'''
Spatial Error with Heteroskedasticity family of models
'''
__author__ = "Luc Anselin luc.anselin@asu.edu, \
Pedro V. Amaral pedro.amaral@asu.edu, \
Daniel Arribas-Bel darribas@asu.edu, \
David C. Folch david.folch@asu.edu \
Ran Wei rwei5@asu.edu"
import numpy as np
import numpy.linalg as la
from . import ols as OLS
from . import user_output as USER
from . import summary_output as SUMMARY
from . import twosls as TSLS
from . import utils as UTILS
from .utils import RegressionPropsY, spdot, set_endog, sphstack
from scipy import sparse as SP
from libpysal.weights.spatial_lag import lag_spatial
__all__ = ["GM_Error_Het", "GM_Endog_Error_Het", "GM_Combo_Het"]
class BaseGM_Error_Het(RegressionPropsY):
"""
GMM method for a spatial error model with heteroskedasticity (note: no
consistency checks, diagnostics or constant added); based on
:cite:`Arraiz2010`, following :cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
w : Sparse matrix
Spatial weights sparse matrix
max_iter : int
Maximum number of iterations of steps 2a and 2b from
:cite:`Arraiz2010`. Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
Attributes
----------
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
xtx : float
X'X
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X.append(db.by_col("CRIME"))
>>> X = np.array(X).T
>>> X = np.hstack((np.ones(y.shape),X))
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
>>> w.transform = 'r'
>>> reg = BaseGM_Error_Het(y, X, w.sparse, step1c=True)
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 47.9963 11.479 ]
[ 0.7105 0.3681]
[ -0.5588 0.1616]
[ 0.4118 0.168 ]]
"""
def __init__(self, y, x, w,
max_iter=1, epsilon=0.00001, step1c=False):
self.step1c = step1c
# 1a. OLS --> \tilde{betas}
ols = OLS.BaseOLS(y=y, x=x)
self.x, self.y, self.n, self.k, self.xtx = ols.x, ols.y, ols.n, ols.k, ols.xtx
wA1 = UTILS.get_A1_het(w)
# 1b. GMM --> \tilde{\lambda1}
moments = UTILS._moments2eqs(wA1, w, ols.u)
lambda1 = UTILS.optim_moments(moments)
if step1c:
# 1c. GMM --> \tilde{\lambda2}
sigma = get_psi_sigma(w, ols.u, lambda1)
vc1 = get_vc_het(w, wA1, sigma)
lambda2 = UTILS.optim_moments(moments, vc1)
else:
lambda2 = lambda1
lambda_old = lambda2
self.iteration, eps = 0, 1
while self.iteration < max_iter and eps > epsilon:
# 2a. reg -->\hat{betas}
xs = UTILS.get_spFilter(w, lambda_old, self.x)
ys = UTILS.get_spFilter(w, lambda_old, self.y)
ols_s = OLS.BaseOLS(y=ys, x=xs)
self.predy = spdot(self.x, ols_s.betas)
self.u = self.y - self.predy
# 2b. GMM --> \hat{\lambda}
sigma_i = get_psi_sigma(w, self.u, lambda_old)
vc_i = get_vc_het(w, wA1, sigma_i)
moments_i = UTILS._moments2eqs(wA1, w, self.u)
lambda3 = UTILS.optim_moments(moments_i, vc_i)
eps = abs(lambda3 - lambda_old)
lambda_old = lambda3
self.iteration += 1
self.iter_stop = UTILS.iter_msg(self.iteration, max_iter)
sigma = get_psi_sigma(w, self.u, lambda3)
vc3 = get_vc_het(w, wA1, sigma)
self.vm = get_vm_het(moments_i[0], lambda3, self, w, vc3)
self.betas = np.vstack((ols_s.betas, lambda3))
self.e_filtered = self.u - lambda3 * w * self.u
self._cache = {}
[docs]class GM_Error_Het(BaseGM_Error_Het):
"""
GMM method for a spatial error model with heteroskedasticity, with results
and diagnostics; based on :cite:`Arraiz2010`, following
:cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
w : pysal W object
Spatial weights object
max_iter : int
Maximum number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
vm : boolean
If True, include variance-covariance matrix in summary
results
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
Attributes
----------
summary : string
Summary of regression results and diagnostics (note: use in
conjunction with the print command)
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
vm : array
Variance covariance matrix (kxk)
std_err : array
1xk array of standard errors of the betas
z_stat : list of tuples
z statistic; each tuple contains the pair (statistic,
p-value), where each is a float
xtx : float
:math:`X'X`
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
title : string
Name of the regression method used
Examples
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import numpy as np
>>> import libpysal
Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open().
