xjtulyc/development-economics

Development Economics Analysis

TL;DR — Analyze RCT data with ITT regressions, test covariate balance with love plots, correct for selective attrition using Lee bounds, estimate treatment effects using propensity score matching, and compute FGT poverty indices from household survey data.

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When to Use

| Situation | Recommended Method | |---|---| | Randomized control trial — intention to treat | OLS with strata fixed effects | | RCT with partial compliance — LATE | IV regression (treatment assignment as instrument) | | Systematic attrition may bias estimates | Lee (2009) sharp bounds | | Observational data with rich covariates | Propensity score matching (PSM) | | Cross-country or household poverty measurement | FGT poverty indices | | Validate randomization quality | Covariate balance table + love plot |

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Background

Intent-to-Treat (ITT) in RCTs

The ITT estimator regresses the outcome on the randomly assigned treatment indicator (not actual take-up), controlling for randomization strata:

Y_i = α + β_ITT Z_i + γ_s strata_i + ε_i

where Z_i ∈ {0, 1} is random assignment. β_ITT is the average effect of being offered treatment, regardless of compliance.

Local Average Treatment Effect (LATE) via IV is the ITT scaled by the compliance rate. Z_i instruments for D_i (actual treatment take-up): LATE = ITT / (First Stage: E[D_i|Z_i=1] - E[D_i|Z_i=0])

Covariate Balance

Balance test: At baseline, treated and control groups should have similar means for all pre-treatment covariates. The standardized mean difference (SMD) is: SMD_k = (X̄_k^1 - X̄_k^0) / SD_pooled_k

Conventional threshold: |SMD| < 0.10 (Rosenbaum and Rubin 1985). A love plot displays all SMDs graphically.

Lee (2009) Sharp Bounds for Selective Attrition

If attrition is higher in treatment or control, the missing data may be non-random (individuals who benefit more may be more likely to remain in the sample). Lee bounds provide sharp bounds on the treatment effect by trimming the tails:

• Upper bound: Trim the bottom (1 - p_min/p_max) fraction from the high-attrition group.
• Lower bound: Trim the top (1 - p_min/p_max) fraction from the high-attrition group.

where p_0 and p_1 are retention rates in control and treatment groups respectively.

Propensity Score Matching

PSM constructs a counterfactual by matching each treated unit to the most similar control unit based on the propensity score e(X) = P(D=1|X).

Steps:

1. Estimate logistic propensity score.
2. Check common support (overlap).
3. Match using nearest-neighbor within a caliper (e.g., 0.25 × SD(log-odds)).
4. Assess balance after matching.
5. Estimate ATT = E[Y_i(1) - Y_i(0) | D_i = 1] on matched sample.

FGT Poverty Indices

Foster-Greer-Thorbecke (1984) family of poverty measures:

P_α = (1/n) Σ_{i: y_i < z} [(z - y_i) / z]^α

• α = 0: P_0 = headcount ratio (share below poverty line)
• α = 1: P_1 = poverty gap (average shortfall as share of poverty line)
• α = 2: P_2 = poverty severity (gives more weight to the poorest)

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Environment Setup

conda create -n devecono python=3.11 -y
conda activate devecono
pip install statsmodels>=0.14 pandas>=1.5 numpy>=1.23 scipy>=1.9 scikit-learn>=1.2 matplotlib>=3.6

python -c "import statsmodels, sklearn; print('OK')"

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Core Workflow

Step 1 — ITT Regression and Covariate Balance

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import statsmodels.api as sm
from scipy import stats

np.random.seed(42)


def generate_rct_data(
    n: int = 1200,
    n_strata: int = 6,
    attrition_rate: float = 0.15,
    true_itt: float = 0.3,
) -> pd.DataFrame:
    """
    Simulate an RCT dataset with stratified randomization and selective attrition.

    Args:
        n:               Total sample size.
        n_strata:        Number of randomization strata.
        attrition_rate:  Overall attrition fraction (higher for treated group).
        true_itt:        True ITT effect size.

