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name labor-economics description Use this Skill for labor economics analysis: Mincer wage equation, Heckman selection, Oaxaca-Blinder decomposition, CPS data cleaning, and unconditional quantile regression.
tags ["economics","labor-economics","Mincer","Heckman-selection","Oaxaca-Blinder","wages"] version 1.0.0 authors [{"name":"awesome-rosetta-skills contributors","github":"@xjtulyc"}] license MIT platforms ["claude-code","codex","gemini-cli","cursor"] dependencies {"python":["statsmodels>=0.14","pandas>=1.5","numpy>=1.23","scipy>=1.9","matplotlib>=3.6"]} last_updated 2026-03-18 status stable
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
Situation Recommended Method Log wage regression with education and experience Mincer wage equation Wages observed only for employed workers Heckman two-step selection correction Wage gap between two groups (e.g., by gender, race) Oaxaca-Blinder decomposition Effect of a policy on wage distribution (not just mean) Unconditional quantile regression Union wage premium, minimum wage study Event study or difference-in-means
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))
rho = selection_bias
u_wage = np.random.randn(n)
u_select = rho * u_wage + np.sqrt(1 - rho**2 ) * np.random.randn(n)
log_wage_latent = (
0.5
+ 0.08 * educ
+ 0.04 * exp
- 0.0006 * exp**2
- 0.15 * female
+ u_wage
)
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 ])
educ_coef = result.params["educ" ]
print (f"\nReturn to education: {educ_coef:.4 f} ({educ_coef*100 :.2 f} % per year)" )
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:.1 f} 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.
"""
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 ])
xb = probit_result.predict(X_sel, linear=True )
df = df.copy()
df["imr" ] = norm.pdf(xb) / norm.cdf(xb)
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:.4 f} p-value: {imr_pval:.4 f} " )
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]
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
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:.4 f} " )
print (f" Group B (value={group_B_value} ) mean log wage: {mean_B:.4 f} " )
print (f" Raw gap: {total_gap:.4 f} ({np.exp(total_gap)-1 :.1 %} in levels)" )
print (f" Explained (endowments): {explained:.4 f} ({explained/total_gap:.1 %} of gap)" )
print (f" Unexplained (returns): {unexplained:.4 f} ({unexplained/total_gap:.1 %} of gap)" )
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:.4 f} " , 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 = 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))
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)
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" ]
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:.4 f} ({(np.exp(naive_premium)-1 )*100 :.2 f} %)" )
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:.4 f} " )
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 Error / Issue Cause Resolution Heckman IMR coefficient not significant Weak exclusion restriction Find stronger exclusion restriction (nonlabor income, spouse's wage) OB decomposition does not sum to total gap Floating-point rounding Check Σ(explained + unexplained) ≈ total_gap with tolerance 1e-6 RIF regression coefs vary widely across quantiles Insufficient bandwidth in KDE Try bw_method='scott'; increase sample size Perfect multicollinearity in Mincer model exp and exp2 highly correlated Use centered exp: exp_c = exp - exp.mean() Probit does not converge Near-perfect prediction Check for perfect separation in selection equation statsmodels Probit slowLarge sample Reduce to 50k rows for initial exploration
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():.3 f} " )
ols_result = mincer_ols(df)
heck = heckman_two_step(df)
print (f"\nOLS educ coef: {ols_result.params['educ' ]:.4 f} " )
print (f"Heckman educ coef: {heck['wage_result' ].params['educ' ]:.4 f} " )
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_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 Version Date Change 1.0.0 2026-03-18 Initial release — Mincer, Heckman two-step, Oaxaca-Blinder, RIF-QR
