# -*- coding: utf-8 -*-
"""Various Statistical Tests
Author: josef-pktd
License: BSD-3
Notes
-----
Almost fully verified against R or Gretl, not all options are the same.
In many cases of Lagrange multiplier tests both the LM test and the F test is
returned. In some but not all cases, R has the option to choose the test
statistic. Some alternative test statistic results have not been verified.
TODO
* refactor to store intermediate results
* how easy is it to attach a test that is a class to a result instance,
for example CompareCox as a method compare_cox(self, other) ?
* StatTestMC has been moved and should be deleted
missing:
* pvalues for breaks_hansen
* additional options, compare with R, check where ddof is appropriate
* new tests:
- breaks_ap, more recent breaks tests
- specification tests against nonparametric alternatives
"""
from __future__ import print_function
from statsmodels.compat.python import iteritems, lrange, map
import numpy as np
from scipy import stats
from statsmodels.regression.linear_model import OLS
from statsmodels.tools.tools import add_constant
from statsmodels.tsa.stattools import acf, adfuller
from statsmodels.tsa.tsatools import lagmat
from statsmodels.compat.numpy import np_matrix_rank
#get the old signature back so the examples work
[docs]def unitroot_adf(x, maxlag=None, trendorder=0, autolag='AIC', store=False):
return adfuller(x, maxlag=maxlag, regression=trendorder, autolag=autolag,
store=store, regresults=False)
#TODO: I like the bunch pattern for this too.
[docs]class ResultsStore(object):
def __str__(self):
return self._str
[docs]class CompareCox(object):
'''Cox Test for non-nested models
Parameters
----------
results_x : Result instance
result instance of first model
results_z : Result instance
result instance of second model
attach : bool
Formulas from Greene, section 8.3.4 translated to code
produces correct results for Example 8.3, Greene
'''
[docs] def run(self, results_x, results_z, attach=True):
'''run Cox test for non-nested models
Parameters
----------
results_x : Result instance
result instance of first model
results_z : Result instance
result instance of second model
attach : bool
If true, then the intermediate results are attached to the instance.
Returns
-------
tstat : float
t statistic for the test that including the fitted values of the
first model in the second model has no effect.
pvalue : float
two-sided pvalue for the t statistic
Notes
-----
Tests of non-nested hypothesis might not provide unambiguous answers.
The test should be performed in both directions and it is possible
that both or neither test rejects. see ??? for more information.
References
----------
???
'''
if not np.allclose(results_x.model.endog, results_z.model.endog):
raise ValueError('endogenous variables in models are not the same')
nobs = results_x.model.endog.shape[0]
x = results_x.model.exog
z = results_z.model.exog
sigma2_x = results_x.ssr/nobs
sigma2_z = results_z.ssr/nobs
yhat_x = results_x.fittedvalues
yhat_z = results_z.fittedvalues
res_dx = OLS(yhat_x, z).fit()
err_zx = res_dx.resid
res_xzx = OLS(err_zx, x).fit()
err_xzx = res_xzx.resid
sigma2_zx = sigma2_x + np.dot(err_zx.T, err_zx)/nobs
c01 = nobs/2. * (np.log(sigma2_z) - np.log(sigma2_zx))
v01 = sigma2_x * np.dot(err_xzx.T, err_xzx) / sigma2_zx**2
q = c01 / np.sqrt(v01)
pval = 2*stats.norm.sf(np.abs(q))
if attach:
self.res_dx = res_dx
self.res_xzx = res_xzx
self.c01 = c01
self.v01 = v01
self.q = q
self.pvalue = pval
self.dist = stats.norm
return q, pval
def __call__(self, results_x, results_z):
return self.run(results_x, results_z, attach=False)
compare_cox = CompareCox()
compare_cox.__doc__ = CompareCox.__doc__
[docs]class CompareJ(object):
'''J-Test for comparing non-nested models
Parameters
----------
results_x : Result instance
result instance of first model
results_z : Result instance
result instance of second model
attach : bool
From description in Greene, section 8.3.3
produces correct results for Example 8.3, Greene - not checked yet
#currently an exception, but I don't have clean reload in python session
check what results should be attached
'''
[docs] def run(self, results_x, results_z, attach=True):
'''run J-test for non-nested models
Parameters
----------
results_x : Result instance
result instance of first model
results_z : Result instance
result instance of second model
attach : bool
If true, then the intermediate results are attached to the instance.
Returns
-------
tstat : float
t statistic for the test that including the fitted values of the
first model in the second model has no effect.
pvalue : float
two-sided pvalue for the t statistic
Notes
-----
Tests of non-nested hypothesis might not provide unambiguous answers.
The test should be performed in both directions and it is possible
that both or neither test rejects. see ??? for more information.
References
----------
???
'''
if not np.allclose(results_x.model.endog, results_z.model.endog):
raise ValueError('endogenous variables in models are not the same')
nobs = results_x.model.endog.shape[0]
y = results_x.model.endog
x = results_x.model.exog
z = results_z.model.exog
#sigma2_x = results_x.ssr/nobs
#sigma2_z = results_z.ssr/nobs
yhat_x = results_x.fittedvalues
#yhat_z = results_z.fittedvalues
res_zx = OLS(y, np.column_stack((yhat_x, z))).fit()
self.res_zx = res_zx #for testing
tstat = res_zx.tvalues[0]
pval = res_zx.pvalues[0]
if attach:
self.res_zx = res_zx
self.dist = stats.t(res_zx.df_resid)
self.teststat = tstat
self.pvalue = pval
return tstat, pval
def __call__(self, results_x, results_z):
return self.run(results_x, results_z, attach=False)
compare_j = CompareJ()
compare_j.__doc__ = CompareJ.__doc__
[docs]def acorr_ljungbox(x, lags=None, boxpierce=False):
'''Ljung-Box test for no autocorrelation
Parameters
----------
x : array_like, 1d
data series, regression residuals when used as diagnostic test
lags : None, int or array_like
If lags is an integer then this is taken to be the largest lag
that is included, the test result is reported for all smaller lag length.
If lags is a list or array, then all lags are included up to the largest
lag in the list, however only the tests for the lags in the list are
reported.
