204 lines
6.8 KiB
Python
204 lines
6.8 KiB
Python
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"""
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Created on Tue Dec 20 20:24:20 2011
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Author: Josef Perktold
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License: BSD-3
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"""
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import numpy as np
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from statsmodels.regression.linear_model import OLS, GLS, WLS
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def atleast_2dcols(x):
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x = np.asarray(x)
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if x.ndim == 1:
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x = x[:,None]
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return x
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class GLSHet2(GLS):
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'''WLS with heteroscedasticity that depends on explanatory variables
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note: mixing GLS sigma and weights for heteroscedasticity might not make
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sense
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I think rewriting following the pattern of GLSAR is better
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stopping criteria: improve in GLSAR also, e.g. change in rho
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'''
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def __init__(self, endog, exog, exog_var, sigma=None):
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self.exog_var = atleast_2dcols(exog_var)
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super(self.__class__, self).__init__(endog, exog, sigma=sigma)
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def fit(self, lambd=1.):
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#maybe iterate
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#preliminary estimate
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res_gls = GLS(self.endog, self.exog, sigma=self.sigma).fit()
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res_resid = OLS(res_gls.resid**2, self.exog_var).fit()
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#or log-link
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#res_resid = OLS(np.log(res_gls.resid**2), self.exog_var).fit()
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#here I could use whiten and current instance instead of delegating
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#but this is easier
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#see pattern of GLSAR, calls self.initialize and self.fit
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res_wls = WLS(self.endog, self.exog, weights=1./res_resid.fittedvalues).fit()
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res_wls._results.results_residual_regression = res_resid
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return res_wls
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class GLSHet(WLS):
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"""
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A regression model with an estimated heteroscedasticity.
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A subclass of WLS, that additionally estimates the weight matrix as a
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function of additional explanatory variables.
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Parameters
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----------
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endog : array_like
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exog : array_like
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exog_var : array_like, 1d or 2d
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regressors, explanatory variables for the variance
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weights : array_like or None
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If weights are given, then they are used in the first step estimation.
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link : link function or None
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If None, then the variance is assumed to be a linear combination of
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the exog_var. If given, then ... not tested yet
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*extra attributes*
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history : dict
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contains the parameter estimates in both regression for each iteration
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result instance has
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results_residual_regression : OLS result instance
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result of heteroscedasticity estimation
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except for fit_iterative all methods are inherited from WLS.
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Notes
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-----
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GLSHet is considered to be experimental.
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`fit` is just standard WLS fit for fixed weights
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`fit_iterative` updates the estimate for weights, see its docstring
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The two alternative for handling heteroscedasticity in the data are to
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use heteroscedasticity robust standard errors or estimating the
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heteroscedasticity
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Estimating heteroscedasticity and using weighted least squares produces
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smaller confidence intervals for the estimated parameters then the
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heteroscedasticity robust standard errors if the heteroscedasticity is
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correctly specified. If the heteroscedasticity is incorrectly specified
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then the estimated covariance is inconsistent.
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Stock and Watson for example argue in favor of using OLS with
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heteroscedasticity robust standard errors instead of GLSHet sind we are
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seldom sure enough about the correct specification (in economics).
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GLSHet has asymptotically the same distribution as WLS if the true
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weights are know. In both cases the asymptotic distribution of the
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parameter estimates is the normal distribution.
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The assumption of the model:
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y = X*beta + u,
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with E(u) = 0, E(X*u)=0, var(u_i) = z_i*gamma
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or for vector of all observations Sigma = diag(Z*gamma)
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where
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y : endog (nobs)
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X : exog (nobs, k_vars)
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Z : exog_var (nobs, k_vars2)
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beta, gamma estimated parameters
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If a link is specified, then the heteroscedasticity is
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var(u_i) = link.inverse(z_i*gamma), or
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link(var(u_i)) = z_i*gamma
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for example for log-linkg
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var(u_i) = exp(z_i*gamma)
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Usage : see example ....
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TODO: test link option
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"""
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def __init__(self, endog, exog, exog_var=None, weights=None, link=None):
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self.exog_var = atleast_2dcols(exog_var)
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if weights is None:
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weights = np.ones(endog.shape)
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if link is not None:
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self.link = link
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self.linkinv = link.inverse #as defined in families.links
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else:
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self.link = lambda x: x #no transformation
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self.linkinv = lambda x: x
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super(self.__class__, self).__init__(endog, exog, weights=weights)
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def iterative_fit(self, maxiter=3):
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"""
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Perform an iterative two-step procedure to estimate a WLS model.
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The model is assumed to have heteroskedastic errors.
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The variance is estimated by OLS regression of the link transformed
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squared residuals on Z, i.e.::
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link(sigma_i) = x_i*gamma.
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Parameters
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----------
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maxiter : int, optional
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the number of iterations
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Notes
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-----
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maxiter=1: returns the estimated based on given weights
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maxiter=2: performs a second estimation with the updated weights,
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this is 2-step estimation
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maxiter>2: iteratively estimate and update the weights
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TODO: possible extension stop iteration if change in parameter
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estimates is smaller than x_tol
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Repeated calls to fit_iterative, will do one redundant pinv_wexog
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calculation. Calling fit_iterative(maxiter) ones does not do any
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redundant recalculations (whitening or calculating pinv_wexog).
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"""
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import collections
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self.history = collections.defaultdict(list) #not really necessary
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res_resid = None #if maxiter < 2 no updating
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for i in range(maxiter):
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#pinv_wexog is cached
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if hasattr(self, 'pinv_wexog'):
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del self.pinv_wexog
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#self.initialize()
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#print 'wls self',
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results = self.fit()
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self.history['self_params'].append(results.params)
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if not i == maxiter-1: #skip for last iteration, could break instead
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#print 'ols',
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self.results_old = results #for debugging
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#estimate heteroscedasticity
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res_resid = OLS(self.link(results.resid**2), self.exog_var).fit()
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self.history['ols_params'].append(res_resid.params)
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#update weights
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self.weights = 1./self.linkinv(res_resid.fittedvalues)
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self.weights /= self.weights.max() #not required
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self.weights[self.weights < 1e-14] = 1e-14 #clip
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#print 'in iter', i, self.weights.var() #debug, do weights change
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self.initialize()
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#note results is the wrapper, results._results is the results instance
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results._results.results_residual_regression = res_resid
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return results
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