"""
Spatial Error Models module
"""
__author__ = "Luc Anselin luc.anselin@asu.edu, \
Daniel Arribas-Bel darribas@asu.edu, \
Pedro V. Amaral pedro.amaral@asu.edu"
import numpy as np
from numpy import linalg as la
from . import ols as OLS
from libpysal.weights.spatial_lag import lag_spatial
from .utils import power_expansion, set_endog, iter_msg, sp_att
from .utils import get_A1_hom, get_A2_hom, get_A1_het, optim_moments, get_spFilter, get_lags, _moments2eqs
from .utils import spdot, RegressionPropsY, set_warn
from . import twosls as TSLS
from . import user_output as USER
from . import summary_output as SUMMARY
__all__ = ["GM_Error", "GM_Endog_Error", "GM_Combo"]
class BaseGM_Error(RegressionPropsY):
"""
GMM method for a spatial error model (note: no consistency checks
diagnostics or constant added); based on Kelejian and Prucha
(1998, 1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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
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
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
sig2 : float
Sigma squared used in computations
Examples
--------
>>> import libpysal
>>> import numpy as np
>>> dbf = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array([dbf.by_col('HOVAL')]).T
>>> x = np.array([dbf.by_col('INC'), dbf.by_col('CRIME')]).T
>>> x = np.hstack((np.ones(y.shape),x))
>>> w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
>>> w.transform='r'
>>> model = BaseGM_Error(y, x, w=w.sparse)
>>> np.around(model.betas, decimals=4)
array([[ 47.6946],
[ 0.7105],
[ -0.5505],
[ 0.3257]])
"""
def __init__(self, y, x, w):
# 1a. OLS --> \tilde{betas}
ols = OLS.BaseOLS(y=y, x=x)
self.n, self.k = ols.x.shape
self.x = ols.x
self.y = ols.y
# 1b. GMM --> \tilde{\lambda1}
moments = _momentsGM_Error(w, ols.u)
lambda1 = optim_moments(moments)
# 2a. OLS -->\hat{betas}
xs = get_spFilter(w, lambda1, self.x)
ys = get_spFilter(w, lambda1, self.y)
ols2 = OLS.BaseOLS(y=ys, x=xs)
# Output
self.predy = spdot(self.x, ols2.betas)
self.u = y - self.predy
self.betas = np.vstack((ols2.betas, np.array([[lambda1]])))
self.sig2 = ols2.sig2n
self.e_filtered = self.u - lambda1 * w * self.u
self.vm = self.sig2 * ols2.xtxi
se_betas = np.sqrt(self.vm.diagonal())
self._cache = {}
[docs]class GM_Error(BaseGM_Error):
"""
GMM method for a spatial error model, with results and diagnostics; based
on Kelejian and Prucha (1998, 1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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 (always needed)
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
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)
sig2 : float
Sigma squared used in computations
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_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 libpysal
>>> import numpy as np
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.
>>> dbf = 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([dbf.by_col('HOVAL')]).T
Extract CRIME (crime) and INC (income) 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.
>>> names_to_extract = ['INC', 'CRIME']
>>> x = np.array([dbf.by_col(name) for name in names_to_extract]).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 use
``columbus.gal``, which contains contiguity relationships between the
observations in the Columbus dataset we are using throughout this example.
Note that, in order to read the file, not only to open it, we need to
append '.read()' at the end of the command.
>>> w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
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 preliminars, 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.
>>> model = GM_Error(y, x, w=w, name_y='hoval', 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. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas).
>>> print model.name_x
['CONSTANT', 'income', 'crime', 'lambda']
>>> np.around(model.betas, decimals=4)
array([[ 47.6946],
[ 0.7105],
[ -0.5505],
[ 0.3257]])
>>> np.around(model.std_err, decimals=4)
array([ 12.412 , 0.5044, 0.1785])
>>> np.around(model.z_stat, decimals=6) #doctest: +SKIP
array([[ 3.84261100e+00, 1.22000000e-04],
[ 1.40839200e+00, 1.59015000e-01],
[ -3.08424700e+00, 2.04100000e-03]])
>>> round(model.sig2,4)
198.5596
"""
[docs] def __init__(self, y, x, w,
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.__init__(self, y=y, x=x_constant, w=w.sparse)
self.title = "SPATIALLY WEIGHTED LEAST SQUARES"
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(reg=self, w=w, vm=vm)
class BaseGM_Endog_Error(RegressionPropsY):
'''
GMM method for a spatial error model with endogenous variables (note: no
consistency checks, diagnostics or constant added); based on Kelejian and
Prucha (1998, 1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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
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
z : array
