statsmodels — quality + safety report

---
name: statsmodels
description: "Statistical modeling toolkit. OLS, GLM, logistic, ARIMA, time series, hypothesis tests, diagnostics, AIC/BIC, for rigorous statistical inference and econometric analysis."
---

# 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)

```python
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)

```python
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)

```python
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)

```python
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:

```python
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(x**2)', 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

```python
# 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)

```python
# 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 bet

… (truncated)