import numpy as np

import statsmodels.api as sm

"""
Explanation: Generalized Least Squares
End of explanation
"""


data = sm.datasets.longley.load()
data.exog = sm.add_constant(data.exog)
print(data.exog.head())

"""
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(
    np.asarray(ols_resid)[1:], sm.add_constant(np.asarray(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 do not 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
"""


