Labor Economics

Labor Economics Analysis

TL;DR — Estimate wage equations, correct for sample selection bias with Heckman two-step, decompose wage gaps between groups with Oaxaca-Blinder, and analyze distributional wage effects with unconditional quantile regression.

When to Use

Background

Mincer Wage Equation

The canonical Mincer (1974) earnings equation:

ln(wage_i) = β_0 + β_1 educ_i + β_2 exp_i + β_3 exp_i² + X_i γ + ε_i

Key coefficients:

• β_1: rate of return to one additional year of education (typically 6–12%)
• β_2: linear return to experience
• β_3: negative coefficient on exp² captures diminishing returns
• Peak experience: exp* = -β_2 / (2 β_3)

Heckman Two-Step Selection Correction

Problem: Wages are only observed for workers who choose to work. If the decision to work is correlated with unobserved wage determinants, OLS on the selected sample is biased (selection bias).

Heckman (1979) two-step procedure:

Step 1 — Selection equation (Probit): P(work_i = 1 | Z_i) = Φ(Z_i δ) Z_i includes exclusion restrictions: variables that affect work participation but not wages (e.g., number of children under 6, non-labor income, spouse's income).

Step 2 — Wage equation with IMR: ln(wage_i) = X_i β + σ λ(Z_i δ̂) + η_i for workers only

where λ(·) = φ(·)/Φ(·) is the Inverse Mills Ratio (IMR). σ = cov(ε, u) / Var(u)^{1/2}. A significant σ (coefficient on IMR) indicates selection bias in the naive OLS.

Oaxaca-Blinder Decomposition

The Oaxaca (1973) and Blinder (1973) decomposition of the mean wage gap between group A (e.g., men) and group B (e.g., women):

Ȳ_A - Ȳ_B = [X̄_A - X̄_B]' β_A    +    X̄_B' [β_A - β_B]
           = "Explained" (endowments)  + "Unexplained" (coefficients/discrimination)

The unexplained component is often interpreted as an upper bound on discrimination, though it also captures differences in unobserved characteristics.

Unconditional Quantile Regression (Firpo-Fortin-Lemieux 2009)

Standard quantile regression (QR) estimates conditional quantile effects: Q_τ(Y | X). But policy questions often concern the unconditional (marginal) distribution of wages.

RIF regression: Replace Y with its recentered influence function (RIF): RIF(y; Q_τ) = Q_τ + (τ - 1{y ≤ Q_τ}) / f_Y(Q_τ)

Then regress RIF(Y; Q_τ) on X using OLS. Coefficients are interpretable as marginal effects on the unconditional τ-th quantile of Y.

Environment Setup

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

python -c "import statsmodels; print('statsmodels', statsmodels.__version__)"

Core Workflow

Step 1 — Mincer Wage Equation and Heckman Correction

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

np.random.seed(42)


def generate_labor_data(
    n: int = 2000,
    selection_bias: float = 0.5,
) -> pd.DataFrame:
    """
    Simulate a labor survey dataset with sample selection.

    Workers with higher unobserved productivity (ε) are both more likely to
    work and earn higher wages — generating selection bias.

    Args:
        n:               Number of individuals in the survey.
        selection_bias:  Correlation between selection and wage unobservables (0–1).

    Returns:
        DataFrame with columns: educ, exp, female, children_u6, nonlabor_inc,
                                 log_wage (NaN if not working), work.
    """
    educ = np.random.randint(8, 21, n).astype(float)
    exp = np.clip(np.random.normal(15, 8, n), 0, 45)
    female = np.random.binomial(1, 0.5, n)
    children_u6 = np.random.poisson(0.5, n)
    nonlabor_inc = np.abs(np.random.normal(1000, 500, n))

    # Correlated errors
    rho = selection_bias
    u_wage = np.random.randn(n)
    u_select = rho * u_wage + np.sqrt(1 - rho**2) * np.random.randn(n)