If lags is None, then the default maxlag is 12*(nobs/100)^{1/4}
boxpierce : {False, True}
If true, then additional to the results of the Ljung-Box test also the
Box-Pierce test results are returned
Returns
-------
lbvalue : float or array
test statistic
pvalue : float or array
p-value based on chi-square distribution
bpvalue : (optionsal), float or array
test statistic for Box-Pierce test
bppvalue : (optional), float or array
p-value based for Box-Pierce test on chi-square distribution
Notes
-----
Ljung-Box and Box-Pierce statistic differ in their scaling of the
autocorrelation function. Ljung-Box test is reported to have better
small sample properties.
TODO: could be extended to work with more than one series
1d or nd ? axis ? ravel ?
needs more testing
''Verification''
Looks correctly sized in Monte Carlo studies.
not yet compared to verified values
Examples
--------
see example script
References
----------
Greene
Wikipedia
'''
x = np.asarray(x)
nobs = x.shape[0]
if lags is None:
lags = lrange(1,41) #TODO: check default; SS: changed to 40
elif isinstance(lags, int):
lags = lrange(1,lags+1)
maxlag = max(lags)
lags = np.asarray(lags)
acfx = acf(x, nlags=maxlag) # normalize by nobs not (nobs-nlags)
# SS: unbiased=False is default now
# acf2norm = acfx[1:maxlag+1]**2 / (nobs - np.arange(1,maxlag+1))
acf2norm = acfx[1:maxlag+1]**2 / (nobs - np.arange(1,maxlag+1))
qljungbox = nobs * (nobs+2) * np.cumsum(acf2norm)[lags-1]
pval = stats.chi2.sf(qljungbox, lags)
if not boxpierce:
return qljungbox, pval
else:
qboxpierce = nobs * np.cumsum(acfx[1:maxlag+1]**2)[lags-1]
pvalbp = stats.chi2.sf(qboxpierce, lags)
return qljungbox, pval, qboxpierce, pvalbp
[docs]def acorr_lm(x, maxlag=None, autolag='AIC', store=False, regresults=False):
'''Lagrange Multiplier tests for autocorrelation
This is a generic Lagrange Multiplier test for autocorrelation. I don't
have a reference for it, but it returns Engle's ARCH test if x is the
squared residual array. A variation on it with additional exogenous
variables is the Breush-Godfrey autocorrelation test.
Parameters
----------
resid : ndarray, (nobs,)
residuals from an estimation, or time series
maxlag : int
highest lag to use
autolag : None or string
If None, then a fixed number of lags given by maxlag is used.
store : bool
If true then the intermediate results are also returned
Returns
-------
lm : float
Lagrange multiplier test statistic
lmpval : float
p-value for Lagrange multiplier test
fval : float
fstatistic for F test, alternative version of the same test based on
F test for the parameter restriction
fpval : float
pvalue for F test
resstore : instance (optional)
a class instance that holds intermediate results. Only returned if
store=True
See Also
--------
het_arch
acorr_breush_godfrey
acorr_ljung_box
'''
if regresults:
store = True
x = np.asarray(x)
nobs = x.shape[0]
if maxlag is None:
#for adf from Greene referencing Schwert 1989
maxlag = int(np.ceil(12. * np.power(nobs/100., 1/4.)))#nobs//4 #TODO: check default, or do AIC/BIC
xdiff = np.diff(x)
#
xdall = lagmat(x[:,None], maxlag, trim='both')
nobs = xdall.shape[0]
xdall = np.c_[np.ones((nobs,1)), xdall]
xshort = x[-nobs:]
if store: resstore = ResultsStore()
if autolag:
#search for lag length with highest information criteria
#Note: I use the same number of observations to have comparable IC
results = {}
for mlag in range(1, maxlag+1):
results[mlag] = OLS(xshort, xdall[:,:mlag+1]).fit()
if autolag.lower() == 'aic':
bestic, icbestlag = min((v.aic,k) for k,v in iteritems(results))
elif autolag.lower() == 'bic':
icbest, icbestlag = min((v.bic,k) for k,v in iteritems(results))
else:
raise ValueError("autolag can only be None, 'AIC' or 'BIC'")
#rerun ols with best ic
xdall = lagmat(x[:,None], icbestlag, trim='both')
nobs = xdall.shape[0]
xdall = np.c_[np.ones((nobs,1)), xdall]
xshort = x[-nobs:]
usedlag = icbestlag
if regresults:
resstore.results = results
else:
usedlag = maxlag
resols = OLS(xshort, xdall[:,:usedlag+1]).fit()
fval = resols.fvalue
fpval = resols.f_pvalue
lm = nobs * resols.rsquared
lmpval = stats.chi2.sf(lm, usedlag)
# Note: degrees of freedom for LM test is nvars minus constant = usedlags
#return fval, fpval, lm, lmpval
if store:
resstore.resols = resols
resstore.usedlag = usedlag
return lm, lmpval, fval, fpval, resstore
else:
return lm, lmpval, fval, fpval
[docs]def het_arch(resid, maxlag=None, autolag=None, store=False, regresults=False,
ddof=0):
'''Engle's Test for Autoregressive Conditional Heteroscedasticity (ARCH)
Parameters
----------
resid : ndarray, (nobs,)
residuals from an estimation, or time series
maxlag : int
highest lag to use
autolag : None or string
If None, then a fixed number of lags given by maxlag is used.
store : bool
If true then the intermediate results are also returned
ddof : int
Not Implemented Yet
If the residuals are from a regression, or ARMA estimation, then there
are recommendations to correct the degrees of freedom by the number
of parameters that have been estimated, for example ddof=p+a for an
ARMA(p,q) (need reference, based on discussion on R finance mailinglist)
Returns
-------
lm : float
Lagrange multiplier test statistic
lmpval : float
p-value for Lagrange multiplier test
fval : float
fstatistic for F test, alternative version of the same test based on
F test for the parameter restriction
fpval : float
pvalue for F test
resstore : instance (optional)
a class instance that holds intermediate results. Only returned if
store=True
Notes
-----
verified agains R:FinTS::ArchTest
'''
return acorr_lm(resid**2, maxlag=maxlag, autolag=autolag, store=store,
regresults=regresults)
[docs]def acorr_breush_godfrey(results, nlags=None, store=False):
'''Breush Godfrey Lagrange Multiplier tests for residual autocorrelation
Parameters
----------
results : Result instance
Estimation results for which the residuals are tested for serial
correlation
nlags : int
Number of lags to include in the auxiliary regression. (nlags is
highest lag)
store : bool
If store is true, then an additional class instance that contains
intermediate results is returned.
