummary::datasummary_balance calls in Step 3 work identically across all three modes.

Default Output Spec — Economics Empirical Paper

This skill defaults to the applied-economics paper convention. Unless the user explicitly asks for a single point estimate, every run produces the full publication-ready output set below. Treat it as the contract of Step 8 — mandatory, not opt-in.

Required tables (always produced)

★ Table 2 is the centerpiece of every economics paper. It is the multi-column regression table that walks the reader from raw correlation (M1) to the fully-specified design (M6: 2-way FE + interacted FE + cluster-robust SE). Do not collapse it into a single column. Do not report only the headline coefficient. The progression is the credibility argument: if M1→M6 is monotone and stable, the design is plausibly identifying; if it collapses on adding FE, that is the result.

>

Canonical 6 columns, in order: 1. M1 raw bivariate (feols(y ~ treat, data)) 2. M2 + demographics (+ age + edu) 3. M3 + sector controls (+ tenure / firm_size) 4. M4 + unit FE (| worker_id) 5. M5 + 2-way FE (| worker_id + year) 6. M6 + interacted FE (| worker_id + year + industry^year) with cluster = ~ worker_id

Required figures (always produced)

Output file layout (default)

project/
├── tables/    table1_balance.xlsx/.docx/.tex  table2_main.xlsx/.docx/.tex
│              table3_mechanism.xlsx/.docx/.tex table4_heterogeneity.xlsx/.docx/.tex
│              table5_robustness.xlsx/.docx/.tex
└── figures/   fig1_trend.png(300dpi)+.pdf      fig2_event_study.png(300dpi)+.pdf
               fig3_coefplot.png(300dpi)+.pdf   fig4_sensitivity.png(300dpi)+.pdf

关键输出规则（必须遵守）：

• 图片格式：所有图片必须同时导出 PNG 格式（≥300 dpi） 和 PDF 格式（用于 LaTeX 排版）
• 表格格式：所有回归表格必须同时导出 Excel（.xlsx）、Word（.docx） 和 LaTeX（.tex） 三种格式
• PNG 用于幻灯片、Markdown 文档、邮件等场景；PDF 用于学术论文排版

When to deviate

• Single quick estimate — produce only the relevant cell, but warn that the standard deliverable is the full set above and offer to run it.
• Design does not support a figure (cross-section → no event study) — skip with a printed message() explaining why; do not silently drop.
• N=1 treated unit (`Synth` / `synthdid`) — replace F1/F2 with the SCM trajectory + placebo distribution; T1–T5 still apply.

Required packages

# Run once on a fresh R install:
install.packages(c(
  # Data
  "tidyverse", "haven", "readxl", "data.table", "janitor",
  "naniar", "VIM", "mice", "validate",
  # Description / tables
  "gtsummary", "tableone", "modelsummary", "kableExtra", "gt",
  "stargazer", "texreg", "flextable", "psych", "summarytools",
  # Tests
  "lmtest", "sandwich", "car", "tseries", "urca", "plm",
  "clubSandwich", "fwildclusterboot",
  # Modeling — workhorses
  "fixest",                                        # panel/IV/DID with HD FE — primary
  "AER",                                           # ivreg
  "ivreg",                                         # alternative IV
  # Modern DID
  "did",                                           # Callaway–Sant'Anna
  "didimputation",                                 # Borusyak–Jaravel–Spiess
  "fixest",                                        # sunab() for Sun–Abraham
  "synthdid",                                      # Synthetic DID
  "bacondecomp", "HonestDiD",
  "DIDmultiplegtDYN",                              # de Chaisemartin–D'Haultfœuille
  # RD
  "rdrobust", "rddensity", "rdmulti",
  # Synthetic control
  "Synth", "gsynth", "tidysynth",
  # Matching / weighting
  "MatchIt", "WeightIt", "cobalt", "ebal",
  # ML causal
  "grf", "DoubleML",
  # Mediation / SEM
  "mediation", "lavaan",
  # Robustness / inference
  "robomit",                                       # Oster delta
  "ri2", "ritools",                                # randomization inference
  "multcomp",
  # Margins / post-estimation
  "marginaleffects",
  # Plotting
  "ggplot2", "ggpubr", "cowplot", "patchwork",
  "binsreg",
  "ggdist", "ggrepel"
))
# fixest's iplot, esttex, etable are bundled.

The 8 Steps — Canonical Pipeline (mapped to AER paper sections)

┌──────────────────────────────────────────────────────────────────────┐
│ Step −1 Pre-Analysis Plan (PAP)  pwr / WebPower / DeclareDesign      │
│ Step 0  Sample log + data contract sample_log/stopifnot/jsonlite     │
│ Step 1  Data import & cleaning   read_csv/read_dta/janitor/naniar/mice│
│ Step 2  Variable construction    mutate/across/winsorize/lag/group_by │
│ Step 2.5 Empirical strategy      equation × ID assumption + pre-reg  │
│ Step 3  Descriptive statistics   gtsummary/datasummary_balance/cor_pmat│
│ Step 3.5 Identification graphics iplot/binsreg/rdplot/cobalt/Synth   │
│ Step 4  Diagnostic tests         shapiro/bptest/dwtest/vif/adf/kpss   │
│ Step 5  Baseline modeling        feols/ivreg/att_gt/synthdid/MatchIt  │
│ Step 6  Robustness battery       bacondecomp/HonestDiD/fwildclusterboot│
│ Step 7  Further analysis         marginaleffects/mediation/grf        │
│ Step 8  Tables & figures         modelsummary/iplot/ggplot2/cowplot   │
└──────────────────────────────────────────────────────────────────────┘

The 8 steps mirror the canonical sections of an applied AER / QJE / AEJ paper. Each step is one paper section and emits a paper-ready artifact on disk:

Paper section               Step  R moves
─────────────────────────── ───── ────────────────────────────────────────────────
Pre-Analysis Plan           −1    pwr / WebPower / DeclareDesign + freeze pap.json
§1. Data                     0    sample_log + 5-check stopifnot → JSON via jsonlite
§1. Data                     1    haven::read_dta · janitor::clean_names · naniar/mice
§1. Data                     2    mutate/across/Winsorize/lag/lead/diff · CPI deflate
§1.1 Descriptives (Table 1)  3    gtsummary::tbl_summary · datasummary_balance
§2. Empirical Strategy       2.5  write equation + ID assumption → strategy.md
§3. Identification graphics  3.5  fixest::iplot · binsreg · rdplot · cobalt::love.plot · Synth
§3.5 Diagnostics             4    bptest · dwtest · car::vif · urca::ur.df · phtest
§4. Main Results (Table 2)   5    fixest::feols progressive (m1...m6) · modelsummary
§5. Heterogeneity (Table 3)  7    feols(... + i(.):X) · marginaleffects::avg_slopes
§6. Mechanisms / Channels    7    mediation::mediate · lavaan · outcome ladder
§7. Robustness gauntlet      6    bacondecomp · HonestDiD · robomit · fwildclusterboot · ri2
§8. Replication package      8    modelsummary("...tex") · gt → docx · result.json

Below is the canonical call at each step. All examples share one running narrative — labor-econ panel where training (treatment) affects log_wage (outcome), with covariates age, edu, tenure, panel keys worker_id/firm_id/year. Variable names and parameter values are illustrative.

When a step has many variants (5 staggered-DID estimators; 4 hetero tests), SKILL.md shows the one you reach for first; deeper variants live in references/NN-<topic>.md.

Paper-ready figure & table inventory (what to produce by section)

A modern AER paper has 5–7 figures and 3–5 main tables + an appendix robustness table. Every step below leaves at least one numbered artifact on disk. Default file names assume parallel .tex / .docx / .xlsx exports (the agent should produce all three so co-authors can edit in Word, the build system can use LaTeX, and editors can edit raw numbers in Excel). 所有图片必须同时保存 PNG（≥300 dpi）和 PDF 两种格式。

Every R estimator above (fixest::feols / AER::ivreg / did::att_gt / grf::causal_forest / synthdid_estimate) returns a result object that can be passed straight into modelsummary(...) / modelplot(...) / etable(...). Don't hand-roll LaTeX from kable(), and don't render Word via flextable directly — modelsummary, etable, and gtsummary apply book-tab borders, AER stars, and the right SE label automatically. For deeper export recipes, see references/08-tables-plots.md.

Export cookbook — LaTeX / Word / Excel in one block

关键规则（必须遵守）：每个表格必须同时导出三种格式——Excel(.xlsx)、Word(.docx)、LaTeX(.tex)。每个图片必须同时保存PNG(≥300dpi)和PDF两种格式。

R has the best publication-table ecosystem of the three languages. Three tiers, picked by scope:

Journal styling — pick the right `stars` and SE label. The AEA convention is c("*"=.1, "**"=.05, "***"=.01) and notes = "Cluster-robust standard errors in parentheses...". Define a wrapper once at the top of master.R:

# Top of master.R — journal house-style wrapper
# 输出三格式：.xlsx（编辑）、.docx（Word）、.tex（LaTeX）
aer_table <- function(models, output, headers = NULL, coef_map = NULL) {
  base <- tools::file_path_sans_ext(output)
  for (ext in c(".xlsx", ".docx", ".tex")) {
    output_file <- paste0(base, ext)
    fmt <- if (ext == ".xlsx") "html" else if (ext == ".docx") "docx" else "latex"
    modelsummary(
      models,
      output    = output_file,
      stars     = c("*" = 0.1, "**" = 0.05, "***" = 0.01),
      gof_omit  = "BIC|AIC|F|Log|Adj",
      coef_map  = coef_map,
      notes     = paste("Cluster-robust standard errors in parentheses.",
                        "* p<0.10, ** p<0.05, *** p<0.01."),
      output_format = fmt
    )
  }
}

For the multi-panel .docx / .xlsx and Quarto cookbook (single-file paper-tables bundle), see references/08-tables-plots.md.

Step −1 — Pre-Analysis Plan (pre-data; AEA RCT Registry style)

Before touching the data, write down (a) the population, (b) the design, (c) the minimum detectable effect (MDE) under the planned sample size and α=0.05, β=0.20. Persist the result as pap.json so a referee can verify the design was powered before, not after, the data were seen.

library(pwr)         # classical power calculations
library(WebPower)    # cluster RCT, longitudinal, mixed designs
library(jsonlite)

# Two-sample MDE for a continuous outcome (Cohen's d framing)
pwr.t.test(d = 0.20, power = 0.80, sig.level = 0.05,
           type = "two.sample", alternative = "two.sided")
# → required n per arm

# Solve for MDE given fixed n
pwr.t.test(n = 2000, power = 0.80, sig.level = 0.05,
           type = "two.sample")$d
# → minimum detectable Cohen's d

# Cluster-randomized RCT — design effect
# Solve via WebPower::wp.crt2arm(...) for clusters / per-cluster size / power triangle
WebPower::wp.crt2arm(f = 0.20, J = NULL, n = 50, icc = 0.05, power = 0.80,
                     alpha = 0.05, alternative = "two.sided")
# → required clusters per arm

# DID power (Frison-Pocock / Bloom 1995): use WebPower::wp.kanova() or simulate
# RD power: simulate via DeclareDesign — see references/05-modeling.md §5.5

# Persist the protocol — referee will ask whether design was powered ex ante
pap <- list(
  population        = "manufacturing workers, 2010–2020",
  treatment         = "training (binary, staggered adoption)",
  outcome           = "log_wage",
  estimand          = "ATT",
  design            = "staggered DID, Callaway-Sant'Anna",
  alpha             = 0.05,
  power_target      = 0.80,
  mde_d             = 0.20,
  n_planned         = 12000,
  frozen_at         = "2026-01-15T09:00:00Z",
  git_sha           = "<paste>"
)
write_json(pap, "artifacts/pap.json", pretty = TRUE, auto_unbox = TRUE)

For richer DAG-aware power analysis (write down the DAG, declare estimands, simulate the design), use `DeclareDesign` — it is the R-native equivalent of EGAP's pre-analysis flow.

