GMM method for a spatial lag and error model with homoskedasticity,
regimes and endogenous variables, with results and diagnostics;
based on Drukker et al. (2013) [Drukker2013], following Anselin (2011)
[Anselin2011].
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’.
 w : pysal W object
Spatial weights object (always needed)
 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.
 regime_err_sep : boolean
If True, a separate regression is run for each 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.
 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).
 max_iter : int
Maximum number of iterations of steps 2a and 2b from Arraiz
et al. Note: epsilon provides an additional stop condition.
 epsilon : float
Minimum change in lambda required to stop iterations of
steps 2a and 2b from Arraiz et al. Note: max_iter provides
an additional stop condition.
 A1 : string
If A1=’het’, then the matrix A1 is defined as in Arraiz et
al. If A1=’hom’, then as in Anselin (2011). If
A1=’hom_sc’, then as in Drukker, Egger and Prucha (2010)
and Drukker, Prucha and Raciborski (2010).
 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_ds : string
Name of dataset for use in output
 name_regimes : string
Name of regime variable for use in the 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
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(pysal.lib.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 combo 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
.
>>> w = pysal.lib.weights.Rook.from_shapefile(pysal.lib.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:
We are all set with the preliminaries, 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.
Example only with spatial lag
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. We can have a
summary of the output by typing: model.summary
Alternatively, we can check the betas:
>>> reg = GM_Combo_Hom_Regimes(y, x, regimes, w=w, A1='hom_sc', name_y=y_var, name_x=x_var, name_regimes=r_var, name_ds='NAT')
>>> print reg.name_z
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', '_Global_W_HR90', 'lambda']
>>> print np.around(reg.betas,4)
[[ 1.4607]
[ 0.9579]
[ 0.5658]
[ 9.1129]
[ 1.1339]
[ 0.6517]
[0.4583]
[ 0.6634]]
This class also allows the user to run a spatial lag+error model with the
extra feature of including nonspatial endogenous regressors. This means
that, in addition to the spatial lag and error, we consider some of the
variables on the righthand side of the equation as endogenous and we
instrument for this. In this case we consider RD90 (resource deprivation)
as an endogenous regressor. We use FP89 (families below poverty)
for this and hence put it in the instruments parameter, ‘q’.
>>> 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 then we can run and explore the model analogously to the previous combo:
>>> reg = GM_Combo_Hom_Regimes(y, x, regimes, yd, q, w=w, A1='hom_sc', name_y=y_var, name_x=x_var, name_yend=yd_var, name_q=q_var, name_regimes=r_var, name_ds='NAT')
>>> print reg.name_z
['0_CONSTANT', '0_PS90', '0_UE90', '1_CONSTANT', '1_PS90', '1_UE90', '0_RD90', '1_RD90', '_Global_W_HR90', 'lambda']
>>> print reg.betas
[[ 3.4196478 ]
[ 1.04065595]
[ 0.16630304]
[ 8.86570777]
[ 1.85134286]
[0.24921597]
[ 2.43007651]
[ 3.61656899]
[ 0.03315061]
[ 0.22636055]]
>>> print np.sqrt(reg.vm.diagonal())
[ 0.53989913 0.13506086 0.06143434 0.77049956 0.18089997 0.07246848
0.29218837 0.25378655 0.06184801 0.06323236]
>>> print 'lambda: ', np.around(reg.betas[1], 4)
lambda: [ 0.2264]
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)
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)
 iter_stop : string
Stop criterion reached during iteration of steps 2a and 2b
from Arraiz et al.
Only available in dictionary ‘multi’ when multiple regressions
(see ‘multi’ below for details)
 iteration : integer
Number of iterations of steps 2a and 2b from Arraiz et al.
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)
 sig2 : float
Sigma squared used in computations (based on filtered
residuals)
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)
 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
 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)
 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_err_sep : boolean
If True, a separate regression is run for each 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.
 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


__init__
(y, x, regimes, yend=None, q=None, w=None, w_lags=1, lag_q=True, cores=False, max_iter=1, epsilon=1e05, A1='het', constant_regi='many', cols2regi='all', regime_err_sep=False, regime_lag_sep=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_ds=None, name_regimes=None)[source]
Initialize self. See help(type(self)) for accurate signature.
Methods
__init__ (y, x, regimes[, yend, q, w, …]) 
Initialize self. 
Attributes