pysal.model.spreg.GM_Lag

class pysal.model.spreg.GM_Lag(y, x, yend=None, q=None, w=None, w_lags=1, lag_q=True, robust=None, gwk=None, sig2n_k=False, spat_diag=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_gwk=None, name_ds=None)[source]

Spatial two stage least squares (S2SLS) with results and diagnostics; 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

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

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).

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. 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

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

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. Since we will need some tests for our model, we also import the diagnostics module.

>>> import numpy as np
>>> import pysal.lib
>>> import pysal.model.spreg.diagnostics as D

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 HOVAL column (home value) 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("HOVAL"))
>>> y = np.reshape(y, (49,1))

Extract INC (income) and CRIME (crime rates) vectors 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, but this can be overridden by passing constant=False.

>>> X = []
>>> X.append(db.by_col("INC"))
>>> X.append(db.by_col("CRIME"))
>>> 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'

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. The default most basic model to be run would be:

>>> reg=GM_Lag(y, X, w=w, w_lags=2, name_x=['inc', 'crime'], name_y='hoval', name_ds='columbus')
>>> reg.betas
array([[ 45.30170561],
       [  0.62088862],
       [ -0.48072345],
       [  0.02836221]])

Once the model is run, we can obtain the standard error of the coefficient estimates by calling the diagnostics module:

>>> D.se_betas(reg)
array([ 17.91278862,   0.52486082,   0.1822815 ,   0.31740089])

But we can also run models that incorporates corrected standard errors following the White procedure. For that, we will have to include the optional parameter robust='white':

>>> reg=GM_Lag(y, X, w=w, w_lags=2, robust='white', name_x=['inc', 'crime'], name_y='hoval', name_ds='columbus')
>>> reg.betas
array([[ 45.30170561],
       [  0.62088862],
       [ -0.48072345],
       [  0.02836221]])

And we can access the standard errors from the model object:

>>> reg.std_err
array([ 20.47077481,   0.50613931,   0.20138425,   0.38028295])

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 assume that CRIME is actually endogenous and we decide to instrument for it with DISCBD (distance to the CBD). We reload the X including INC only and define CRIME as endogenous and DISCBD as instrument:

>>> X = np.array(db.by_col("INC"))
>>> X = np.reshape(X, (49,1))
>>> yd = np.array(db.by_col("CRIME"))
>>> yd = np.reshape(yd, (49,1))
>>> q = np.array(db.by_col("DISCBD"))
>>> q = np.reshape(q, (49,1))

And we can run the model again:

>>> reg=GM_Lag(y, X, w=w, yend=yd, q=q, w_lags=2, name_x=['inc'], name_y='hoval', name_yend=['crime'], name_q=['discbd'], name_ds='columbus')
>>> reg.betas
array([[ 100.79359082],
       [  -0.50215501],
       [  -1.14881711],
       [  -0.38235022]])

Once the model is run, we can obtain the standard error of the coefficient estimates by calling the diagnostics module:

>>> D.se_betas(reg)
array([ 53.0829123 ,   1.02511494,   0.57589064,   0.59891744])
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)

kstar : integer

Number of endogenous variables.

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

q : array

Two dimensional array with n rows and one column for each external exogenous variable used as instruments

z : array

nxk array of variables (combination of x and yend)

h : array

nxl array of instruments (combination of x and q)

robust : string

Adjustment for robust standard errors

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))

utu : float

Sum of squared residuals

sig2 : float

Sigma squared used in computations

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

ak_test : tuple

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

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

title : string

Name of the regression method used

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

hthi : float

(H’H)^-1

varb : array

(Z’H (H’H)^-1 H’Z)^-1

zthhthi : array

Z’H(H’H)^-1

pfora1a2 : array

n(zthhthi)’varb

__init__(y, x, yend=None, q=None, w=None, w_lags=1, lag_q=True, robust=None, gwk=None, sig2n_k=False, spat_diag=False, vm=False, name_y=None, name_x=None, name_yend=None, name_q=None, name_w=None, name_gwk=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
pfora1a2
sig2n
sig2n_k
std_y
utu
vm