This is the DBF associated with the Columbus shapefile. Note that
libpysal.io.open() also reads data in CSV format; since the actual class
requires data to be passed in as numpy arrays, the user can read their
data in using any method.
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
Extract the HOVAL column (home values) from the DBF file and make it the
dependent variable for the regression. Note that PySAL requires this to be
an numpy array of shape (n, 1) as opposed to the also common shape of (n, )
that other packages accept.
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
Extract INC (income) and CRIME (crime) vectors from the DBF to be used as
independent variables in the regression. Note that PySAL requires this to
be an nxj numpy array, where j is the number of independent variables (not
including a constant). By default this class adds a vector of ones to the
independent variables passed in.
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X.append(db.by_col("CRIME"))
>>> X = np.array(X).T
Since we want to run a spatial error model, we need to specify the spatial
weights matrix that includes the spatial configuration of the observations
into the error component of the model. To do that, we can open an already
existing gal file or create a new one. In this case, we will create one
from ``columbus.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
Unless there is a good reason not to do it, the weights have to be
row-standardized so every row of the matrix sums to one. Among other
things, his allows to interpret the spatial lag of a variable as the
average value of the neighboring observations. In PySAL, this can be
easily performed in the following way:
>>> w.transform = 'r'
We are all set with the preliminaries, we are good to run the model. In this
case, we will need the variables and the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> reg = GM_Error_Het(y, X, w=w, step1c=True, name_y='home value', name_x=['income', 'crime'], name_ds='columbus')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. This class offers an error model that explicitly accounts
for heteroskedasticity and that unlike the models from
``spreg.error_sp``, it allows for inference on the spatial
parameter.
>>> print reg.name_x
['CONSTANT', 'income', 'crime', 'lambda']
Hence, we find the same number of betas as of standard errors,
which we calculate taking the square root of the diagonal of the
variance-covariance matrix:
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 47.9963 11.479 ]
[ 0.7105 0.3681]
[ -0.5588 0.1616]
[ 0.4118 0.168 ]]
"""
[docs] def __init__(self, y, x, w,
max_iter=1, epsilon=0.00001, step1c=False,
vm=False, name_y=None, name_x=None,
name_w=None, name_ds=None):
n = USER.check_arrays(y, x)
USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
x_constant = USER.check_constant(x)
BaseGM_Error_Het.__init__(
self, y, x_constant, w.sparse, max_iter=max_iter,
step1c=step1c, epsilon=epsilon)
self.title = "SPATIALLY WEIGHTED LEAST SQUARES (HET)"
self.name_ds = USER.set_name_ds(name_ds)
self.name_y = USER.set_name_y(name_y)
self.name_x = USER.set_name_x(name_x, x)
self.name_x.append('lambda')
self.name_w = USER.set_name_w(name_w, w)
SUMMARY.GM_Error_Het(reg=self, w=w, vm=vm)
class BaseGM_Endog_Error_Het(RegressionPropsY):
"""
GMM method for a spatial error model with heteroskedasticity and
endogenous variables (note: no consistency checks, diagnostics or constant
added); based on :cite:`Arraiz2010`, following :cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
w : Sparse matrix
Spatial weights sparse matrix
max_iter : int
Maximum number of iterations of steps 2a and 2b from
:cite:`Arraiz2010`. Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
inv_method : string
If "power_exp", then compute inverse using the power
expansion. If "true_inv", then compute the true inverse.
Note that true_inv will fail for large n.