Returns
-------
lm : float
Lagrange multiplier test statistic
lmpval : float
p-value for Lagrange multiplier test
fval : float
fstatistic for F test, alternative version of the same test based on
F test for the parameter restriction
fpval : float
pvalue for F test
resstore : instance (optional)
a class instance that holds intermediate results. Only returned if
store=True
Notes
-----
BG adds lags of residual to exog in the design matrix for the auxiliary
regression with residuals as endog,
see Greene 12.7.1.
References
----------
Greene Econometrics, 5th edition
'''
x = np.asarray(results.resid)
exog_old = results.model.exog
nobs = x.shape[0]
if nlags is None:
#for adf from Greene referencing Schwert 1989
nlags = np.trunc(12. * np.power(nobs/100., 1/4.))#nobs//4 #TODO: check default, or do AIC/BIC
x = np.concatenate((np.zeros(nlags), x))
#xdiff = np.diff(x)
#
xdall = lagmat(x[:,None], nlags, trim='both')
nobs = xdall.shape[0]
xdall = np.c_[np.ones((nobs,1)), xdall]
xshort = x[-nobs:]
exog = np.column_stack((exog_old, xdall))
k_vars = exog.shape[1]
if store: resstore = ResultsStore()
resols = OLS(xshort, exog).fit()
ft = resols.f_test(np.eye(nlags, k_vars, k_vars - nlags))
fval = ft.fvalue
fpval = ft.pvalue
fval = np.squeeze(fval)[()] #TODO: fix this in ContrastResults
fpval = np.squeeze(fpval)[()]
lm = nobs * resols.rsquared
lmpval = stats.chi2.sf(lm, nlags)
# Note: degrees of freedom for LM test is nvars minus constant = usedlags
#return fval, fpval, lm, lmpval
if store:
resstore.resols = resols
resstore.usedlag = nlags
return lm, lmpval, fval, fpval, resstore
else:
return lm, lmpval, fval, fpval
[docs]def het_breushpagan(resid, exog_het):
'''Breush-Pagan Lagrange Multiplier test for heteroscedasticity
The tests the hypothesis that the residual variance does not depend on
the variables in x in the form
:math: \sigma_i = \\sigma * f(\\alpha_0 + \\alpha z_i)
Homoscedasticity implies that $\\alpha=0$
Parameters
----------
resid : arraylike, (nobs,)
For the Breush-Pagan test, this should be the residual of a regression.
If an array is given in exog, then the residuals are calculated by
the an OLS regression or resid on exog. In this case resid should
contain the dependent variable. Exog can be the same as x.
TODO: I dropped the exog option, should I add it back?
exog_het : array_like, (nobs, nvars)
This contains variables that might create data dependent
heteroscedasticity.
Returns
-------
lm : float
lagrange multiplier statistic
lm_pvalue :float
p-value of lagrange multiplier test
fvalue : float
f-statistic of the hypothesis that the error variance does not depend
on x
f_pvalue : float
p-value for the f-statistic
Notes
-----
Assumes x contains constant (for counting dof and calculation of R^2).
In the general description of LM test, Greene mentions that this test
exaggerates the significance of results in small or moderately large
samples. In this case the F-statistic is preferrable.
*Verification*
Chisquare test statistic is exactly (<1e-13) the same result as bptest
in R-stats with defaults (studentize=True).
Implementation
This is calculated using the generic formula for LM test using $R^2$
(Greene, section 17.6) and not with the explicit formula
(Greene, section 11.4.3).
The degrees of freedom for the p-value assume x is full rank.
References
----------
http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test
Greene 5th edition
Breush, Pagan article
'''
x = np.asarray(exog_het)
y = np.asarray(resid)**2
nobs, nvars = x.shape
resols = OLS(y, x).fit()
fval = resols.fvalue
fpval = resols.f_pvalue
lm = nobs * resols.rsquared
# Note: degrees of freedom for LM test is nvars minus constant
return lm, stats.chi2.sf(lm, nvars-1), fval, fpval
[docs]def het_white(resid, exog, retres=False):
'''White's Lagrange Multiplier Test for Heteroscedasticity
Parameters
----------
resid : array_like
residuals, square of it is used as endogenous variable
exog : array_like
possible explanatory variables for variance, squares and interaction
terms are included in the auxilliary regression.
resstore : instance (optional)
a class instance that holds intermediate results. Only returned if
store=True
Returns
-------
lm : float
lagrange multiplier statistic
lm_pvalue :float
p-value of lagrange multiplier test
fvalue : float
f-statistic of the hypothesis that the error variance does not depend
on x. This is an alternative test variant not the original LM test.
f_pvalue : float
p-value for the f-statistic
Notes
-----
assumes x contains constant (for counting dof)
question: does f-statistic make sense? constant ?
References
----------
Greene section 11.4.1 5th edition p. 222
now test statistic reproduces Greene 5th, example 11.3
'''
x = np.asarray(exog)
y = np.asarray(resid)
if x.ndim == 1:
raise ValueError('x should have constant and at least one more variable')
nobs, nvars0 = x.shape
i0,i1 = np.triu_indices(nvars0)
exog = x[:,i0]*x[:,i1]
nobs, nvars = exog.shape
assert nvars == nvars0*(nvars0-1)/2. + nvars0
resols = OLS(y**2, exog).fit()
fval = resols.fvalue
fpval = resols.f_pvalue
lm = nobs * resols.rsquared
# Note: degrees of freedom for LM test is nvars minus constant
#degrees of freedom take possible reduced rank in exog into account
#df_model checks the rank to determine df
#extra calculation that can be removed:
assert resols.df_model == np_matrix_rank(exog) - 1
lmpval = stats.chi2.sf(lm, resols.df_model)
return lm, lmpval, fval, fpval
def _het_goldfeldquandt2_old(y, x, idx, split=None, retres=False):
'''test whether variance is the same in 2 subsamples
Parameters
----------
y : array_like
endogenous variable
x : array_like
exogenous variable, regressors
idx : integer
column index of variable according to which observations are
sorted for the split
split : None or integer or float in intervall (0,1)
index at which sample is split.