    Returns:
        DataFrame with columns: id, strata, Z (assignment), D (take-up),
                                 age, female, baseline_y, endline_y (NaN if attrited).
    """
    strata = np.random.randint(0, n_strata, n)
    # Stratified randomization: 50% treated in each stratum
    Z = np.zeros(n, dtype=int)
    for s in range(n_strata):
        idx = np.where(strata == s)[0]
        np.random.shuffle(idx)
        Z[idx[:len(idx) // 2]] = 1

    # Covariates
    age = np.random.normal(35, 8, n)
    female = np.random.binomial(1, 0.55, n)
    baseline_y = np.random.normal(1000, 200, n) + 50 * age * 0.01

    # Compliance: 80% of treated take up
    D = np.where(Z == 1, np.random.binomial(1, 0.8, n), 0)

    # Selective attrition: treated with lower baseline outcomes more likely to attrite
    attrit_prob_control = attrition_rate * 0.5
    attrit_prob_treat = attrition_rate * 1.5
    attrit = np.where(
        Z == 1,
        np.random.binomial(1, attrit_prob_treat, n),
        np.random.binomial(1, attrit_prob_control, n),
    )

    # Endline outcome
    endline_y = baseline_y + true_itt * 1000 * Z + np.random.normal(0, 150, n)
    endline_y = np.where(attrit == 1, np.nan, endline_y)

    return pd.DataFrame({
        "id": np.arange(n),
        "strata": strata,
        "Z": Z,
        "D": D,
        "age": age,
        "female": female,
        "baseline_y": baseline_y,
        "endline_y": endline_y,
        "attrited": attrit,
    })


def covariate_balance_table(
    df: pd.DataFrame,
    treatment_col: str = "Z",
    covariates: list = None,
    output_path: str = None,
) -> pd.DataFrame:
    """
    Compute standardized mean differences and produce a love plot.

    Args:
        df:            RCT dataset.
        treatment_col: Binary treatment assignment column.
        covariates:    List of covariate names to check.
        output_path:   If provided, save love plot.

    Returns:
        DataFrame with columns: covariate, mean_treated, mean_control, smd, p_value.
    """
    if covariates is None:
        covariates = ["age", "female", "baseline_y"]

    treated = df[df[treatment_col] == 1]
    control = df[df[treatment_col] == 0]

    records = []
    for cov in covariates:
        m1 = treated[cov].mean()
        m0 = control[cov].mean()
        s1 = treated[cov].std()
        s0 = control[cov].std()
        pooled_sd = np.sqrt((s1**2 + s0**2) / 2)
        smd = (m1 - m0) / (pooled_sd + 1e-10)

        t_stat, p_val = stats.ttest_ind(treated[cov].dropna(), control[cov].dropna())
        records.append({
            "covariate": cov,
            "mean_treated": round(m1, 4),
            "mean_control": round(m0, 4),
            "smd": round(smd, 4),
            "p_value": round(p_val, 4),
            "balanced": abs(smd) < 0.10,
        })

    bal_df = pd.DataFrame(records)
    print("Covariate Balance Table:")
    print(bal_df.to_string(index=False))
    n_unbalanced = (abs(bal_df["smd"]) >= 0.10).sum()
    print(f"\nUnbalanced covariates (|SMD| >= 0.10): {n_unbalanced} / {len(covariates)}")

    # Love plot
    fig, ax = plt.subplots(figsize=(8, max(4, len(covariates) * 0.5)))
    colors = ["#E74C3C" if not b else "#2ECC71" for b in bal_df["balanced"]]
    ax.scatter(bal_df["smd"], range(len(bal_df)), color=colors, s=80, zorder=3)
    ax.axvline(-0.10, color="gray", linestyle="--", linewidth=1)
    ax.axvline(0.10, color="gray", linestyle="--", linewidth=1, label="±0.10 threshold")
    ax.axvline(0, color="black", linewidth=0.8)
    ax.set_yticks(range(len(bal_df)))
    ax.set_yticklabels(bal_df["covariate"])
    ax.set_xlabel("Standardized Mean Difference (SMD)")
    ax.set_title("Love Plot — Covariate Balance")
    ax.legend()
    fig.tight_layout()
    if output_path:
        fig.savefig(output_path, dpi=150)
        print(f"Saved love plot to {output_path}")
    plt.show()

    return bal_df


def itt_regression(
    df: pd.DataFrame,
    outcome: str = "endline_y",
    treatment: str = "Z",
    strata_col: str = "strata",
    controls: list = None,
) -> dict:
    """
    ITT regression with strata fixed effects and optional covariate controls.

    Args:
        df:          RCT dataset (complete cases for outcome).
        outcome:     Endline outcome column.
        treatment:   Random assignment indicator.
        strata_col:  Randomization strata column (included as dummies).
        controls:    Additional baseline controls for efficiency.

