# -*- coding: utf-8 -*-
"""Sandwich covariance estimators
Created on Sun Nov 27 14:10:57 2011
Author: Josef Perktold
Author: Skipper Seabold for HCxxx in linear_model.RegressionResults
License: BSD-3
Notes
-----
for calculating it, we have two versions
version 1: use pinv
pinv(x) scale pinv(x) used currently in linear_model, with scale is
1d (or diagonal matrix)
(x'x)^(-1) x' scale x (x'x)^(-1), scale in general is (nobs, nobs) so
pretty large general formulas for scale in cluster case are in [4],
which can be found (as of 2017-05-20) at
http://www.tandfonline.com/doi/abs/10.1198/jbes.2010.07136
This paper also has the second version.
version 2:
(x'x)^(-1) S (x'x)^(-1) with S = x' scale x, S is (kvar,kvars),
(x'x)^(-1) is available as normalized_covparams.
S = sum (x*u) dot (x*u)' = sum x*u*u'*x' where sum here can aggregate
over observations or groups. u is regression residual.
x is (nobs, k_var)
u is (nobs, 1)
x*u is (nobs, k_var)
For cluster robust standard errors, we first sum (x*w) over other groups
(including time) and then take the dot product (sum of outer products)
S = sum_g(x*u)' dot sum_g(x*u)
For HAC by clusters, we first sum over groups for each time period, and then
use HAC on the group sums of (x*w).
If we have several groups, we have to sum first over all relevant groups, and
then take the outer product sum. This can be done by summing using indicator
functions or matrices or with explicit loops. Alternatively we calculate
separate covariance matrices for each group, sum them and subtract the
duplicate counted intersection.
Not checked in details yet: degrees of freedom or small sample correction
factors, see (two) references (?)
This is the general case for MLE and GMM also
in MLE hessian H, outerproduct of jacobian S, cov_hjjh = HJJH,
which reduces to the above in the linear case, but can be used
generally, e.g. in discrete, and is misnomed in GenericLikelihoodModel
in GMM it's similar but I would have to look up the details, (it comes
out in sandwich form by default, it's in the sandbox), standard Newey
West or similar are on the covariance matrix of the moment conditions
quasi-MLE: MLE with mis-specified model where parameter estimates are
fine (consistent ?) but cov_params needs to be adjusted similar or
same as in sandwiches. (I didn't go through any details yet.)
TODO
----
* small sample correction factors, Done for cluster, not yet for HAC
* automatic lag-length selection for Newey-West HAC,
-> added: nlag = floor[4(T/100)^(2/9)] Reference: xtscc paper, Newey-West
note this will not be optimal in the panel context, see Peterson
* HAC should maybe return the chosen nlags
* get consistent notation, varies by paper, S, scale, sigma?
* replace diag(hat_matrix) calculations in cov_hc2, cov_hc3
References
----------
[1] John C. Driscoll and Aart C. Kraay, “Consistent Covariance Matrix Estimation
with Spatially Dependent Panel Data,” Review of Economics and Statistics 80,
no. 4 (1998): 549-560.
[2] Daniel Hoechle, "Robust Standard Errors for Panel Regressions with
Cross-Sectional Dependence", The Stata Journal
[3] Mitchell A. Petersen, “Estimating Standard Errors in Finance Panel Data
Sets: Comparing Approaches,” Review of Financial Studies 22, no. 1
(January 1, 2009): 435 -480.
[4] A. Colin Cameron, Jonah B. Gelbach, and Douglas L. Miller, “Robust Inference
With Multiway Clustering,” Journal of Business and Economic Statistics 29
(April 2011): 238-249.
not used yet:
A.C. Cameron, J.B. Gelbach, and D.L. Miller, “Bootstrap-based improvements
for inference with clustered errors,” The Review of Economics and
Statistics 90, no. 3 (2008): 414–427.
"""
from statsmodels.compat.python import range
import numpy as np
from statsmodels.tools.grouputils import combine_indices, group_sums
from statsmodels.stats.moment_helpers import se_cov
__all__ = ['cov_cluster', 'cov_cluster_2groups', 'cov_hac', 'cov_nw_panel',
'cov_white_simple',
'cov_hc0', 'cov_hc1', 'cov_hc2', 'cov_hc3',
'se_cov', 'weights_bartlett', 'weights_uniform']
#----------- from linear_model.RegressionResults
'''
HC0_se
White's (1980) heteroskedasticity robust standard errors.
