Linear Regression

Linear models with independently and identically distributed errors, and for errors with heteroscedasticity or autocorrelation. This module allows estimation by ordinary least squares (OLS), weighted least squares (WLS), generalized least squares (GLS), and feasible generalized least squares with autocorrelated AR(p) errors.

See Module Reference for commands and arguments.

Examples

# Load modules and data
In [1]: import numpy as np

In [2]: import statsmodels.api as sm

In [3]: spector_data = sm.datasets.spector.load()

In [4]: spector_data.exog = sm.add_constant(spector_data.exog, prepend=False)

# Fit and summarize OLS model
In [5]: mod = sm.OLS(spector_data.endog, spector_data.exog)

In [6]: res = mod.fit()

In [7]: print(res.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.416
Model:                            OLS   Adj. R-squared:                  0.353
Method:                 Least Squares   F-statistic:                     6.646
Date:                Mon, 14 May 2018   Prob (F-statistic):            0.00157
Time:                        21:48:12   Log-Likelihood:                -12.978
No. Observations:                  32   AIC:                             33.96
Df Residuals:                      28   BIC:                             39.82
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
x1             0.4639      0.162      2.864      0.008       0.132       0.796
x2             0.0105      0.019      0.539      0.594      -0.029       0.050
x3             0.3786      0.139      2.720      0.011       0.093       0.664
const         -1.4980      0.524     -2.859      0.008      -2.571      -0.425
==============================================================================
Omnibus:                        0.176   Durbin-Watson:                   2.346
Prob(Omnibus):                  0.916   Jarque-Bera (JB):                0.167
Skew:                           0.141   Prob(JB):                        0.920
Kurtosis:                       2.786   Cond. No.                         176.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Detailed examples can be found here:

• OLS
• WLS
• GLS
• Recursive LS

Technical Documentation

The statistical model is assumed to be

\(Y = X\beta + \mu\), where \(\mu\sim N\left(0,\Sigma\right).\)

Depending on the properties of \(\Sigma\), we have currently four classes available:

• GLS : generalized least squares for arbitrary covariance \(\Sigma\)
• OLS : ordinary least squares for i.i.d. errors \(\Sigma=\textbf{I}\)
• WLS : weighted least squares for heteroskedastic errors \(\text{diag}\left (\Sigma\right)\)
• GLSAR : feasible generalized least squares with autocorrelated AR(p) errors \(\Sigma=\Sigma\left(\rho\right)\)

All regression models define the same methods and follow the same structure, and can be used in a similar fashion. Some of them contain additional model specific methods and attributes.

GLS is the superclass of the other regression classes except for RecursiveLS.

References

General reference for regression models:

• D.C. Montgomery and E.A. Peck. “Introduction to Linear Regression Analysis.” 2nd. Ed., Wiley, 1992.

Econometrics references for regression models:

• R.Davidson and J.G. MacKinnon. “Econometric Theory and Methods,” Oxford, 2004.
• W.Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

Attributes

The following is more verbose description of the attributes which is mostly common to all regression classes

pinv_wexog : array

The p x n Moore-Penrose pseudoinverse of the whitened design matrix. It is approximately equal to \(\left(X^{T}\Sigma^{-1}X\right)^{-1}X^{T}\Psi\), where \(\Psi\) is defined such that \(\Psi\Psi^{T}=\Sigma^{-1}\).

cholsimgainv : array

The n x n upper triangular matrix \(\Psi^{T}\) that satisfies \(\Psi\Psi^{T}=\Sigma^{-1}\).

df_model : float

The model degrees of freedom. This is equal to p - 1, where p is the number of regressors. Note that the intercept is not counted as using a degree of freedom here.

df_resid : float

The residual degrees of freedom. This is equal n - p where n is the number of observations and p is the number of parameters. Note that the intercept is counted as using a degree of freedom here.

llf : float

The value of the likelihood function of the fitted model.

nobs : float

The number of observations n

normalized_cov_params : array

A p x p array equal to \((X^{T}\Sigma^{-1}X)^{-1}\).

sigma : array

The n x n covariance matrix of the error terms: \(\mu\sim N\left(0,\Sigma\right)\).

wexog : array

The whitened design matrix \(\Psi^{T}X\).

wendog : array

The whitened response variable \(\Psi^{T}Y\).

Module Reference

Model Classes

OLS(endog[, exog, missing, hasconst])	A simple ordinary least squares model.
GLS(endog, exog[, sigma, missing, hasconst])	Generalized least squares model with a general covariance structure.
WLS(endog, exog[, weights, missing, hasconst])	A regression model with diagonal but non-identity covariance structure.
GLSAR(endog[, exog, rho, missing])	A regression model with an AR(p) covariance structure.
yule_walker(X[, order, method, df, inv, demean])	Estimate AR(p) parameters from a sequence X using Yule-Walker equation.

QuantReg(endog, exog, **kwargs)

Quantile Regression

RecursiveLS(endog, exog, **kwargs)

Recursive least squares

Results Classes

Fitting a linear regression model returns a results class. OLS has a specific results class with some additional methods compared to the results class of the other linear models.

RegressionResults(model, params[, …])	This class summarizes the fit of a linear regression model.
OLSResults(model, params[, …])	Results class for for an OLS model.
PredictionResults(predicted_mean, …[, df, …])

QuantRegResults(model, params[, …])

Results instance for the QuantReg model

RecursiveLSResults(model, params, filter_results)

Class to hold results from fitting a recursive least squares model.