Commit artifacts/pap.json in the repo before Step 1. AEA RCT Registry / OSF preregistration tools accept it as the analysis-plan exhibit.

Step 0 — Sample-construction log & 5-check data contract

An AER §1 Data section has three jobs: (a) describe sources, (b) document every sample restriction (the "footnote 4" sample log), (c) lock the panel structure.

0.1 Sample-construction log (footnote 4)

library(tidyverse); library(jsonlite)

sample_log <- tibble::tibble(step = character(), n = integer())

df_raw <- read_dta("raw/panel.dta") %>% janitor::clean_names()
sample_log <- sample_log %>% add_row(step = "0. raw",                    n = nrow(df_raw))

df1 <- df_raw %>% drop_na(wage)
sample_log <- sample_log %>% add_row(step = "1. drop missing wage",       n = nrow(df1))

df2 <- df1 %>% filter(between(age, 18, 65))
sample_log <- sample_log %>% add_row(step = "2. drop age outside 18-65",  n = nrow(df2))

df3 <- df2 %>% filter(industry %in% c("manuf", "construction", "transport"))
sample_log <- sample_log %>% add_row(step = "3. keep target industries",  n = nrow(df3))

df <- df3
print(sample_log)
write_json(sample_log, "artifacts/sample_construction.json", pretty = TRUE)

Paste the printed tibble verbatim as footnote 4 of the paper.

0.2 Five-check data contract (go / no-go gate)

library(validate); library(assertr)

data_contract <- function(df, y, treatment, id = NULL, time = NULL, covariates = c()) {
  keys <- c(y, treatment, id, time, covariates)
  contract <- list(
    n_obs            = nrow(df),                                            # 1. shape
    dtypes           = sapply(df[keys], function(x) class(x)[1]),           # 2. dtypes
    n_missing        = sapply(df[keys], function(x) sum(is.na(x))),         # 3. missingness
    n_dupes_on_keys  = if (!is.null(id) && !is.null(time))
                         sum(duplicated(df[, c(id, time)])) else 0,          # 4. duplicates
    panel_balanced   = NULL,
    cohort_sizes     = NULL
  )

  if (!is.null(id) && !is.null(time)) {
    bal <- df %>% count(.data[[id]])
    contract$panel_balanced <- all(bal$n == max(bal$n))                      # 5. balance
    contract$n_dropped_by_balance <- sum(bal$n != max(bal$n))

    if ("first_treat" %in% names(df)) {
      contract$cohort_sizes <- df %>% distinct(.data[[id]], .keep_all = TRUE) %>%
                                count(first_treat) %>% deframe()
    }
  }

  contract$y_range         <- range(df[[y]],         na.rm = TRUE)
  contract$treatment_share <- mean(df[[treatment]],  na.rm = TRUE)

  # MCAR sniff test (Rubin) — if missing(y) is associated with covariates,
  # listwise deletion biases the estimate. Use mice / IPW instead.
  miss_y <- is.na(df[[y]])
  contract$mcar_hint <- "likely MCAR (listwise OK)"
  if (any(miss_y) && any(!miss_y)) {
    for (cov in covariates) {
      if (is.numeric(df[[cov]])) {
        p <- t.test(df[[cov]][miss_y], df[[cov]][!miss_y])$p.value
        if (p < 0.05) {
          contract$mcar_hint <- sprintf("NOT MCAR (y-miss differs on %s, p=%.3f) → use mice / IPW",
                                         cov, p)
          break
        }
      }
    }
  }
  contract
}

contract <- data_contract(df, y = "wage", treatment = "training",
                          id = "worker_id", time = "year",
                          covariates = c("age", "edu", "tenure"))

stopifnot(contract$n_dupes_on_keys == 0)
stopifnot(all(contract$n_missing == 0))

write_json(contract, "artifacts/data_contract.json",
           pretty = TRUE, auto_unbox = TRUE)

If any stopifnot fires, stop and fix it in dplyr first. R estimators silently drop NA rows downstream — this contract is the cheapest insurance against "why did N drop from 12,000 to 9,800 between Table 1 and Table 2?" referee questions.

Step 1 — Data import & cleaning

Deeper patterns: references/01-data-cleaning.md — every format (haven/readxl/data.table::fread/arrow::read_parquet/DBI), janitor::clean_names, naniar missingness viz, MCAR/MAR/MNAR triage with mice, validation with validate/assertr, panel structure checks.

library(tidyverse)
library(haven)        # .dta / .sav / .sas7bdat
library(janitor)      # clean_names()
library(naniar)       # missing-data viz
library(skimr)        # one-line dataset summary

# 1a. Load + first look
df <- read_dta("raw/panel.dta") %>%
  clean_names()                       # standardize to snake_case

skim(df)                              # rich one-line-per-var summary
naniar::miss_var_summary(df)
naniar::vis_miss(df)                  # missingness heatmap

# 1b. Dtypes
df <- df %>%
  mutate(
    year   = as.integer(year),
    wage   = as.numeric(wage),
    gender = as.factor(gender),
    date   = as.Date(date)
  )

# 1c. Missing values — decide PER VARIABLE
key_vars <- c("wage", "training", "worker_id", "year")
df <- df %>%
  drop_na(all_of(key_vars))
cat("After dropping NA on keys:", nrow(df), "rows\n")

df <- df %>%
  mutate(
    tenure_missing = is.na(tenure),
    tenure         = if_else(is.na(tenure), median(tenure, na.rm = TRUE), tenure),
    union          = fct_explicit_na(as.factor(union), na_level = "unknown")
  )

# 1d. Outliers — flag, don't drop yet
df <- df %>%
  mutate(wage_z = scale(wage)[,1],
         outlier_z4 = abs(wage_z) > 4)
cat("|z|>4 on wage:", sum(df$outlier_z4, na.rm = TRUE), "\n")

# 1e. Deduplicate panel key
stopifnot(nrow(df %>% distinct(worker_id, year)) == nrow(df))

# 1f. Merge with assertion
firm_chars <- read_dta("raw/firm_chars.dta")
n_before <- nrow(df)
df <- df %>%
  left_join(firm_chars, by = "firm_id", relationship = "many-to-one")
stopifnot(nrow(df) == n_before)       # no row inflation

# 1g. Panel structure
df %>% count(year)                    # per-year
df %>% count(worker_id) %>% summary() # per-unit

Key principle: dplyr + explicit stopifnot() assertions. No silent row drops downstream.