Attributes
----------
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable used as instruments
z : array
nxk array of variables (combination of x and yend)
h : array
nxl array of instruments (combination of x and q)
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
hth : float
:math:`H'H`
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
>>> X = np.hstack((np.ones(y.shape),X))
>>> yd = []
>>> yd.append(db.by_col("CRIME"))
>>> yd = np.array(yd).T
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
>>> w.transform = 'r'
>>> reg = BaseGM_Endog_Error_Het(y, X, yd, q, w=w.sparse, step1c=True)
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 55.3971 28.8901]
[ 0.4656 0.7731]
[ -0.6704 0.468 ]
[ 0.4114 0.1777]]
"""
def __init__(self, y, x, yend, q, w,
max_iter=1, epsilon=0.00001,
step1c=False, inv_method='power_exp'):
self.step1c = step1c
# 1a. reg --> \tilde{betas}
tsls = TSLS.BaseTSLS(y=y, x=x, yend=yend, q=q)
self.x, self.z, self.h, self.y = tsls.x, tsls.z, tsls.h, tsls.y
self.yend, self.q, self.n, self.k, self.hth = tsls.yend, tsls.q, tsls.n, tsls.k, tsls.hth
wA1 = UTILS.get_A1_het(w)
# 1b. GMM --> \tilde{\lambda1}
moments = UTILS._moments2eqs(wA1, w, tsls.u)
lambda1 = UTILS.optim_moments(moments)
if step1c:
# 1c. GMM --> \tilde{\lambda2}
self.u = tsls.u
zs = UTILS.get_spFilter(w, lambda1, self.z)
vc1 = get_vc_het_tsls(w, wA1, self, lambda1,
tsls.pfora1a2, zs, inv_method, filt=False)
lambda2 = UTILS.optim_moments(moments, vc1)
else:
lambda2 = lambda1
lambda_old = lambda2
self.iteration, eps = 0, 1
while self.iteration < max_iter and eps > epsilon:
# 2a. reg -->\hat{betas}
xs = UTILS.get_spFilter(w, lambda_old, self.x)
ys = UTILS.get_spFilter(w, lambda_old, self.y)
yend_s = UTILS.get_spFilter(w, lambda_old, self.yend)
tsls_s = TSLS.BaseTSLS(ys, xs, yend_s, h=self.h)
self.predy = spdot(self.z, tsls_s.betas)
self.u = self.y - self.predy
# 2b. GMM --> \hat{\lambda}
vc2 = get_vc_het_tsls(w, wA1, self, lambda_old,
tsls_s.pfora1a2, sphstack(xs, yend_s), inv_method)
moments_i = UTILS._moments2eqs(wA1, w, self.u)
lambda3 = UTILS.optim_moments(moments_i, vc2)
eps = abs(lambda3 - lambda_old)
lambda_old = lambda3
self.iteration += 1
self.iter_stop = UTILS.iter_msg(self.iteration, max_iter)
zs = UTILS.get_spFilter(w, lambda3, self.z)
P = get_P_hat(self, tsls.hthi, zs)
vc3 = get_vc_het_tsls(w, wA1, self, lambda3, P,
zs, inv_method, save_a1a2=True)
self.vm = get_Omega_GS2SLS(w, lambda3, self, moments_i[0], vc3, P)
self.betas = np.vstack((tsls_s.betas, lambda3))
self.e_filtered = self.u - lambda3 * w * self.u
self._cache = {}
[docs]class GM_Endog_Error_Het(BaseGM_Endog_Error_Het):
"""
GMM method for a spatial error model with heteroskedasticity and
endogenous variables, with results and diagnostics; based on
:cite:`Arraiz2010`, following :cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
w : pysal W object
Spatial weights object
max_iter : int
Maximum number of iterations of steps 2a and 2b from
:cite:`Arraiz2010`. Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
inv_method : string
If "power_exp", then compute inverse using the power
expansion. If "true_inv", then compute the true inverse.
Note that true_inv will fail for large n.
vm : boolean
If True, include variance-covariance matrix in summary
results
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_yend : list of strings
Names of endogenous variables for use in output
name_q : list of strings
Names of instruments for use in output
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
Attributes
----------
summary : string
Summary of regression results and diagnostics (note: use in
conjunction with the print command)
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable used as instruments
z : array
nxk array of variables (combination of x and yend)
h : array
nxl array of instruments (combination of x and q)
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
std_err : array
1xk array of standard errors of the betas
z_stat : list of tuples
z statistic; each tuple contains the pair (statistic,
p-value), where each is a float
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_yend : list of strings
Names of endogenous variables for use in output
name_z : list of strings
Names of exogenous and endogenous variables for use in
output
name_q : list of strings
Names of external instruments
name_h : list of strings
Names of all instruments used in ouput
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
title : string
Name of the regression method used
hth : float
:math:`H'H`
Examples
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import numpy as np
>>> import libpysal
Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open().
This is the DBF associated with the Columbus shapefile. Note that
libpysal.io.open() also reads data in CSV format; since the actual class
requires data to be passed in as numpy arrays, the user can read their
data in using any method.