If 0<split<1 then split is interpreted as fraction of the observations
in the first sample
retres : boolean
if true, then an instance of a result class is returned,
otherwise 2 numbers, fvalue and p-value, are returned
Returns
-------
(fval, pval) or res
fval : float
value of the F-statistic
pval : float
p-value of the hypothesis that the variance in one subsample is larger
than in the other subsample
res : instance of result class
The class instance is just a storage for the intermediate and final
results that are calculated
Notes
-----
TODO:
add resultinstance - DONE
maybe add drop-middle as option
maybe allow for several breaks
recommendation for users: use this function as pattern for more flexible
split in tests, e.g. drop middle.
can do Chow test for structural break in same way
ran sanity check
'''
x = np.asarray(x)
y = np.asarray(y)
nobs, nvars = x.shape
if split is None:
split = nobs//2
elif (0<split) and (split<1):
split = int(nobs*split)
xsortind = np.argsort(x[:,idx])
y = y[xsortind]
x = x[xsortind,:]
resols1 = OLS(y[:split], x[:split]).fit()
resols2 = OLS(y[split:], x[split:]).fit()
fval = resols1.mse_resid/resols2.mse_resid
if fval>1:
fpval = stats.f.sf(fval, resols1.df_resid, resols2.df_resid)
ordering = 'larger'
else:
fval = 1./fval;
fpval = stats.f.sf(fval, resols2.df_resid, resols1.df_resid)
ordering = 'smaller'
if retres:
res = ResultsStore()
res.__doc__ = 'Test Results for Goldfeld-Quandt test of heterogeneity'
res.fval = fval
res.fpval = fpval
res.df_fval = (resols2.df_resid, resols1.df_resid)
res.resols1 = resols1
res.resols2 = resols2
res.ordering = ordering
res.split = split
#res.__str__
res._str = '''The Goldfeld-Quandt test for null hypothesis that the
variance in the second subsample is %s than in the first subsample:
F-statistic =%8.4f and p-value =%8.4f''' % (ordering, fval, fpval)
return res
else:
return fval, fpval
[docs]class HetGoldfeldQuandt(object):
'''test whether variance is the same in 2 subsamples
Parameters
----------
y : array_like
endogenous variable
x : array_like
exogenous variable, regressors
idx : integer
column index of variable according to which observations are
sorted for the split
split : None or integer or float in intervall (0,1)
index at which sample is split.
If 0<split<1 then split is interpreted as fraction of the observations
in the first sample
drop : None, float or int
If this is not None, then observation are dropped from the middle part
of the sorted series. If 0<split<1 then split is interpreted as fraction
of the number of observations to be dropped.
Note: Currently, observations are dropped between split and
split+drop, where split and drop are the indices (given by rounding if
specified as fraction). The first sample is [0:split], the second
sample is [split+drop:]
alternative : string, 'increasing', 'decreasing' or 'two-sided'
default is increasing. This specifies the alternative for the p-value
calculation.
Returns
-------
(fval, pval) or res
fval : float
value of the F-statistic
pval : float
p-value of the hypothesis that the variance in one subsample is larger
than in the other subsample
res : instance of result class
The class instance is just a storage for the intermediate and final
results that are calculated
Notes
-----
The Null hypothesis is that the variance in the two sub-samples are the
same. The alternative hypothesis, can be increasing, i.e. the variance in
the second sample is larger than in the first, or decreasing or two-sided.
Results are identical R, but the drop option is defined differently.
(sorting by idx not tested yet)
'''
#TODO: can do Chow test for structural break in same way
[docs] def run(self, y, x, idx=None, split=None, drop=None,
alternative='increasing', attach=True):
'''see class docstring'''
x = np.asarray(x)
y = np.asarray(y)#**2
nobs, nvars = x.shape
if split is None:
split = nobs//2
elif (0<split) and (split<1):
split = int(nobs*split)
if drop is None:
start2 = split
elif (0<drop) and (drop<1):
start2 = split + int(nobs*drop)
else:
start2 = split + drop
if not idx is None:
xsortind = np.argsort(x[:,idx])
y = y[xsortind]
x = x[xsortind,:]
resols1 = OLS(y[:split], x[:split]).fit()
resols2 = OLS(y[start2:], x[start2:]).fit()
fval = resols2.mse_resid/resols1.mse_resid
#if fval>1:
if alternative.lower() in ['i', 'inc', 'increasing']:
fpval = stats.f.sf(fval, resols1.df_resid, resols2.df_resid)
ordering = 'increasing'
elif alternative.lower() in ['d', 'dec', 'decreasing']:
fval = fval;
fpval = stats.f.sf(1./fval, resols2.df_resid, resols1.df_resid)
ordering = 'decreasing'
elif alternative.lower() in ['2', '2-sided', 'two-sided']:
fpval_sm = stats.f.cdf(fval, resols2.df_resid, resols1.df_resid)
fpval_la = stats.f.sf(fval, resols2.df_resid, resols1.df_resid)
fpval = 2*min(fpval_sm, fpval_la)
ordering = 'two-sided'
else:
raise ValueError('invalid alternative')
if attach:
res = self
res.__doc__ = 'Test Results for Goldfeld-Quandt test of heterogeneity'
res.fval = fval
res.fpval = fpval
res.df_fval = (resols2.df_resid, resols1.df_resid)
res.resols1 = resols1
res.resols2 = resols2
res.ordering = ordering
res.split = split
#res.__str__
#TODO: check if string works
res._str = '''The Goldfeld-Quandt test for null hypothesis that the
variance in the second subsample is %s than in the first subsample:
F-statistic =%8.4f and p-value =%8.4f''' % (ordering, fval, fpval)
return fval, fpval, ordering
#return self
def __str__(self):
try:
return self._str
except AttributeError:
return repr(self)
#TODO: missing the alternative option in call
def __call__(self, y, x, idx=None, split=None, drop=None,
alternative='increasing'):
return self.run(y, x, idx=idx, split=split, drop=drop, attach=False,
alternative=alternative)
het_goldfeldquandt = HetGoldfeldQuandt()
het_goldfeldquandt.__doc__ = het_goldfeldquandt.run.__doc__
[docs]def linear_harvey_collier(res):
'''Harvey Collier test for linearity
The Null hypothesis is that the regression is correctly modeled as linear.
Parameters
----------
res : Result instance
Returns
-------
tvalue : float
test statistic, based on ttest_1sample
pvalue : float
pvalue of the test
Notes
-----
TODO: add sort_by option
This test is a t-test that the mean of the recursive ols residuals is zero.
Calculating the recursive residuals might take some time for large samples.
