import numpy as np
import multiprocessing as mp
import pandas as pd
from . import regimes as REGI
from . import user_output as USER
from .utils import set_warn, RegressionProps_basic, spdot, sphstack, get_lags
from .twosls import BaseTSLS
from .robust import hac_multi
from .output import output, _spat_diag_out
"""
Two-stage Least Squares estimation with regimes.
"""
__author__ = "Luc Anselin, Pedro V. Amaral, David C. Folch"
[docs]class TSLS_Regimes(BaseTSLS, REGI.Regimes_Frame):
"""
Two stage least squares (2SLS) with regimes.
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)
regimes : list
List of n values with the mapping of each
observation to a regime. Assumed to be aligned with 'x'.
constant_regi: string
Switcher controlling the constant term setup. It may take
the following values:
* 'one': a vector of ones is appended to x and held constant across regimes.
* 'many': a vector of ones is appended to x and considered different per regime (default).
cols2regi : list, 'all'
Argument indicating whether each
column of x should be considered as different per regime
or held constant across regimes (False).
If a list, k booleans indicating for each variable the
option (True if one per regime, False to be held constant).
If 'all' (default), all the variables vary by regime.
regime_err_sep: boolean
If True, a separate regression is run for each regime.
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.
If 'ogmm', then Optimal GMM is used to estimate
betas and the variance-covariance matrix.
Default set to None.
gwk : pysal W object
Kernel spatial weights needed for HAC estimation. Note:
matrix must have ones along the main diagonal.
slx_lags : integer
Number of spatial lags of X to include in the model specification.
If slx_lags>0, the specification becomes of the SLX type.
Note: WX is computed using the complete weights matrix
sig2n_k : boolean
If True, then use n-k to estimate sigma^2. If False, use n.
vm : boolean
If True, include variance-covariance matrix in summary
cores : boolean
Specifies if multiprocessing is to be used
Default: no multiprocessing, cores = False
Note: Multiprocessing may not work on all platforms.
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_regimes : string
Name of regimes variable 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
latex : boolean
Specifies if summary is to be printed in latex format
Attributes
----------
output : dataframe
regression results pandas dataframe
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
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
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
yend : array
Two dimensional array with n rows and one column for each
endogenous variable
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
q : array
Two dimensional array with n rows and one column for each
external exogenous variable used as instruments
Only available in dictionary 'multi' when multiple regressions
(see 'multi' below for details)
vm : array
Variance covariance matrix (kxk)
regimes : list
List of n values with the mapping of each
observation to a regime. Assumed to be aligned with 'x'.
constant_regi: [False, 'one', 'many']
Ignored if regimes=False. Constant option for regimes.
Switcher controlling the constant term setup. It may take
the following values:
* 'one': a vector of ones is appended to x and held constant across regimes.
* 'many': a vector of ones is appended to x and considered different per regime (default).
cols2regi : list, 'all'
Ignored if regimes=False. Argument indicating whether each
column of x should be considered as different per regime
or held constant across regimes (False).
If a list, k booleans indicating for each variable the
option (True if one per regime, False to be held constant).
If 'all', all the variables vary by regime.
regime_err_sep: boolean
If True, a separate regression is run for each regime.
kr : int
Number of variables/columns to be "regimized" or subject
to change by regime. These will result in one parameter
estimate by regime for each variable (i.e. nr parameters per
variable)
kf : int
Number of variables/columns to be considered fixed or
global across regimes and hence only obtain one parameter
estimate
nr : int
Number of different regimes in the 'regimes' list
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_regimes : string
Name of regimes variable 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
multi : dictionary
Only available when multiple regressions are estimated,
i.e. when regime_err_sep=True and no variable is fixed
across regimes.
Contains all attributes of each individual regression
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
>>> from libpysal.examples import load_example
>>> from libpysal.weights import Rook
Open data on NCOVR US County Homicides (3085 areas) using libpysal.io.open().
This is the DBF associated with the NAT 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.
>>> nat = load_example('Natregimes')
>>> db = libpysal.io.open(nat.get_path('natregimes.dbf'), 'r')
Extract the HR90 column (homicide rates in 1990) 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_var = 'HR90'
>>> y = np.array([db.by_col(y_var)]).reshape(3085,1)
Extract UE90 (unemployment rate) and PS90 (population structure) vectors from
the DBF to be used as independent variables in the regression. Other variables
can be inserted by adding their names to x_var, such as x_var = ['Var1','Var2','...]