    # True log-wage equation
    log_wage_latent = (
        0.5
        + 0.08 * educ
        + 0.04 * exp
        - 0.0006 * exp**2
        - 0.15 * female
        + u_wage
    )

    # Selection equation (probit)
    index_select = (
        -1.0
        + 0.05 * educ
        + 0.02 * exp
        - 0.3 * female
        - 0.2 * children_u6
        - 0.0003 * nonlabor_inc
        + u_select
    )
    work = (index_select > 0).astype(int)

    log_wage = np.where(work == 1, log_wage_latent, np.nan)

    return pd.DataFrame({
        "educ": educ,
        "exp": exp,
        "exp2": exp**2,
        "female": female,
        "children_u6": children_u6,
        "nonlabor_inc": nonlabor_inc,
        "log_wage": log_wage,
        "work": work,
    })


def mincer_ols(df: pd.DataFrame) -> sm.regression.linear_model.RegressionResultsWrapper:
    """
    Estimate the Mincer wage equation via OLS (for workers only).

    Args:
        df: Labor dataset with log_wage, educ, exp, exp2, female columns.

    Returns:
        OLS results object.
    """
    workers = df.dropna(subset=["log_wage"]).copy()
    X = sm.add_constant(workers[["educ", "exp", "exp2", "female"]])
    model = sm.OLS(workers["log_wage"], X)
    result = model.fit(cov_type="HC3")
    print("Mincer OLS (workers only):")
    print(result.summary().tables[1])

    # Return to education
    educ_coef = result.params["educ"]
    print(f"\nReturn to education: {educ_coef:.4f} ({educ_coef*100:.2f}% per year)")
    # Peak experience
    b_exp = result.params["exp"]
    b_exp2 = result.params["exp2"]
    if b_exp2 < 0:
        peak_exp = -b_exp / (2 * b_exp2)
        print(f"Peak experience: {peak_exp:.1f} years")

    return result


def heckman_two_step(df: pd.DataFrame) -> dict:
    """
    Heckman two-step selection correction.

    Step 1: Probit selection equation (uses exclusion restrictions:
            children_u6, nonlabor_inc).
    Step 2: OLS wage equation with Inverse Mills Ratio.

    Args:
        df: Labor dataset.

    Returns:
        Dictionary with keys: selection_result, wage_result, imr_significant.
    """
    # Step 1: Probit on full sample
    X_sel = sm.add_constant(df[["educ", "exp", "exp2", "female",
                                  "children_u6", "nonlabor_inc"]])
    probit = sm.Probit(df["work"], X_sel)
    probit_result = probit.fit(disp=False)
    print("Step 1 — Selection Probit:")
    print(probit_result.summary().tables[1])

    # Compute Inverse Mills Ratio for all observations
    xb = probit_result.predict(X_sel, linear=True)
    df = df.copy()
    df["imr"] = norm.pdf(xb) / norm.cdf(xb)

    # Step 2: Wage OLS with IMR (workers only)
    workers = df.dropna(subset=["log_wage"]).copy()
    X_wage = sm.add_constant(workers[["educ", "exp", "exp2", "female", "imr"]])
    wage_model = sm.OLS(workers["log_wage"], X_wage)
    wage_result = wage_model.fit(cov_type="HC3")
    print("\nStep 2 — Wage Equation with IMR:")
    print(wage_result.summary().tables[1])

    imr_pval = float(wage_result.pvalues["imr"])
    imr_coef = float(wage_result.params["imr"])
    print(f"\nIMR coefficient: {imr_coef:.4f}  p-value: {imr_pval:.4f}")
    if imr_pval < 0.05:
        direction = "upward" if imr_coef > 0 else "downward"
        print(f"  Significant selection bias: OLS is biased {direction}.")
    else:
        print("  No significant selection bias detected.")

    return {
        "selection_result": probit_result,
        "wage_result": wage_result,
        "imr_significant": imr_pval < 0.05,
        "df_with_imr": df,
    }

Step 2 — Oaxaca-Blinder Decomposition

def oaxaca_blinder(
    df: pd.DataFrame,
    group_col: str = "female",
    group_A_value: int = 0,
    group_B_value: int = 1,
    outcome: str = "log_wage",
    covariates: list = None,
    output_path: str = None,
) -> dict:
    """
    Oaxaca-Blinder wage gap decomposition.

