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# Regression

## Introduction

Regression contains linear models with independently and identically distributed errors and for errors with heteroscedasticity or autocorrelation

The statistical model is assumed to be

> , where

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

- GLS : generalized least squares for arbitrary covariance
- OLS : ordinary least squares for i.i.d. errors
- WLS : weighted least squares for heteroskedastic errors
- GLSAR : feasible generalized least squares with autocorrelated AR(p) errors

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 spedific methods and attributes.

GLS is the superclass of the other regression classes.

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.

### 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
  where is given by
cholsimgainv : array
  n x n upper triangular matrix such that
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
sigma : array
  sigma is the n x n strucutre of the covariance matrix of the error terms
wexog : array
  wexog is the whitened design matrix.
wendog : array
  The whitened response variable.

### References

General reference for regression models:

4. 3. Montgomery and E.A. Peck. “Introduction to Linear Regression

   > Analysis.” 2nd. Ed., Wiley, 1992.

Econometrics references for regression models:

18. Davidson and J.G. MacKinnon. “Econometric Theory and Methods,” Oxford,

23. Green. “Econometric Analysis,” 5th ed., Pearson, 2003.

## Examples

see also the examples and the tests folders

## Module Reference

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

## Technical Documentation

- Technical Documentation

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