nxk array of variables (combination of x and yend)
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
sig2 : float
Sigma squared used in computations
Examples
--------
>>> import libpysal
>>> import numpy as np
>>> from spreg import BaseGM_Endog_Error
>>> dbf = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array([dbf.by_col('CRIME')]).T
>>> x = np.array([dbf.by_col('INC')]).T
>>> x = np.hstack((np.ones(y.shape),x))
>>> yend = np.array([dbf.by_col('HOVAL')]).T
>>> q = np.array([dbf.by_col('DISCBD')]).T
>>> w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
>>> w.transform='r'
>>> model = BaseGM_Endog_Error(y, x, yend, q, w=w.sparse)
>>> np.around(model.betas, decimals=4)
array([[ 82.573 ],
[ 0.581 ],
[ -1.4481],
[ 0.3499]])
'''
def __init__(self, y, x, yend, q, w):
# 1a. TSLS --> \tilde{betas}
tsls = TSLS.BaseTSLS(y=y, x=x, yend=yend, q=q)
self.n, self.k = tsls.z.shape
self.x = tsls.x
self.y = tsls.y
self.yend, self.z = tsls.yend, tsls.z
# 1b. GMM --> \tilde{\lambda1}
moments = _momentsGM_Error(w, tsls.u)
lambda1 = optim_moments(moments)
# 2a. 2SLS -->\hat{betas}
xs = get_spFilter(w, lambda1, self.x)
ys = get_spFilter(w, lambda1, self.y)
yend_s = get_spFilter(w, lambda1, self.yend)
tsls2 = TSLS.BaseTSLS(ys, xs, yend_s, h=tsls.h)
# Output
self.betas = np.vstack((tsls2.betas, np.array([[lambda1]])))
self.predy = spdot(tsls.z, tsls2.betas)
self.u = y - self.predy
self.sig2 = float(np.dot(tsls2.u.T, tsls2.u)) / self.n
self.e_filtered = self.u - lambda1 * w * self.u
self.vm = self.sig2 * tsls2.varb
self._cache = {}
[docs]class GM_Endog_Error(BaseGM_Endog_Error):
'''
GMM method for a spatial error model with endogenous variables, with
results and diagnostics; based on Kelejian and Prucha (1998,
1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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)
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
z : array
nxk array of variables (combination of x and yend)
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)
sig2 : float
Sigma squared used in computations
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
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 libpysal
>>> import numpy as np
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.
>>> dbf = libpysal.io.open(libpysal.examples.get_path("columbus.dbf"),'r')
Extract the CRIME column (crime rates) 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([dbf.by_col('CRIME')]).T
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 model adds a vector of ones to the
independent variables passed in.
>>> x = np.array([dbf.by_col('INC')]).T
In this case we consider HOVAL (home value) is an endogenous regressor.
We tell the model that this is so by passing it in a different parameter
from the exogenous variables (x).
>>> yend = np.array([dbf.by_col('HOVAL')]).T
Because we have endogenous variables, to obtain a correct estimate of the
model, we need to instrument for HOVAL. We use DISCBD (distance to the
CBD) for this and hence put it in the instruments parameter, 'q'.
>>> q = np.array([dbf.by_col('DISCBD')]).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 use
``columbus.gal``, which contains contiguity relationships between the
observations in the Columbus dataset we are using throughout this example.
Note that, in order to read the file, not only to open it, we need to
append '.read()' at the end of the command.
>>> w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
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, this 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 preliminars, 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.
>>> from spreg import GM_Endog_Error
>>> model = GM_Endog_Error(y, x, yend, q, w=w, name_x=['inc'], name_y='crime', name_yend=['hoval'], 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. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas). Also, this regression uses a two stage least
squares estimation method that accounts for the endogeneity created by the
endogenous variables included.
>>> print model.name_z
['CONSTANT', 'inc', 'hoval', 'lambda']
>>> np.around(model.betas, decimals=4)
array([[ 82.573 ],
[ 0.581 ],
[ -1.4481],
[ 0.3499]])
>>> np.around(model.std_err, decimals=4)
array([ 16.1381, 1.3545, 0.7862])
'''
[docs] def __init__(self, y, x, yend, q, w,
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.__init__(
self, y=y, x=x_constant, w=w.sparse, yend=yend, q=q)
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES"
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')
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(reg=self, w=w, vm=vm)
class BaseGM_Combo(BaseGM_Endog_Error):
"""
GMM method for a spatial lag and error model, with endogenous variables
(note: no consistency checks, diagnostics or constant added); based on
Kelejian and Prucha (1998, 1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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).