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_var = ['PS90','UE90']
>>> x = np.array([db.by_col(name) for name in x_var]).T
In this case we consider RD90 (resource deprivation) as an endogenous regressor.
We tell the model that this is so by passing it in a different parameter
from the exogenous variables (x).
>>> yd_var = ['RD90']
>>> yd = np.array([db.by_col(name) for name in yd_var]).T
Because we have endogenous variables, to obtain a correct estimate of the
model, we need to instrument for RD90. We use FP89 (families below poverty)
for this and hence put it in the instruments parameter, 'q'.
>>> q_var = ['FP89']
>>> q = np.array([db.by_col(name) for name in q_var]).T
The different regimes in this data are given according to the North and
South dummy (SOUTH).
>>> r_var = 'SOUTH'
>>> regimes = db.by_col(r_var)
Since we want to perform tests for spatial dependence, 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 ``NAT.shp``.
>>> w = Rook.from_shapefile(nat.get_path("natregimes.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'
We can now run the regression and then have a summary of the output
by typing: model.summary
Alternatively, we can just check the betas and standard errors of the
parameters:
>>> from spreg import TSLS_Regimes
>>> tslsr = TSLS_Regimes(y, x, yd, q, regimes, w=w, constant_regi='many', spat_diag=False, name_y=y_var, name_x=x_var, name_yend=yd_var, name_q=q_var, name_regimes=r_var, name_ds='NAT', name_w='NAT.shp')
>>> tslsr.betas
array([[ 3.66973562],
[ 1.06950466],
[ 0.14680946],
[ 2.45864196],
[ 9.55873243],
[ 1.94666348],
[-0.30810214],
[ 3.68718119]])
>>> np.sqrt(tslsr.vm.diagonal())
array([0.38389901, 0.09963973, 0.04672091, 0.22725012, 0.49181223,
0.19630774, 0.07784587, 0.25529011])
>>> print(tslsr.summary)
REGRESSION RESULTS
------------------
<BLANKLINE>
SUMMARY OF OUTPUT: TWO STAGE LEAST SQUARES ESTIMATION - REGIME 0
----------------------------------------------------------------
Data set : NAT
Weights matrix : NAT.shp
Dependent Variable : 0_HR90 Number of Observations: 1673
Mean dependent var : 3.3416 Number of Variables : 4
S.D. dependent var : 4.6795 Degrees of Freedom : 1669
Pseudo R-squared : 0.2092
<BLANKLINE>
------------------------------------------------------------------------------------
Variable Coefficient Std.Error z-Statistic Probability
------------------------------------------------------------------------------------
0_CONSTANT 3.6697356 0.3838990 9.5591172 0.0000000
0_PS90 1.0695047 0.0996397 10.7337170 0.0000000
0_UE90 0.1468095 0.0467209 3.1422643 0.0016765
0_RD90 2.4586420 0.2272501 10.8191009 0.0000000
------------------------------------------------------------------------------------
Instrumented: 0_RD90
Instruments: 0_FP89
Regimes variable: SOUTH
<BLANKLINE>
SUMMARY OF OUTPUT: TWO STAGE LEAST SQUARES ESTIMATION - REGIME 1
----------------------------------------------------------------
Data set : NAT
Weights matrix : NAT.shp
Dependent Variable : 1_HR90 Number of Observations: 1412
Mean dependent var : 9.5493 Number of Variables : 4
S.D. dependent var : 7.0389 Degrees of Freedom : 1408
Pseudo R-squared : 0.2987
<BLANKLINE>
------------------------------------------------------------------------------------
Variable Coefficient Std.Error z-Statistic Probability
------------------------------------------------------------------------------------
1_CONSTANT 9.5587324 0.4918122 19.4357356 0.0000000
1_PS90 1.9466635 0.1963077 9.9163867 0.0000000
1_UE90 -0.3081021 0.0778459 -3.9578483 0.0000756
