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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Generalized Least Squares
Step1:  The Longley dataset is a time series dataset
Step2:  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
Step3:  Assume that the error terms follow an AR(1) process with a trend
Step4:  While we don't have strong evidence that the errors follow an AR(1)
 process we continue
Step5:  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
Step6:  so that our error covariance structure is actually rho**order
 which defines an autocorrelation structure
Step7:  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
Step8:  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.
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Python Code:
from __future__ import print_function
import statsmodels.api as sm
import numpy as np
from statsmodels.iolib.table import (SimpleTable, default_txt_fmt)
Explanation: Generalized Least Squares
End of explanation
data = sm.datasets.longley.load()
data.exog = sm.add_constant(data.exog)
print(data.exog[:5])
Explanation: The Longley dataset is a time series dataset:
End of explanation
ols_resid = sm.OLS(data.endog, data.exog).fit().resid
Explanation: 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
End of explanation
resid_fit = sm.OLS(ols_resid[1:], sm.add_constant(ols_resid[:-1])).fit()
print(resid_fit.tvalues[1])
print(resid_fit.pvalues[1])
Explanation: 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
End of explanation
rho = resid_fit.params[1]
Explanation: While we don't have strong evidence that the errors follow an AR(1)
 process we continue
End of explanation
from scipy.linalg import toeplitz
toeplitz(range(5))
order = toeplitz(range(len(ols_resid)))
Explanation: 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
End of explanation
sigma = rho**order
gls_model = sm.GLS(data.endog, data.exog, sigma=sigma)
gls_results = gls_model.fit()
Explanation: so that our error covariance structure is actually rho**order
 which defines an autocorrelation structure
End of explanation
glsar_model = sm.GLSAR(data.endog, data.exog, 1)
glsar_results = glsar_model.iterative_fit(1)
print(glsar_results.summary())
Explanation: 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:
End of explanation
print(gls_results.params)
print(glsar_results.params)
print(gls_results.bse)
print(glsar_results.bse)
Explanation: 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.
End of explanation
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