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name: "statsmodels-statistical-modeling" description: "Python statistical modeling: regression (OLS, WLS, GLM), discrete (Logit, Poisson, NegBin), time series (ARIMA, SARIMAX, VAR), with rigorous inference, diagnostics, and hypothesis tests. Use scikit-learn for ML; statistical-analysis for test choice." license: "BSD-3-Clause"

statsmodels

Overview

Statsmodels provides classical statistical modeling with rigorous inference for Python. It covers linear models, generalized linear models, discrete choice, time series, and comprehensive diagnostics. Unlike scikit-learn (prediction-focused), statsmodels emphasizes coefficient interpretation, p-values, confidence intervals, and model diagnostics.

When to Use

• Fitting linear regression (OLS, WLS, GLS) with detailed coefficient tables and diagnostics
• Running logistic regression with odds ratios and marginal effects for clinical/epidemiological studies
• Analyzing count data with Poisson or negative binomial regression
• Time series forecasting with ARIMA, SARIMAX, or exponential smoothing
• Performing ANOVA, t-tests, or non-parametric tests with proper corrections
• Testing model assumptions (heteroskedasticity, autocorrelation, normality of residuals)
• Model comparison using AIC/BIC or likelihood ratio tests
• Using R-style formula interface (y ~ x1 + x2 + C(group)) for intuitive model specification
• For prediction-focused ML with cross-validation and hyperparameter tuning, use scikit-learn instead
• For Bayesian modeling with posterior inference, use pymc instead

Prerequisites

• Python packages: statsmodels, numpy, pandas, scipy
• Optional: matplotlib (for diagnostic plots), patsy (for formula API, included with statsmodels)
• Data: Tabular data as pandas DataFrames or NumPy arrays

pip install statsmodels numpy pandas matplotlib

Quick Start

import statsmodels.api as sm
import statsmodels.formula.api as smf
import pandas as pd
import numpy as np

# Generate sample data
np.random.seed(42)
n = 100
df = pd.DataFrame({
    "x1": np.random.randn(n),
    "x2": np.random.randn(n),
    "group": np.random.choice(["A", "B"], n)
})
df["y"] = 2 + 3 * df["x1"] - 1.5 * df["x2"] + np.random.randn(n)

# OLS with formula API (R-style)
results = smf.ols("y ~ x1 + x2 + C(group)", data=df).fit()
print(results.summary())
print(f"R²: {results.rsquared:.3f}, AIC: {results.aic:.1f}")

Core API

Module 1: Linear Regression (OLS, WLS, GLS)

Standard linear models with comprehensive diagnostics.

import statsmodels.api as sm
import numpy as np

# Generate data
np.random.seed(42)
X = np.random.randn(200, 3)
y = 1 + 2*X[:, 0] - 0.5*X[:, 1] + np.random.randn(200)

# ALWAYS add constant for intercept
X_const = sm.add_constant(X)
results = sm.OLS(y, X_const).fit()

print(results.summary())
print(f"\nCoefficients: {results.params}")
print(f"P-values: {results.pvalues}")
print(f"R²: {results.rsquared:.4f}")

# Predictions with confidence intervals
pred = results.get_prediction(X_const[:5])
print(pred.summary_frame())

# Robust standard errors (heteroskedasticity-consistent)
results_robust = sm.OLS(y, X_const).fit(cov_type="HC3")
print("Robust SEs:", results_robust.bse)

# Weighted Least Squares
weights = 1 / np.abs(results.resid + 0.1)  # Example weights
results_wls = sm.WLS(y, X_const, weights=weights).fit()
print(f"WLS R²: {results_wls.rsquared:.4f}")

Module 2: Generalized Linear Models (GLM)

Extend regression to non-normal outcomes (binary, count, continuous-positive).

import statsmodels.api as sm
import numpy as np

# Poisson regression for count data
np.random.seed(42)
X = np.random.randn(200, 2)
X_const = sm.add_constant(X)
y_counts = np.random.poisson(np.exp(0.5 + 0.3*X[:, 0]))

model = sm.GLM(y_counts, X_const, family=sm.families.Poisson())
results = model.fit()
print(results.summary())

# Rate ratios
rate_ratios = np.exp(results.params)
print(f"Rate ratios: {rate_ratios}")

# Check overdispersion
overdispersion = results.pearson_chi2 / results.df_resid
print(f"Overdispersion ratio: {overdispersion:.2f}")
if overdispersion > 1.5:
    print("→ Consider Negative Binomial model")

Module 3: Discrete Choice Models (Logit, Probit, Count)