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
z : array
nxk array of variables (combination of x and yend)
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
sig2 : float
Sigma squared used in computations
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> import spreg
>>> from spreg import BaseGM_Combo
>>> db = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'),'r')
>>> y = np.array(db.by_col("CRIME"))
>>> 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(y, X, yend=yd2, q=q2, w=w.sparse)
Print the betas
>>> print np.around(np.hstack((reg.betas[:-1],np.sqrt(reg.vm.diagonal()).reshape(3,1))),3)
[[ 39.059 11.86 ]
[ -1.404 0.391]
[ 0.467 0.2 ]]
And lambda
>>> print 'Lamda: ', np.around(reg.betas[-1], 3)
Lamda: [-0.048]
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("HOVAL"))
>>> 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(y, X, yd2, q2, w=w.sparse)
>>> betas = np.array([['CONSTANT'],['INC'],['HOVAL'],['W_CRIME']])
>>> print np.hstack((betas, np.around(np.hstack((reg.betas[:-1], np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)))
[['CONSTANT' '50.0944' '14.3593']
['INC' '-0.2552' '0.5667']
['HOVAL' '-0.6885' '0.3029']
['W_CRIME' '0.4375' '0.2314']]
"""
def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True):
BaseGM_Endog_Error.__init__(self, y=y, x=x, w=w, yend=yend, q=q)
[docs]class GM_Combo(BaseGM_Combo):
"""
GMM method for a spatial lag and error model with endogenous variables,
with results and diagnostics; based on Kelejian and Prucha (1998,
1999) :cite:`Kelejian1998` :cite:`Kelejian1999`.
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).
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
z : array
nxk array of variables (combination of x and yend)
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))
sig2 : float
Sigma squared used in computations (based on filtered
residuals)
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
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 CRIME column (crime rates) 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("CRIME"))
>>> 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 model 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, this 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(y, X, w=w, name_y='crime', 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. Note that because we are running the classical GMM error
model from 1998/99, the spatial parameter is obtained as a point estimate, so
although you get a value for it (there are for coefficients under
model.betas), you cannot perform inference on it (there are only three
values in model.se_betas). Also, this regression uses a two stage least
squares estimation method that accounts for the endogeneity created by the
spatial lag of the dependent variable. We can check the betas:
>>> print reg.name_z
['CONSTANT', 'income', 'W_crime', 'lambda']
>>> print np.around(np.hstack((reg.betas[:-1],np.sqrt(reg.vm.diagonal()).reshape(3,1))),3)
[[ 39.059 11.86 ]
[ -1.404 0.391]
[ 0.467 0.2 ]]
And lambda:
>>> print 'lambda: ', np.around(reg.betas[-1], 3)
lambda: [-0.048]
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 HOVAL (home value) 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("HOVAL"))
>>> 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(y, X, yd, q, w=w, name_x=['inc'], name_y='crime', name_yend=['hoval'], name_q=['discbd'], name_ds='columbus')
>>> print reg.name_z
['CONSTANT', 'inc', 'hoval', 'W_crime', 'lambda']
>>> names = np.array(reg.name_z).reshape(5,1)
>>> print np.hstack((names[0:4,:], np.around(np.hstack((reg.betas[:-1], np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)))
[['CONSTANT' '50.0944' '14.3593']
['inc' '-0.2552' '0.5667']
['hoval' '-0.6885' '0.3029']
['W_crime' '0.4375' '0.2314']]
>>> print 'lambda: ', np.around(reg.betas[-1], 3)
lambda: [ 0.254]
"""
[docs] def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True,
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.__init__(
self, y=y, x=x_constant, w=w.sparse, yend=yend2, q=q2,
w_lags=w_lags, lag_q=lag_q)
self.rho = self.betas[-2]
self.predy_e, self.e_pred, warn = sp_att(w, self.y,
self.predy, yend2[:, -1].reshape(self.n, 1), self.rho)
set_warn(self, warn)
self.title = "SPATIALLY WEIGHTED TWO STAGE LEAST SQUARES"
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')
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(reg=self, w=w, vm=vm)
def _momentsGM_Error(w, u):
try:
wsparse = w.sparse
except:
wsparse = w
n = wsparse.shape[0]
u2 = np.dot(u.T, u)
wu = wsparse * u
uwu = np.dot(u.T, wu)
wu2 = np.dot(wu.T, wu)
wwu = wsparse * wu
uwwu = np.dot(u.T, wwu)
wwu2 = np.dot(wwu.T, wwu)
wuwwu = np.dot(wu.T, wwu)
wtw = wsparse.T * wsparse
trWtW = np.sum(wtw.diagonal())
g = np.array([[u2[0][0], wu2[0][0], uwu[0][0]]]).T / n
G = np.array(
[[2 * uwu[0][0], -wu2[0][0], n], [2 * wuwwu[0][0], -wwu2[0][0], trWtW],
[uwwu[0][0] + wu2[0][0], -wuwwu[0][0], 0.]]) / n
return [G, g]
def _test():
import doctest
start_suppress = np.get_printoptions()['suppress']
np.set_printoptions(suppress=True)
doctest.testmod()
np.set_printoptions(suppress=start_suppress)
if __name__ == '__main__':
_test()
import libpysal
import numpy as np
dbf = libpysal.io.open(libpysal.examples.get_path('columbus.dbf'), 'r')
y = np.array([dbf.by_col('HOVAL')]).T
names_to_extract = ['INC', 'CRIME']
x = np.array([dbf.by_col(name) for name in names_to_extract]).T
w = libpysal.io.open(libpysal.examples.get_path("columbus.gal"), 'r').read()
w.transform = 'r'
model = GM_Error(y, x, w, name_y='hoval',
name_x=['income', 'crime'], name_ds='columbus')
print(model.summary)