Defined as sqrt(diag(X.T X)^(-1)X.T diag(e_i^(2)) X(X.T X)^(-1)
where e_i = resid[i]
HC0_se is a property. It is not evaluated until it is called.
When it is called the RegressionResults instance will then have
another attribute cov_HC0, which is the full heteroskedasticity
consistent covariance matrix and also `het_scale`, which is in
this case just resid**2. HCCM matrices are only appropriate for OLS.
HC1_se
MacKinnon and White's (1985) alternative heteroskedasticity robust
standard errors.
Defined as sqrt(diag(n/(n-p)*HC_0)
HC1_se is a property. It is not evaluated until it is called.
When it is called the RegressionResults instance will then have
another attribute cov_HC1, which is the full HCCM and also `het_scale`,
which is in this case n/(n-p)*resid**2. HCCM matrices are only
appropriate for OLS.
HC2_se
MacKinnon and White's (1985) alternative heteroskedasticity robust
standard errors.
Defined as (X.T X)^(-1)X.T diag(e_i^(2)/(1-h_ii)) X(X.T X)^(-1)
where h_ii = x_i(X.T X)^(-1)x_i.T
HC2_se is a property. It is not evaluated until it is called.
When it is called the RegressionResults instance will then have
another attribute cov_HC2, which is the full HCCM and also `het_scale`,
which is in this case is resid^(2)/(1-h_ii). HCCM matrices are only
appropriate for OLS.
HC3_se
MacKinnon and White's (1985) alternative heteroskedasticity robust
standard errors.
Defined as (X.T X)^(-1)X.T diag(e_i^(2)/(1-h_ii)^(2)) X(X.T X)^(-1)
where h_ii = x_i(X.T X)^(-1)x_i.T
HC3_se is a property. It is not evaluated until it is called.
When it is called the RegressionResults instance will then have
another attribute cov_HC3, which is the full HCCM and also `het_scale`,
which is in this case is resid^(2)/(1-h_ii)^(2). HCCM matrices are
only appropriate for OLS.
'''
# Note: HCCM stands for Heteroskedasticity Consistent Covariance Matrix
def _HCCM(results, scale):
'''
sandwich with pinv(x) * diag(scale) * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs,)
'''
H = np.dot(results.model.pinv_wexog,
scale[:,None]*results.model.pinv_wexog.T)
return H
[docs]def cov_hc0(results):
"""
See statsmodels.RegressionResults
"""
het_scale = results.resid**2 # or whitened residuals? only OLS?
cov_hc0 = _HCCM(results, het_scale)
return cov_hc0
[docs]def cov_hc1(results):
"""
See statsmodels.RegressionResults
"""
het_scale = results.nobs/(results.df_resid)*(results.resid**2)
cov_hc1 = _HCCM(results, het_scale)
return cov_hc1
[docs]def cov_hc2(results):
"""
See statsmodels.RegressionResults
"""
# probably could be optimized
h = np.diag(np.dot(results.model.exog,
np.dot(results.normalized_cov_params,
results.model.exog.T)))
het_scale = results.resid**2/(1-h)
cov_hc2_ = _HCCM(results, het_scale)
return cov_hc2_
[docs]def cov_hc3(results):
"""
See statsmodels.RegressionResults
"""
# above probably could be optimized to only calc the diag
h = np.diag(np.dot(results.model.exog,
np.dot(results.normalized_cov_params,
results.model.exog.T)))
het_scale=(results.resid/(1-h))**2
cov_hc3_ = _HCCM(results, het_scale)
return cov_hc3_
#---------------------------------------
def _get_sandwich_arrays(results, cov_type=''):
"""Helper function to get scores from results
Parameters
"""
if isinstance(results, tuple):
# assume we have jac and hessian_inv
jac, hessian_inv = results
xu = jac = np.asarray(jac)
hessian_inv = np.asarray(hessian_inv)
elif hasattr(results, 'model'):
if hasattr(results, '_results'):
# remove wrapper
results = results._results
# assume we have a results instance
if hasattr(results.model, 'jac'):
xu = results.model.jac(results.params)
hessian_inv = np.linalg.inv(results.model.hessian(results.params))
elif hasattr(results.model, 'score_obs'):
xu = results.model.score_obs(results.params)
hessian_inv = np.linalg.inv(results.model.hessian(results.params))
