Linear Regression ¶

# 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:                  GRADE   R-squared:                       0.416
Model:                            OLS   Adj. R-squared:                  0.353
Method:                 Least Squares   F-statistic:                     6.646
Date:                Fri, 05 May 2023   Prob (F-statistic):            0.00157
Time:                        13:59:54   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]
------------------------------------------------------------------------------
GPA            0.4639      0.162      2.864      0.008       0.132       0.796
TUCE           0.0105      0.019      0.539      0.594      -0.029       0.050
PSI            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.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Detailed examples can be found here:
Recursive LS
Rolling 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,
RollingWLS and RollingOLS.
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_wexogarray
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 \(\Psi\Psi^{T}=\Sigma^{-1}\).
cholsimgainvarray
The n x n upper triangular matrix \(\Psi^{T}\) that satisfies
\(\Psi\Psi^{T}=\Sigma^{-1}\).
df_modelfloat
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_residfloat
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.
llffloat
The value of the likelihood function of the fitted model.
nobsfloat
The number of observations n
normalized_cov_paramsarray
A p x p array equal to \((X^{T}\Sigma^{-1}X)^{-1}\).
sigmaarray
The n x n covariance matrix of the error terms:
\(\mu\sim N\left(0,\Sigma\right)\).
wexogarray
The whitened design matrix \(\Psi^{T}X\).
wendogarray
The whitened response variable \(\Psi^{T}Y\).
Module Reference¶
Model Classes¶
OLS(endog[, exog, missing, hasconst])
Ordinary Least Squares
GLS(endog, exog[, sigma, missing, hasconst])
Generalized Least Squares
WLS(endog, exog[, weights, missing, hasconst])
Weighted Least Squares
GLSAR(endog[, exog, rho, missing, hasconst])
Generalized Least Squares with AR covariance structure
yule_walker(x[, order, method, df, inv, demean])
Estimate AR(p) parameters from a sequence using the Yule-Walker equations.
burg(endog[, order, demean])
Compute Burg's AP(p) parameter estimator.
An implementation of ProcessCovariance using the Gaussian kernel.
ProcessMLE(endog, exog, exog_scale, ...[, cov])
Fit a Gaussian mean/variance regression model.
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, ...])
Results class for predictions.