Linear Regression Models Reference

This document provides detailed guidance on linear regression models in statsmodels, including OLS, GLS, WLS, quantile regression, and specialized variants.

Core Model Classes

OLS (Ordinary Least Squares)

OLS estimates a linear conditional mean. Conventional covariance assumes homoscedastic, uncorrelated errors; normal errors support exact small-sample t/F inference. Robust covariance changes uncertainty, not an omitted-variable or endogeneity bias.

When to use:

• Standard regression analysis
• Errors are independent and have constant variance
• No autocorrelation or heteroscedasticity
• Most common starting point

Basic usage:

import statsmodels.api as sm
import numpy as np

# Array interface: add a constant if an intercept is intended
X = sm.add_constant(X_data)  # Adds column of 1s for intercept

# Fit model
model = sm.OLS(y, X)
results = model.fit()

# View results
print(results.summary())

Key results attributes:

results.params           # Coefficients
results.bse              # Standard errors
results.tvalues          # T-statistics
results.pvalues          # P-values
results.rsquared         # R-squared
results.rsquared_adj     # Adjusted R-squared
results.fittedvalues     # Fitted values (predictions on training data)
results.resid            # Residuals
results.conf_int()       # Confidence intervals for parameters

Prediction with confidence/prediction intervals:

# For in-sample predictions
pred = results.get_prediction(X)
pred_summary = pred.summary_frame()
print(pred_summary)  # Contains mean, std, confidence intervals

# For out-of-sample predictions
X_new = sm.add_constant(X_new_data, has_constant="add")
pred_new = results.get_prediction(X_new)
pred_summary = pred_new.summary_frame()

# Access intervals
mean_ci_lower = pred_summary["mean_ci_lower"]
mean_ci_upper = pred_summary["mean_ci_upper"]
obs_ci_lower = pred_summary["obs_ci_lower"]  # Prediction intervals
obs_ci_upper = pred_summary["obs_ci_upper"]

Formula API (R-style):

import statsmodels.formula.api as smf

# Automatic handling of categorical variables and interactions
formula = 'y ~ x1 + x2 + C(category) + x1:x2'
results = smf.ols(formula, data=df).fit()

WLS (Weighted Least Squares)

Handles heteroscedastic errors (diagonal Σ) where variance differs across observations.

When to use:

• Known heteroscedasticity (non-constant error variance)
• Different observations have different reliability
• Weights are known or can be estimated

Usage:

# If you know the weights (inverse variance)
weights = 1 / error_variance
model = sm.WLS(y, X, weights=weights)
results = model.fit()

# Common weight patterns:
# - 1/variance: when variance is known
# - n_i: sample size for grouped data
# - 1/x: when variance proportional to x

Feasible WLS (estimating weights):

# Step 1: Fit OLS
ols_results = sm.OLS(y, X).fit()

# Step 2: Model squared residuals to estimate variance
abs_resid = np.abs(ols_results.resid)
variance_model = sm.OLS(np.log(np.maximum(abs_resid**2, np.finfo(float).tiny)), X).fit()

# Step 3: Use estimated variance as weights
weights = 1 / np.exp(variance_model.fittedvalues)
wls_results = sm.WLS(y, X, weights=weights).fit()

GLS (Generalized Least Squares)

Handles arbitrary covariance structure (Σ). Superclass for other regression methods.

When to use:

• Known covariance structure
• Correlated errors
• More general than WLS

Usage:

# Specify covariance structure
# Sigma should be (n x n) covariance matrix
model = sm.GLS(y, X, sigma=Sigma)
results = model.fit()

GLSAR (GLS with Autoregressive Errors)

Feasible generalized least squares with AR(p) errors for time series data.

When to use:

• Time series regression with autocorrelated errors
• Need to account for serial correlation
• Violations of error independence

Usage:

# AR(1) errors
model = sm.GLSAR(y, X, rho=1)  # rho=1 for AR(1), rho=2 for AR(2), etc.
results = model.iterative_fit(maxiter=20)  # Default maxiter=3 may be insufficient

print(results.summary())
print(f"Estimated rho: {results.model.rho}")

RLS (Recursive Least Squares)

Expanding-window estimates of a fixed-coefficient regression, plus stability diagnostics; this is not a stochastic time-varying-coefficient model.

When to use:

• Assess coefficient stability as observations accumulate
• Inspect recursive residuals and CUSUM
• Refit with additional data; do not assume a streaming update interface

Usage:

from statsmodels.regression.recursive_ls import RecursiveLS

model = RecursiveLS(y, X)
results = model.fit()

# Expanding-window filtered/smoothed coefficient estimates
params_over_time = results.recursive_coefficients.filtered
cusum = results.cusum  # CUSUM statistic for structural breaks

Rolling Regressions

Compute estimates across moving windows for time-varying parameter detection.