else:
xu = results.model.wexog * results.wresid[:, None]
hessian_inv = np.asarray(results.normalized_cov_params)
# experimental support for freq_weights
if hasattr(results.model, 'freq_weights') and not cov_type == 'clu':
# we don't want to square the weights in the covariance calculations
# assumes that freq_weights are incorporated in score_obs or equivalent
# assumes xu/score_obs is 2D
# temporary asarray
xu /= np.sqrt(np.asarray(results.model.freq_weights)[:, None])
else:
raise ValueError('need either tuple of (jac, hessian_inv) or results' +
'instance')
return xu, hessian_inv
def _HCCM1(results, scale):
'''
sandwich with pinv(x) * scale * pinv(x).T
where pinv(x) = (X'X)^(-1) X
and scale is (nobs, nobs), or (nobs,) with diagonal matrix diag(scale)
Parameters
----------
results : result instance
need to contain regression results, uses results.model.pinv_wexog
scale : ndarray (nobs,) or (nobs, nobs)
scale matrix, treated as diagonal matrix if scale is one-dimensional
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates
'''
if scale.ndim == 1:
H = np.dot(results.model.pinv_wexog,
scale[:,None]*results.model.pinv_wexog.T)
else:
H = np.dot(results.model.pinv_wexog,
np.dot(scale, results.model.pinv_wexog.T))
return H
def _HCCM2(hessian_inv, scale):
'''
sandwich with (X'X)^(-1) * scale * (X'X)^(-1)
scale is (kvars, kvars)
this uses results.normalized_cov_params for (X'X)^(-1)
Parameters
----------
results : result instance
need to contain regression results, uses results.normalized_cov_params
scale : ndarray (k_vars, k_vars)
scale matrix
Returns
-------
H : ndarray (k_vars, k_vars)
robust covariance matrix for the parameter estimates
'''
if scale.ndim == 1:
scale = scale[:,None]
xxi = hessian_inv
H = np.dot(np.dot(xxi, scale), xxi.T)
return H
#TODO: other kernels, move ?
def weights_bartlett(nlags):
'''Bartlett weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for Bartlett kernel
'''
#with lag zero
return 1 - np.arange(nlags+1)/(nlags+1.)
def weights_uniform(nlags):
'''uniform weights for HAC
this will be moved to another module
Parameters
----------
nlags : int
highest lag in the kernel window, this does not include the zero lag
Returns
-------
kernel : ndarray, (nlags+1,)
weights for uniform kernel
'''
#with lag zero
return np.ones(nlags+1)
kernel_dict = {'bartlett': weights_bartlett,
'uniform': weights_uniform}
def S_hac_simple(x, nlags=None, weights_func=weights_bartlett):
'''inner covariance matrix for HAC (Newey, West) sandwich
assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor(4(T/100)^(2/9)) is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
used by cov_hac_simple
options might change when other kernels besides Bartlett are available.
'''
if x.ndim == 1:
x = x[:,None]
n_periods = x.shape[0]
if nlags is None:
nlags = int(np.floor(4 * (n_periods / 100.)**(2./9.)))
weights = weights_func(nlags)
S = weights[0] * np.dot(x.T, x) #weights[0] just for completeness, is 1
for lag in range(1, nlags+1):
s = np.dot(x[lag:].T, x[:-lag])
S += weights[lag] * (s + s.T)
return S
def S_white_simple(x):
'''inner covariance matrix for White heteroscedastistity sandwich
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
Notes
-----
this is just dot(X.T, X)
'''
if x.ndim == 1:
x = x[:,None]
return np.dot(x.T, x)
def S_hac_groupsum(x, time, nlags=None, weights_func=weights_bartlett):
'''inner covariance matrix for HAC over group sums sandwich
This assumes we have complete equal spaced time periods.
The number of time periods per group need not be the same, but we need
at least one observation for each time period
For a single categorical group only, or a everything else but time
dimension. This first aggregates x over groups for each time period, then
applies HAC on the sum per period.
