def __init__(self, iv, w, case='nosp'):
if case == 'gen':
cache = spDcache(iv, w)
self.mi, self.ak, self.p = akTest(iv, w, cache)
elif case == 'nosp':
cache = spDcache(iv, w)
self.mi = get_mI(iv, w, cache)
self.ak, self.p = lmErr(iv, w, cache)
else:
print """\n
def akTest(iv, w, spDcache):
"""
Computes AK-test for the general case (end. reg. + sp. lag)
...
Parameters
----------
iv : STSLS_dev
Instance from spatial 2SLS regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Attributes
----------
mi : float
Moran's I statistic for IV residuals
ak : float
Square of corrected Moran's I for residuals::
.. math::
ak = \dfrac{N \times I^*}{\phi^2}
p : float
P-value of the test
ToDo:
* Code in as Nancy
* Compare both
"""
mi = get_mI(iv, w, spDcache)
# Phi2
etwz = np.dot(iv.u.T, (w.sparse * iv.z))
a = np.dot(etwz, np.dot(iv.varb, etwz.T))
s12 = (w.s0 / w.n)**2
phi2 = (spDcache.t + (4.0 / iv.sig2n) * a) / (s12 * w.n)
ak = w.n * mi**2 / phi2
pval = chisqprob(ak, 1)
return (mi, ak[0][0], pval[0][0])
def akTest(iv, w, spDcache):
"""
Computes AK-test for the general case (end. reg. + sp. lag)
...
Parameters
----------
iv : STSLS_dev
Instance from spatial 2SLS regression
w : W
Spatial weights instance
spDcache : spDcache
Instance of spDcache class
Attributes
----------
mi : float
Moran's I statistic for IV residuals
ak : float
Square of corrected Moran's I for residuals::
.. math::
ak = \dfrac{N \times I^*}{\phi^2}
p : float
P-value of the test
ToDo:
* Code in as Nancy
* Compare both
"""
mi = get_mI(iv, w, spDcache)
# Phi2
etwz = np.dot(iv.u.T, (w.sparse * iv.z))
a = np.dot(etwz,np.dot(iv.varb,etwz.T))
s12 = (w.s0 / w.n)**2
phi2 = ( spDcache.t + (4.0 / iv.sig2n) * a ) / (s12 * w.n)
ak = w.n * mi**2 / phi2
pval = chisqprob(ak, 1)
return (mi, ak[0][0], pval[0][0])
class AKtest:
"""
Moran's I test of spatial autocorrelation for IV estimation.
Implemented following the original reference Anselin and Kelejian
(1997) [1]_
...
Parameters
----------
iv : TSLS
Regression object from TSLS class
w : W
Spatial weights instance
case : string
Flag for special cases (default to 'nosp'):
* 'nosp': Only NO spatial end. reg.
* 'gen': General case (spatial lag + end. reg.)
Attributes
----------
mi : float
Moran's I statistic for IV residuals
ak : float
Square of corrected Moran's I for residuals::
.. math::
ak = \dfrac{N \times I^*}{\phi^2}
Note: if case='nosp' then it simplifies to the LMerror
p : float
P-value of the test
References
----------
.. [1] Anselin, L., Kelejian, H. (1997) "Testing for spatial error
autocorrelation in the presence of endogenous regressors". Interregional
Regional Science Review, 20, 1.
.. [2] Kelejian, H.H., Prucha, I.R. and Yuzefovich, Y. (2004)
"Instrumental variable estimation of a spatial autorgressive model with
autoregressive disturbances: large and small sample results". Advances in
Econometrics, 18, 163-198.
Examples
--------
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 pysal
>>> from twosls import TSLS
>>> from twosls_sp import GM_Lag
Open data on Columbus neighborhood crime (49 areas) using pysal.open().
This is the DBF associated with the Columbus shapefile. Note that
pysal.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.open(pysal.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 = pysal.rook_from_shapefile(pysal.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\tP-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\tP-value: %f'%(ak_sp.ak, ak_sp.p))
AK test: 1.157593 P-value: 0.281965
"""
def __init__(self, iv, w, case='nosp'):
if case == 'gen':
cache = spDcache(iv, w)
self.mi, self.ak, self.p = akTest(iv, w, cache)
elif case == 'nosp':
cache = spDcache(iv, w)
self.mi = get_mI(iv, w, cache)
self.ak, self.p = lmErr(iv, w, cache)
else:
print """\n