    Decomposes mean wage gap (Group A - Group B) into:
      - Explained: difference in endowments valued at Group A's coefficients
      - Unexplained: difference in coefficients applied to Group B's endowments

    Args:
        df:              Labor dataset (workers only for wage regressions).
        group_col:       Column identifying the two groups.
        group_A_value:   Value of group_col for Group A (reference, higher-wage).
        group_B_value:   Value of group_col for Group B.
        outcome:         Log wage column.
        covariates:      List of control variables (excluding group_col).
        output_path:     If provided, save decomposition bar chart.

    Returns:
        Dictionary with keys: total_gap, explained, unexplained, explained_share,
                              unexplained_share, coef_detail_A, coef_detail_B.
    """
    workers = df.dropna(subset=[outcome]).copy()

    if covariates is None:
        covariates = ["educ", "exp", "exp2"]

    df_A = workers[workers[group_col] == group_A_value]
    df_B = workers[workers[group_col] == group_B_value]

    # OLS for each group separately
    def ols_group(data, cols, dep):
        X = sm.add_constant(data[cols])
        return sm.OLS(data[dep], X).fit()

    res_A = ols_group(df_A, covariates, outcome)
    res_B = ols_group(df_B, covariates, outcome)

    X_A_mean = np.array([1.0] + [df_A[c].mean() for c in covariates])
    X_B_mean = np.array([1.0] + [df_B[c].mean() for c in covariates])
    beta_A = res_A.params.values
    beta_B = res_B.params.values

    mean_A = float(df_A[outcome].mean())
    mean_B = float(df_B[outcome].mean())
    total_gap = mean_A - mean_B

    # Oaxaca decomposition (reference = Group A coefficients)
    explained = float((X_A_mean - X_B_mean) @ beta_A)
    unexplained = float(X_B_mean @ (beta_A - beta_B))

    print(f"\nOaxaca-Blinder Decomposition")
    print(f"  Group A (value={group_A_value}) mean log wage: {mean_A:.4f}")
    print(f"  Group B (value={group_B_value}) mean log wage: {mean_B:.4f}")
    print(f"  Raw gap: {total_gap:.4f} ({np.exp(total_gap)-1:.1%} in levels)")
    print(f"  Explained (endowments): {explained:.4f} ({explained/total_gap:.1%} of gap)")
    print(f"  Unexplained (returns):  {unexplained:.4f} ({unexplained/total_gap:.1%} of gap)")

    # Bar chart
    fig, ax = plt.subplots(figsize=(7, 5))
    bars = ax.bar(["Explained\n(Endowments)", "Unexplained\n(Returns)", "Total Gap"],
                  [explained, unexplained, total_gap],
                  color=["#3498DB", "#E74C3C", "#2ECC71"],
                  edgecolor="white", width=0.5)
    ax.axhline(0, color="black", linewidth=0.8)
    ax.set_ylabel("Log Wage Gap")
    ax.set_title("Oaxaca-Blinder Decomposition")
    for bar, val in zip(bars, [explained, unexplained, total_gap]):
        ax.text(bar.get_x() + bar.get_width() / 2, val + 0.005,
                f"{val:.4f}", ha="center", va="bottom", fontsize=9)
    fig.tight_layout()
    if output_path:
        fig.savefig(output_path, dpi=150)
        print(f"Saved decomposition chart to {output_path}")
    plt.show()

    return {
        "total_gap": total_gap,
        "explained": explained,
        "unexplained": unexplained,
        "explained_share": explained / total_gap if total_gap != 0 else np.nan,
        "unexplained_share": unexplained / total_gap if total_gap != 0 else np.nan,
    }

Step 3 — Unconditional Quantile Regression (RIF)

from scipy.stats import gaussian_kde


def rif_quantile_regression(
    df: pd.DataFrame,
    outcome: str = "log_wage",
    covariates: list = None,
    quantiles: list = None,
    output_path: str = None,
) -> pd.DataFrame:
    """
    Recentered Influence Function (RIF) unconditional quantile regression.