Step 2 — Variable construction & transformation

Deeper patterns: references/02-data-transformation.md — log/IHS/Box–Cox via MASS::boxcox, group winsorization with dplyr, scale() and bestNormalize, factor handling, lag/lead with dplyr::lag, panel timing.

library(DescTools)        # Winsorize()

df <- df %>%
  mutate(
    # 2a. Log / IHS
    log_wage   = log(pmax(wage, 1)),
    ihs_assets = asinh(assets),

    # 2b. Winsorize 1/99
    wage_w1 = DescTools::Winsorize(wage, probs = c(0.01, 0.99), na.rm = TRUE),

    # 2c. Standardize
    age_std = as.numeric(scale(age)),

    # 2d. Polynomial / interaction (or use formula syntax in fixest)
    age_sq        = age^2,
    trt_x_edu     = training * edu
  ) %>%

  # 2e. Within-group winsorize
  group_by(industry, year) %>%
  mutate(wage_w1_iy = DescTools::Winsorize(wage, probs = c(0.01, 0.99),
                                           na.rm = TRUE)) %>%
  ungroup() %>%

  # 2f. Panel operators (always arrange first to make lag deterministic)
  arrange(worker_id, year) %>%
  group_by(worker_id) %>%
  mutate(
    log_wage_l1 = lag(log_wage, 1),
    log_wage_f1 = lead(log_wage, 1),
    d_log_wage  = log_wage - lag(log_wage, 1),
    wage_mean_i = mean(log_wage, na.rm = TRUE),
    log_wage_dm = log_wage - wage_mean_i
  ) %>%
  ungroup() %>%

  # 2g. Staggered-DID timing
  group_by(worker_id) %>%
  mutate(first_treat = ifelse(any(training == 1),
                              min(year[training == 1]), NA_real_)) %>%
  ungroup() %>%
  mutate(rel_time      = year - first_treat,
         never_treated = is.na(first_treat))

# 2h. CPI deflation
cpi <- read_csv("raw/cpi.csv")
df <- df %>%
  left_join(cpi, by = "year") %>%
  mutate(cpi_base = cpi[year == 2010][1],
         wage_real     = wage * cpi_base / cpi,
         log_wage_real = log(pmax(wage_real, 1)))

Step 2.5 — Empirical strategy (write the equation + identifying assumption)

This is the heart of an AER paper. Before any code, write down the equation explicitly and state the identifying assumption. Vague identification language is the single most common reason a referee rejects an applied paper. Persist the strategy as strategy.md so it is a dated, version-controlled artifact — not a post-hoc rationalization written after seeing the coefficient.

#### Equation × identifying assumption × R estimator (decision table)

#### Design picker (when the user is unsure)

                 ┌─ running var + cutoff ───────────────── RDD       (rdrobust)
                 │
                 ├─ exogenous instrument Z ─────────────── IV/2SLS   (feols  / AER::ivreg)
data + question ─┤
                 ├─ pre/post × treat/control ─┬ 2 periods  ── 2×2 DID (feols + i())
                 │                            └ staggered  ── CS / SA / BJS  (att_gt / sunab / did_imputation)
                 │
                 ├─ 1 treated unit + donor pool + long pre ── SCM    (Synth / gsynth / synthdid)
                 │
                 ├─ high-dim X, selection-on-observables ── ML causal (DoubleML / grf — see §B)
                 │
                 └─ none of the above ──────────────────── matching + sensitivity (MatchIt + EValue)

#### Pre-registration strategy.md template

strategy <- "\\
# Empirical Strategy (pre-registration)

**Frozen**: 2026-01-15  (Git SHA: <paste>)
**Population**: manufacturing workers, 2010–2020, balanced panel
**Treatment**: training (binary, staggered adoption)
**Outcome**:   log_wage (CPI-deflated 2010 USD)
**Estimand**:  ATT on the treated, dynamic horizon -4..+4

## Estimating equation (paste from §2.5 row that matches the design)

  log_wage_it = α_i + λ_t + Σ_{e≠-1} β_e · 1{t - G_i = e} + ε_it

## Identifying assumption

1. No anticipation:   E[Y_it(0) | t < G_i] = E[Y_it(0) | never-treated]
2. Group-time PT:     Δ E[Y_it(0)] is the same across treatment cohorts

## Auto-flagged threats (must defend in §2)

- Selection of G_i on Y_i(0)              → bacondecomp + HonestDiD sensitivity
- Spillover within firm                    → cluster at firm_id, also try firm_id × year
- Anticipation in pre-period               → include lead in event study

## Fallback estimators (Step 6 robustness)

- Sun–Abraham via `feols(y ~ sunab(G, t) | i + t, data)`
- Borusyak-Jaravel-Spiess via `didimputation::did_imputation`
- Synthetic DID via `synthdid::synthdid_estimate`
"
writeLines(strategy, "artifacts/strategy.md")

Commit artifacts/strategy.md in the repo before running Step 5 / Step 6. The git log of this file is the analysis plan.