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
Extract the HOVAL column (home values) from the DBF file and make it the
dependent variable for the regression. Note that PySAL requires this to be
an numpy array of shape (n, 1) as opposed to the also common shape of (n, )
that other packages accept.
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
Extract INC (income) vector from the DBF to be used as
independent variables in the regression. Note that PySAL requires this to
be an nxj numpy array, where j is the number of independent variables (not
including a constant). By default this class adds a vector of ones to the
independent variables passed in.
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
In this case we consider CRIME (crime rates) is an endogenous regressor.
We tell the model that this is so by passing it in a different parameter
from the exogenous variables (x).
>>> yd = []
>>> yd.append(db.by_col("CRIME"))
>>> yd = np.array(yd).T
Because we have endogenous variables, to obtain a correct estimate of the
model, we need to instrument for CRIME. We use DISCBD (distance to the
CBD) for this and hence put it in the instruments parameter, 'q'.
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
Since we want to run a spatial error model, we need to specify the spatial
weights matrix that includes the spatial configuration of the observations
into the error component of the model. To do that, we can open an already
existing gal file or create a new one. In this case, we will create one
from ``columbus.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
Unless there is a good reason not to do it, the weights have to be
row-standardized so every row of the matrix sums to one. Among other
things, his allows to interpret the spatial lag of a variable as the
average value of the neighboring observations. In PySAL, this can be
easily performed in the following way:
>>> w.transform = 'r'
We are all set with the preliminaries, we are good to run the model. In this
case, we will need the variables (exogenous and endogenous), the
instruments and the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> reg = GM_Endog_Error_Het(y, X, yd, q, w=w, step1c=True, name_x=['inc'], name_y='hoval', name_yend=['crime'], name_q=['discbd'], name_ds='columbus')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. This class offers an error model that explicitly accounts
for heteroskedasticity and that unlike the models from
``spreg.error_sp``, it allows for inference on the spatial
parameter. Hence, we find the same number of betas as of standard errors,
which we calculate taking the square root of the diagonal of the
variance-covariance matrix:
>>> print reg.name_z
['CONSTANT', 'inc', 'crime', 'lambda']
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 55.3971 28.8901]
[ 0.4656 0.7731]
[ -0.6704 0.468 ]
[ 0.4114 0.1777]]
"""
[docs] def __init__(self, y, x, yend, q, w,
max_iter=1, epsilon=0.00001,
step1c=False, inv_method='power_exp',
vm=False, name_y=None, name_x=None,
name_yend=None, name_q=None,
name_w=None, name_ds=None):
n = USER.check_arrays(y, x, yend, q)
USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
x_constant = USER.check_constant(x)
BaseGM_Endog_Error_Het.__init__(self, y=y, x=x_constant, yend=yend,
q=q, w=w.sparse, max_iter=max_iter,
step1c=step1c, epsilon=epsilon, inv_method=inv_method)
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES (HET)"
self.name_ds = USER.set_name_ds(name_ds)
self.name_y = USER.set_name_y(name_y)
self.name_x = USER.set_name_x(name_x, x)
self.name_yend = USER.set_name_yend(name_yend, yend)
self.name_z = self.name_x + self.name_yend
self.name_z.append('lambda') # listing lambda last
self.name_q = USER.set_name_q(name_q, q)
self.name_h = USER.set_name_h(self.name_x, self.name_q)
self.name_w = USER.set_name_w(name_w, w)
SUMMARY.GM_Endog_Error_Het(reg=self, w=w, vm=vm)
class BaseGM_Combo_Het(BaseGM_Endog_Error_Het):
"""
GMM method for a spatial lag and error model with heteroskedasticity and
endogenous variables (note: no consistency checks, diagnostics or constant
added); based on :cite:`Arraiz2010`, following :cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
w : Sparse matrix
Spatial weights sparse matrix
w_lags : integer
Orders of W to include as instruments for the spatially
lagged dependent variable. For example, w_lags=1, then
instruments are WX; if w_lags=2, then WX, WWX; and so on.
lag_q : boolean
If True, then include spatial lags of the additional
instruments (q).
max_iter : int
Maximum number of iterations of steps 2a and 2b from
:cite:`Arraiz2010`. Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
inv_method : string
If "power_exp", then compute inverse using the power
expansion. If "true_inv", then compute the true inverse.