Binary, multinomial, and count outcome models.

import statsmodels.api as sm
import numpy as np

# Logistic regression
np.random.seed(42)
X = np.random.randn(300, 2)
X_const = sm.add_constant(X)
prob = 1 / (1 + np.exp(-(0.5 + X[:, 0] - 0.5*X[:, 1])))
y_binary = np.random.binomial(1, prob)

logit_results = sm.Logit(y_binary, X_const).fit()
print(logit_results.summary())

# Odds ratios
odds_ratios = np.exp(logit_results.params)
print(f"Odds ratios: {odds_ratios}")

# Marginal effects (at means)
margeff = logit_results.get_margeff()
print(margeff.summary())

# Predicted probabilities
probs = logit_results.predict(X_const[:5])
print(f"Predicted P(Y=1): {probs}")

Module 4: Time Series (ARIMA, SARIMAX)

Univariate and multivariate time series modeling and forecasting.

import statsmodels.api as sm
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.stattools import adfuller
import numpy as np
import pandas as pd

# Generate time series
np.random.seed(42)
dates = pd.date_range("2020-01-01", periods=200, freq="D")
y = np.cumsum(np.random.randn(200)) + 50
ts = pd.Series(y, index=dates)

# Stationarity test
adf_result = adfuller(ts)
print(f"ADF statistic: {adf_result[0]:.4f}, p-value: {adf_result[1]:.4f}")
print("Stationary" if adf_result[1] < 0.05 else "Non-stationary → difference")

# Fit ARIMA
model = ARIMA(ts, order=(1, 1, 1))
results = model.fit()
print(results.summary())

# Forecast with confidence intervals
forecast = results.get_forecast(steps=30)
forecast_df = forecast.summary_frame()
print(f"30-day forecast:\n{forecast_df.head()}")

# Seasonal ARIMA (SARIMAX)
from statsmodels.tsa.statespace.sarimax import SARIMAX

# Monthly data with yearly seasonality
model_sarima = SARIMAX(ts, order=(1, 1, 1), seasonal_order=(1, 1, 1, 12))
results_sarima = model_sarima.fit(disp=False)
print(f"AIC: {results_sarima.aic:.1f}")

# Diagnostic plots
results_sarima.plot_diagnostics(figsize=(12, 8))

Module 5: Statistical Tests and Diagnostics

Assumption tests, hypothesis tests, and model validation.

import statsmodels.api as sm
from statsmodels.stats.diagnostic import het_breuschpagan, acorr_ljungbox
from statsmodels.stats.stattools import jarque_bera
import numpy as np

# Fit a model first
np.random.seed(42)
X = sm.add_constant(np.random.randn(200, 2))
y = 1 + 2*X[:, 1] + np.random.randn(200) * X[:, 1]  # Heteroskedastic
results = sm.OLS(y, X).fit()

# Heteroskedasticity test (Breusch-Pagan)
bp_stat, bp_p, _, _ = het_breuschpagan(results.resid, X)
print(f"Breusch-Pagan p-value: {bp_p:.4f} {'→ heteroskedastic' if bp_p < 0.05 else '→ OK'}")

# Normality test (Jarque-Bera)
jb_stat, jb_p, _, _ = jarque_bera(results.resid)
print(f"Jarque-Bera p-value: {jb_p:.4f} {'→ non-normal' if jb_p < 0.05 else '→ OK'}")

# Autocorrelation test (Ljung-Box)
lb_result = acorr_ljungbox(results.resid, lags=[10], return_df=True)
print(f"Ljung-Box p-value (lag 10): {lb_result['lb_pvalue'].values[0]:.4f}")

# Variance Inflation Factor (multicollinearity)
from statsmodels.stats.outliers_influence import variance_inflation_factor

vif_data = pd.DataFrame({
    "Variable": [f"x{i}" for i in range(X.shape[1])],
    "VIF": [variance_inflation_factor(X, i) for i in range(X.shape[1])]
})
print(vif_data)  # VIF > 10 suggests multicollinearity

Module 6: Formula API (R-style)

Intuitive model specification using formulas with automatic dummy coding.

import statsmodels.formula.api as smf
import pandas as pd
import numpy as np

np.random.seed(42)
df = pd.DataFrame({
    "y": np.random.randn(100),
    "x1": np.random.randn(100),
    "x2": np.random.randn(100),
    "group": np.random.choice(["A", "B", "C"], 100),
})

# Formula with categoricals (auto dummy-coded)
res = smf.ols("y ~ x1 + x2 + C(group)", data=df).fit()
print(res.summary())

