statsmodels

Statsmodels: Statistical Modeling and Econometrics

Overview

Statsmodels is Python's premier library for statistical modeling, providing tools for estimation, inference, and diagnostics across a wide range of statistical methods. Apply this skill for rigorous statistical analysis, from simple linear regression to complex time series models and econometric analyses.

When to Use This Skill

This skill should be used when:

Fitting regression models (OLS, WLS, GLS, quantile regression)

Performing generalized linear modeling (logistic, Poisson, Gamma, etc.)

Analyzing discrete outcomes (binary, multinomial, count, ordinal)

Conducting time series analysis (ARIMA, SARIMAX, VAR, forecasting)

Running statistical tests and diagnostics

Testing model assumptions (heteroskedasticity, autocorrelation, normality)

Detecting outliers and influential observations

Comparing models (AIC/BIC, likelihood ratio tests)

Estimating causal effects

Producing publication-ready statistical tables and inference

Quick Start Guide

Linear Regression (OLS)

import statsmodels.api as sm
import numpy as np
import pandas as pd
Prepare data - ALWAYS add constant for intercept

X = sm.add_constant(X_data)
Fit OLS model

model = sm.OLS(y, X)
results = model.fit()
View comprehensive results

print(results.summary())
Key results

print(f"R-squared: {results.rsquared:.4f}")
print(f"Coefficients:\\n{results.params}")
print(f"P-values:\\n{results.pvalues}")
Predictions with confidence intervals

predictions = results.get_prediction(X_new)
pred_summary = predictions.summary_frame()
print(pred_summary)  # includes mean, CI, prediction intervals
Diagnostics

from statsmodels.stats.diagnostic import het_breuschpagan
bp_test = het_breuschpagan(results.resid, X)
print(f"Breusch-Pagan p-value: {bp_test[1]:.4f}")
Visualize residuals

import matplotlib.pyplot as plt
plt.scatter(results.fittedvalues, results.resid)
plt.axhline(y=0, color='r', linestyle='--')
plt.xlabel('Fitted values')
plt.ylabel('Residuals')
plt.show()

Logistic Regression (Binary Outcomes)

from statsmodels.discrete.discrete_model import Logit
Add constant

X = sm.add_constant(X_data)
Fit logit model

model = Logit(y_binary, X)
results = model.fit()
print(results.summary())
Odds ratios

odds_ratios = np.exp(results.params)
print("Odds ratios:\\n", odds_ratios)
Predicted probabilities

probs = results.predict(X)
Binary predictions (0.5 threshold)

predictions = (probs > 0.5).astype(int)
Model evaluation

from sklearn.metrics import classification_report, roc_auc_score
print(classification_report(y_binary, predictions))
print(f"AUC: {roc_auc_score(y_binary, probs):.4f}")
Marginal effects

marginal = results.get_margeff()
print(marginal.summary())

Time Series (ARIMA)

from statsmodels.tsa.arima.model import ARIMA
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
Check stationarity

from statsmodels.tsa.stattools import adfuller
adf_result = adfuller(y_series)
print(f"ADF p-value: {adf_result[1]:.4f}")
if adf_result[1] > 0.05:
    # Series is non-stationary, difference it
    y_diff = y_series.diff().dropna()
Plot ACF/PACF to identify p, q

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))
plot_acf(y_diff, lags=40, ax=ax1)
plot_pacf(y_diff, lags=40, ax=ax2)
plt.show()
Fit ARIMA(p,d,q)

model = ARIMA(y_series, order=(1, 1, 1))
results = model.fit()
print(results.summary())
Forecast

forecast = results.forecast(steps=10)
forecast_obj = results.get_forecast(steps=10)
forecast_df = forecast_obj.summary_frame()
print(forecast_df)  # includes mean and confidence intervals
Residual diagnostics

results.plot_diagnostics(figsize=(12, 8))
plt.show()

Generalized Linear Models (GLM)

import statsmodels.api as sm
Poisson regression for count data

X = sm.add_constant(X_data)
model = sm.GLM(y_counts, X, family=sm.families.Poisson())
results = model.fit()
print(results.summary())
Rate ratios (for Poisson with log link)

rate_ratios = np.exp(results.params)
print("Rate ratios:\\n", rate_ratios)
Check overdispersion

overdispersion = results.pearson_chi2 / results.df_resid
print(f"Overdispersion: {overdispersion:.2f}")
if overdispersion > 1.5:
    # Use Negative Binomial instead
    from statsmodels.discrete.count_model import NegativeBinomial
    nb_model = NegativeBinomial(y_counts, X)
    nb_results = nb_model.fit()
    print(nb_results.summary())

Core Statistical Modeling Capabilities

1. Linear Regression Models

Comprehensive suite of linear models for continuous outcomes with various error structures.

Available models:

OLS: Standard linear regression with i.i.d. errors

WLS: Weighted least squares for heteroskedastic errors

GLS: Generalized least squares for arbitrary covariance structure

GLSAR: GLS with autoregressive errors for time series

Quantile Regression: Conditional quantiles (robust to outliers)

Mixed Effects: Hierarchical/multilevel models with random effects

Recursive/Rolling: Time-varying parameter estimation

Key features:

Comprehensive diagnostic tests

Robust standard errors (HC, HAC, cluster-robust)

Influence statistics (Cook's distance, leverage, DFFITS)

Hypothesis testing (F-tests, Wald tests)

Model comparison (AIC, BIC, likelihood ratio tests)

Prediction with confidence and prediction intervals

When to use: Continuous outcome variable, want inference on coefficients, need diagnostics

Reference: See references/linear_models.md for detailed guidance on model selection, diagnostics, and best practices.

2. Generalized Linear Models (GLM)

Flexible framework extending linear models to non-normal distributions.

Distribution families:

Binomial: Binary outcomes or proportions (logistic regression)

Poisson: Count data

Negative Binomial: Overdispersed counts

Gamma: Positive continuous, right-skewed data

Inverse Gaussian: Positive continuous with specific variance structure

Gaussian: Equivalent to OLS

Tweedie: Flexible family for semi-continuous data

Link functions:

Logit, Probit, Log, Identity, Inverse, Sqrt, CLogLog, Power

Choose based on interpretation needs and model fit

Key features:

Maximum likelihood estimation via IRLS

Deviance and Pearson residuals

Goodness-of-fit statistics

Pseudo R-squared measures

Robust standard errors

When to use: Non-normal outcomes, need flexible variance and link specifications

Reference: See references/glm.md for family selection, link functions, interpretation, and diagnostics.

3. Discrete Choice Models

Models for categorical and count outcomes.

Binary models:

Logit: Logistic regression (odds ratios)

Probit: Probit regression (normal distribution)

Multinomial models:

MNLogit: Unordered categories (3+ levels)

Conditional Logit: Choice models with alternative-specific variables

Ordered Model: Ordinal outcomes (ordered categories)

Count models:

Poisson: Standard count model

Negative Binomial: Overdispersed counts

Zero-Inflated: Excess zeros (ZIP, ZINB)

Hurdle Models: Two-stage models for zero-heavy data

Key features:

Maximum likelihood estimation

Marginal effects at means or average marginal effects

Model comparison via AIC/BIC

Predicted probabilities and classification

Goodness-of-fit tests

When to use: Binary, categorical, or count outcomes

Reference: See references/discrete_choice.md for model selection, interpretation, and evaluation.

4. Time Series Analysis

Comprehensive time series modeling and forecasting capabilities.

Univariate models:

AutoReg (AR): Autoregressive models

ARIMA: Autoregressive integrated moving average

SARIMAX: Seasonal ARIMA with exogenous variables

Exponential Smoothing: Simple, Holt, Holt-Winters

ETS: Innovations state space models

Multivariate models:

VAR: Vector autoregression

VARMAX: VAR with MA and exogenous variables

Dynamic Factor Models: Extract common factors

VECM: Vector error correction models (cointegration)

Advanced models:

State Space: Kalman filtering, custom specifications

Regime Switching: Markov switching models

ARDL: Autoregressive distributed lag

Key features:

ACF/PACF analysis for model identification

Stationarity tests (ADF, KPSS)

Forecasting with prediction intervals

Residual diagnostics (Ljung-Box, heteroskedasticity)

Granger causality testing

Impulse response functions (IRF)

Forecast error variance decomposition (FEVD)

When to use: Time-ordered data, forecasting, understanding temporal dynamics

Reference: See references/time_series.md for model selection, diagnostics, and forecasting methods.

5. Statistical Tests and Diagnostics

Extensive testing and diagnostic capabilities for model validation.

Residual diagnostics:

Autocorrelation tests (Ljung-Box, Durbin-Watson, Breusch-Godfrey)

Heteroskedasticity tests (Breusch-Pagan, White, ARCH)

Normality tests (Jarque-Bera, Omnibus, Anderson-Darling, Lilliefors)

Specification tests (RESET, Harvey-Collier)

Influence and outliers:

Leverage (hat values)

Cook's distance

DFFITS and DFBETAs

Studentized residuals

Influence plots

Hypothesis testing:

t-tests (one-sample, two-sample, paired)

Proportion tests

Chi-square tests

Non-parametric tests (Mann-Whitney, Wilcoxon, Kruskal-Wallis)

ANOVA (one-way, two-way, repeated measures)

Multiple comparisons:

Tukey's HSD

Bonferroni correction

False Discovery Rate (FDR)

Effect sizes and power:

Cohen's d, eta-squared

Power analysis for t-tests, proportions

Sample size calculations

Robust inference:

Heteroskedasticity-consistent SEs (HC0-HC3)

HAC standard errors (Newey-West)

Cluster-robust standard errors

When to use: Validating assumptions, detecting problems, ensuring robust inference

Reference: See references/stats_diagnostics.md for comprehensive testing and diagnostic procedures.

Formula API (R-style)

Statsmodels supports R-style formulas for intuitive model specification:

import statsmodels.formula.api as smf
OLS with formula

results = smf.ols('y ~ x1 + x2 + x1:x2', data=df).fit()
Categorical variables (automatic dummy coding)

results = smf.ols('y ~ x1 + C(category)', data=df).fit()
Interactions

results = smf.ols('y ~ x1  x2', data=df).fit()  # x1 + x2 + x1:x2
Polynomial terms

results = smf.ols('y ~ x + I(x2)', data=df).fit()
Logit

results = smf.logit('y ~ x1 + x2 + C(group)', data=df).fit()
Poisson

results = smf.poisson('count ~ x1 + x2', data=df).fit()
ARIMA (not available via formula, use regular API)

Model Selection and Comparison

Information Criteria

# Compare models using AIC/BIC
models = {
    'Model 1': model1_results,
    'Model 2': model2_results,
    'Model 3': model3_results
}
comparison = pd.DataFrame({
    'AIC': {name: res.aic for name, res in models.items()},
    'BIC': {name: res.bic for name, res in models.items()},
    'Log-Likelihood': {name: res.llf for name, res in models.items()}
})
print(comparison.sort_values('AIC'))
Lower AIC/BIC indicates better model

Likelihood Ratio Test (Nested Models)

# For nested models (one is subset of the other)
from scipy import stats
lr_stat = 2  (full_model.llf - reduced_model.llf)
df = full_model.df_model - reduced_model.df_model
p_value = 1 - stats.chi2.cdf(lr_stat, df)
print(f"LR statistic: {lr_stat:.4f}")
print(f"p-value: {p_value:.4f}")
if p_value < 0.05:
    print("Full model significantly better")
else:
    print("Reduced model preferred (parsimony)")

Cross-Validation

from sklearn.model_selection import KFold
from sklearn.metrics import mean_squared_error
kf = KFold(n_splits=5, shuffle=True, random_state=42)
cv_scores = []
for train_idx, val_idx in kf.split(X):
    X_train, X_val = X.iloc[train_idx], X.iloc[val_idx]
    y_train, y_val = y.iloc[train_idx], y.iloc[val_idx]
    # Fit model
    model = sm.OLS(y_train, X_train).fit()
    # Predict
    y_pred = model.predict(X_val)
    # Score
    rmse = np.sqrt(mean_squared_error(y_val, y_pred))
    cv_scores.append(rmse)
print(f"CV RMSE: {np.mean(cv_scores):.4f} ± {np.std(cv_scores):.4f}")

Best Practices

Data Preparation

Always add constant: Use sm.add_constant() unless excluding intercept

Check for missing values: Handle or impute before fitting

Scale if needed: Improves convergence, interpretation (but not required for tree models)

Encode categoricals: Use formula API or manual dummy coding

Model Building

Start simple: Begin with basic model, add complexity as needed

Check assumptions: Test residuals, heteroskedasticity, autocorrelation

Use appropriate model: Match model to outcome type (binary→Logit, count→Poisson)

Consider alternatives: If assumptions violated, use robust methods or different model

Inference

Report effect sizes: Not just p-values

Use robust SEs: When heteroskedasticity or clustering present

Multiple comparisons: Correct when testing many hypotheses

Confidence intervals: Always report alongside point estimates

Model Evaluation

Check residuals: Plot residuals vs fitted, Q-Q plot

Influence diagnostics: Identify and investigate influential observations

Out-of-sample validation: Test on holdout set or cross-validate

Compare models: Use AIC/BIC for non-nested, LR test for nested

Reporting

Comprehensive summary: Use .summary() for detailed output

Document decisions: Note transformations, excluded observations

Interpret carefully: Account for link functions (e.g., exp(β) for log link)

Visualize: Plot predictions, confidence intervals, diagnostics

Common Workflows

Workflow 1: Linear Regression Analysis

Explore data (plots, descriptives)

Fit initial OLS model

Check residual diagnostics

Test for heteroskedasticity, autocorrelation

Check for multicollinearity (VIF)

Identify influential observations

Refit with robust SEs if needed

Interpret coefficients and inference

Validate on holdout or via CV

Workflow 2: Binary Classification

Fit logistic regression (Logit)

Check for convergence issues

Interpret odds ratios

Calculate marginal effects

Evaluate classification performance (AUC, confusion matrix)

Check for influential observations

Compare with alternative models (Probit)

Validate predictions on test set

Workflow 3: Count Data Analysis

Fit Poisson regression

Check for overdispersion

If overdispersed, fit Negative Binomial

Check for excess zeros (consider ZIP/ZINB)

Interpret rate ratios

Assess goodness of fit

Compare models via AIC

Validate predictions

Workflow 4: Time Series Forecasting

Plot series, check for trend/seasonality

Test for stationarity (ADF, KPSS)

Difference if non-stationary

Identify p, q from ACF/PACF

Fit ARIMA or SARIMAX

Check residual diagnostics (Ljung-Box)

Generate forecasts with confidence intervals

Evaluate forecast accuracy on test set

Reference Documentation

This skill includes comprehensive reference files for detailed guidance:

references/linear_models.md

Detailed coverage of linear regression models including:

OLS, WLS, GLS, GLSAR, Quantile Regression

Mixed effects models

Recursive and rolling regression

Comprehensive diagnostics (heteroskedasticity, autocorrelation, multicollinearity)

Influence statistics and outlier detection

Robust standard errors (HC, HAC, cluster)

Hypothesis testing and model comparison

references/glm.md

Complete guide to generalized linear models:

All distribution families (Binomial, Poisson, Gamma, etc.)

Link functions and when to use each

Model fitting and interpretation

Pseudo R-squared and goodness of fit

Diagnostics and residual analysis

Applications (logistic, Poisson, Gamma regression)

references/discrete_choice.md

Comprehensive guide to discrete outcome models:

Binary models (Logit, Probit)

Multinomial models (MNLogit, Conditional Logit)

Count models (Poisson, Negative Binomial, Zero-Inflated, Hurdle)

Ordinal models

Marginal effects and interpretation

Model diagnostics and comparison

references/time_series.md

In-depth time series analysis guidance:

Univariate models (AR, ARIMA, SARIMAX, Exponential Smoothing)

Multivariate models (VAR, VARMAX, Dynamic Factor)

State space models

Stationarity testing and diagnostics

Forecasting methods and evaluation

Granger causality, IRF, FEVD

references/stats_diagnostics.md

Comprehensive statistical testing and diagnostics:

Residual diagnostics (autocorrelation, heteroskedasticity, normality)

Influence and outlier detection

Hypothesis tests (parametric and non-parametric)

ANOVA and post-hoc tests

Multiple comparisons correction

Robust covariance matrices

Power analysis and effect sizes

When to reference:

Need detailed parameter explanations

Choosing between similar models

Troubleshooting convergence or diagnostic issues

Understanding specific test statistics

Looking for code examples for advanced features

Search patterns:

# Find information about specific models
grep -r "Quantile Regression" references/
Find diagnostic tests

grep -r "Breusch-Pagan" references/stats_diagnostics.md
Find time series guidance

grep -r "SARIMAX" references/time_series.md

Common Pitfalls to Avoid

Forgetting constant term: Always use sm.add_constant() unless no intercept desired

Ignoring assumptions: Check residuals, heteroskedasticity, autocorrelation

Wrong model for outcome type: Binary→Logit/Probit, Count→Poisson/NB, not OLS

Not checking convergence: Look for optimization warnings

Misinterpreting coefficients: Remember link functions (log, logit, etc.)

Using Poisson with overdispersion: Check dispersion, use Negative Binomial if needed

Not using robust SEs: When heteroskedasticity or clustering present

Overfitting: Too many parameters relative to sample size

Data leakage: Fitting on test data or using future information

Not validating predictions: Always check out-of-sample performance

Comparing non-nested models: Use AIC/BIC, not LR test

Ignoring influential observations: Check Cook's distance and leverage

Multiple testing: Correct p-values when testing many hypotheses

Not differencing time series: Fit ARIMA on non-stationary data

Confusing prediction vs confidence intervals: Prediction intervals are wider

Getting Help

For detailed documentation and examples:

Official docs: https://www.statsmodels.org/stable/

User guide: https://www.statsmodels.org/stable/user-guide.html

Examples: https://www.statsmodels.org/stable/examples/index.html

API reference: https://www.statsmodels.org/stable/api.html

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