Note that true_inv will fail for large n.
Attributes
----------
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
predy : array
nx1 array of predicted y values
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable used as instruments
z : array
nxk array of variables (combination of x and yend)
h : array
nxl array of instruments (combination of x and q)
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
hth : float
:math:`H'H`
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
>>> w.transform = 'r'
>>> w_lags = 1
>>> yd2, q2 = spreg.utils.set_endog(y, X, w, None, None, w_lags, True)
>>> X = np.hstack((np.ones(y.shape),X))
Example only with spatial lag
>>> reg = BaseGM_Combo_Het(y, X, yend=yd2, q=q2, w=w.sparse, step1c=True)
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 9.9753 14.1435]
[ 1.5742 0.374 ]
[ 0.1535 0.3978]
[ 0.2103 0.3924]]
Example with both spatial lag and other endogenous variables
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
>>> yd = []
>>> yd.append(db.by_col("CRIME"))
>>> yd = np.array(yd).T
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
>>> yd2, q2 = spreg.utils.set_endog(y, X, w, yd, q, w_lags, True)
>>> X = np.hstack((np.ones(y.shape),X))
>>> reg = BaseGM_Combo_Het(y, X, yd2, q2, w=w.sparse, step1c=True)
>>> betas = np.array([['CONSTANT'],['inc'],['crime'],['lag_hoval'],['lambda']])
>>> print np.hstack((betas, np.around(np.hstack((reg.betas, np.sqrt(reg.vm.diagonal()).reshape(5,1))),5)))
[['CONSTANT' '113.91292' '64.38815']
['inc' '-0.34822' '1.18219']
['crime' '-1.35656' '0.72482']
['lag_hoval' '-0.57657' '0.75856']
['lambda' '0.65608' '0.15719']]
"""
def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True,
max_iter=1, epsilon=0.00001,
step1c=False, inv_method='power_exp'):
BaseGM_Endog_Error_Het.__init__(
self, y=y, x=x, w=w, yend=yend, q=q, max_iter=max_iter,
step1c=step1c, epsilon=epsilon, inv_method=inv_method)
[docs]class GM_Combo_Het(BaseGM_Combo_Het):
"""
GMM method for a spatial lag and error model with heteroskedasticity and
endogenous variables, with results and diagnostics; based on
:cite:`Arraiz2010`, following :cite:`Anselin2011`.
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, excluding the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable to use as instruments (note:
this should not contain any variables from x)
w : pysal W object
Spatial weights object (always needed)
w_lags : integer
Orders of W to include as instruments for the spatially
lagged dependent variable. For example, w_lags=1, then
instruments are WX; if w_lags=2, then WX, WWX; and so on.
lag_q : boolean
If True, then include spatial lags of the additional
instruments (q).
max_iter : int
Maximum number of iterations of steps 2a and 2b from
:cite:`Arraiz2010`. Note: epsilon provides an additional
stop condition.
epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from :cite:`Arraiz2010`. Note: max_iter provides
an additional stop condition.
step1c : boolean
If True, then include Step 1c from :cite:`Arraiz2010`.
inv_method : string
If "power_exp", then compute inverse using the power
expansion. If "true_inv", then compute the true inverse.
Note that true_inv will fail for large n.
vm : boolean
If True, include variance-covariance matrix in summary
results
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_yend : list of strings
Names of endogenous variables for use in output
name_q : list of strings
Names of instruments for use in output
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
Attributes
----------
summary : string
Summary of regression results and diagnostics (note: use in
conjunction with the print command)
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of residuals
e_filtered : array
nx1 array of spatially filtered residuals
e_pred : array
nx1 array of residuals (using reduced form)
predy : array
nx1 array of predicted y values
predy_e : array
nx1 array of predicted y values (using reduced form)
n : integer
Number of observations
k : integer
Number of variables for which coefficients are estimated
(including the constant)
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable, including the constant
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
q : array
Two dimensional array with n rows and one column for each
external exogenous variable used as instruments
z : array
nxk array of variables (combination of x and yend)
h : array
nxl array of instruments (combination of x and q)
iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from :cite:`Arraiz2010`.
iteration : integer
Number of iterations of steps 2a and 2b from :cite:`Arraiz2010`.