'''
#I think this has different ddof than
#B.H. Baltagi, Econometrics, 2011, chapter 8
#but it matches Gretl and R:lmtest, pvalue at decimal=13
rr = recursive_olsresiduals(res, skip=3, alpha=0.95)
from scipy import stats
return stats.ttest_1samp(rr[3][3:], 0)
[docs]def linear_rainbow(res, frac = 0.5):
'''Rainbow test for linearity
The Null hypothesis is that the regression is correctly modelled as linear.
The alternative for which the power might be large are convex, check
Parameters
----------
res : Result instance
Returns
-------
fstat : float
test statistic based of F test
pvalue : float
pvalue of the test
'''
nobs = res.nobs
endog = res.model.endog
exog = res.model.exog
lowidx = np.ceil(0.5 * (1 - frac) * nobs).astype(int)
uppidx = np.floor(lowidx + frac * nobs).astype(int)
mi_sl = slice(lowidx, uppidx)
res_mi = OLS(endog[mi_sl], exog[mi_sl]).fit()
nobs_mi = res_mi.model.endog.shape[0]
ss_mi = res_mi.ssr
ss = res.ssr
fstat = (ss - ss_mi) / (nobs-nobs_mi) / ss_mi * res_mi.df_resid
from scipy import stats
pval = stats.f.sf(fstat, nobs - nobs_mi, res_mi.df_resid)
return fstat, pval
[docs]def linear_lm(resid, exog, func=None):
'''Lagrange multiplier test for linearity against functional alternative
limitations: Assumes currently that the first column is integer.
Currently it doesn't check whether the transformed variables contain NaNs,
for example log of negative number.
Parameters
----------
resid : ndarray
residuals of a regression
exog : ndarray
exogenous variables for which linearity is tested
func : callable
If func is None, then squares are used. func needs to take an array
of exog and return an array of transformed variables.
Returns
-------
lm : float
Lagrange multiplier test statistic
lm_pval : float
p-value of Lagrange multiplier tes
ftest : ContrastResult instance
the results from the F test variant of this test
Notes
-----
written to match Gretl's linearity test.
The test runs an auxilliary regression of the residuals on the combined
original and transformed regressors.
The Null hypothesis is that the linear specification is correct.
'''
from scipy import stats
if func is None:
func = lambda x: np.power(x, 2)
exog_aux = np.column_stack((exog, func(exog[:,1:])))
nobs, k_vars = exog.shape
ls = OLS(resid, exog_aux).fit()
ftest = ls.f_test(np.eye(k_vars - 1, k_vars * 2 - 1, k_vars))
lm = nobs * ls.rsquared
lm_pval = stats.chi2.sf(lm, k_vars - 1)
return lm, lm_pval, ftest
def _neweywestcov(resid, x):
'''
Did not run yet
from regstats2 ::
if idx(29) % HAC (Newey West)
L = round(4*(nobs/100)^(2/9));
% L = nobs^.25; % as an alternative
hhat = repmat(residuals',p,1).*X';
xuux = hhat*hhat';
for l = 1:L;
za = hhat(:,(l+1):nobs)*hhat(:,1:nobs-l)';
w = 1 - l/(L+1);
xuux = xuux + w*(za+za');
end
d = struct;
d.covb = xtxi*xuux*xtxi;
'''
nobs = resid.shape[0] #TODO: check this can only be 1d
nlags = int(round(4*(nobs/100.)**(2/9.)))
hhat = resid * x.T
xuux = np.dot(hhat, hhat.T)
for lag in range(nlags):
za = np.dot(hhat[:,lag:nobs], hhat[:,:nobs-lag].T)
w = 1 - lag/(nobs + 1.)
xuux = xuux + np.dot(w, za+za.T)
xtxi = np.linalg.inv(np.dot(x.T, x)) #QR instead?
covbNW = np.dot(xtxi, np.dot(xuux, xtxi))
return covbNW
def _recursive_olsresiduals2(olsresults, skip):
'''this is my original version based on Greene and references
keep for now for comparison and benchmarking
'''
y = olsresults.model.endog
x = olsresults.model.exog
nobs, nvars = x.shape
rparams = np.nan * np.zeros((nobs,nvars))
rresid = np.nan * np.zeros((nobs))
rypred = np.nan * np.zeros((nobs))
rvarraw = np.nan * np.zeros((nobs))
#XTX = np.zeros((nvars,nvars))
#XTY = np.zeros((nvars))
x0 = x[:skip]
y0 = y[:skip]
XTX = np.dot(x0.T, x0)
XTY = np.dot(x0.T, y0) #xi * y #np.dot(xi, y)
beta = np.linalg.solve(XTX, XTY)
rparams[skip-1] = beta
yipred = np.dot(x[skip-1], beta)
rypred[skip-1] = yipred
rresid[skip-1] = y[skip-1] - yipred
rvarraw[skip-1] = 1+np.dot(x[skip-1],np.dot(np.linalg.inv(XTX),x[skip-1]))
for i in range(skip,nobs):
xi = x[i:i+1,:]
yi = y[i]
xxT = np.dot(xi.T, xi) #xi is 2d 1 row
xy = (xi*yi).ravel() # XTY is 1d #np.dot(xi, yi) #np.dot(xi, y)
print(xy.shape, XTY.shape)
print(XTX)
print(XTY)
beta = np.linalg.solve(XTX, XTY)
rparams[i-1] = beta #this is beta based on info up to t-1
yipred = np.dot(xi, beta)
rypred[i] = yipred
rresid[i] = yi - yipred
rvarraw[i] = 1 + np.dot(xi,np.dot(np.linalg.inv(XTX),xi.T))
XTX += xxT
XTY += xy
i = nobs
beta = np.linalg.solve(XTX, XTY)
rparams[i-1] = beta
rresid_scaled = rresid/np.sqrt(rvarraw) #this is N(0,sigma2) distributed
nrr = nobs-skip
sigma2 = rresid_scaled[skip-1:].var(ddof=1)
rresid_standardized = rresid_scaled/np.sqrt(sigma2) #N(0,1) distributed
rcusum = rresid_standardized[skip-1:].cumsum()
#confidence interval points in Greene p136 looks strange?