    Returns:
        Dictionary with keys: itt_coef, se, pvalue, ci, late_coef, result.
    """
    dfc = df.dropna(subset=[outcome]).copy()

    # Strata dummies
    strata_dummies = pd.get_dummies(dfc[strata_col], prefix="strata", drop_first=True)
    X_cols = [treatment]
    if controls:
        X_cols += controls

    X = pd.concat([dfc[X_cols], strata_dummies], axis=1)
    X = sm.add_constant(X)

    model = sm.OLS(dfc[outcome], X.astype(float))
    result = model.fit(cov_type="HC3")

    itt = float(result.params[treatment])
    se = float(result.bse[treatment])
    pval = float(result.pvalues[treatment])
    ci_low = float(result.conf_int().loc[treatment, 0])
    ci_high = float(result.conf_int().loc[treatment, 1])

    print(f"\nITT: {itt:.4f}  SE: {se:.4f}  p: {pval:.4f}  95% CI: [{ci_low:.4f}, {ci_high:.4f}]")

    # LATE via IV (using D compliance and Z instrument)
    if "D" in dfc.columns:
        from statsmodels.sandbox.regression.gmm import IV2SLS
        X_iv = sm.add_constant(pd.concat([dfc[["D"] + (controls or [])], strata_dummies], axis=1).astype(float))
        Z_instr = sm.add_constant(pd.concat([dfc[[treatment] + (controls or [])], strata_dummies], axis=1).astype(float))
        iv_model = IV2SLS(dfc[outcome].values, X_iv.values, Z_instr.values)
        iv_result = iv_model.fit()
        late = float(iv_result.params[1])
        print(f"LATE (IV): {late:.4f}")
    else:
        late = None

    return {"itt_coef": itt, "se": se, "pvalue": pval,
            "ci": (ci_low, ci_high), "late_coef": late, "result": result}

Step 2 — Lee Bounds for Selective Attrition

def lee_bounds(
    df: pd.DataFrame,
    outcome: str = "endline_y",
    treatment: str = "Z",
) -> dict:
    """
    Lee (2009) sharp bounds for selective attrition.

    Computes lower and upper bounds on the ATT when attrition is non-random.
    Assumes: attrition in control is random; trims from appropriate tail in treatment.

    Args:
        df:        Dataset with outcome (NaN = attrited) and treatment assignment.
        outcome:   Outcome column (NaN means attrited/missing).
        treatment: Binary treatment assignment.

    Returns:
        Dictionary with keys: p0, p1, trim_fraction, lb_estimate,
                              ub_estimate, lb_se, ub_se.
    """
    treated = df[df[treatment] == 1].copy()
    control = df[df[treatment] == 0].copy()

    p1 = treated[outcome].notna().mean()
    p0 = control[outcome].notna().mean()

    print(f"Retention rate: Treated = {p1:.4f}, Control = {p0:.4f}")

    # Trim fraction from higher-retention group
    trim_fraction = (p1 - p0) / p1 if p1 > p0 else (p0 - p1) / p0

    # Lee bounds: trim from treated group (if p1 > p0)
    y_treated_obs = treated[outcome].dropna().sort_values()
    n_trim = int(np.ceil(trim_fraction * len(y_treated_obs)))

    y_lower = y_treated_obs.iloc[n_trim:].values   # trim from bottom (lower bound)
    y_upper = y_treated_obs.iloc[:len(y_treated_obs) - n_trim].values  # trim from top (upper bound)
    y_control = control[outcome].dropna().values

    lb_estimate = float(np.mean(y_lower) - np.mean(y_control))
    ub_estimate = float(np.mean(y_upper) - np.mean(y_control))

    # Bootstrap SE for bounds
    n_boot = 500
    lb_boots, ub_boots = [], []
    for _ in range(n_boot):
        y_tr_b = np.random.choice(y_treated_obs.values, size=len(y_treated_obs), replace=True)
        y_ctrl_b = np.random.choice(y_control, size=len(y_control), replace=True)
        y_tr_b_sorted = np.sort(y_tr_b)

        lb_boots.append(float(np.mean(y_tr_b_sorted[n_trim:]) - np.mean(y_ctrl_b)))
        ub_boots.append(float(np.mean(y_tr_b_sorted[:max(1, len(y_tr_b_sorted) - n_trim)]) - np.mean(y_ctrl_b)))

    lb_se = float(np.std(lb_boots))
    ub_se = float(np.std(ub_boots))

    print(f"\nLee Sharp Bounds (trim fraction = {trim_fraction:.4f}):")
    print(f"  Lower bound: {lb_estimate:.4f}  (SE = {lb_se:.4f})")
    print(f"  Upper bound: {ub_estimate:.4f}  (SE = {ub_se:.4f})")
    if lb_estimate > 0:
        print("  Both bounds positive — treatment effect significantly positive.")

    return {
        "p0": p0,
        "p1": p1,
        "trim_fraction": trim_fraction,
        "lb_estimate": lb_estimate,
        "ub_estimate": ub_estimate,
        "lb_se": lb_se,
        "ub_se": ub_se,
    }

Step 3 — Propensity Score Matching and FGT Poverty Indices

from sklearn.linear_model import LogisticRegression
from sklearn.neighbors import NearestNeighbors


def propensity_score_matching(
    df: pd.DataFrame,
    treatment_col: str = "Z",
    outcome_col: str = "endline_y",
    covariate_cols: list = None,
    caliper: float = 0.25,
) -> dict:
    """
    Propensity score matching (nearest neighbor, 1-to-1, with caliper).

