class spreg.AKtest(iv, w, case='nosp')[source]

Moran’s I test of spatial autocorrelation for IV estimation. Implemented following the original reference [AK97]


Regression object from TSLS class


Spatial weights instance


Flag for special cases (default to ‘nosp’):

  • ‘nosp’: Only NO spatial end. reg.

  • ‘gen’: General case (spatial lag + end. reg.)


We first need to import the needed modules. Numpy is needed to convert the data we read into arrays that spreg understands and pysal to perform all the analysis. The TSLS is required to run the model on which we will perform the tests.

>>> import numpy as np
>>> import libpysal
>>> from spreg import TSLS, GM_Lag, AKtest

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')

Before being able to apply the diagnostics, we have to run a model and, for that, we need the input variables. 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, but this can be overridden by passing constant=False.

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

In this case, we consider HOVAL (home value) as an endogenous regressor, so we acknowledge that by reading it in a different category.

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

In order to properly account for the endogeneity, we have to pass in the instruments. Let us consider DISCBD (distance to the CBD) is a good one:

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

Now we are good to run the model. It is an easy one line task.

>>> reg = TSLS(y, X, yd, q=q)

Now we are concerned with whether our non-spatial model presents spatial autocorrelation in the residuals. To assess this possibility, we can run the Anselin-Kelejian test, which is a version of the classical LM error test adapted for the case of residuals from an instrumental variables (IV) regression. First we need an extra object, the weights matrix, which 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 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'

We are good to run the test. It is a very simple task:

>>> ak = AKtest(reg, w)

And explore the information obtained:

>>> print('AK test: %f      P-value: %f'%(ak.ak, ak.p))
AK test: 4.642895      P-value: 0.031182

The test also accomodates the case when the residuals come from an IV regression that includes a spatial lag of the dependent variable. The only requirement needed is to modify the case parameter when we call AKtest. First, let us run a spatial lag model:

>>> reg_lag = GM_Lag(y, X, yd, q=q, w=w)

And now we can run the AK test and obtain similar information as in the non-spatial model.

>>> ak_sp = AKtest(reg, w, case='gen')
>>> print('AK test: %f      P-value: %f'%(ak_sp.ak, ak_sp.p))
AK test: 1.157593      P-value: 0.281965

Moran’s I statistic for IV residuals


Square of corrected Moran’s I for residuals \(ak = \dfrac{N imes I^*}{\phi^2}\). Note: if case=’nosp’ then it simplifies to the LMerror


P-value of the test

__init__(iv, w, case='nosp')[source]

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


__init__(iv, w[, case])

Initialize self.