#this assumes sum of independent standard normal
#rcusumci = np.sqrt(np.arange(skip,nobs+1))*np.array([[-1.],[+1.]])*stats.norm.sf(0.025)
a = 1.143 #for alpha=0.99 =0.948 for alpha=0.95
#following taken from Ploberger,
crit = a*np.sqrt(nrr)
rcusumci = (a*np.sqrt(nrr) + a*np.arange(0,nobs-skip)/np.sqrt(nrr)) \
* np.array([[-1.],[+1.]])
return (rresid, rparams, rypred, rresid_standardized, rresid_scaled,
rcusum, rcusumci)
[docs]def recursive_olsresiduals(olsresults, skip=None, lamda=0.0, alpha=0.95):
'''calculate recursive ols with residuals and cusum test statistic
Parameters
----------
olsresults : instance of RegressionResults
uses only endog and exog
skip : int or None
number of observations to use for initial OLS, if None then skip is
set equal to the number of regressors (columns in exog)
lamda : float
weight for Ridge correction to initial (X'X)^{-1}
alpha : {0.95, 0.99}
confidence level of test, currently only two values supported,
used for confidence interval in cusum graph
Returns
-------
rresid : array
recursive ols residuals
rparams : array
recursive ols parameter estimates
rypred : array
recursive prediction of endogenous variable
rresid_standardized : array
recursive residuals standardized so that N(0,sigma2) distributed, where
sigma2 is the error variance
rresid_scaled : array
recursive residuals normalize so that N(0,1) distributed
rcusum : array
cumulative residuals for cusum test
rcusumci : array
confidence interval for cusum test, currently hard coded for alpha=0.95
Notes
-----
It produces same recursive residuals as other version. This version updates
the inverse of the X'X matrix and does not require matrix inversion during
updating. looks efficient but no timing
Confidence interval in Greene and Brown, Durbin and Evans is the same as
in Ploberger after a little bit of algebra.
References
----------
jplv to check formulas, follows Harvey
BigJudge 5.5.2b for formula for inverse(X'X) updating
Greene section 7.5.2
Brown, R. L., J. Durbin, and J. M. Evans. “Techniques for Testing the
Constancy of Regression Relationships over Time.”
Journal of the Royal Statistical Society. Series B (Methodological) 37,
no. 2 (1975): 149-192.
'''
y = olsresults.model.endog
x = olsresults.model.exog
nobs, nvars = x.shape
if skip is None:
skip = nvars
rparams = np.nan * np.zeros((nobs,nvars))
rresid = np.nan * np.zeros((nobs))
rypred = np.nan * np.zeros((nobs))
rvarraw = np.nan * np.zeros((nobs))
#intialize with skip observations
x0 = x[:skip]
y0 = y[:skip]
#add Ridge to start (not in jplv
XTXi = np.linalg.inv(np.dot(x0.T, x0)+lamda*np.eye(nvars))
XTY = np.dot(x0.T, y0) #xi * y #np.dot(xi, y)
#beta = np.linalg.solve(XTX, XTY)
beta = np.dot(XTXi, XTY)
#print('beta', beta
rparams[skip-1] = beta
yipred = np.dot(x[skip-1], beta)
rypred[skip-1] = yipred
rresid[skip-1] = y[skip-1] - yipred
rvarraw[skip-1] = 1 + np.dot(x[skip-1],np.dot(XTXi, x[skip-1]))
for i in range(skip,nobs):
xi = x[i:i+1,:]
yi = y[i]
#xxT = np.dot(xi.T, xi) #xi is 2d 1 row
xy = (xi*yi).ravel() # XTY is 1d #np.dot(xi, yi) #np.dot(xi, y)
#print(xy.shape, XTY.shape
#print(XTX
#print(XTY
# get prediction error with previous beta
yipred = np.dot(xi, beta)
rypred[i] = yipred
residi = yi - yipred
rresid[i] = residi
#update beta and inverse(X'X)
tmp = np.dot(XTXi, xi.T)
ft = 1 + np.dot(xi, tmp)
XTXi = XTXi - np.dot(tmp,tmp.T) / ft #BigJudge equ 5.5.15
#print('beta', beta
beta = beta + (tmp*residi / ft).ravel() #BigJudge equ 5.5.14
# #version for testing
# XTY += xy
# beta = np.dot(XTXi, XTY)
# print((tmp*yipred / ft).shape
# print('tmp.shape, ft.shape, beta.shape', tmp.shape, ft.shape, beta.shape
rparams[i] = beta
rvarraw[i] = ft
i = nobs
#beta = np.linalg.solve(XTX, XTY)
#rparams[i] = beta
rresid_scaled = rresid/np.sqrt(rvarraw) #this is N(0,sigma2) distributed
nrr = nobs-skip
#sigma2 = rresid_scaled[skip-1:].var(ddof=1) #var or sum of squares ?
#Greene has var, jplv and Ploberger have sum of squares (Ass.:mean=0)
#Gretl uses: by reverse engineering matching their numbers
sigma2 = rresid_scaled[skip:].var(ddof=1)
rresid_standardized = rresid_scaled/np.sqrt(sigma2) #N(0,1) distributed
rcusum = rresid_standardized[skip-1:].cumsum()
#confidence interval points in Greene p136 looks strange. Cleared up
#this assumes sum of independent standard normal, which does not take into
#account that we make many tests at the same time
#rcusumci = np.sqrt(np.arange(skip,nobs+1))*np.array([[-1.],[+1.]])*stats.norm.sf(0.025)
if alpha == 0.95:
a = 0.948 #for alpha=0.95
elif alpha == 0.99:
a = 1.143 #for alpha=0.99
elif alpha == 0.90:
a = 0.850
else:
raise ValueError('alpha can only be 0.9, 0.95 or 0.99')
#following taken from Ploberger,
crit = a*np.sqrt(nrr)
rcusumci = (a*np.sqrt(nrr) + 2*a*np.arange(0,nobs-skip)/np.sqrt(nrr)) \
* np.array([[-1.],[+1.]])
return (rresid, rparams, rypred, rresid_standardized, rresid_scaled,
rcusum, rcusumci)
[docs]def breaks_hansen(olsresults):
'''test for model stability, breaks in parameters for ols, Hansen 1992
Parameters
----------
olsresults : instance of RegressionResults
uses only endog and exog
Returns
-------
teststat : float
Hansen's test statistic
crit : structured array
critical values at alpha=0.95 for different nvars
pvalue Not yet
ft, s : arrays
temporary return for debugging, will be removed
Notes
-----
looks good in example, maybe not very powerful for small changes in
parameters
According to Greene, distribution of test statistics depends on nvar but
not on nobs.