    Args:
        df:              Dataset (complete cases).
        treatment_col:   Binary treatment indicator.
        outcome_col:     Outcome variable.
        covariate_cols:  Matching covariates (pre-treatment).
        caliper:         Caliper width = caliper × SD(logit propensity score).

    Returns:
        Dictionary with keys: att, se, n_matched, balance_after.
    """
    dfc = df.dropna(subset=[outcome_col, treatment_col]).copy()
    if covariate_cols is None:
        covariate_cols = ["age", "female", "baseline_y"]

    X = dfc[covariate_cols].values
    T = dfc[treatment_col].values
    Y = dfc[outcome_col].values

    # Estimate propensity score
    lr = LogisticRegression(max_iter=500, random_state=0)
    lr.fit(X, T)
    ps = lr.predict_proba(X)[:, 1]
    logit_ps = np.log(ps / (1 - ps + 1e-10))

    caliper_width = caliper * np.std(logit_ps)
    dfc["ps"] = ps
    dfc["logit_ps"] = logit_ps

    treated_idx = np.where(T == 1)[0]
    control_idx = np.where(T == 0)[0]

    # Nearest-neighbor matching with caliper
    nn = NearestNeighbors(n_neighbors=1, metric="euclidean")
    nn.fit(logit_ps[control_idx].reshape(-1, 1))
    distances, indices = nn.kneighbors(logit_ps[treated_idx].reshape(-1, 1))

    matched_pairs = []
    for i, (dist, ctrl_local_idx) in enumerate(zip(distances[:, 0], indices[:, 0])):
        if dist <= caliper_width:
            t_idx = treated_idx[i]
            c_idx = control_idx[ctrl_local_idx]
            matched_pairs.append((t_idx, c_idx))

    n_matched = len(matched_pairs)
    print(f"Matched pairs: {n_matched} / {len(treated_idx)} treated units")

    if n_matched == 0:
        print("Warning: no matches within caliper. Widen caliper.")
        return {"att": np.nan, "se": np.nan, "n_matched": 0, "balance_after": None}

    y_treated_matched = Y[[p[0] for p in matched_pairs]]
    y_control_matched = Y[[p[1] for p in matched_pairs]]
    diffs = y_treated_matched - y_control_matched

    att = float(np.mean(diffs))
    se = float(np.std(diffs) / np.sqrt(n_matched))
    ci = (att - 1.96 * se, att + 1.96 * se)
    print(f"ATT: {att:.4f}  SE: {se:.4f}  95% CI: [{ci[0]:.4f}, {ci[1]:.4f}]")

    return {"att": att, "se": se, "n_matched": n_matched, "ci": ci}


def fgt_poverty_indices(
    income: np.ndarray,
    poverty_line: float,
    alphas: list = None,
) -> dict:
    """
    Compute Foster-Greer-Thorbecke (1984) poverty measures.

    Args:
        income:       Array of household income or consumption per capita.
        poverty_line: Absolute poverty threshold (same units as income).
        alphas:       List of FGT alpha parameters. Default: [0, 1, 2].

    Returns:
        Dictionary with keys P0 (headcount), P1 (gap), P2 (severity),
                             poverty_line, n_poor, n_total.
    """
    if alphas is None:
        alphas = [0, 1, 2]
    n = len(income)
    poor = income < poverty_line

    result = {"poverty_line": poverty_line, "n_poor": int(poor.sum()), "n_total": n}
    for alpha in alphas:
        gaps = np.maximum(0, (poverty_line - income) / poverty_line)
        p_alpha = float(np.mean(gaps ** alpha))
        result[f"P{alpha}"] = round(p_alpha, 6)

    print(f"\nFGT Poverty Indices (poverty line = {poverty_line}):")
    print(f"  P0 (headcount ratio):  {result['P0']:.4f}  ({result['P0']*100:.2f}% below line)")
    print(f"  P1 (poverty gap):      {result.get('P1', 'N/A')}")
    print(f"  P2 (poverty severity): {result.get('P2', 'N/A')}")
    print(f"  Poor households: {result['n_poor']} / {n}")
    return result

---

Advanced Usage

Sensitivity Analysis: FGT Across Poverty Line Values

def fgt_sensitivity(
    income: np.ndarray,
    poverty_lines: np.ndarray = None,
    alpha: int = 0,
    output_path: str = None,
) -> pd.DataFrame:
    """
    Plot FGT P_alpha across a range of poverty lines for robustness.