When to use:

• Parameters vary over time
• Want to detect structural changes
• Time series with evolving relationships

Usage:

from statsmodels.regression.rolling import RollingOLS, RollingWLS

# Rolling OLS with 60-period window
rolling_model = RollingOLS(y, X, window=60)
rolling_results = rolling_model.fit()

# Extract time-varying parameters
rolling_params = pd.DataFrame(rolling_results.params)  # Also handles NumPy inputs
rolling_rsquared = rolling_results.rsquared

# Plot parameter evolution
import matplotlib.pyplot as plt
rolling_params.plot()
plt.title('Time-Varying Coefficients')
plt.show()

Quantile Regression

Analyzes conditional quantiles rather than conditional mean.

When to use:

• Interest in quantiles (median, 90th percentile, etc.)
• Robust to outliers (median regression)
• Distributional effects across quantiles
• Heterogeneous effects

Usage:

from statsmodels.regression.quantile_regression import QuantReg

# Median regression (50th percentile)
model = QuantReg(y, X)
results_median = model.fit(q=0.5)

# Multiple quantiles
quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
results_dict = {}
for q in quantiles:
    results_dict[q] = model.fit(q=q)

# Plot quantile-varying effects
import matplotlib.pyplot as plt
coef_dict = {q: res.params for q, res in results_dict.items()}
coef_df = pd.DataFrame(coef_dict).T
coef_df.plot()
plt.xlabel('Quantile')
plt.ylabel('Coefficient')
plt.show()

Mixed Effects Models

For hierarchical/nested data with random effects.

When to use:

• Clustered/grouped data (students in schools, patients in hospitals)
• Repeated measures
• Need random effects to account for grouping

Usage:

from statsmodels.regression.mixed_linear_model import MixedLM

# Random intercept model
model = MixedLM(y, X, groups=group_ids)
results = model.fit()

# Random intercept and slope
model = MixedLM(y, X, groups=group_ids, exog_re=X_random)
results = model.fit()

print(results.summary())

Use ML (fit(reml=False)) when comparing mixed models with different fixed effects; REML likelihoods are not comparable across those designs. Check convergence, singular random-effect covariance and boundary variance estimates before inference.

Diagnostics and Model Assessment

The following diagnostics use an OLS result with its original full-rank design. VIF, Cook's distance and condition-number thresholds are screening heuristics, not automatic exclusion rules; conditioning also depends on predictor units. Robust SEs do not validate classical OLS observation intervals under heteroskedasticity.

Residual Analysis

# Basic residual plots
import matplotlib.pyplot as plt

# Residuals vs fitted
plt.scatter(results.fittedvalues, results.resid)
plt.xlabel('Fitted values')
plt.ylabel('Residuals')
plt.axhline(y=0, color='r', linestyle='--')
plt.title('Residuals vs Fitted')
plt.show()

# Q-Q plot for normality
from statsmodels.graphics.gofplots import qqplot
qqplot(results.resid, line='s')
plt.show()

# Histogram of residuals
plt.hist(results.resid, bins=30, edgecolor='black')
plt.xlabel('Residuals')
plt.ylabel('Frequency')
plt.title('Distribution of Residuals')
plt.show()

Specification Tests

from statsmodels.stats.diagnostic import het_breuschpagan, het_white
from statsmodels.stats.stattools import durbin_watson, jarque_bera

# Heteroscedasticity tests
lm_stat, lm_pval, f_stat, f_pval = het_breuschpagan(results.resid, X)
print(f"Breusch-Pagan test p-value: {lm_pval}")

# White test
white_test = het_white(results.resid, X)
print(f"White test p-value: {white_test[1]}")

# Autocorrelation
dw_stat = durbin_watson(results.resid)
print(f"Durbin-Watson statistic: {dw_stat}")
# DW ~ 2 indicates no autocorrelation
# DW < 2 suggests positive autocorrelation
# DW > 2 suggests negative autocorrelation

# Normality test
jb_stat, jb_pval, skew, kurtosis = jarque_bera(results.resid)
print(f"Jarque-Bera test p-value: {jb_pval}")

Multicollinearity

from statsmodels.stats.outliers_influence import variance_inflation_factor

# Calculate VIF for each variable
vif_data = pd.DataFrame()
vif_data["Variable"] = X.columns
vif_data["VIF"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]

print(vif_data)
# VIF > 10 indicates problematic multicollinearity
# VIF > 5 suggests moderate multicollinearity

