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statsmodels Statistical models library for Python. Use when you need specific model classes (OLS, GLM, mixed models, ARIMA) with detailed diagnostics, residuals, and inference. Best for econometrics, time series, rigorous inference with coefficient tables. For guided statistical test selection with APA reporting use statistical-analysis.
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name statsmodels description Statistical models library for Python. Use when you need specific model classes (OLS, GLM, mixed models, ARIMA) with detailed diagnostics, residuals, and inference. Best for econometrics, time series, rigorous inference with coefficient tables. For guided statistical test selection with APA reporting use statistical-analysis. category coding license BSD-3-Clause license metadata {"skill-author":"Synthetic Sciences"} version 1.0.0 author Synthetic Sciences tags ["Statistics","Econometrics","Time Series","Regression"] dependencies ["statsmodels>=0.14.0","numpy>=1.25.0","scipy>=1.12.0"]
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
X = sm.add_constant(X_data)
model = sm.OLS(y, X)
results = model.fit()
print (results.summary())
print (f"R-squared: {results.rsquared:.4 f} " )
print (f"Coefficients:\\n{results.params} " )
print (f"P-values:\\n{results.pvalues} " )
predictions = results.get_prediction(X_new)
pred_summary = predictions.summary_frame()
print (pred_summary)
from
import
print
f"Breusch-Pagan p-value: {bp_test[1 ]:.4 f} "
import
as
0
'r'
'--'
'Fitted values'
'Residuals'
Logistic Regression (Binary Outcomes) from statsmodels.discrete.discrete_model import Logit
X = sm.add_constant(X_data)
model = Logit(y_binary, X)
results = model.fit()
print (results.summary())
odds_ratios = np.exp(results.params)
print ("Odds ratios:\\n" , odds_ratios)
probs = results.predict(X)
predictions = (probs > 0.5 ).astype(int )
from sklearn.metrics import classification_report, roc_auc_score
print (classification_report(y_binary, predictions))
print (f"AUC: {roc_auc_score(y_binary, probs):.4 f} " )
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
from statsmodels.tsa.stattools import adfuller
adf_result = adfuller(y_series)
print (f"ADF p-value: {adf_result[1 ]:.4 f} " )
if adf_result[1 ] > 0.05 :
y_diff = y_series.diff().dropna()
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()
model = ARIMA(y_series, order=(1 , 1 , 1 ))
results = model.fit()
print (results.summary())
forecast = results.forecast(steps=10 )
forecast_obj = results.get_forecast(steps=10 )
forecast_df = forecast_obj.summary_frame()
print (forecast_df)
results.plot_diagnostics(figsize=(12 , 8 ))
plt.show()
Generalized Linear Models (GLM) import statsmodels.api as sm
X = sm.add_constant(X_data)
model = sm.GLM(y_counts, X, family=sm.families.Poisson())
results = model.fit()
print (results.summary())
rate_ratios = np.exp(results.params)
print ("Rate ratios:\\n" , rate_ratios)
overdispersion = results.pearson_chi2 / results.df_resid
print (f"Overdispersion: {overdispersion:.2 f} " )
if overdispersion > 1.5 :
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.
OLS : Standard linear regression with i.i.d. errorsWLS : Weighted least squares for heteroskedastic errorsGLS : Generalized least squares for arbitrary covariance structureGLSAR : GLS with autoregressive errors for time seriesQuantile Regression : Conditional quantiles (robust to outliers)Mixed Effects : Hierarchical/multilevel models with random effectsRecursive/Rolling : Time-varying parameter estimation
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.
Binomial : Binary outcomes or proportions (logistic regression)Poisson : Count dataNegative Binomial : Overdispersed countsGamma : Positive continuous, right-skewed dataInverse Gaussian : Positive continuous with specific variance structureGaussian : Equivalent to OLSTweedie : Flexible family for semi-continuous data
Logit, Probit, Log, Identity, Inverse, Sqrt, CLogLog, Power
Choose based on interpretation needs and model fit
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.