    Args:
        income:        Household income/consumption array.
        poverty_lines: Array of threshold values. Default: 50th percentile ± 50%.
        alpha:         FGT parameter (0, 1, or 2).
        output_path:   If provided, save sensitivity plot.

    Returns:
        DataFrame with columns: poverty_line, p_alpha.
    """
    if poverty_lines is None:
        median_income = np.median(income)
        poverty_lines = np.linspace(0.5 * median_income, 1.5 * median_income, 50)

    records = []
    for z in poverty_lines:
        gaps = np.maximum(0, (z - income) / z)
        records.append({"poverty_line": z, "p_alpha": float(np.mean(gaps ** alpha))})

    df_sens = pd.DataFrame(records)

    fig, ax = plt.subplots(figsize=(9, 5))
    ax.plot(df_sens["poverty_line"], df_sens["p_alpha"], color="#8E44AD", linewidth=2)
    ax.set_xlabel("Poverty Line")
    ax.set_ylabel(f"P{alpha}")
    ax.set_title(f"FGT P{alpha} Sensitivity to Poverty Line Choice")
    fig.tight_layout()
    if output_path:
        fig.savefig(output_path, dpi=150)
        print(f"Saved FGT sensitivity plot to {output_path}")
    plt.show()

    return df_sens

---

Troubleshooting

| Error / Issue | Cause | Resolution | |---|---|---| | ITT coefficient imprecise (wide CI) | Small sample or high variance outcome | Use baseline covariate controls to reduce residual variance | | Lee bounds span zero | Moderate attrition; ambiguous effect direction | Improve study design to reduce attrition; report both bounds | | PSM few matched pairs | Propensity scores outside common support | Widen caliper; trim extreme propensity scores; check overlap plot | | PSM ATT biased after matching | Poor covariate balance after matching | Use genetic matching or reweighting (IPW) instead | | IV2SLS AttributeError | Old statsmodels API change | Use linearmodels.IV2SLS instead | | FGT P1/P2 undefined | All income above poverty line | Check poverty line choice; verify income units |

---

External Resources

• Lee, D.S. (2009). "Training, Wages, and Sample Selection." Review of Economic Studies, 76(3), 1071–1102.
• Foster, J., Greer, J., Thorbecke, E. (1984). "A Class of Decomposable Poverty Measures." Econometrica, 52(3), 761–766.
• Duflo, E., Glennerster, R., Kremer, M. (2007). "Using Randomization in Development Economics Research." Handbook of Development Economics, Vol. 4.
• World Bank LSMS data: https://microdata.worldbank.org/index.php/catalog/lsms
• J-PAL methods guides: https://www.povertyactionlab.org/research-resources/research-methods

---

Examples

Example 1 — RCT ITT with Balance Check

df = generate_rct_data(n=1500, n_strata=6, attrition_rate=0.15, true_itt=0.3)

# Balance check at baseline
bal = covariate_balance_table(df, treatment_col="Z",
                               covariates=["age", "female", "baseline_y"],
                               output_path="love_plot.png")

# ITT regression
itt_out = itt_regression(df, outcome="endline_y", treatment="Z",
                          strata_col="strata", controls=["age", "female", "baseline_y"])
print(f"True ITT = 0.3 × 1000 = 300.0  |  Estimated ITT = {itt_out['itt_coef']:.2f}")

Example 2 — Lee Bounds + PSM + Poverty Indices

df = generate_rct_data(n=2000, attrition_rate=0.20, true_itt=0.3)

# Lee bounds for attrition
bounds = lee_bounds(df, outcome="endline_y", treatment="Z")

# PSM on observed sample
psm_out = propensity_score_matching(
    df.dropna(subset=["endline_y"]),
    treatment_col="Z",
    outcome_col="endline_y",
    covariate_cols=["age", "female", "baseline_y"],
    caliper=0.25,
)

# Poverty indices on baseline income distribution
income = df["baseline_y"].values
poverty_line = np.percentile(income, 40)   # 40th percentile as poverty line
fgt = fgt_poverty_indices(income, poverty_line)
fgt_sensitivity(income, alpha=0, output_path="fgt_sensitivity.png")

---

Changelog

| Version | Date | Change | |---|---|---| | 1.0.0 | 2026-03-18 | Initial release — ITT, covariate balance, Lee bounds, PSM, FGT poverty |