import numpy as np
import numpy.linalg as la
from . import summary_output as SUMMARY
from . import robust as ROBUST
from . import user_output as USER
from .utils import spdot, sphstack, RegressionPropsY, RegressionPropsVM, set_warn
__author__ = "Luc Anselin luc.anselin@asu.edu, David C. Folch david.folch@asu.edu, Jing Yao jingyao@asu.edu"
__all__ = ["TSLS"]
class BaseTSLS(RegressionPropsY, RegressionPropsVM):
"""
Two stage least squares (2SLS) (note: no consistency checks,
diagnostics or constant added)
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
h : array
Two dimensional array with n rows and one column for each
exogenous variable to use as instruments (note: this
can contain variables from x); cannot be used in
combination with 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
: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
:math:`n(zthhthi)'varb`
Examples
--------
>>> import numpy as np
>>> import libpysal
>>> import spreg
>>> 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
>>> X = np.hstack((np.ones(y.shape),X))
>>> yd = []
>>> yd.append(db.by_col("HOVAL"))
>>> yd = np.array(yd).T
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
>>> reg = spreg.twosls.BaseTSLS(y, X, yd, q=q)
>>> print(reg.betas)
[[88.46579584]
[ 0.5200379 ]
[-1.58216593]]
>>> reg = spreg.twosls.BaseTSLS(y, X, yd, q=q, robust="white")
"""
def __init__(self, y, x, yend, q=None, h=None,
robust=None, gwk=None, sig2n_k=False):
if issubclass(type(q), np.ndarray) and issubclass(type(h), np.ndarray):
raise Exception("Please do not provide 'q' and 'h' together")
if q is None and h is None:
raise Exception("Please provide either 'q' or 'h'")
self.y = y
self.n = y.shape[0]
self.x = x
self.kstar = yend.shape[1]
# including exogenous and endogenous variables
z = sphstack(self.x, yend)
if type(h).__name__ not in ['ndarray', 'csr_matrix']:
# including exogenous variables and instrument
h = sphstack(self.x, q)
self.z = z
self.h = h
self.q = q
self.yend = yend
# k = number of exogenous variables and endogenous variables
self.k = z.shape[1]
hth = spdot(h.T, h)
hthi = la.inv(hth)
zth = spdot(z.T, h)
hty = spdot(h.T, y)
factor_1 = np.dot(zth, hthi)
factor_2 = np.dot(factor_1, zth.T)
# this one needs to be in cache to be used in AK
varb = la.inv(factor_2)
factor_3 = np.dot(varb, factor_1)
betas = np.dot(factor_3, hty)
self.betas = betas
self.varb = varb
self.zthhthi = factor_1
# predicted values
self.predy = spdot(z, betas)
# residuals
u = y - self.predy
self.u = u
# attributes used in property
self.hth = hth # Required for condition index
self.hthi = hthi # Used in error models
self.htz = zth.T
if robust:
self.vm = ROBUST.robust_vm(reg=self, gwk=gwk, sig2n_k=sig2n_k)
if sig2n_k:
self.sig2 = self.sig2n_k
else:
self.sig2 = self.sig2n
@property
def pfora1a2(self):
if 'pfora1a2' not in self._cache:
self._cache['pfora1a2'] = self.n * \
np.dot(self.zthhthi.T, self.varb)
return self._cache['pfora1a2']
@property
def vm(self):
try:
return self._cache['vm']
except AttributeError:
self._cache = {}
self._cache['vm'] = np.dot(self.sig2, self.varb)
except KeyError:
self._cache['vm'] = np.dot(self.sig2, self.varb)
return self._cache['vm']
@vm.setter
def vm(self, val):
try:
self._cache['vm'] = val
except AttributeError:
self._cache = {}
self._cache['vm'] = val
except KeyError:
self._cache['vm'] = val
[docs]class TSLS(BaseTSLS):
"""
Two stage least squares with results and diagnostics.
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 (required if running spatial
diagnostics)
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 (requires w)
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
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)
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)
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
:math:`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.
>>> 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, but this can be overridden by passing
constant=False.
>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).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).
>>> yd = []
>>> yd.append(db.by_col("HOVAL"))
>>> yd = np.array(yd).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 = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T
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) and the
instruments. 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 TSLS
>>> reg = TSLS(y, X, yd, q, name_x=['inc'], name_y='crime', name_yend=['hoval'], name_q=['discbd'], name_ds='columbus')
>>> print(reg.betas)
[[88.46579584]
[ 0.5200379 ]
[-1.58216593]]
"""
[docs] def __init__(self, y, x, yend, q,
w=None,
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(y, x, yend, q)
y = USER.check_y(y, n)
USER.check_weights(w, y)
USER.check_robust(robust, gwk)
USER.check_spat_diag(spat_diag, w)
x_constant,name_x,warn = USER.check_constant(x,name_x)
set_warn(self, warn)
BaseTSLS.__init__(self, y=y, x=x_constant, yend=yend, q=q,
robust=robust, gwk=gwk, sig2n_k=sig2n_k)
self.title = "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_z = self.name_x + self.name_yend
self.name_q = USER.set_name_q(name_q, 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.TSLS(reg=self, vm=vm, w=w, 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 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'
tsls = TSLS(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(tsls.summary)