'''
Spatial Two Stages Least Squares
'''
__author__ = "Luc Anselin luc.anselin@asu.edu, David C. Folch david.folch@asu.edu"
import numpy as np
from . import twosls as TSLS
from . import user_output as USER
from . import summary_output as SUMMARY
from .utils import set_endog, sp_att, set_warn
__all__ = ["GM_Lag"]
class BaseGM_Lag(TSLS.BaseTSLS):
"""
Spatial two stage least squares (S2SLS) (note: no consistency checks,
diagnostics or constant added); Anselin (1988) [Anselin1988]_
Parameters
----------
y : array
nx1 array for dependent variable
x : array
Two dimensional array with n rows and one column for each
independent (exogenous) variable; assumes the constant is
in column 0.
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); cannot be
used in combination with h
w : Pysal weights matrix
Spatial weights 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).
robust : string
If 'white', then a White consistent estimator of the
variance-covariance matrix is given. If 'hac', then a
HAC consistent estimator of the variance-covariance
matrix is given. Default set to None.
gwk : pysal W object
Kernel spatial weights needed for HAC estimation. Note:
matrix must have ones along the main diagonal.
sig2n_k : boolean
If True, then use n-k to estimate sigma^2. If False, use n.
Attributes
----------
betas : array
kx1 array of estimated coefficients
u : array
nx1 array of 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)
kstar : integer
Number of endogenous variables.
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)
mean_y : float
Mean of dependent variable
std_y : float
Standard deviation of dependent variable
vm : array
Variance covariance matrix (kxk)
utu : float
Sum of squared residuals
sig2 : float
Sigma squared used in computations
sig2n : float
Sigma squared (computed with n in the denominator)
sig2n_k : float
Sigma squared (computed with n-k in the denominator)
hth : float
H'H
hthi : float
(H'H)^-1
varb : array
(Z'H (H'H)^-1 H'Z)^-1
zthhthi : array
Z'H(H'H)^-1
pfora1a2 : array
n(zthhthi)'varb
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> import spreg
>>> np.set_printoptions(suppress=True) #prevent scientific format
>>> w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
>>> w.transform = 'r'
>>> db = libpysal.io.open(libpysal.examples.get_path("columbus.dbf"),'r')
>>> y = np.array(db.by_col("HOVAL"))
>>> y = np.reshape(y, (49,1))
>>> # no non-spatial endogenous variables
>>> 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))
>>> reg = spreg.twosls_sp.BaseGM_Lag(y, X, w=w, w_lags=2)
>>> reg.betas
array([[45.30170561],
[ 0.62088862],
[-0.48072345],
[ 0.02836221]])
>>> spreg.se_betas(reg)
array([17.91278862, 0.52486082, 0.1822815 , 0.31740089])
>>> reg = spreg.twosls_sp.BaseGM_Lag(y, X, w=w, w_lags=2, robust='white')
>>> reg.betas
array([[45.30170561],
[ 0.62088862],
[-0.48072345],
[ 0.02836221]])
>>> spreg.se_betas(reg)
array([20.47077481, 0.50613931, 0.20138425, 0.38028295])
>>> # instrument for HOVAL with DISCBD
>>> X = np.array(db.by_col("INC"))
>>> X = np.reshape(X, (49,1))
>>> yd = np.array(db.by_col("CRIME"))
>>> yd = np.reshape(yd, (49,1))
>>> q = np.array(db.by_col("DISCBD"))
>>> q = np.reshape(q, (49,1))
>>> X = np.hstack((np.ones(y.shape),X))
>>> reg = spreg.twosls_sp.BaseGM_Lag(y, X, w=w, yend=yd, q=q, w_lags=2)
>>> reg.betas
array([[100.79359082],
[ -0.50215501],
[ -1.14881711],
[ -0.38235022]])
>>> spreg.se_betas(reg)
array([53.0829123 , 1.02511494, 0.57589064, 0.59891744])
"""
def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True,
robust=None, gwk=None, sig2n_k=False):
yend2, q2 = set_endog(y, x[:,1:], w, yend, q, w_lags, lag_q) #assumes constant in first column
TSLS.BaseTSLS.__init__(self, y=y, x=x, yend=yend2, q=q2,
robust=robust, gwk=gwk, sig2n_k=sig2n_k)
[docs]class GM_Lag(BaseGM_Lag):
"""
Spatial two stage least squares (S2SLS) with results and diagnostics;
Anselin (1988) :cite:`Anselin1988`
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); cannot be
used in combination with h
w : pysal W object
Spatial weights object
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).
robust : string
If 'white', then a White consistent estimator of the
variance-covariance matrix is given. If 'hac', then a
HAC consistent estimator of the variance-covariance
matrix is given. Default set to None.
gwk : pysal W object
Kernel spatial weights needed for HAC estimation. Note:
matrix must have ones along the main diagonal.
sig2n_k : boolean
If True, then use n-k to estimate sigma^2. If False, use n.
spat_diag : boolean
If True, then compute Anselin-Kelejian test
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_gwk : string
Name of kernel 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_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)
kstar : integer
Number of endogenous variables.