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
pr2 : float
Pseudo R squared (squared correlation between y and ypred)
pr2_e : float
Pseudo R squared (squared correlation between y and ypred_e
(using reduced form))
std_err : array
1xk array of standard errors of the betas
z_stat : list of tuples
z statistic; each tuple contains the pair (statistic,
p-value), where each is a float
name_y : string
Name of dependent variable for use in output
name_x : list of strings
Names of independent variables for use in output
name_yend : list of strings
Names of endogenous variables for use in output
name_z : list of strings
Names of exogenous and endogenous variables for use in
output
name_q : list of strings
Names of external instruments
name_h : list of strings
Names of all instruments used in ouput
name_w : string
Name of weights matrix for use in output
name_ds : string
Name of dataset for use in output
title : string
Name of the regression method used
hth : float
:math:`H'H`
Examples
--------
We first need to import the needed modules, namely numpy to convert the
data we read into arrays that ``spreg`` understands and ``pysal`` to
perform all the analysis.
>>> import numpy as np
>>> import libpysal
Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open().
This is the DBF associated with the Columbus shapefile. Note that
libpysal.io.open() also reads data in CSV format; since the actual class
requires data to be passed in as numpy arrays, the user can read their
data in using any method.
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
Extract the HOVAL column (home values) from the DBF file and make it the
dependent variable for the regression. Note that PySAL requires this to be
an numpy array of shape (n, 1) as opposed to the also common shape of (n, )
that other packages accept.
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
Extract INC (income) vector from the DBF to be used as
independent variables in the regression. Note that PySAL requires this to
be an nxj numpy array, where j is the number of independent variables (not
including a constant). By default this class adds a vector of ones to the
independent variables passed in.
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T
Since we want to run a spatial error model, we need to specify the spatial
weights matrix that includes the spatial configuration of the observations
into the error component of the model. To do that, we can open an already
existing gal file or create a new one. In this case, we will create one
from ``columbus.shp``.
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
Unless there is a good reason not to do it, the weights have to be
row-standardized so every row of the matrix sums to one. Among other
things, his allows to interpret the spatial lag of a variable as the
average value of the neighboring observations. In PySAL, this can be
easily performed in the following way:
>>> w.transform = 'r'
The Combo class runs an SARAR model, that is a spatial lag+error model.
In this case we will run a simple version of that, where we have the
spatial effects as well as exogenous variables. Since it is a spatial
model, we have to pass in the weights matrix. If we want to
have the names of the variables printed in the output summary, we will
have to pass them in as well, although this is optional.
>>> reg = GM_Combo_Het(y, X, w=w, step1c=True, name_y='hoval', name_x=['income'], name_ds='columbus')
Once we have run the model, we can explore a little bit the output. The
regression object we have created has many attributes so take your time to
discover them. This class offers an error model that explicitly accounts
for heteroskedasticity and that unlike the models from
``spreg.error_sp``, it allows for inference on the spatial
parameter. Hence, we find the same number of betas as of standard errors,
which we calculate taking the square root of the diagonal of the
variance-covariance matrix:
>>> print reg.name_z
['CONSTANT', 'income', 'W_hoval', 'lambda']
>>> print np.around(np.hstack((reg.betas,np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)
[[ 9.9753 14.1435]
[ 1.5742 0.374 ]
[ 0.1535 0.3978]
[ 0.2103 0.3924]]
This class also allows the user to run a spatial lag+error model with the
extra feature of including non-spatial endogenous regressors. This means
that, in addition to the spatial lag and error, we consider some of the
variables on the right-hand side of the equation as endogenous and we
