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

.. ipython:: python

# Load modules and data
import numpy as np
import statsmodels.api as sm
spector_data = sm.datasets.spector.load()
spector_data.exog = sm.add_constant(spector_data.exog, prepend=False)

# Fit and summarize OLS model
mod = sm.OLS(spector_data.endog, spector_data.exog)
res = mod.fit()
print(res.summary())

Detailed examples can be found here:

OLS <examples/notebooks/generated/ols.ipynb>_
WLS <examples/notebooks/generated/wls.ipynb>_
GLS <examples/notebooks/generated/gls.ipynb>_
Recursive LS <examples/notebooks/generated/recursive_ls.ipynb>_
Rolling LS <examples/notebooks/generated/rolling_ls.ipynb>_

Technical Documentation

The statistical model is assumed to be

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

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

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

.. Class hierachy: TODO

.. yule_walker is not a full model class, but a function that estimate the .. parameters of a univariate autoregressive process, AR(p). It is used in GLSAR, .. but it can also be used independently of any models. yule_walker only .. calculates the estimates and the standard deviation of the lag parameters but .. not the additional regression statistics. We hope to include yule-walker in .. future in a separate univariate time series class. A similar result can be .. obtained with GLSAR if only the constant is included as regressors. In this .. case the parameter estimates of the lag estimates are not reported, however .. additional statistics, for example aic, become available.

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.

.. toctree:: .. :maxdepth: 1 .. .. regression_techn1

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 :math:\left(X^{T}\Sigma^{-1}X\right)^{-1}X^{T}\Psi, where :math:\Psi is defined such that :math:\Psi\Psi^{T}=\Sigma^{-1}. cholsimgainv : array The n x n upper triangular matrix :math:\Psi^{T} that satisfies :math:\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 :math:(X^{T}\Sigma^{-1}X)^{-1}. sigma : array The n x n covariance matrix of the error terms: :math:\epsilon\sim N\left(0,\Sigma\right). wexog : array The whitened design matrix :math:\Psi^{T}X. wendog : array The whitened response variable :math:\Psi^{T}Y.