# Interactions
res2 = smf.ols("y ~ x1 * x2", data=df).fit()  # x1 + x2 + x1:x2
print(f"Interaction term p-value: {res2.pvalues['x1:x2']:.4f}")

# Logit via formula
df["binary"] = (df["y"] > 0).astype(int)
logit_res = smf.logit("binary ~ x1 + x2 + C(group)", data=df).fit()
print(f"Logit AIC: {logit_res.aic:.1f}")

Common Workflows

Workflow 1: Complete Regression Analysis

Goal: Fit OLS, validate assumptions, use robust SEs if needed.

import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.stats.diagnostic import het_breuschpagan
from statsmodels.stats.outliers_influence import variance_inflation_factor
import numpy as np
import pandas as pd

# 1. Fit initial model
np.random.seed(42)
df = pd.DataFrame({"y": np.random.randn(200), "x1": np.random.randn(200), "x2": np.random.randn(200)})
df["y"] = 2 + 3*df["x1"] - df["x2"] + np.random.randn(200)

results = smf.ols("y ~ x1 + x2", data=df).fit()

# 2. Check heteroskedasticity
bp_stat, bp_p, _, _ = het_breuschpagan(results.resid, results.model.exog)
print(f"Breusch-Pagan p: {bp_p:.4f}")

# 3. If heteroskedastic, use robust SEs
if bp_p < 0.05:
    results = smf.ols("y ~ x1 + x2", data=df).fit(cov_type="HC3")
    print("Using HC3 robust standard errors")

# 4. Check multicollinearity
X = results.model.exog
for i in range(1, X.shape[1]):  # skip constant
    print(f"VIF x{i}: {variance_inflation_factor(X, i):.2f}")

# 5. Final results
print(results.summary())
print(f"\nAIC: {results.aic:.1f}, BIC: {results.bic:.1f}")

Workflow 2: Model Comparison

Goal: Compare nested and non-nested models using appropriate criteria.

import statsmodels.formula.api as smf
from scipy import stats
import pandas as pd
import numpy as np

np.random.seed(42)
df = pd.DataFrame({"y": np.random.randn(200), "x1": np.random.randn(200),
                    "x2": np.random.randn(200), "x3": np.random.randn(200)})
df["y"] = 1 + 2*df["x1"] - df["x2"] + 0.1*df["x3"] + np.random.randn(200)

# Fit nested models
m1 = smf.ols("y ~ x1", data=df).fit()
m2 = smf.ols("y ~ x1 + x2", data=df).fit()
m3 = smf.ols("y ~ x1 + x2 + x3", data=df).fit()

# Compare via AIC/BIC (lower = better)
comparison = pd.DataFrame({
    "R²": [m.rsquared for m in [m1, m2, m3]],
    "AIC": [m.aic for m in [m1, m2, m3]],
    "BIC": [m.bic for m in [m1, m2, m3]],
}, index=["y~x1", "y~x1+x2", "y~x1+x2+x3"])
print(comparison)

# Likelihood ratio test (nested: m2 vs m3)
lr_stat = 2 * (m3.llf - m2.llf)
p_val = 1 - stats.chi2.cdf(lr_stat, df=m3.df_model - m2.df_model)
print(f"\nLR test (m3 vs m2): stat={lr_stat:.2f}, p={p_val:.4f}")

Workflow 3: Time Series Forecasting Pipeline

Goal: Test stationarity, identify model order, forecast.

from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.stattools import adfuller
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# Generate data
np.random.seed(42)
ts = pd.Series(np.cumsum(np.random.randn(200)) + 100,
               index=pd.date_range("2020-01-01", periods=200, freq="D"))

# 1. Test stationarity
adf_p = adfuller(ts)[1]
print(f"ADF p-value: {adf_p:.4f} → {'stationary' if adf_p < 0.05 else 'non-stationary'}")

# 2. Identify order from ACF/PACF (on differenced series)
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 6))
plot_acf(ts.diff().dropna(), lags=20, ax=ax1)
plot_pacf(ts.diff().dropna(), lags=20, ax=ax2)
plt.savefig("acf_pacf.png", dpi=150, bbox_inches="tight")

# 3. Fit and forecast
model = ARIMA(ts[:180], order=(1, 1, 1))
results = model.fit()
forecast = results.get_forecast(steps=20)
fc_df = forecast.summary_frame()
print(f"ARIMA AIC: {results.aic:.1f}")
print(f"Forecast (first 5 days):\n{fc_df.head()}")

Key Parameters

| Parameter | Module | Default | Range /