instrument for this. As an example, we will include CRIME (crime rates) as
endogenous and will instrument with DISCBD (distance to the CSB). We first
need to read in the variables:
>>> yd = []
>>> yd.append(db.by_col("CRIME"))
>>> yd = np.array(yd).T
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
And then we can run and explore the model analogously to the previous combo:
>>> reg = GM_Combo_Het(y, X, yd, q, w=w, step1c=True, name_x=['inc'], name_y='hoval', name_yend=['crime'], name_q=['discbd'], name_ds='columbus')
>>> print reg.name_z
['CONSTANT', 'inc', 'crime', 'W_hoval', 'lambda']
>>> print np.round(reg.betas,4)
[[ 113.9129]
[ -0.3482]
[ -1.3566]
[ -0.5766]
[ 0.6561]]
"""
[docs] def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True,
max_iter=1, epsilon=0.00001,
step1c=False, inv_method='power_exp',
vm=False, name_y=None, name_x=None,
name_yend=None, name_q=None,
name_w=None, name_ds=None):
n = USER.check_arrays(y, x, yend, q)
USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
yend2, q2 = set_endog(y, x, w, yend, q, w_lags, lag_q)
x_constant = USER.check_constant(x)
BaseGM_Combo_Het.__init__(self, y=y, x=x_constant, yend=yend2, q=q2,
w=w.sparse, w_lags=w_lags,
max_iter=max_iter, step1c=step1c, lag_q=lag_q,
epsilon=epsilon, inv_method=inv_method)
self.rho = self.betas[-2]
self.predy_e, self.e_pred, warn = UTILS.sp_att(w, self.y, self.predy,
yend2[:, -1].reshape(self.n, 1), self.rho)
UTILS.set_warn(self, warn)
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES (HET)"
self.name_ds = USER.set_name_ds(name_ds)
self.name_y = USER.set_name_y(name_y)
self.name_x = USER.set_name_x(name_x, x)
self.name_yend = USER.set_name_yend(name_yend, yend)
self.name_yend.append(USER.set_name_yend_sp(self.name_y))
self.name_z = self.name_x + self.name_yend
self.name_z.append('lambda') # listing lambda last
self.name_q = USER.set_name_q(name_q, q)
self.name_q.extend(
USER.set_name_q_sp(self.name_x, w_lags, self.name_q, lag_q))
self.name_h = USER.set_name_h(self.name_x, self.name_q)
self.name_w = USER.set_name_w(name_w, w)
SUMMARY.GM_Combo_Het(reg=self, w=w, vm=vm)
# Functions
def get_psi_sigma(w, u, lamb):
"""
Computes the Sigma matrix needed to compute Psi
Parameters
----------
w : Sparse matrix
Spatial weights sparse matrix
u : array
nx1 vector of residuals
lamb : float
Lambda
"""
e = (u - lamb * (w * u)) ** 2
E = SP.dia_matrix((e.flat, 0), shape=(w.shape[0], w.shape[0]))
return E.tocsr()
def get_vc_het(w, wA1, E):
"""
Computes the VC matrix Psi based on lambda as in Arraiz et al :cite:`Arraiz2010`:
..math::
\tilde{Psi} = \left(\begin{array}{c c}
\psi_{11} & \psi_{12} \\
\psi_{21} & \psi_{22} \\
\end{array} \right)
NOTE: psi12=psi21
...
Parameters
----------
w : Sparse matrix
Spatial weights sparse matrix
E : sparse matrix
Sigma
Returns
-------
Psi : array
2x2 array with estimator of the variance-covariance matrix
"""
aPatE = 2 * wA1 * E
wPwtE = (w + w.T) * E
psi11 = aPatE * aPatE
psi12 = aPatE * wPwtE
psi22 = wPwtE * wPwtE
psi = list(map(np.sum, [psi11.diagonal(), psi12.diagonal(), psi22.diagonal()]))
return np.array([[psi[0], psi[1]], [psi[1], psi[2]]]) / (2. * w.shape[0])
def get_vm_het(G, lamb, reg, w, psi):
"""
Computes the variance-covariance matrix Omega as in Arraiz et al
:cite:`Arraiz2010`:
...
Parameters
----------
G : array
G from moments equations
lamb : float
Final lambda from spHetErr estimation
reg : regression object
output instance from a regression model
u : array
nx1 vector of residuals
w : Sparse matrix
Spatial weights sparse matrix
psi : array
2x2 array with the variance-covariance matrix of the moment equations
Returns
-------
vm : array
(k+1)x(k+1) array with the variance-covariance matrix of the parameters
"""
J = np.dot(G, np.array([[1], [2 * lamb]]))
Zs = UTILS.get_spFilter(w, lamb, reg.x)
ZstEZs = spdot((Zs.T * get_psi_sigma(w, reg.u, lamb)), Zs)
ZsZsi = la.inv(spdot(Zs.T, Zs))
omega11 = w.shape[0] * np.dot(np.dot(ZsZsi, ZstEZs), ZsZsi)
omega22 = la.inv(np.dot(np.dot(J.T, la.inv(psi)), J))
zero = np.zeros((reg.k, 1), float)
vm = np.vstack((np.hstack((omega11, zero)), np.hstack((zero.T, omega22)))) / \
w.shape[0]
return vm
def get_P_hat(reg, hthi, zf):
"""
P_hat from Appendix B, used for a1 a2, using filtered Z
"""
htzf = spdot(reg.h.T, zf)
P1 = spdot(hthi, htzf)
P2 = spdot(htzf.T, P1)
P2i = la.inv(P2)
return reg.n * np.dot(P1, P2i)
def get_a1a2(w, wA1, reg, lambdapar, P, zs, inv_method, filt):
"""
Computes the a1 in psi assuming residuals come from original regression.