Test statistic is verified against R:strucchange
References
----------
Greene section 7.5.1, notation follows Greene
'''
y = olsresults.model.endog
x = olsresults.model.exog
resid = olsresults.resid
nobs, nvars = x.shape
resid2 = resid**2
ft = np.c_[x*resid[:,None], (resid2 - resid2.mean())]
s = ft.cumsum(0)
assert (np.abs(s[-1]) < 1e10).all() #can be optimized away
F = nobs*(ft[:,:,None]*ft[:,None,:]).sum(0)
S = (s[:,:,None]*s[:,None,:]).sum(0)
H = np.trace(np.dot(np.linalg.inv(F), S))
crit95 = np.array([(2,1.9),(6,3.75),(15,3.75),(19,4.52)],
dtype = [('nobs',int), ('crit', float)])
#TODO: get critical values from Bruce Hansens' 1992 paper
return H, crit95, ft, s
[docs]def breaks_cusumolsresid(olsresidual, ddof=0):
'''cusum test for parameter stability based on ols residuals
Parameters
----------
olsresiduals : ndarray
array of residuals from an OLS estimation
ddof : int
number of parameters in the OLS estimation, used as degrees of freedom
correction for error variance.
Returns
-------
sup_b : float
test statistic, maximum of absolute value of scaled cumulative OLS
residuals
pval : float
Probability of observing the data under the null hypothesis of no
structural change, based on asymptotic distribution which is a Brownian
Bridge
crit: list
tabulated critical values, for alpha = 1%, 5% and 10%
Notes
-----
tested agains R:strucchange
Not clear: Assumption 2 in Ploberger, Kramer assumes that exog x have
asymptotically zero mean, x.mean(0) = [1, 0, 0, ..., 0]
Is this really necessary? I don't see how it can affect the test statistic
under the null. It does make a difference under the alternative.
Also, the asymptotic distribution of test statistic depends on this.
From examples it looks like there is little power for standard cusum if
exog (other than constant) have mean zero.
References
----------
Ploberger, Werner, and Walter Kramer. “The Cusum Test with Ols Residuals.”
Econometrica 60, no. 2 (March 1992): 271-285.
'''
resid = olsresidual.ravel()
nobs = len(resid)
nobssigma2 = (resid**2).sum()
if ddof > 0:
#print('ddof', ddof, 1. / (nobs - ddof) * nobs
nobssigma2 = nobssigma2 / (nobs - ddof) * nobs
#B is asymptotically a Brownian Bridge
B = resid.cumsum()/np.sqrt(nobssigma2) # use T*sigma directly
sup_b = np.abs(B).max() #asymptotically distributed as standard Brownian Bridge
crit = [(1,1.63), (5, 1.36), (10, 1.22)]
#Note stats.kstwobign.isf(0.1) is distribution of sup.abs of Brownian Bridge
#>>> stats.kstwobign.isf([0.01,0.05,0.1])
#array([ 1.62762361, 1.35809864, 1.22384787])
pval = stats.kstwobign.sf(sup_b)
return sup_b, pval, crit
#def breaks_cusum(recolsresid):
# '''renormalized cusum test for parameter stability based on recursive residuals
#
#
# still incorrect: in PK, the normalization for sigma is by T not T-K
# also the test statistic is asymptotically a Wiener Process, Brownian motion
# not Brownian Bridge
# for testing: result reject should be identical as in standard cusum version
#
# References
# ----------
# Ploberger, Werner, and Walter Kramer. “The Cusum Test with Ols Residuals.”
# Econometrica 60, no. 2 (March 1992): 271-285.
#
# '''
# resid = recolsresid.ravel()
# nobssigma2 = (resid**2).sum()
# #B is asymptotically a Brownian Bridge
# B = resid.cumsum()/np.sqrt(nobssigma2) # use T*sigma directly
# nobs = len(resid)
# denom = 1. + 2. * np.arange(nobs)/(nobs-1.) #not sure about limits
# sup_b = np.abs(B/denom).max()
# #asymptotically distributed as standard Brownian Bridge
# crit = [(1,1.63), (5, 1.36), (10, 1.22)]
# #Note stats.kstwobign.isf(0.1) is distribution of sup.abs of Brownian Bridge
# #>>> stats.kstwobign.isf([0.01,0.05,0.1])
# #array([ 1.62762361, 1.35809864, 1.22384787])
# pval = stats.kstwobign.sf(sup_b)
# return sup_b, pval, crit
[docs]def breaks_AP(endog, exog, skip):
'''supLM, expLM and aveLM by Andrews, and Andrews,Ploberger
p-values by B Hansen
just idea for computation of sequence of tests with given change point
(Chow tests)
run recursive ols both forward and backward, match the two so they form a
split of the data, calculate sum of squares for residuals and get test
statistic for each breakpoint between skip and nobs-skip
need to put recursive ols (residuals) into separate function
alternative: B Hansen loops over breakpoints only once and updates
x'x and xe'xe
update: Andrews is based on GMM estimation not OLS, LM test statistic is
easy to compute because it only requires full sample GMM estimate (p.837)
with GMM the test has much wider applicability than just OLS
for testing loop over single breakpoint Chow test function
'''
pass
#delete when testing is finished
[docs]class StatTestMC(object):
"""class to run Monte Carlo study on a statistical test'''
TODO
print(summary, for quantiles and for histogram
draft in trying out script log
this has been copied to tools/mctools.py, with improvements
"""
[docs] def __init__(self, dgp, statistic):
self.dgp = dgp #staticmethod(dgp) #no self
self.statistic = statistic # staticmethod(statistic) #no self
[docs] def run(self, nrepl, statindices=None, dgpargs=[], statsargs=[]):
'''run the actual Monte Carlo and save results
'''
self.nrepl = nrepl
self.statindices = statindices
self.dgpargs = dgpargs
self.statsargs = statsargs
dgp = self.dgp
statfun = self.statistic # name ?