# Condition number (from summary)
print(f"Condition number: {results.condition_number}")
# Condition number > 20 suggests multicollinearity
# Condition number > 30 indicates serious problems

Influence Statistics

from statsmodels.stats.outliers_influence import OLSInfluence

influence = results.get_influence()

# Leverage (hat values)
leverage = influence.hat_matrix_diag
# High leverage: > 2*p/n (p=predictors, n=observations)

# Cook's distance
cooks_d = influence.cooks_distance[0]
# Influential if Cook's D > 4/n

# DFFITS
dffits = influence.dffits[0]
# Influential if |DFFITS| > 2*sqrt(p/n)

# Create influence plot
from statsmodels.graphics.regressionplots import influence_plot
fig, ax = plt.subplots(figsize=(12, 8))
influence_plot(results, ax=ax)
plt.show()

Hypothesis Testing

# Test single coefficient
# H0: beta_i = 0 (automatically in summary)

# Test multiple restrictions using F-test
# Example: Test beta_1 = beta_2 = 0
R = [[0, 1, 0, 0], [0, 0, 1, 0]]  # Restriction matrix
f_test = results.f_test(R)
print(f_test)

# Formula-based hypothesis testing
f_test = results.f_test("x1 = x2 = 0")
print(f_test)

# Test linear combination: beta_1 + beta_2 = 1
r_matrix = [[0, 1, 1, 0]]
q_matrix = [1]  # RHS value
f_test = results.f_test((r_matrix, q_matrix))
print(f_test)

# Wald test (equivalent to F-test for linear restrictions)
wald_test = results.wald_test(R)
print(wald_test)

Model Comparison

# Classical nested OLS F-test: same observations, larger model second
from statsmodels.stats.anova import anova_lm

# Fit restricted and unrestricted models
model_restricted = sm.OLS(y, X_restricted).fit()
model_full = sm.OLS(y, X_full).fit()

# ANOVA table for model comparison
anova_results = anova_lm(model_restricted, model_full)
print(anova_results)

# AIC/BIC for non-nested model comparison
print(f"Model 1 AIC: {model1.aic}, BIC: {model1.bic}")
print(f"Model 2 AIC: {model2.aic}, BIC: {model2.bic}")
# Lower AIC/BIC indicates better model

Robust Standard Errors

Handle heteroscedasticity or clustering without reweighting.

# Heteroscedasticity-robust (HC) standard errors
results_hc = results.get_robustcov_results(cov_type='HC0')  # White's
results_hc1 = results.get_robustcov_results(cov_type='HC1')
results_hc2 = results.get_robustcov_results(cov_type='HC2')
results_hc3 = results.get_robustcov_results(cov_type='HC3')  # Leverage-adjusted

# Newey-West HAC (Heteroscedasticity and Autocorrelation Consistent)
results_hac = results.get_robustcov_results(cov_type='HAC', maxlags=4)

# Cluster-robust standard errors
results_cluster = results.get_robustcov_results(cov_type='cluster',
                                                groups=cluster_ids)

# View robust results
print(results_hc3.summary())

Best Practices

1. Specify intercept once: Array models need it explicitly; formulas include it by default. Keep prediction columns in the training order.
2. Check assumptions: Run diagnostic tests (heteroscedasticity, autocorrelation, normality)
3. Use formula API for categorical variables: smf.ols() handles categorical variables automatically
4. Robust standard errors: Use when heteroscedasticity detected but model specification is correct
5. Model selection: Use AIC/BIC for non-nested models, F-test/likelihood ratio for nested models
6. Outliers and influence: Always check Cook's distance and leverage
7. Multicollinearity: Check VIF and condition number before interpretation
8. Time series: Use GLSAR or robust HAC standard errors for autocorrelated errors
9. Grouped data: Consider mixed effects models or cluster-robust standard errors
10. Quantile regression: Use for robust estimation or when interested in distributional effects

Common Pitfalls

1. Forgetting to add constant: Results in no-intercept model
2. Ignoring heteroscedasticity: Use WLS or robust standard errors
3. Using OLS with autocorrelated errors: Use GLSAR or HAC standard errors
4. Over-interpreting with multicollinearity: Check VIF first
5. Not checking residuals: Always plot residuals vs fitted values
6. Rank deficiency: Check design rank and singular values; a pseudoinverse fit can return coefficients for a non-identifiable design
7. Confusing prediction vs confidence intervals: Prediction intervals are wider
8. Not handling categorical variables properly: Use formula API or manual dummy coding
9. Comparing models with different sample sizes: Ensure same observations used
10. Ignoring influential observations: Check Cook's distance and DFFITS