Logit : Logistic regression (odds ratios)Probit : Probit regression (normal distribution)
MNLogit : Unordered categories (3+ levels)Conditional Logit : Choice models with alternative-specific variablesOrdered Model : Ordinal outcomes (ordered categories)
Poisson : Standard count modelNegative Binomial : Overdispersed countsZero-Inflated : Excess zeros (ZIP, ZINB)Hurdle Models : Two-stage models for zero-heavy data
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.
AutoReg (AR) : Autoregressive modelsARIMA : Autoregressive integrated moving averageSARIMAX : Seasonal ARIMA with exogenous variablesExponential Smoothing : Simple, Holt, Holt-WintersETS : Innovations state space models
VAR : Vector autoregressionVARMAX : VAR with MA and exogenous variablesDynamic Factor Models : Extract common factorsVECM : Vector error correction models (cointegration)
State Space : Kalman filtering, custom specificationsRegime Switching : Markov switching modelsARDL : Autoregressive distributed lag
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.
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)
Leverage (hat values)
Cook's distance
DFFITS and DFBETAs
Studentized residuals
Influence plots
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)
Tukey's HSD
Bonferroni correction
False Discovery Rate (FDR)
Cohen's d, eta-squared
Power analysis for t-tests, proportions
Sample size calculations
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
results = smf.ols('y ~ x1 + x2 + x1:x2' , data=df).fit()
results = smf.ols('y ~ x1 + C(category)' , data=df).fit()
results = smf.ols('y ~ x1 * x2' , data=df).fit()
results = smf.ols('y ~ x + I(x**2)' , data=df).fit()
results = smf.logit('y ~ x1 + x2 + C(group)' , data=df).fit()
results = smf.poisson('count ~ x1 + x2' , data=df).fit()
Model Selection and Comparison
Information Criteria
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' ))
Likelihood Ratio Test (Nested Models)
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:.4 f} " )
print (f"p-value: {p_value:.4 f} " )
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]
model = sm.OLS(y_train, X_train).fit()
y_pred = model.predict(X_val)
rmse = np.sqrt(mean_squared_error(y_val, y_pred))
cv_scores.append(rmse)
print (f"CV RMSE: {np.mean(cv_scores):.4 f} ± {np.std(cv_scores):.4 f} " )
Best Practices
Data Preparation
Always add constant : Use sm.add_constant() unless excluding interceptCheck for missing values : Handle or impute before fittingScale 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 neededCheck assumptions : Test residuals, heteroskedasticity, autocorrelationUse 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-valuesUse robust SEs : When heteroskedasticity or clustering presentMultiple comparisons : Correct when testing many hypothesesConfidence intervals : Always report alongside point estimates
Model Evaluation
Check residuals : Plot residuals vs fitted, Q-Q plotInfluence diagnostics : Identify and investigate influential observationsOut-of-sample validation : Test on holdout set or cross-validateCompare models : Use AIC/BIC for non-nested, LR test for nested
Reporting
Comprehensive summary : Use .summary() for detailed outputDocument decisions : Note transformations, excluded observationsInterpret 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
Need detailed parameter explanations
Choosing between similar models
Troubleshooting convergence or diagnostic issues
Understanding specific test statistics
Looking for code examples for advanced features
grep -r "Quantile Regression" references/
grep -r "Breusch-Pagan" references/stats_diagnostics.md
grep -r "SARIMAX" references/time_series.md
Common Pitfalls to Avoid
Forgetting constant term : Always use sm.add_constant() unless no intercept desiredIgnoring assumptions : Check residuals, heteroskedasticity, autocorrelationWrong model for outcome type : Binary→Logit/Probit, Count→Poisson/NB, not OLSNot checking convergence : Look for optimization warningsMisinterpreting coefficients : Remember link functions (log, logit, etc.)Using Poisson with overdispersion : Check dispersion, use Negative Binomial if neededNot using robust SEs : When heteroskedasticity or clustering presentOverfitting : Too many parameters relative to sample sizeData leakage : Fitting on test data or using future informationNot validating predictions : Always check out-of-sample performanceComparing non-nested models : Use AIC/BIC, not LR testIgnoring influential observations : Check Cook's distance and leverageMultiple testing : Correct p-values when testing many hypothesesNot differencing time series : Fit ARIMA on non-stationary dataConfusing prediction vs confidence intervals : Prediction intervals are wider
Getting Help For detailed documentation and examples:
