pysal.model.spreg.GM_Lag_Regimes

class pysal.model.spreg.GM_Lag_Regimes(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]

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 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.

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

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 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.

>>> 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 non-spatial 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, p-value), where each is a float Only available in dictionary ‘multi’ when multiple regressions (see ‘multi’ below for details)

ak_test : tuple

Anselin-Kelejian test; tuple contains the pair (statistic, p-value) 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 n-k 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