Step 3 — Descriptive statistics & Table 1

Deeper patterns: references/03-descriptive-stats.md — gtsummary::tbl_summary (the modern Table 1 standard), modelsummary::datasummary_balance with SMDs, tableone::CreateTableOne, correlation matrices with significance via corrplot / psych::corr.test, distribution plots via ggplot2.

library(gtsummary)
library(modelsummary)

# 3a. Full-sample summary — one line, publication ready
df %>%
  select(log_wage, age, edu, tenure, training) %>%
  datasummary_skim()

# Or
df %>%
  select(log_wage, age, edu, tenure, training) %>%
  tbl_summary(
    type  = list(all_continuous() ~ "continuous2"),
    statistic = all_continuous() ~ c("{N_nonmiss}", "{mean} ({sd})",
                                      "{min} – {median} – {max}")
  ) %>%
  bold_labels() %>%
  as_kable_extra() %>%
  kableExtra::save_kable("tables/table1_full.tex")

# 3b. Stratified Table 1 (treated vs control, with SMDs + p-values)
df %>%
  select(log_wage, age, edu, tenure, female, training) %>%
  tbl_summary(by = training, missing = "ifany") %>%
  add_p() %>%
  add_difference() %>%
  add_n() %>%
  modify_header(label = "**Variable**") %>%
  bold_labels() %>%
  as_gt() %>%
  gt::gtsave("tables/table1_balance.html")

# Or via modelsummary (writes LaTeX/Word/HTML)
datasummary_balance(~ training,
                    data = df %>% select(training, age, edu, tenure, female),
                    output = "tables/table1_balance.tex")

# 3c. Correlation matrix with stars
library(corrplot); library(psych)
corr_obj <- corr.test(df %>% select(log_wage, age, edu, tenure, training),
                       method = "pearson")
corrplot(corr_obj$r, method = "color", type = "upper",
         p.mat = corr_obj$p, sig.level = 0.05, insig = "blank",
         addCoef.col = "black", number.cex = 0.7,
         tl.col = "black", tl.srt = 45,
         col = colorRampPalette(c("#B2182B","white","#2166AC"))(200))

# 3d. Distribution plots
library(ggplot2)
p1 <- ggplot(df, aes(log_wage, fill = factor(training))) +
  geom_density(alpha = 0.5) +
  scale_fill_manual(values = c("0" = "darkred", "1" = "navy"),
                    labels = c("Control", "Treated"), name = "") +
  labs(x = "Log wage", y = "Density",
       title = "Log-wage density by treatment") +
  theme_classic()

p2 <- ggplot(df, aes(sample = log_wage)) +
  stat_qq() + stat_qq_line() +
  labs(title = "Normal Q-Q") + theme_classic()

cowplot::plot_grid(p1, p2, labels = "auto") %>%
  ggsave("figures/distributions.pdf", plot = ., width = 10, height = 4)

# 3e. Time-trend (DID motivation)
df %>%
  group_by(year, training) %>%
  summarise(mean_log_wage = mean(log_wage, na.rm = TRUE), .groups = "drop") %>%
  ggplot(aes(year, mean_log_wage, color = factor(training))) +
  geom_line(linewidth = 1) + geom_point(size = 2) +
  geom_vline(xintercept = policy_year, linetype = "dashed") +
  scale_color_manual(values = c("0" = "darkred", "1" = "navy"),
                     labels = c("Control","Treated"), name = "") +
  labs(x = "Year", y = "Mean log wage") + theme_classic()
ggsave("figures/trend_did.pdf", width = 7, height = 4)

Step 3.5 — Identification graphics (Section "Identification, graphical evidence")

AER convention: the identification figure precedes the regression table. The reader should see graphical evidence that PT holds / first stage is strong / RD jumps cleanly before you ask them to trust your point estimate.

#### 3.5.1 Event-study figure + numerical pre-trends test (DID identification)

Pre-period coefficients ≈ 0 (with the −1 reference period normalized to zero) is the visual evidence for parallel trends. Pair the figure with a numerical pre-trends test so reviewers don't have to eyeball it.

library(fixest); library(ggplot2)

# (a) Sun-Abraham via fixest::sunab — the modern primary for staggered DID
es <- feols(log_wage ~ sunab(first_treat, year) | worker_id + year,
            data = df, cluster = ~ worker_id)

# (b) Coefficient figure
iplot(es,
      xlab = "Years relative to treatment",
      ylab = "Coefficient (ATT, 95% CI)",
      main = "Figure 2a. Event-study coefficients (95% CI; ref. e = -1)")
ggsave("figures/fig2a_event_study.pdf", width = 7, height = 4)
ggsave("figures/fig2a_event_study.png", width = 7, height = 4, dpi = 300)

# (c) Numerical pre-trends Wald test (joint zero on the leads)
pre_idx <- grep("year::-", names(coef(es)))[!grepl("ref", names(coef(es)))]
W <- wald(es, names(coef(es))[pre_idx])
cat(sprintf("Pre-trends Wald χ² = %.2f, p = %.3f\n", W$stat, W$p))

# (d) Bacon decomposition (Goodman-Bacon 2021) — TWFE diagnostic
library(bacondecomp)
bd <- bacon(log_wage ~ training, data = df,
            id_var = "worker_id", time_var = "year")
ggplot(bd, aes(weight, estimate, color = type, shape = type)) +
  geom_point(size = 2) +
  labs(title = "Figure 2a-bis. Goodman-Bacon decomposition",
       x = "Weight", y = "Estimate")
ggsave("figures/fig2a_bacon.pdf", width = 7, height = 4)

# (e) Callaway-Sant'Anna dynamic ATT (when att_gt is the main estimator)
library(did)
cs <- att_gt(yname = "log_wage", tname = "year", idname = "worker_id",
             gname = "first_treat", data = df,
             control_group = "nevertreated", est_method = "dr",
             clustervars = "firm_id")
ggdid(aggte(cs, type = "dynamic")) +
  labs(title = "Figure 2a-ter. Dynamic ATT (Callaway-Sant'Anna)")
ggsave("figures/fig2a_csdid.pdf", width = 7, height = 4)