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)
robust : string
Adjustment for robust standard errors
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))
utu : float
Sum of squared residuals
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
ak_test : tuple
Anselin-Kelejian test; tuple contains the pair (statistic,
p-value)
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_gwk : string
Name of kernel weights matrix for use in output
name_ds : string
Name of dataset for use in output
title : string
Name of the regression method used
sig2n : float
Sigma squared (computed with n in the denominator)
sig2n_k : float
Sigma squared (computed with n-k in the denominator)
hth : float
:math:`H'H`
hthi : float
:math:`(H'H)^{-1}`
varb : array
:math:`(Z'H (H'H)^{-1} H'Z)^{-1}`
zthhthi : array
:math:`Z'H(H'H)^{-1}`
pfora1a2 : array
n(zthhthi)'varb
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. Since we will need some tests for our
model, we also import the diagnostics module.
>>> import numpy as np
>>> import libpysal
>>> import spreg
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 value) 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 rates) 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 model adds a vector of ones to the
independent variables passed in, but this can be overridden by passing
constant=False.
>>> 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, 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'
This class runs a lag model, which means that includes the spatial lag of
the dependent variable on the right-hand side of the equation. 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. The default most basic model to be run would be:
>>> from spreg import GM_Lag
>>> np.set_printoptions(suppress=True) #prevent scientific format
>>> reg=GM_Lag(y, X, w=w, w_lags=2, name_x=['inc', 'crime'], name_y='hoval', name_ds='columbus')
>>> reg.betas
array([[45.30170561],
[ 0.62088862],
[-0.48072345],
[ 0.02836221]])
Once the model is run, we can obtain the standard error of the coefficient
estimates by calling the diagnostics module:
>>> spreg.se_betas(reg)
array([17.91278862, 0.52486082, 0.1822815 , 0.31740089])
But we can also run models that incorporates corrected standard errors
following the White procedure. For that, we will have to include the
optional parameter ``robust='white'``:
>>> reg=GM_Lag(y, X, w=w, w_lags=2, robust='white', name_x=['inc', 'crime'], name_y='hoval', name_ds='columbus')
>>> reg.betas
array([[45.30170561],
[ 0.62088862],
[-0.48072345],
[ 0.02836221]])
And we can access the standard errors from the model object:
>>> reg.std_err
array([20.47077481, 0.50613931, 0.20138425, 0.38028295])
The class is flexible enough to accomodate a spatial lag model that,
besides the spatial lag of the dependent variable, includes other
non-spatial endogenous regressors. As an example, we will assume that
CRIME is actually endogenous and we decide to instrument for it with
DISCBD (distance to the CBD). We reload the X including INC only and
define CRIME as endogenous and DISCBD as instrument:
>>> X = np.array(db.by_col("INC"))
>>> X = np.reshape(X, (49,1))
>>> yd = np.array(db.by_col("CRIME"))
>>> yd = np.reshape(yd, (49,1))
>>> q = np.array(db.by_col("DISCBD"))
>>> q = np.reshape(q, (49,1))
And we can run the model again:
>>> reg=GM_Lag(y, X, w=w, yend=yd, q=q, w_lags=2, name_x=['inc'], name_y='hoval', name_yend=['crime'], name_q=['discbd'], name_ds='columbus')
>>> reg.betas
array([[100.79359082],
[ -0.50215501],
[ -1.14881711],
[ -0.38235022]])
Once the model is run, we can obtain the standard error of the coefficient
estimates by calling the diagnostics module:
>>> spreg.se_betas(reg)
array([53.0829123 , 1.02511494, 0.57589064, 0.59891744])
"""
[docs] def __init__(self, y, x, yend=None, q=None,
w=None, w_lags=1, lag_q=True,
robust=None, gwk=None, sig2n_k=False,
spat_diag=False,
vm=False, name_y=None, name_x=None,
name_yend=None, name_q=None,
name_w=None, name_gwk=None, name_ds=None):
n = USER.check_arrays(x, yend, q)
y = USER.check_y(y, n)
USER.check_weights(w, y, w_required=True)
USER.check_robust(robust, gwk)
x_constant,name_x,warn = USER.check_constant(x,name_x)
set_warn(self, warn)
BaseGM_Lag.__init__(
self, y=y, x=x_constant, w=w, yend=yend, q=q,
w_lags=w_lags, robust=robust, gwk=gwk,
lag_q=lag_q, sig2n_k=sig2n_k)
self.rho = self.betas[-1]
self.predy_e, self.e_pred, warn = sp_att(w, self.y, self.predy,
self.yend[:, -1].reshape(self.n, 1), self.rho)
set_warn(self, warn)
self.title = "SPATIAL 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_constant)
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_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.robust = USER.set_robust(robust)
self.name_w = USER.set_name_w(name_w, w)
self.name_gwk = USER.set_name_w(name_gwk, gwk)
SUMMARY.GM_Lag(reg=self, w=w, vm=vm, spat_diag=spat_diag)
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 numpy as np
import libpysal
db = libpysal.io.open(libpysal.examples.get_path("columbus.dbf"), 'r')
y_var = 'CRIME'
y = np.array([db.by_col(y_var)]).reshape(49, 1)
x_var = ['INC']
x = np.array([db.by_col(name) for name in x_var]).T
yd_var = ['HOVAL']
yd = np.array([db.by_col(name) for name in yd_var]).T
q_var = ['DISCBD']
q = np.array([db.by_col(name) for name in q_var]).T
w = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.shp"))
w.transform = 'r'
model = GM_Lag(
y, x, yd, q, w=w, spat_diag=True, name_y=y_var, name_x=x_var,
name_yend=yd_var, name_q=q_var, name_ds='columbus', name_w='columbus.gal')
print(model.summary)