Spatial two stage least squares (S2SLS) with regimes;
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, excluding the constant
 regimes : list
List of n values with the mapping of each
observation to a regime. Assumed to be aligned with ‘x’.
 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
 constant_regi: [‘one’, ‘many’]
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.
 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).
 regime_lag_sep: boolean
If True (default), the spatial parameter for spatial lag is also
computed according to different regimes. If False,
the spatial parameter is fixed accross regimes.
Option valid only when regime_err_sep=True
 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
variancecovariance matrix is given.
If ‘hac’, then a HAC consistent estimator of the
variancecovariance matrix is given.
If ‘ogmm’, then Optimal GMM is used to estimate
betas and the variancecovariance 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.
 sig2n_k : boolean
If True, then use nk to estimate sigma^2. If False, use n.
 spat_diag : boolean
If True, then compute AnselinKelejian test
 vm : boolean
If True, include variancecovariance matrix in summary
results
 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_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
 name_regimes : string
Name of regimes variable for use in output

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 pysal.lib
>>> from pysal.lib import examples
Open data on NCOVR US County Homicides (3085 areas) using pysal.lib.io.open().
This is the DBF associated with the NAT shapefile. Note that
pysal.lib.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 = pysal.lib.io.open(examples.get_path("NAT.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
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 run a spatial lag model, we need to specify
the spatial weights matrix that includes the spatial configuration of the
observations. 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
.
>>> from pysal.lib import weights
>>> w = weights.Rook.from_shapefile(examples.get_path("NAT.shp"))
Unless there is a good reason not to do it, the weights have to be
rowstandardized 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:
This class runs a lag model, which means that includes the spatial lag of
the dependent variable on the righthand 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.
>>> model=GM_Lag_Regimes(y, x, regimes, w=w, regime_lag_sep=False, regime_err_sep=False, name_y=y_var, name_x=x_var, name_regimes=r_var, name_ds='NAT', name_w='NAT.shp')
>>> model.betas
array([[ 1.28897623],
[ 0.79777722],
[ 0.56366891],
[ 8.73327838],
[ 1.30433406],
[ 0.62418643],
[0.39993716]])
Once the model is run, we can have a summary of the output by typing:
model.summary . Alternatively, we can obtain the standard error of
the coefficient estimates by calling:
>>> model.std_err
array([ 0.44682888, 0.14358192, 0.05655124, 1.06044865, 0.20184548,
0.06118262, 0.12387232])
In the example above, all coefficients but the spatial lag vary
according to the regime. It is also possible to have the spatial lag
varying according to the regime, which effective will result in an
independent spatial lag model estimated for each regime. To run these
models, the argument regime_lag_sep must be set to True:
>>> model=GM_Lag_Regimes(y, x, regimes, w=w, regime_lag_sep=True, name_y=y_var, name_x=x_var, name_regimes=r_var, name_ds='NAT', name_w='NAT.shp')
>>> print np.hstack((np.array(model.name_z).reshape(8,1),model.betas,np.sqrt(model.vm.diagonal().reshape(8,1))))
[['0_CONSTANT' '1.36584769' '0.39854720']
['0_PS90' '0.80875730' '0.11324884']
['0_UE90' '0.56946813' '0.04625087']
['0_W_HR90' '0.4342438' '0.13350159']
['1_CONSTANT' '7.90731073' '1.63601874']
['1_PS90' '1.27465703' '0.24709870']
['1_UE90' '0.60167693' '0.07993322']
['1_W_HR90' '0.2960338' '0.19934459']]
Alternatively, we can type: ‘model.summary’ to see the organized results output.
The class is flexible enough to accomodate a spatial lag model that,
besides the spatial lag of the dependent variable, includes other
nonspatial endogenous regressors. As an example, we will add the endogenous
variable RD90 (resource deprivation) and we decide to instrument for it with
FP89 (families below poverty):
>>> yd_var = ['RD90']
>>> yd = np.array([db.by_col(name) for name in yd_var]).T
>>> q_var = ['FP89']
>>> q = np.array([db.by_col(name) for name in q_var]).T
And we can run the model again:
>>> model = GM_Lag_Regimes(y, x, regimes, yend=yd, q=q, w=w, regime_lag_sep=False, regime_err_sep=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')
>>> model.betas
array([[ 3.42195202],
[ 1.03311878],
[ 0.14308741],
[ 8.99740066],
[ 1.91877758],
[0.32084816],
[ 2.38918212],
[ 3.67243761],
[ 0.06959139]])
Once the model is run, we can obtain the standard error of the coefficient
estimates. Alternatively, we can have a summary of the output by typing:
model.summary
>>> model.std_err
array([ 0.49163311, 0.12237382, 0.05633464, 0.72555909, 0.17250521,
0.06749131, 0.27370369, 0.25106224, 0.05804213])
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)
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 kstar : integer
Number of endogenous variables.
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 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)
 z : array
nxk array of variables (combination of x and yend)
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 h : array
nxl array of instruments (combination of x and q)
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 robust : string
Adjustment for robust standard errors
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 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)
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 pr2_e : float
Pseudo R squared (squared correlation between y and ypred_e
(using reduced form))
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 utu : float
Sum of squared residuals
 sig2 : float
Sigma squared used in computations
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 std_err : array
1xk array of standard errors of the betas
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 z_stat : list of tuples
z statistic; each tuple contains the pair (statistic,
pvalue), where each is a float
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 ak_test : tuple
AnselinKelejian test; tuple contains the pair (statistic,
pvalue)
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 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
 name_regimes : string
Name of regimes variable for use in output
 title : string
Name of the regression method used
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 sig2n : float
Sigma squared (computed with n in the denominator)
 sig2n_k : float
Sigma squared (computed with nk in the denominator)
 hth : float
H’H
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 hthi : float
(H’H)^1
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 varb : array
(Z’H (H’H)^1 H’Z)^1
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 zthhthi : array
Z’H(H’H)^1
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 pfora1a2 : array
n(zthhthi)’varb
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 regimes : list
List of n values with the mapping of each
observation to a regime. Assumed to be aligned with ‘x’.
 constant_regi: [‘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
 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_lag_sep : boolean
If True, the spatial parameter for spatial lag is also
computed according to different regimes. If False (default),
the spatial parameter is fixed accross regimes.
 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
 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

Methods
GM_Lag_Regimes_Multi 

sp_att_reg 


__init__
(y, x, regimes, yend=None, q=None, w=None, w_lags=1, lag_q=True, robust=None, gwk=None, sig2n_k=False, spat_diag=False, constant_regi='many', cols2regi='all', regime_lag_sep=False, regime_err_sep=True, cores=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_regimes=None, name_w=None, name_gwk=None, name_ds=None)[source]
Initialize self. See help(type(self)) for accurate signature.
Methods
GM_Lag_Regimes_Multi (y, x, w_i, w, regi_ids) 

__init__ (y, x, regimes[, yend, q, w, …]) 
Initialize self. 
sp_att_reg (w_i, regi_ids, wy) 

Attributes
mean_y 

pfora1a2 

sig2n 

sig2n_k 

std_y 

utu 

vm 