Parameters
----------
x : ndarray (nobs,) or (nobs, k_var)
data, for HAC this is array of x_i * u_i
time : ndarray, (nobs,)
timeindes, assumed to be integers range(n_periods)
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
S : ndarray, (k_vars, k_vars)
inner covariance matrix for sandwich
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay
'''
#needs groupsums
x_group_sums = group_sums(x, time).T #TODO: transpose return in grou_sum
return S_hac_simple(x_group_sums, nlags=nlags, weights_func=weights_func)
def S_crosssection(x, group):
'''inner covariance matrix for White on group sums sandwich
I guess for a single categorical group only,
categorical group, can also be the product/intersection of groups
This is used by cov_cluster and indirectly verified
'''
x_group_sums = group_sums(x, group).T #TODO: why transposed
return S_white_simple(x_group_sums)
def cov_crosssection_0(results, group):
'''this one is still wrong, use cov_cluster instead'''
#TODO: currently used version of groupsums requires 2d resid
scale = S_crosssection(results.resid[:,None], group)
scale = np.squeeze(scale)
cov = _HCCM1(results, scale)
return cov
[docs]def cov_cluster(results, group, use_correction=True):
'''cluster robust covariance matrix
Calculates sandwich covariance matrix for a single cluster, i.e. grouped
variables.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates
Notes
-----
same result as Stata in UCLA example and same as Peterson
'''
#TODO: currently used version of groupsums requires 2d resid
xu, hessian_inv = _get_sandwich_arrays(results, cov_type='clu')
if not hasattr(group, 'dtype') or group.dtype != np.dtype('int'):
clusters, group = np.unique(group, return_inverse=True)
else:
clusters = np.unique(group)
scale = S_crosssection(xu, group)
nobs, k_params = xu.shape
n_groups = len(clusters) #replace with stored group attributes if available
cov_c = _HCCM2(hessian_inv, scale)
if use_correction:
cov_c *= (n_groups / (n_groups - 1.) *
((nobs-1.) / float(nobs - k_params)))
return cov_c
[docs]def cov_cluster_2groups(results, group, group2=None, use_correction=True):
'''cluster robust covariance matrix for two groups/clusters
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
use_correction : bool
If true (default), then the small sample correction factor is used.
Returns
-------
cov_both : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates, for both
clusters
cov_0 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for first
cluster
cov_1 : ndarray, (k_vars, k_vars)
cluster robust covariance matrix for parameter estimates for second
cluster
Notes
-----
verified against Peterson's table, (4 decimal print precision)
'''
if group2 is None:
if group.ndim !=2 or group.shape[1] != 2:
raise ValueError('if group2 is not given, then groups needs to be ' +
'an array with two columns')
group0 = group[:, 0]
group1 = group[:, 1]
else:
group0 = group
group1 = group2
group = (group0, group1)
cov0 = cov_cluster(results, group0, use_correction=use_correction)
#[0] because we get still also returns bse
cov1 = cov_cluster(results, group1, use_correction=use_correction)
# cov of cluster formed by intersection of two groups
cov01 = cov_cluster(results,
combine_indices(group)[0],
use_correction=use_correction)
#robust cov matrix for union of groups
cov_both = cov0 + cov1 - cov01
#return all three (for now?)
return cov_both, cov0, cov1
[docs]def cov_white_simple(results, use_correction=True):
'''
heteroscedasticity robust covariance matrix (White)
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
Returns
-------
cov : ndarray, (k_vars, k_vars)
heteroscedasticity robust covariance matrix for parameter estimates
Notes
-----
This produces the same result as cov_hc0, and does not include any small
sample correction.
verified (against LinearRegressionResults and Peterson)
See Also
--------
cov_hc1, cov_hc2, cov_hc3 : heteroscedasticity robust covariance matrices
with small sample corrections
'''
xu, hessian_inv = _get_sandwich_arrays(results)
sigma = S_white_simple(xu)
cov_w = _HCCM2(hessian_inv, sigma) #add bread to sandwich
if use_correction:
nobs, k_params = xu.shape
cov_w *= nobs / float(nobs - k_params)
return cov_w
def cov_hac_simple(results, nlags=None, weights_func=weights_bartlett,
use_correction=True):
'''
heteroscedasticity and autocorrelation robust covariance matrix (Newey-West)
Assumes we have a single time series with zero axis consecutive, equal
spaced time periods
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
highest lag to include in kernel window. If None, then
nlags = floor[4(T/100)^(2/9)] is used.
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
verified only for nlags=0, which is just White
just guessing on correction factor, need reference
options might change when other kernels besides Bartlett are available.