#### 3.5.2 First-stage F-statistic + scatter (IV identification)

Rule of thumb: first-stage F ≥ 10 for OLS-style inference; F ≥ 23 for AR-equivalent inference (Stock–Yogo / Lee 2022). fixest::feols reports F automatically; AER::ivreg requires summary(..., diagnostics = TRUE).

iv <- feols(log_wage ~ age + edu | training ~ Z1 + Z2,
            data = df, cluster = ~ firm_id)
summary(iv, stage = 1)
fitstat(iv, ~ ivf + ivwald + sargan + cd)        # CD / KP / Sargan / first-stage F

# Binscatter for the first-stage scatter (residualized on age + edu)
library(binsreg)
binsreg(y = df$training, x = df$Z1, w = df[, c("age","edu")],
        nbins = 20, polyreg = 2, ci = c(3, 3))
ggsave("figures/fig2b_first_stage.pdf", width = 7, height = 4)

#### 3.5.3 RD: McCrary density + canonical RD plot

The signature RD figure is rdplot (CCT-style binned scatter with local-polynomial fit on each side), paired with the McCrary manipulation test.

library(rdrobust); library(rddensity)

# (a) Canonical RD plot — binned means + local poly on each side
rdplot(y = df$outcome, x = df$running_var, c = 0,
       p = 4, kernel = "triangular", binselect = "esmv",
       title = "Figure 2c. RD plot")
ggsave("figures/fig2c_rdplot.pdf", width = 7, height = 4)

# (b) McCrary density (Cattaneo-Jansson-Ma 2018)
rdd <- rddensity(X = df$running_var, c = 0)
print(summary(rdd))
rdplotdensity(rdd, X = df$running_var,
              title = "Figure 2c-bis. McCrary density (manipulation test)")
ggsave("figures/fig2c_mccrary.pdf", width = 7, height = 4)

#### 3.5.4 Matching: love plot (standardized differences pre vs post)

library(MatchIt); library(cobalt)

m.out <- matchit(training ~ age + edu + tenure + firm_size,
                 data = df, method = "nearest", ratio = 1)
love.plot(m.out, threshold = 0.10,
          var.order = "unadjusted", abs = TRUE,
          title = "Figure 2d. Love plot — |SMD| pre vs post matching")
ggsave("figures/fig2d_loveplot.pdf", width = 7, height = 4)

#### 3.5.5 SCM: synthetic-control trajectory + gap plot

For synthetic-control designs the canonical Figure 2 is the treated-vs-synthetic time series with treatment time annotated.

library(tidysynth)
sc <- df %>%
  synthetic_control(outcome = log_wage, unit = unit_id, time = year,
                    i_unit = "treated_unit_name", i_time = 2015) %>%
  generate_predictor(time_window = 2010:2014,
                     mean_age = mean(age, na.rm = TRUE),
                     mean_edu = mean(edu, na.rm = TRUE)) %>%
  generate_weights() %>% generate_control()
plot_trends(sc); ggsave("figures/fig2e_synth_trajectory.pdf", width = 7, height = 4)
plot_differences(sc); ggsave("figures/fig2e_synth_gap.pdf", width = 7, height = 4)

# Synthetic DID
library(synthdid)
sdid_setup <- panel.matrices(df, unit = "worker_id", time = "year",
                              outcome = "log_wage", treatment = "training")
sdid_fit <- synthdid_estimate(sdid_setup$Y, sdid_setup$N0, sdid_setup$T0)
plot(sdid_fit, control.name = "Synthetic DiD")
ggsave("figures/fig2e_sdid.pdf", width = 7, height = 4)

Identification-specific checks (PT for DID, weak-IV F, density for RD, common support for matching) are also auto-run inside the Step-5 estimators — don't duplicate the numerics here, but DO produce the figures: a referee scans the figures first.

Step 4 — Diagnostic statistical tests

Deeper patterns: references/04-statistical-tests.md — every classical test. lmtest/sandwich/car/tseries/urca/plm.

library(lmtest)
library(sandwich)
library(car)
library(tseries)
library(urca)

# Fit baseline OLS for diagnostics
ols <- lm(log_wage ~ training + age + edu + tenure, data = df)

# 4a. Normality of residuals
shapiro.test(sample(residuals(ols), min(5000, length(residuals(ols)))))
tseries::jarque.bera.test(residuals(ols))

# 4b. Heteroskedasticity
bptest(ols)                                  # Breusch-Pagan
bptest(ols, ~ I(fitted(ols)^2) + ., data = df)  # White-style

# 4c. Autocorrelation (time series / panel)
dwtest(ols)                                   # Durbin-Watson
bgtest(ols, order = 4)                        # Breusch-Godfrey
Box.test(residuals(ols), lag = 8, type = "Ljung-Box")

# Panel-specific
library(plm)
pdata <- pdata.frame(df, index = c("worker_id", "year"))
plm_fe  <- plm(log_wage ~ training + age + edu, data = pdata, model = "within")
pbgtest(plm_fe)                               # Wooldridge serial correlation
pcdtest(plm_fe, test = "cd")                  # Pesaran cross-sectional dependence

# 4d. Multicollinearity
vif(ols)                                       # VIFs
kappa(model.matrix(ols), exact = TRUE)         # condition number

# 4e. Stationarity (time series — assumes a single y over time)
adf.test(df$log_wage, k = 4)                   # ADF
kpss.test(df$log_wage, null = "Level")         # KPSS

# 4f. Hausman (FE vs RE)
plm_re <- plm(log_wage ~ training + age + edu, data = pdata, model = "random")
phtest(plm_fe, plm_re)

# 4g. Specification — RESET
resettest(ols, power = 2:3, type = "fitted")

Decision table:

Step 5 — Baseline empirical modeling (Section 4: Main Results)

Deeper patterns: references/05-modeling.md — every estimator. fixest is the workhorse.

