pysal.model.spreg.GM_Combo

class pysal.model.spreg.GM_Combo(y, x, yend=None, q=None, w=None, w_lags=1, lag_q=True, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_ds=None)[source]

GMM method for a spatial lag and error model with endogenous variables, with results and diagnostics; based on Kelejian and Prucha (1998, 1999) [Kelejian1998] [Kelejian1999].

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 (always needed)

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).

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_ds : string

Name of dataset 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

Open data on Columbus neighborhood crime (49 areas) using pysal.lib.io.open(). This is the DBF associated with the Columbus 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("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.

>>> X = []
>>> X.append(db.by_col("INC"))
>>> X = np.array(X).T

Since we want to run a spatial error model, 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 columbus.shp.

>>> w = pysal.lib.weights.Rook.from_shapefile(pysal.lib.examples.get_path("columbus.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'

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.

>>> reg = GM_Combo(y, X, w=w, name_y='crime', name_x=['income'], name_ds='columbus')

Once we have run the model, we can explore a little bit the output. The regression object we have created has many attributes so take your time to discover them. Note that because we are running the classical GMM error model from 1998/99, the spatial parameter is obtained as a point estimate, so although you get a value for it (there are for coefficients under model.betas), you cannot perform inference on it (there are only three values in model.se_betas). Also, this regression uses a two stage least squares estimation method that accounts for the endogeneity created by the spatial lag of the dependent variable. We can check the betas:

>>> print reg.name_z
['CONSTANT', 'income', 'W_crime', 'lambda']
>>> print np.around(np.hstack((reg.betas[:-1],np.sqrt(reg.vm.diagonal()).reshape(3,1))),3)
[[ 39.059  11.86 ]
 [ -1.404   0.391]
 [  0.467   0.2  ]]

And lambda:

>>> print 'lambda: ', np.around(reg.betas[-1], 3)
lambda:  [-0.048]

This class also allows the user to run a spatial lag+error model with the extra feature of including non-spatial endogenous regressors. This means that, in addition to the spatial lag and error, we consider some of the variables on the right-hand side of the equation as endogenous and we instrument for this. As an example, we will include HOVAL (home value) as endogenous and will instrument with DISCBD (distance to the CSB). We first need to read in the variables:

>>> yd = []
>>> yd.append(db.by_col("HOVAL"))
>>> yd = np.array(yd).T
>>> q = []
>>> q.append(db.by_col("DISCBD"))
>>> q = np.array(q).T

And then we can run and explore the model analogously to the previous combo:

>>> reg = GM_Combo(y, X, yd, q, w=w, name_x=['inc'], name_y='crime', name_yend=['hoval'], name_q=['discbd'], name_ds='columbus')
>>> print reg.name_z
['CONSTANT', 'inc', 'hoval', 'W_crime', 'lambda']
>>> names = np.array(reg.name_z).reshape(5,1)
>>> print np.hstack((names[0:4,:], np.around(np.hstack((reg.betas[:-1], np.sqrt(reg.vm.diagonal()).reshape(4,1))),4)))
[['CONSTANT' '50.0944' '14.3593']
 ['inc' '-0.2552' '0.5667']
 ['hoval' '-0.6885' '0.3029']
 ['W_crime' '0.4375' '0.2314']]
>>> print 'lambda: ', np.around(reg.betas[-1], 3)
lambda:  [ 0.254]
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)

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

z : array

nxk array of variables (combination of x and yend)

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)

pr2_e : float

Pseudo R squared (squared correlation between y and ypred_e (using reduced form))

sig2 : float

Sigma squared used in computations (based on filtered residuals)

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

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

title : string

Name of the regression method used

__init__(y, x, yend=None, q=None, w=None, w_lags=1, lag_q=True, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_ds=None)[source]

Initialize self. See help(type(self)) for accurate signature.

Methods

__init__(y, x[, yend, q, w, w_lags, lag_q, …]) Initialize self.

Attributes

mean_y
std_y