1_RD90 3.6871812 0.2552901 14.4431026 0.0000000
------------------------------------------------------------------------------------
Instrumented: 1_RD90
Instruments: 1_FP89
Regimes variable: SOUTH
------------------------------------------------------------------------------------
GLOBAL DIAGNOSTICS
<BLANKLINE>
REGIMES DIAGNOSTICS - CHOW TEST
VARIABLE DF VALUE PROB
CONSTANT 1 89.093 0.0000
PS90 1 15.876 0.0001
UE90 1 25.106 0.0000
RD90 1 12.920 0.0003
Global test 4 201.237 0.0000
================================ END OF REPORT =====================================
"""
[docs] def __init__(
self,
y,
x,
yend,
q,
regimes,
w=None,
robust=None,
gwk=None,
slx_lags=0,
sig2n_k=True,
spat_diag=False,
vm=False,
constant_regi="many",
cols2regi="all",
regime_err_sep=True,
name_y=None,
name_x=None,
cores=False,
name_yend=None,
name_q=None,
name_regimes=None,
name_w=None,
name_gwk=None,
name_ds=None,
summ=True,
latex=False,
):
n = USER.check_arrays(y, x)
y = USER.check_y(y, n)
USER.check_robust(robust, gwk)
if robust == "hac":
if regime_err_sep:
set_warn(
self,
"Error by regimes is not available for HAC estimation. The error by regimes has been disabled for this model.",
)
regime_err_sep = False
if spat_diag:
set_warn(
self,
"Spatial diagnostics are not available for HAC estimation. The spatial diagnostics have been disabled for this model.",
)
spat_diag = False
USER.check_spat_diag(spat_diag, w)
x_constant, name_x, warn = USER.check_constant(x, name_x, just_rem=True)
set_warn(self, warn)
name_x = USER.set_name_x(name_x, x_constant, constant=True)
if slx_lags > 0:
USER.check_weights(w, y, w_required=True)
lag_x = get_lags(w, x_constant, slx_lags)
x_constant = np.hstack((x_constant, lag_x))
name_x += USER.set_name_spatial_lags(name_x, slx_lags)
else:
USER.check_weights(w, y, w_required=False)
self.constant_regi = constant_regi
self.cols2regi = cols2regi
self.name_ds = USER.set_name_ds(name_ds)
self.name_regimes = USER.set_name_ds(name_regimes)
self.name_w = USER.set_name_w(name_w, w)
self.name_gwk = USER.set_name_w(name_gwk, gwk)
self.name_y = USER.set_name_y(name_y)
name_yend = USER.set_name_yend(name_yend, yend)
name_q = USER.set_name_q(name_q, q)
self.name_x_r = USER.set_name_x(name_x, x_constant) + name_yend
self.n = n
cols2regi = REGI.check_cols2regi(
constant_regi, cols2regi, x_constant, yend=yend, add_cons=False
)
self.regimes_set = REGI._get_regimes_set(regimes)
self.regimes = regimes
USER.check_regimes(self.regimes_set, self.n, x_constant.shape[1])
self.regime_err_sep = regime_err_sep
if (
regime_err_sep == True
and set(cols2regi) == set([True])
and constant_regi == "many"
):
self.y = y
regi_ids = dict(
(r, list(np.where(np.array(regimes) == r)[0])) for r in self.regimes_set
)
self._tsls_regimes_multi(
x_constant,
yend,
q,
w,
regi_ids,
cores,
gwk,
slx_lags,
sig2n_k,
robust,
spat_diag,
vm,
name_x,
name_yend,
name_q,
summ,
latex
)
else:
q, self.name_q = REGI.Regimes_Frame.__init__(
self, q, regimes, constant_regi=None, cols2regi="all", names=name_q
)
x, self.name_x, x_rlist = REGI.Regimes_Frame.__init__(
self,
x_constant,
regimes,
constant_regi,
cols2regi=cols2regi,
names=name_x,
rlist=True
)
yend, self.name_yend, yend_rlist = REGI.Regimes_Frame.__init__(
self,
yend,
regimes,
constant_regi=None,
cols2regi=cols2regi,
yend=True,
names=name_yend,
rlist=True
)
self.output = pd.DataFrame(self.name_x+self.name_yend,
columns=['var_names'])
self.output['var_type'] = ['x']*len(self.name_x)+['yend']*len(self.name_yend)
self.output['regime'] = x_rlist+yend_rlist
self.output['equation'] = 0
BaseTSLS.__init__(