#single return statistic
if statindices is None:
self.nreturn = nreturns = 1
mcres = np.zeros(nrepl)
for ii in range(nrepl-1):
x = dgp(*dgpargs) #(1e-4+np.random.randn(nobs)).cumsum()
mcres[ii] = statfun(x, *statsargs) #unitroot_adf(x, 2,trendorder=0, autolag=None)
#more than one return statistic
else:
self.nreturn = nreturns = len(statindices)
self.mcres = mcres = np.zeros((nrepl, nreturns))
for ii in range(nrepl-1):
x = dgp(*dgpargs) #(1e-4+np.random.randn(nobs)).cumsum()
ret = statfun(x, *statsargs)
mcres[ii] = [ret[i] for i in statindices]
self.mcres = mcres
[docs] def histogram(self, idx=None, critval=None):
'''calculate histogram values
does not do any plotting
'''
if self.mcres.ndim == 2:
if not idx is None:
mcres = self.mcres[:,idx]
else:
raise ValueError('currently only 1 statistic at a time')
else:
mcres = self.mcres
if critval is None:
histo = np.histogram(mcres, bins=10)
else:
if not critval[0] == -np.inf:
bins=np.r_[-np.inf, critval, np.inf]
if not critval[0] == -np.inf:
bins=np.r_[bins, np.inf]
histo = np.histogram(mcres,
bins=np.r_[-np.inf, critval, np.inf])
self.histo = histo
self.cumhisto = np.cumsum(histo[0])*1./self.nrepl
self.cumhistoreversed = np.cumsum(histo[0][::-1])[::-1]*1./self.nrepl
return histo, self.cumhisto, self.cumhistoreversed
[docs] def quantiles(self, idx=None, frac=[0.01, 0.025, 0.05, 0.1, 0.975]):
'''calculate quantiles of Monte Carlo results
'''
if self.mcres.ndim == 2:
if not idx is None:
mcres = self.mcres[:,idx]
else:
raise ValueError('currently only 1 statistic at a time')
else:
mcres = self.mcres
self.frac = frac = np.asarray(frac)
self.mcressort = mcressort = np.sort(self.mcres)
return frac, mcressort[(self.nrepl*frac).astype(int)]
if __name__ == '__main__':
examples = ['adf']
if 'adf' in examples:
x = np.random.randn(20)
print(acorr_ljungbox(x,4))
print(unitroot_adf(x))
nrepl = 100
nobs = 100
mcres = np.zeros(nrepl)
for ii in range(nrepl-1):
x = (1e-4+np.random.randn(nobs)).cumsum()
mcres[ii] = unitroot_adf(x, 2,trendorder=0, autolag=None)[0]
print((mcres<-2.57).sum())
print(np.histogram(mcres))
mcressort = np.sort(mcres)
for ratio in [0.01, 0.025, 0.05, 0.1]:
print(ratio, mcressort[int(nrepl*ratio)])
print('critical values in Green table 20.5')
print('sample size = 100')
print('with constant')
print('0.01: -19.8, 0.025: -16.3, 0.05: -13.7, 0.01: -11.0, 0.975: 0.47')
print('0.01: -3.50, 0.025: -3.17, 0.05: -2.90, 0.01: -2.58, 0.975: 0.26')
crvdg = dict([map(float,s.split(':')) for s in ('0.01: -19.8, 0.025: -16.3, 0.05: -13.7, 0.01: -11.0, 0.975: 0.47'.split(','))])
crvd = dict([map(float,s.split(':')) for s in ('0.01: -3.50, 0.025: -3.17, 0.05: -2.90, 0.01: -2.58, 0.975: 0.26'.split(','))])
'''
>>> crvd
{0.050000000000000003: -13.699999999999999, 0.97499999999999998: 0.46999999999999997, 0.025000000000000001: -16.300000000000001, 0.01: -11.0}
>>> sorted(crvd.values())
[-16.300000000000001, -13.699999999999999, -11.0, 0.46999999999999997]
'''
#for trend = 0
crit_5lags0p05 =-4.41519 + (-14.0406)/nobs + (-12.575)/nobs**2
print(crit_5lags0p05)
adfstat, _,_,resstore = unitroot_adf(x, 2,trendorder=0, autolag=None, store=1)
print((mcres>crit_5lags0p05).sum())
print(resstore.resols.model.exog[-5:])
print(x[-5:])
print(np.histogram(mcres, bins=[-np.inf, -3.5, -3.17, -2.9 , -2.58, 0.26, np.inf]))
print(mcressort[(nrepl*(np.array([0.01, 0.025, 0.05, 0.1, 0.975]))).astype(int)])
def randwalksim(nobs=100, drift=0.0):
return (drift+np.random.randn(nobs)).cumsum()
def normalnoisesim(nobs=500, loc=0.0):
return (loc+np.random.randn(nobs))
def adf20(x):
return unitroot_adf(x, 2,trendorder=0, autolag=None)[:2]
print('\nResults with MC class')
mc1 = StatTestMC(randwalksim, adf20)
mc1.run(1000, statindices=[0,1])
print(mc1.histogram(0, critval=[-3.5, -3.17, -2.9 , -2.58, 0.26]))
print(mc1.quantiles(0))
print('\nLjung Box')
def lb4(x):
s,p = acorr_ljungbox(x, lags=4)
return s[-1], p[-1]
def lb4(x):
s,p = acorr_ljungbox(x, lags=1)
return s[0], p[0]
print('Results with MC class')
mc1 = StatTestMC(normalnoisesim, lb4)
mc1.run(1000, statindices=[0,1])
print(mc1.histogram(1, critval=[0.01, 0.025, 0.05, 0.1, 0.975]))
print(mc1.quantiles(1))
print(mc1.quantiles(0))
print(mc1.histogram(0))
nobs = 100
x = np.ones((nobs,2))
x[:,1] = np.arange(nobs)/20.
y = x.sum(1) + 1.01*(1+1.5*(x[:,1]>10))*np.random.rand(nobs)
print(het_goldfeldquandt(y,x, 1))
y = x.sum(1) + 1.01*(1+0.5*(x[:,1]>10))*np.random.rand(nobs)
print(het_goldfeldquandt(y,x, 1))
y = x.sum(1) + 1.01*(1-0.5*(x[:,1]>10))*np.random.rand(nobs)
print(het_goldfeldquandt(y,x, 1))
print(het_breushpagan(y,x))
print(het_white(y,x))
f, fp, fo = het_goldfeldquandt(y,x, 1)
print(f, fp)
resgq = het_goldfeldquandt(y,x, 1, retres=True)
print(resgq)
#this is just a syntax check:
print(_neweywestcov(y, x))
resols1 = OLS(y, x).fit()
print(_neweywestcov(resols1.resid, x))
print(resols1.cov_params())
print(resols1.HC0_se)
print(resols1.cov_HC0)
y = x.sum(1) + 10.*(1-0.5*(x[:,1]>10))*np.random.rand(nobs)
print(HetGoldfeldQuandt().run(y,x, 1, alternative='dec'))