:cite:`Anselin2011`
Parameters
----------
w : Sparse matrix
Spatial weights sparse matrix
reg : TSLS
Two stage least quare regression instance
lambdapar : float
Spatial autoregressive parameter
Returns
-------
[a1, a2] : list
a1 and a2 are two nx1 array in psi equation
"""
us = UTILS.get_spFilter(w, lambdapar, reg.u)
alpha1 = (-2.0 / w.shape[0]) * (np.dot(spdot(zs.T, wA1), us))
alpha2 = (-1.0 / w.shape[0]) * (np.dot(spdot(zs.T, (w + w.T)), us))
a1 = np.dot(spdot(reg.h, P), alpha1)
a2 = np.dot(spdot(reg.h, P), alpha2)
if not filt:
a1 = UTILS.inverse_prod(
w, a1, lambdapar, post_multiply=True, inv_method=inv_method).T
a2 = UTILS.inverse_prod(
w, a2, lambdapar, post_multiply=True, inv_method=inv_method).T
return [a1, a2]
def get_vc_het_tsls(w, wA1, reg, lambdapar, P, zs, inv_method, filt=True, save_a1a2=False):
sigma = get_psi_sigma(w, reg.u, lambdapar)
vc1 = get_vc_het(w, wA1, sigma)
a1, a2 = get_a1a2(w, wA1, reg, lambdapar, P, zs, inv_method, filt)
a1s = a1.T * sigma
a2s = a2.T * sigma
psi11 = float(np.dot(a1s, a1))
psi12 = float(np.dot(a1s, a2))
psi21 = float(np.dot(a2s, a1))
psi22 = float(np.dot(a2s, a2))
psi0 = np.array([[psi11, psi12], [psi21, psi22]]) / w.shape[0]
if save_a1a2:
psi = (vc1 + psi0, a1, a2)
else:
psi = vc1 + psi0
return psi
def get_Omega_GS2SLS(w, lamb, reg, G, psi, P):
"""
Computes the variance-covariance matrix for GS2SLS:
Parameters
----------
w : Sparse matrix
Spatial weights sparse matrix
lamb : float
Spatial autoregressive parameter
reg : GSTSLS
Generalized Spatial two stage least quare regression instance
G : array
Moments
psi : array
Weighting matrix
Returns
-------
omega : array
(k+1)x(k+1)
"""
psi, a1, a2 = psi
sigma = get_psi_sigma(w, reg.u, lamb)
psi_dd_1 = (1.0 / w.shape[0]) * reg.h.T * sigma
psi_dd = spdot(psi_dd_1, reg.h)
psi_dl = spdot(psi_dd_1, np.hstack((a1, a2)))
psi_o = np.hstack(
(np.vstack((psi_dd, psi_dl.T)), np.vstack((psi_dl, psi))))
psii = la.inv(psi)
j = np.dot(G, np.array([[1.], [2 * lamb]]))
jtpsii = np.dot(j.T, psii)
jtpsiij = np.dot(jtpsii, j)
jtpsiiji = la.inv(jtpsiij)
omega_1 = np.dot(jtpsiiji, jtpsii)
omega_2 = np.dot(np.dot(psii, j), jtpsiiji)
om_1_s = omega_1.shape
om_2_s = omega_2.shape
p_s = P.shape
omega_left = np.hstack((np.vstack((P.T, np.zeros((om_1_s[0], p_s[0])))),
np.vstack((np.zeros((p_s[1], om_1_s[1])), omega_1))))
omega_right = np.hstack((np.vstack((P, np.zeros((om_2_s[0], p_s[1])))),
np.vstack((np.zeros((p_s[0], om_2_s[1])), omega_2))))
omega = np.dot(np.dot(omega_left, psi_o), omega_right)
return omega / w.shape[0]
def _test():
import doctest
doctest.testmod()
if __name__ == '__main__':
_test()