    For each quantile τ, construct RIF(Y; Q_τ) and regress on X via OLS.
    Coefficients measure marginal effects on the unconditional τ-th quantile.

    Args:
        df:          Workers dataset with outcome variable.
        outcome:     Log wage column.
        covariates:  Control variables for the RIF regression.
        quantiles:   List of quantiles (0–1). Default: [0.1, 0.25, 0.5, 0.75, 0.9].
        output_path: If provided, save coefficient plot.

    Returns:
        DataFrame with rows = quantiles, columns = covariate coefficients.
    """
    workers = df.dropna(subset=[outcome]).copy()
    y = workers[outcome].values

    if covariates is None:
        covariates = ["educ", "exp", "exp2", "female"]
    if quantiles is None:
        quantiles = [0.10, 0.25, 0.50, 0.75, 0.90]

    # KDE for density estimation at sample quantiles
    kde = gaussian_kde(y, bw_method="silverman")

    records = []
    for tau in quantiles:
        q_tau = np.quantile(y, tau)
        f_q_tau = float(kde.evaluate(q_tau))

        # RIF = Q_τ + (τ - 1{Y <= Q_τ}) / f(Q_τ)
        indicator = (y <= q_tau).astype(float)
        rif = q_tau + (tau - indicator) / (f_q_tau + 1e-10)

        X_rif = sm.add_constant(workers[covariates])
        rif_result = sm.OLS(rif, X_rif).fit(cov_type="HC3")

        row = {"quantile": tau}
        for var in covariates:
            row[f"{var}_coef"] = float(rif_result.params[var])
            row[f"{var}_se"] = float(rif_result.bse[var])
        records.append(row)

    rif_df = pd.DataFrame(records)

    # Plot education coefficient across quantiles
    fig, ax = plt.subplots(figsize=(9, 5))
    educ_coefs = [r["educ_coef"] for r in records]
    educ_ses = [r["educ_se"] for r in records]
    quantile_vals = [r["quantile"] for r in records]

    ax.plot(quantile_vals, educ_coefs, "o-", color="#2980B9", linewidth=2,
            label="Education RIF coefficient")
    ax.fill_between(quantile_vals,
                    [c - 1.96 * s for c, s in zip(educ_coefs, educ_ses)],
                    [c + 1.96 * s for c, s in zip(educ_coefs, educ_ses)],
                    alpha=0.2, color="#2980B9")
    ax.axhline(0, color="black", linewidth=0.8, linestyle="--")
    ax.set_xlabel("Quantile (τ)")
    ax.set_ylabel("Effect on Unconditional Quantile")
    ax.set_title("Unconditional Quantile Regression: Return to Education")
    ax.legend()
    fig.tight_layout()

    if output_path:
        fig.savefig(output_path, dpi=150)
        print(f"Saved RIF plot to {output_path}")
    plt.show()

    print("\nRIF Quantile Regression — Education Coefficients:")
    print(rif_df[["quantile", "educ_coef", "educ_se"]].round(4).to_string(index=False))

    return rif_df

Advanced Usage

Union Wage Premium with Selection Correction

def union_wage_premium(
    df: pd.DataFrame,
    union_col: str = "union",
    wage_col: str = "log_wage",
    controls: list = None,
) -> dict:
    """
    Estimate union wage premium: naive OLS and selection-corrected version.

    Union membership is not random — workers may self-select into unions.
    Compare naive OLS estimate with Heckman-corrected estimate.