This is the densest section of an applied paper. A modern AER §4 typically contains 2–3 multi-regression tables and one coefficient plot:

• Table 2 (main): progressive controls, 4–6 columns — Pattern A below
• Table 2-bis (design horse race): same coefficient under OLS / IV / DID / DML — Pattern B
• Table 2-ter (multi-outcome): same treatment, several outcomes side-by-side — Pattern C
• Figure 3 (coefplot): visual summary of β̂ and 95% CI across specs

Estimator routing (memorize this — getting it wrong silently produces nonsense): - No FE / single low-card FE → feols(y ~ X, data, cluster = ~i) - High-dim FE → feols(y ~ X | fe1 + fe2, data, cluster = ~i) - Two-way cluster → feols(..., cluster = ~ firm_id + year) - 2SLS / IV → feols(y ~ X | D ~ Z, data, cluster = ~ firm_id) (or AER::ivreg for diagnostics) - DID / event-study → feols(y ~ sunab(G, t) | i + t, data) (SA) · did::att_gt (CS) · didimputation::did_imputation (BJS)

Pick by identification strategy:

Cross-section, selection on observables  →  feols  |  MatchIt + lm  |  WeightIt
Panel + policy shock + parallel trends   →  feols / did::att_gt / sunab / didimputation / synthdid
Exogenous instrument                     →  feols(... | endog ~ z)  |  AER::ivreg
Discontinuity                            →  rdrobust + rddensity + rdmc
N=1 treated, long panel                  →  Synth / gsynth / synthdid
Selection on observables + heterogeneity →  WeightIt + cobalt; grf::causal_forest
Binary outcome                           →  feglm or glm(family=binomial)
Count outcome                            →  fepois

Canonical calls (the eight patterns A–H below are the AER table cookbook — modelsummary(...) and fixest::etable(...) are the two workhorses, equivalent to Stata outreg2/esttab and Python pf.etable/Stargazer).

#### 5.A Pattern A — Progressive controls (the canonical Table 2) Stable β̂ across columns ⇒ less concern that selection on observables is driving the estimate (Oster 2019 selection-stability logic; quantified in Step 6).

library(fixest); library(modelsummary)

m1 <- feols(log_wage ~ training,                                                 data = df, cluster = ~ firm_id)
m2 <- feols(log_wage ~ training + age + edu,                                     data = df, cluster = ~ firm_id)
m3 <- feols(log_wage ~ training + age + edu + tenure + firm_size,                data = df, cluster = ~ firm_id)
m4 <- feols(log_wage ~ training + age + edu + tenure + firm_size | industry + year,
            data = df, cluster = ~ firm_id)
m5 <- feols(log_wage ~ training + age + edu + tenure + firm_size | worker_id + year,
            data = df, cluster = ~ firm_id)
m6 <- feols(log_wage ~ training + age + edu + tenure + firm_size | worker_id + year + industry^year,
            data = df, cluster = ~ firm_id)

modelsummary(
  list("(1) Baseline"    = m1,
       "(2) +Demog"      = m2,
       "(3) +Labor-mkt"  = m3,
       "(4) Ind×Yr FE"   = m4,
       "(5) Worker FE"   = m5,
       "(6) Worker FE+Ind×Yr" = m6),
  output    = "tables/table2_main.tex",
  stars     = c("*" = 0.1, "**" = 0.05, "***" = 0.01),
  gof_omit  = "BIC|AIC|F|Log|Adj",
  coef_map  = c("training" = "Job training",
                "age" = "Age", "edu" = "Education",
                "tenure" = "Tenure", "firm_size" = "Firm size"),
  notes     = c("Cluster-robust SE in parentheses, clustered at firm_id.",
                "* p<0.10, ** p<0.05, *** p<0.01.")
)
modelsummary(list("(1)"=m1,"(2)"=m2,"(3)"=m3,"(4)"=m4,"(5)"=m5,"(6)"=m6),
             output = "tables/table2_main.docx")

AER convention: show ALL controls (and the intercept). Pass NEITHER keep = NOR coef_omit = so every parameter is visible. Use coef_map = c("training" = "Training") (single mapping) only when a focal-coefficient-only table is intentional (interaction-form heterogeneity, IV first-stage triplet); use coef_omit = "Intercept" only when you want to suppress the constant for paper aesthetics.

#### 5.B Pattern B — Design horse race (Table 2-bis) Show the same coefficient of interest under multiple identification strategies. This is the AER credibility move: convergent evidence across designs each making different identifying assumptions.

library(fixest); library(AER); library(did); library(MatchIt); library(WeightIt)

ols  <- feols(log_wage ~ training + age + edu + tenure | industry + year,
              data = df, cluster = ~ firm_id)
iv   <- feols(log_wage ~ age + edu + tenure | training ~ Z1 + Z2,
              data = df, cluster = ~ firm_id)
cs   <- att_gt(yname = "log_wage", tname = "year", idname = "worker_id",
                gname = "first_treat", data = df,
                control_group = "nevertreated", est_method = "dr",
                clustervars = "firm_id")
psm  <- matchit(training ~ age + edu + tenure, data = df,
                method = "nearest", ratio = 1)
psm_lm <- lm(log_wage ~ training + age + edu + tenure,
             data = match.data(psm), weights = weights)
ebal <- weightit(training ~ age + edu + tenure, data = df, method = "ebal")
ebal_lm <- lm(log_wage ~ training + age + edu + tenure,
              data = df, weights = ebal$weights)

modelsummary(
  list("(1) OLS+FE"     = ols,
       "(2) 2SLS"       = iv,
       "(3) CS-DID"     = aggte(cs, type = "simple"),
       "(4) PSM"        = psm_lm,
       "(5) Entropy bal." = ebal_lm),
  output    = "tables/table2b_designs.tex",
  stars     = c("*" = 0.1, "**" = 0.05, "***" = 0.01),
  coef_map  = c("training" = "Job training (β̂)"),
  gof_omit  = "BIC|AIC|F|Log|Adj",
  notes     = "Convergent evidence: same β̂ under five identification strategies."
)

#### 5.C Pattern C — Multi-outcome table (same X, several Y's)

ys <- c("log_wage", "weeks_employed", "left_firm", "promoted")
multi_y <- lapply(ys, function(y)
  feols(as.formula(paste(y, "~ training + age + edu + tenure | industry + year")),
        data = df, cluster = ~ firm_id))
names(multi_y) <- ys

modelsummary(multi_y,
             output = "tables/table2c_multi_outcome.tex",
             stars  = c("*" = 0.1, "**" = 0.05, "***" = 0.01),
             coef_map = c("training" = "Training"),
             notes  = "Each column is a separate regression on the labelled outcome.")

