Skip to content

Latest commit

 

History

History
174 lines (129 loc) · 5.13 KB

regression.rst

File metadata and controls

174 lines (129 loc) · 5.13 KB
Raw

statsmodels.regression.linear_model

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
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:

examples/notebooks/generated/ols examples/notebooks/generated/wls examples/notebooks/generated/gls

Technical Documentation

The statistical model is assumed to be

Y = Xβ + μ, where μ ∼ N(0,σ2Σ)

depending on the assumption on Σ, we have currently four classes available

  • GLS : generalized least squares for arbitrary covariance Σ
  • OLS : ordinary least squares for i.i.d. errors Σ = I
  • WLS : weighted least squares for heteroskedastic errors diag(Σ)
  • GLSAR : feasible generalized least squares with autocorrelated AR(p) errors Σ = Σ(ρ)

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.

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
pinv_wexog is the p x n Moore-Penrose pseudoinverse of the
whitened design matrix. Approximately equal to
(XTΣ − 1X) − 1XTΨ
where Ψ is given by ΨΨT = Σ − 1
cholsimgainv : array
n x n upper triangular matrix such that
ΨΨT = Σ − 1
cholsigmainv = ΨT
df_model : float

The model degrees of freedom 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 is equal to the number of observations n less the number of parameters p. 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
(XTΣ − 1X) − 1
sigma : array
sigma is the n x n strucutre of the covariance matrix of the error terms
μ ∼ N(0,σ2Σ)
wexog : array
wexog is the whitened design matrix.
ΨTX
wendog : array
The whitened response variable.
ΨTY

Module Reference

Model Classes

OLS GLS WLS GLSAR yule_walker

statsmodels.regression.quantile_regression

QuantReg

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.

statsmodels.regression.linear_model

RegressionResults OLSResults

statsmodels.regression.quantile_regression

QuantRegResults