spreg.GM_Lag¶

class
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) [Ans88]
 Parameters
 yarray
nx1 array for dependent variable
 xarray
Two dimensional array with n rows and one column for each independent (exogenous) variable, excluding the constant
 yendarray
Two dimensional array with n rows and one column for each endogenous variable
 qarray
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
 wpysal W object
Spatial weights object
 w_lagsinteger
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_qboolean
If True, then include spatial lags of the additional instruments (q).
 robuststring
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. Default set to None.
 gwkpysal W object
Kernel spatial weights needed for HAC estimation. Note: matrix must have ones along the main diagonal.
 sig2n_kboolean
If True, then use nk to estimate sigma^2. If False, use n.
 spat_diagboolean
If True, then compute AnselinKelejian test
 vmboolean
If True, include variancecovariance matrix in summary results
 name_ystring
Name of dependent variable for use in output
 name_xlist of strings
Names of independent variables for use in output
 name_yendlist of strings
Names of endogenous variables for use in output
 name_qlist of strings
Names of instruments for use in output
 name_wstring
Name of weights matrix for use in output
 name_gwkstring
Name of kernel weights matrix for use in output
 name_dsstring
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 andpysal
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 libpysal >>> import spreg.diagnostics as D
Open data on Columbus neighborhood crime (49 areas) using libpysal.io.open(). This is the DBF associated with the Columbus shapefile. Note that libpysal.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 = libpysal.io.open(libpysal.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 = libpysal.weights.Rook.from_shapefile(libpysal.examples.get_path("columbus.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:
>>> w.transform = 'r'
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. 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 nonspatial 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
 summarystring
Summary of regression results and diagnostics (note: use in conjunction with the print command)
 betasarray
kx1 array of estimated coefficients
 uarray
nx1 array of residuals
 e_predarray
nx1 array of residuals (using reduced form)
 predyarray
nx1 array of predicted y values
 predy_earray
nx1 array of predicted y values (using reduced form)
 ninteger
Number of observations
 kinteger
Number of variables for which coefficients are estimated (including the constant)
 kstarinteger
Number of endogenous variables.
 yarray
nx1 array for dependent variable
 xarray
Two dimensional array with n rows and one column for each independent (exogenous) variable, including the constant
 yendarray
Two dimensional array with n rows and one column for each endogenous variable
 qarray
Two dimensional array with n rows and one column for each external exogenous variable used as instruments
 zarray
nxk array of variables (combination of x and yend)
 harray
nxl array of instruments (combination of x and q)
 robuststring
Adjustment for robust standard errors
 mean_yfloat
Mean of dependent variable
 std_yfloat
Standard deviation of dependent variable
 vmarray
Variance covariance matrix (kxk)
 pr2float
Pseudo R squared (squared correlation between y and ypred)
 pr2_efloat
Pseudo R squared (squared correlation between y and ypred_e (using reduced form))
 utufloat
Sum of squared residuals
 sig2float
Sigma squared used in computations
 std_errarray
1xk array of standard errors of the betas
 z_statlist of tuples
z statistic; each tuple contains the pair (statistic, pvalue), where each is a float
 ak_testtuple
AnselinKelejian test; tuple contains the pair (statistic, pvalue)
 name_ystring
Name of dependent variable for use in output
 name_xlist of strings
Names of independent variables for use in output
 name_yendlist of strings
Names of endogenous variables for use in output
 name_zlist of strings
Names of exogenous and endogenous variables for use in output
 name_qlist of strings
Names of external instruments
 name_hlist of strings
Names of all instruments used in ouput
 name_wstring
Name of weights matrix for use in output
 name_gwkstring
Name of kernel weights matrix for use in output
 name_dsstring
Name of dataset for use in output
 titlestring
Name of the regression method used
 sig2nfloat
Sigma squared (computed with n in the denominator)
 sig2n_kfloat
Sigma squared (computed with nk in the denominator)
 hthfloat
\(H'H\)
 hthifloat
\((H'H)^{1}\)
 varbarray
\((Z'H (H'H)^{1} H'Z)^{1}\)
 zthhthiarray
\(Z'H(H'H)^{1}\)
 pfora1a2array
n(zthhthi)’varb

__init__
(self, 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__
(self, y, x[, yend, q, w, w_lags, …])Initialize self.
Attributes
mean_y
pfora1a2
sig2n
sig2n_k
std_y
utu
vm