#### 5.D Pattern D — Stacked Panel A / Panel B table Same model family, two horizons (short-run / long-run) or two samples. Use gt::gt_group() to stack two modelsummary blocks with panel headers.

library(gt)

panelA <- list(
  "(1) Industry FE" = feols(wage_t1 ~ training + X | industry + year,  data = df, cluster = ~ firm_id),
  "(2) Worker FE"   = feols(wage_t1 ~ training + X | worker_id + year, data = df, cluster = ~ firm_id))
panelB <- list(
  "(1) Industry FE" = feols(wage_t5 ~ training + X | industry + year,  data = df, cluster = ~ firm_id),
  "(2) Worker FE"   = feols(wage_t5 ~ training + X | worker_id + year, data = df, cluster = ~ firm_id))

ms_A <- modelsummary(panelA, output = "gt") %>%
  tab_header(title = "Panel A. Short-run (1 year)")
ms_B <- modelsummary(panelB, output = "gt") %>%
  tab_header(title = "Panel B. Long-run (5 years)")

gt_group(ms_A, ms_B) %>%
  gtsave("tables/table2d_horizons.tex")
gt_group(ms_A, ms_B) %>%
  gtsave("tables/table2d_horizons.docx")

#### 5.E Pattern E — IV reporting triplet (first-stage / reduced-form / 2SLS) The textbook AER IV table presents the first stage, the reduced form, and the 2SLS in three columns so the reader can verify Wald-ratio = RF / FS.

fs  <- feols(training ~ Z + age + edu | industry + year, data = df, cluster = ~ firm_id)
rf  <- feols(log_wage ~ Z + age + edu | industry + year, data = df, cluster = ~ firm_id)
iv2 <- feols(log_wage ~ age + edu | training ~ Z, data = df, cluster = ~ firm_id)

modelsummary(
  list("(1) First stage"   = fs,
       "(2) Reduced form"  = rf,
       "(3) 2SLS"          = iv2),
  output     = "tables/table2e_iv_triplet.tex",
  stars      = c("*" = 0.1, "**" = 0.05, "***" = 0.01),
  coef_map   = c("Z" = "Instrument Z", "training" = "Training (endog.)"),
  gof_map    = list(list(raw = "ivf", clean = "First-stage F", fmt = 2)),
  notes      = "Wald ratio: $\\hat\\beta_{2SLS} = \\hat\\beta_{RF} / \\hat\\pi_{FS}$."
)

IV triplet is intentionally focal: show only Z + endogenous regressor so the reader can eyeball the Wald ratio. Drop coef_map= only if a referee asks for the full coefficient list.

#### 5.F Pattern F — Causal-orchestrator main via did::att_gt / synthdid / grf::causal_forest For DID / SCM / matching / forest mains, the modern R estimator returns a self-contained estimate + automatic placebos / pre-trends / overlap diagnostics. Pipe into modelsummary via the auto-tidiers.

# CS-DID with full diagnostics
cs <- att_gt(yname = "log_wage", tname = "year", idname = "worker_id",
             gname = "first_treat", data = df,
             control_group = "nevertreated", est_method = "dr",
             clustervars = "firm_id")
print(aggte(cs, type = "group"))                           # ATT(g) summary
print(aggte(cs, type = "dynamic", min_e = -4, max_e = 4))  # event-study aggregation

# Synthetic DID
library(synthdid)
sdid_setup <- panel.matrices(df, unit="worker_id", time="year",
                              outcome="log_wage", treatment="training")
sdid_fit <- synthdid_estimate(sdid_setup$Y, sdid_setup$N0, sdid_setup$T0)
print(summary(sdid_fit))

# Causal forest with overlap + variable importance
library(grf)
cf <- causal_forest(X = as.matrix(df[, c("age","edu","tenure","firm_size")]),
                    Y = df$log_wage, W = df$training, num.trees = 4000)
average_treatment_effect(cf, target.sample = "treated")
test_calibration(cf)
variable_importance(cf)

#### 5.G Pattern G — Subgroup modelsummary (Table 3, see Step 7) One column per subgroup. Detailed code in §Step 7 — Heterogeneity.

#### 5.H Pattern H — Robustness master (Table A1, see Step 6) Stack every robustness specification next to the baseline. Detailed code in §Step 6.

#### Canonical estimator commands (the underlying primitives)

library(fixest)

# 5a. OLS with cluster-robust SEs — feols is the modern primary
ols <- feols(log_wage ~ training + age + edu + tenure,
             data = df, cluster = ~ firm_id)
summary(ols)

# 5b. Two-way FE — single line
fe <- feols(log_wage ~ training + age + edu + tenure | worker_id + year,
            data = df, cluster = ~ worker_id)

# Multi-way clustering
fe_mw <- feols(log_wage ~ training | worker_id + year,
               data = df, cluster = ~ worker_id + firm_id)

# High-dim interaction FE
fe_hd <- feols(log_wage ~ training | worker_id + industry^year,
               data = df, cluster = ~ firm_id)

# 5c. 2×2 DID
did22 <- feols(log_wage ~ i(treated, post, ref = 0) + age + edu,