Generalized Least Squares

import statsmodels.api as sm

The Longley dataset is a time series dataset:

data = sm.datasets.longley.load(as_pandas=False)
data.exog = sm.add_constant(data.exog)
print(data.exog[:5])

[[1.00000e+00 8.30000e+01 2.34289e+05 2.35600e+03 1.59000e+03 1.07608e+05
  1.94700e+03]
 [1.00000e+00 8.85000e+01 2.59426e+05 2.32500e+03 1.45600e+03 1.08632e+05
  1.94800e+03]
 [1.00000e+00 8.82000e+01 2.58054e+05 3.68200e+03 1.61600e+03 1.09773e+05
  1.94900e+03]
 [1.00000e+00 8.95000e+01 2.84599e+05 3.35100e+03 1.65000e+03 1.10929e+05
  1.95000e+03]
 [1.00000e+00 9.62000e+01 3.28975e+05 2.09900e+03 3.09900e+03 1.12075e+05
  1.95100e+03]]

Let’s assume that the data is heteroskedastic and that we know the nature of the heteroskedasticity. We can then define sigma and use it to give us a GLS model

First we will obtain the residuals from an OLS fit

ols_resid = sm.OLS(data.endog, data.exog).fit().resid

Assume that the error terms follow an AR(1) process with a trend:

\(\epsilon_i = \beta_0 + \rho\epsilon_{i-1} + \eta_i\)

where \(\eta \sim N(0,\Sigma^2)\)

and that \(\rho\) is simply the correlation of the residual a consistent estimator for rho is to regress the residuals on the lagged residuals

resid_fit = sm.OLS(ols_resid[1:], sm.add_constant(ols_resid[:-1])).fit()
print(resid_fit.tvalues[1])
print(resid_fit.pvalues[1])

-1.4390229839731974
0.17378444788825434

While we do not have strong evidence that the errors follow an AR(1) process we continue

rho = resid_fit.params[1]

As we know, an AR(1) process means that near-neighbors have a stronger relation so we can give this structure by using a toeplitz matrix

from scipy.linalg import toeplitz

toeplitz(range(5))

array([[0, 1, 2, 3, 4],
       [1, 0, 1, 2, 3],
       [2, 1, 0, 1, 2],
       [3, 2, 1, 0, 1],
       [4, 3, 2, 1, 0]])

order = toeplitz(range(len(ols_resid)))

so that our error covariance structure is actually rho**order which defines an autocorrelation structure

sigma = rho**order
gls_model = sm.GLS(data.endog, data.exog, sigma=sigma)
gls_results = gls_model.fit()

Of course, the exact rho in this instance is not known so it it might make more sense to use feasible gls, which currently only has experimental support.

We can use the GLSAR model with one lag, to get to a similar result:

glsar_model = sm.GLSAR(data.endog, data.exog, 1)
glsar_results = glsar_model.iterative_fit(1)
print(glsar_results.summary())

                           GLSAR Regression Results
==============================================================================
Dep. Variable:                      y   R-squared:                       0.996
Model:                          GLSAR   Adj. R-squared:                  0.992
Method:                 Least Squares   F-statistic:                     295.2
Date:                Tue, 02 Feb 2021   Prob (F-statistic):           6.09e-09
Time:                        07:02:21   Log-Likelihood:                -102.04
No. Observations:                  15   AIC:                             218.1
Df Residuals:                       8   BIC:                             223.0
Df Model:                           6
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const      -3.468e+06   8.72e+05     -3.979      0.004   -5.48e+06   -1.46e+06
x1            34.5568     84.734      0.408      0.694    -160.840     229.953
x2            -0.0343      0.033     -1.047      0.326      -0.110       0.041
x3            -1.9621      0.481     -4.083      0.004      -3.070      -0.854
x4            -1.0020      0.211     -4.740      0.001      -1.489      -0.515
x5            -0.0978      0.225     -0.435      0.675      -0.616       0.421
x6          1823.1829    445.829      4.089      0.003     795.100    2851.266
==============================================================================
Omnibus:                        1.960   Durbin-Watson:                   2.554
Prob(Omnibus):                  0.375   Jarque-Bera (JB):                1.423
Skew:                           0.713   Prob(JB):                        0.491
Kurtosis:                       2.508   Cond. No.                     4.80e+09
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.8e+09. This might indicate that there are
strong multicollinearity or other numerical problems.

/home/travis/miniconda/envs/statsmodels-test/lib/python3.7/site-packages/scipy/stats/stats.py:1604: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=15
  "anyway, n=%i" % int(n))

Comparing gls and glsar results, we see that there are some small differences in the parameter estimates and the resulting standard errors of the parameter estimate. This might be do to the numerical differences in the algorithm, e.g. the treatment of initial conditions, because of the small number of observations in the longley dataset.

print(gls_results.params)
print(glsar_results.params)
print(gls_results.bse)
print(glsar_results.bse)

[-3.79785490e+06 -1.27656454e+01 -3.80013250e-02 -2.18694871e+00
 -1.15177649e+00 -6.80535580e-02  1.99395293e+03]
[-3.46796063e+06  3.45567846e+01 -3.43410090e-02 -1.96214395e+00
 -1.00197296e+00 -9.78045986e-02  1.82318289e+03]
[6.70688699e+05 6.94308073e+01 2.62476822e-02 3.82393151e-01
 1.65252692e-01 1.76428334e-01 3.42634628e+02]
[8.71584052e+05 8.47337145e+01 3.28032450e-02 4.80544865e-01
 2.11383871e-01 2.24774369e-01 4.45828748e+02]