self, y=y, x=x, yend=yend, q=q, robust=robust, gwk=gwk, sig2n_k=sig2n_k
)
if slx_lags == 0:
self.title = "TWO STAGE LEAST SQUARES - REGIMES"
else:
self.title = "TWO STAGE LEAST SQUARES WITH SPATIALLY LAGGED X (2SLS-SLX) - REGIMES"
if robust == "ogmm":
_optimal_weight(self, sig2n_k)
self.name_z = self.name_x + self.name_yend
self.name_h = USER.set_name_h(self.name_x, self.name_q)
self.chow = REGI.Chow(self)
self.robust = USER.set_robust(robust)
if summ:
if spat_diag:
diag_out = _spat_diag_out(self, w, 'yend')
else:
diag_out = None
output(reg=self, vm=vm, robust=robust, other_end=diag_out, latex=latex)
def _tsls_regimes_multi(
self,
x,
yend,
q,
w,
regi_ids,
cores,
gwk,
slx_lags,
sig2n_k,
robust,
spat_diag,
vm,
name_x,
name_yend,
name_q,
summ,
latex
):
results_p = {}
"""
for r in self.regimes_set:
if system() != 'Windows':
is_win = True
results_p[r] = _work(*(self.y,x,w,regi_ids,r,yend,q,robust,sig2n_k,self.name_ds,self.name_y,name_x,name_yend,name_q,self.name_w,self.name_regimes))
else:
pool = mp.Pool(cores)
results_p[r] = pool.apply_async(_work,args=(self.y,x,w,regi_ids,r,yend,q,robust,sig2n_k,self.name_ds,self.name_y,name_x,name_yend,name_q,self.name_w,self.name_regimes))
is_win = False
"""
x_constant, name_x = REGI.check_const_regi(self, x, name_x, regi_ids)
self.name_x_r = name_x + name_yend
for r in self.regimes_set:
if cores:
pool = mp.Pool(None)
results_p[r] = pool.apply_async(
_work,
args=(
self.y,
x_constant,
w,
regi_ids,
r,
yend,
q,
robust,
sig2n_k,
self.name_ds,
self.name_y,
name_x,
name_yend,
name_q,
self.name_w,
self.name_regimes,
slx_lags
),
)
else:
results_p[r] = _work(
*(
self.y,
x_constant,
w,
regi_ids,
r,
yend,
q,
robust,
sig2n_k,
self.name_ds,
self.name_y,
name_x,
name_yend,
name_q,
self.name_w,
self.name_regimes,
slx_lags
)
)
self.kryd = 0
self.kr = x_constant.shape[1] + yend.shape[1]
self.kf = 0
self.nr = len(self.regimes_set)
self.vm = np.zeros((self.nr * self.kr, self.nr * self.kr), float)
self.betas = np.zeros((self.nr * self.kr, 1), float)
self.u = np.zeros((self.n, 1), float)
self.predy = np.zeros((self.n, 1), float)
"""
if not is_win:
pool.close()
pool.join()
"""
if cores:
pool.close()
pool.join()
results = {}
(
self.name_y,
self.name_x,
self.name_yend,
self.name_q,
self.name_z,
self.name_h,
) = ([], [], [], [], [], [])
counter = 0
self.output = pd.DataFrame(columns=['var_names', 'var_type', 'regime', 'equation'])
for r in self.regimes_set:
"""
if is_win:
results[r] = results_p[r]
else:
results[r] = results_p[r].get()
"""
if not cores:
results[r] = results_p[r]
else:
results[r] = results_p[r].get()
self.vm[
(counter * self.kr) : ((counter + 1) * self.kr),
(counter * self.kr) : ((counter + 1) * self.kr),
] = results[r].vm
self.betas[
(counter * self.kr) : ((counter + 1) * self.kr),
] = results[r].betas
self.u[
regi_ids[r],
] = results[r].u
self.predy[
regi_ids[r],
] = results[r].predy
self.name_y += results[r].name_y
self.name_x += results[r].name_x
self.name_yend += results[r].name_yend
self.name_q += results[r].name_q
self.name_z += results[r].name_z
self.name_h += results[r].name_h
self.output = pd.concat([self.output, pd.DataFrame({'var_names': results[r].name_x+results[r].name_yend,
'var_type': ['x']*len(results[r].name_x)+['yend']*len(results[r].name_yend),
'regime': r, 'equation': r})], ignore_index=True)
counter += 1
self.multi = results
self.hac_var = sphstack(x_constant[:, 1:], q)
if robust == "hac":
hac_multi(self, gwk)
if robust == "ogmm":
set_warn(
self,
"Residuals treated as homoskedastic for the purpose of diagnostics.",
)
self.chow = REGI.Chow(self)
if spat_diag:
self._get_spat_diag_props(results, regi_ids, x_constant, yend, q)
diag_out = _spat_diag_out(self, w, 'yend')
else:
diag_out = None
if summ:
self.output.sort_values(by='regime', inplace=True)
output(reg=self, vm=vm, robust=robust, other_end=diag_out, latex=latex)
def _get_spat_diag_props(self, results, regi_ids, x, yend, q):
self._cache = {}
x = REGI.regimeX_setup(x, self.regimes, [True] * x.shape[1], self.regimes_set)
self.z = sphstack(
x,
REGI.regimeX_setup(
yend, self.regimes, [True] * yend.shape[1], self.regimes_set
),
)
self.h = sphstack(
x,
REGI.regimeX_setup(q, self.regimes, [True] * q.shape[1], self.regimes_set),
)
hthi = np.linalg.inv(spdot(self.h.T, self.h))
zth = spdot(self.z.T, self.h)
self.varb = np.linalg.inv(spdot(spdot(zth, hthi), zth.T))
def _work(
y,
x,
w,
regi_ids,
r,
yend,
q,
robust,
sig2n_k,
name_ds,
name_y,
name_x,
name_yend,
name_q,
name_w,
name_regimes,
slx_lags,
):
y_r = y[regi_ids[r]]
x_r = x[regi_ids[r]]
yend_r = yend[regi_ids[r]]
q_r = q[regi_ids[r]]
if robust == "hac" or robust == "ogmm":
robust2 = None
else:
robust2 = robust
model = BaseTSLS(y_r, x_r, yend_r, q_r, robust=robust2, sig2n_k=sig2n_k)
if slx_lags == 0:
model.title = "TWO STAGE LEAST SQUARES ESTIMATION - REGIME %s" % r
else:
model.title = "TWO STAGE LEAST SQUARES ESTIMATION WITH SLX - REGIME %s" % r
if robust == "ogmm":
_optimal_weight(model, sig2n_k, warn=False)
model.robust = USER.set_robust(robust)
model.name_ds = name_ds
model.name_y = "%s_%s" % (str(r), name_y)
model.name_x = ["%s_%s" % (str(r), i) for i in name_x]
model.name_yend = ["%s_%s" % (str(r), i) for i in name_yend]
model.name_z = model.name_x + model.name_yend
model.name_q = ["%s_%s" % (str(r), i) for i in name_q]
model.name_h = model.name_x + model.name_q
model.name_w = name_w
model.name_regimes = name_regimes
if w:
w_r, warn = REGI.w_regime(w, regi_ids[r], r, transform=True)
set_warn(model, warn)
model.w = w_r
return model
def _optimal_weight(reg, sig2n_k, warn=True):
try:
Hu = reg.h.toarray() * reg.u ** 2
except:
Hu = reg.h * reg.u ** 2
if sig2n_k:
S = spdot(reg.h.T, Hu, array_out=True) / (reg.n - reg.k)
else:
S = spdot(reg.h.T, Hu, array_out=True) / reg.n
Si = np.linalg.inv(S)
ZtH = spdot(reg.z.T, reg.h)
ZtHSi = spdot(ZtH, Si)
fac2 = np.linalg.inv(spdot(ZtHSi, ZtH.T, array_out=True))
fac3 = spdot(ZtHSi, spdot(reg.h.T, reg.y), array_out=True)
betas = np.dot(fac2, fac3)
if sig2n_k:
vm = fac2 * (reg.n - reg.k)
else:
vm = fac2 * reg.n
RegressionProps_basic(reg, betas=betas, vm=vm, sig2=False)
#reg.title += " (Optimal-Weighted GMM)"
if warn:
set_warn(
reg, "Residuals treated as homoskedastic for the purpose of diagnostics."
)
return
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
from libpysal.examples import load_example
nat = load_example("Natregimes")
db = libpysal.io.open(nat.get_path("natregimes.dbf"), "r")
y_var = "HR60"
y = np.array([db.by_col(y_var)]).T
x_var = ["PS60", "DV60", "RD60"]
x = np.array([db.by_col(name) for name in x_var]).T
yd_var = ["UE60"]
yd = np.array([db.by_col(name) for name in yd_var]).T
q_var = ["FP59", "MA60"]
q = np.array([db.by_col(name) for name in q_var]).T
r_var = "SOUTH"
regimes = db.by_col(r_var)
w = libpysal.weights.Rook.from_shapefile(nat.get_path("natregimes.shp"))
w.transform = "r"
tslsr = TSLS_Regimes(
y,
x,
yd,
q,
regimes,
w = w,
constant_regi="many",
spat_diag=True,
name_y=y_var,
name_x=x_var,
name_yend=yd_var,
name_q=q_var,
name_regimes=r_var,
#cols2regi=[False, True, True, False],
sig2n_k=False,
regime_err_sep = True,
#robust = 'hac',
vm = False
)
print(tslsr.output)
print(tslsr.summary)