    Args:
        df:         Dataset with union indicator, log wages, and controls.
        union_col:  Binary union membership column.
        wage_col:   Log wage column.
        controls:   Control variables. Defaults to educ, exp, exp2, female.

    Returns:
        Dictionary with naive_premium, corrected_premium, selection_bias.
    """
    workers = df.dropna(subset=[wage_col]).copy()
    if controls is None:
        controls = ["educ", "exp", "exp2", "female"]

    # Naive OLS
    X_naive = sm.add_constant(workers[[union_col] + controls])
    ols_naive = sm.OLS(workers[wage_col], X_naive).fit(cov_type="HC3")
    naive_premium = float(ols_naive.params[union_col])
    print(f"Naive OLS union premium: {naive_premium:.4f} ({(np.exp(naive_premium)-1)*100:.2f}%)")

    # Heckman correction for union selection (if exclusion restriction available)
    # Here we use nonlabor_inc as a proxy exclusion restriction (for illustration)
    if "nonlabor_inc" in df.columns:
        X_sel = sm.add_constant(workers[controls + ["nonlabor_inc"]])
        probit_union = sm.Probit(workers[union_col], X_sel).fit(disp=False)
        xb = probit_union.predict(X_sel, linear=True)
        workers["imr_union"] = norm.pdf(xb) / norm.cdf(xb)

        X_corr = sm.add_constant(workers[[union_col] + controls + ["imr_union"]])
        ols_corr = sm.OLS(workers[wage_col], X_corr).fit(cov_type="HC3")
        corrected_premium = float(ols_corr.params[union_col])
        print(f"Selection-corrected union premium: {corrected_premium:.4f}")
    else:
        corrected_premium = None

    return {
        "naive_premium": naive_premium,
        "corrected_premium": corrected_premium,
        "selection_bias": (naive_premium - corrected_premium
                           if corrected_premium is not None else None),
    }

Troubleshooting

External Resources

• Mincer, J. (1974). Schooling, Experience, and Earnings. NBER.
• Heckman, J. (1979). "Sample Selection Bias as a Specification Error." Econometrica, 47(1), 153–161.
• Oaxaca, R. (1973). "Male-Female Wage Differentials in Urban Labor Markets." International Economic Review, 14(3), 693–709.
• Firpo, S., Fortin, N., Lemieux, T. (2009). "Unconditional Quantile Regressions." Econometrica, 77(3), 953–973.
• statsmodels Heckman docs: https://www.statsmodels.org/stable/generated/statsmodels.regression.linear_model.OLS.html

Examples

Example 1 — Mincer + Heckman on Simulated Data

df = generate_labor_data(n=3000, selection_bias=0.6)
workers = df.dropna(subset=["log_wage"])
print(f"Labor force participation rate: {df['work'].mean():.3f}")

# Step 1: Naive Mincer OLS
ols_result = mincer_ols(df)

# Step 2: Heckman correction
heck = heckman_two_step(df)
print(f"\nOLS educ coef: {ols_result.params['educ']:.4f}")
print(f"Heckman educ coef: {heck['wage_result'].params['educ']:.4f}")

Example 2 — Oaxaca-Blinder Gender Wage Gap

df = generate_labor_data(n=5000, selection_bias=0.4)

ob = oaxaca_blinder(
    df.dropna(subset=["log_wage"]),
    group_col="female",
    group_A_value=0,
    group_B_value=1,
    outcome="log_wage",
    covariates=["educ", "exp", "exp2"],
    output_path="oaxaca_blinder.png",
)

# RIF unconditional quantile regression
rif_df = rif_quantile_regression(
    df.dropna(subset=["log_wage"]),
    outcome="log_wage",
    covariates=["educ", "exp", "exp2", "female"],
    quantiles=[0.10, 0.25, 0.50, 0.75, 0.90],
    output_path="rif_quantile.png",
)

Changelog