'''
xu, hessian_inv = _get_sandwich_arrays(results)
sigma = S_hac_simple(xu, nlags=nlags, weights_func=weights_func)
cov_hac = _HCCM2(hessian_inv, sigma)
if use_correction:
nobs, k_params = xu.shape
cov_hac *= nobs / float(nobs - k_params)
return cov_hac
cov_hac = cov_hac_simple #alias for users
#---------------------- use time lags corrected for groups
#the following were copied from a different experimental script,
#groupidx is tuple, observations assumed to be stacked by group member and
#sorted by time, equal number of periods is not required, but equal spacing is.
#I think this is pure within group HAC: apply HAC to each group member
#separately
def lagged_groups(x, lag, groupidx):
'''
assumes sorted by time, groupidx is tuple of start and end values
not optimized, just to get a working version, loop over groups
'''
out0 = []
out_lagged = []
for l,u in groupidx:
if l+lag < u: #group is longer than lag
out0.append(x[l+lag:u])
out_lagged.append(x[l:u-lag])
if out0 == []:
raise ValueError('all groups are empty taking lags')
#return out0, out_lagged
return np.vstack(out0), np.vstack(out_lagged)
def S_nw_panel(xw, weights, groupidx):
'''inner covariance matrix for HAC for panel data
no denominator nobs used
no reference for this, just accounting for time indices
'''
nlags = len(weights)-1
S = weights[0] * np.dot(xw.T, xw) #weights just for completeness
for lag in range(1, nlags+1):
xw0, xwlag = lagged_groups(xw, lag, groupidx)
s = np.dot(xw0.T, xwlag)
S += weights[lag] * (s + s.T)
return S
[docs]def cov_nw_panel(results, nlags, groupidx, weights_func=weights_bartlett,
use_correction='hac'):
'''Panel HAC robust covariance matrix
Assumes we have a panel of time series with consecutive, equal spaced time
periods. Data is assumed to be in long format with time series of each
individual stacked into one array. Panel can be unbalanced.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
groupidx : list of tuple
each tuple should contain the start and end index for an individual.
(groupidx might change in future).
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'cluster' (default), then the same correction as in cov_cluster is
used.
If 'hac', then the same correction as in single time series, cov_hac
is used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
For nlags=0, this is just White covariance, cov_white.
If kernel is uniform, `weights_uniform`, with nlags equal to the number
of observations per unit in a balance panel, then cov_cluster and
cov_hac_panel are identical.
Tested against STATA `newey` command with same defaults.
Options might change when other kernels besides Bartlett and uniform are
available.
'''
if nlags == 0: #so we can reproduce HC0 White
weights = [1, 0] #to avoid the scalar check in hac_nw
else:
weights = weights_func(nlags)
xu, hessian_inv = _get_sandwich_arrays(results)
S_hac = S_nw_panel(xu, weights, groupidx)
cov_hac = _HCCM2(hessian_inv, S_hac)
if use_correction:
nobs, k_params = xu.shape
if use_correction == 'hac':
cov_hac *= nobs / float(nobs - k_params)
elif use_correction in ['c', 'clu', 'cluster']:
n_groups = len(groupidx)
cov_hac *= n_groups / (n_groups - 1.)
cov_hac *= ((nobs-1.) / float(nobs - k_params))
return cov_hac
[docs]def cov_nw_groupsum(results, nlags, time, weights_func=weights_bartlett,
use_correction=0):
'''Driscoll and Kraay Panel robust covariance matrix
Robust covariance matrix for panel data of Driscoll and Kraay.
Assumes we have a panel of time series where the time index is available.
The time index is assumed to represent equal spaced periods. At least one
observation per period is required.
Parameters
----------
results : result instance
result of a regression, uses results.model.exog and results.resid
TODO: this should use wexog instead
nlags : int or None
Highest lag to include in kernel window. Currently, no default
because the optimal length will depend on the number of observations
per cross-sectional unit.
time : ndarray of int
this should contain the coding for the time period of each observation.
time periods should be integers in range(maxT) where maxT is obs of i
weights_func : callable
weights_func is called with nlags as argument to get the kernel
weights. default are Bartlett weights
use_correction : 'cluster' or 'hac' or False
If False, then no small sample correction is used.
If 'hac' (default), then the same correction as in single time series, cov_hac
is used.
If 'cluster', then the same correction as in cov_cluster is
used.
Returns
-------
cov : ndarray, (k_vars, k_vars)
HAC robust covariance matrix for parameter estimates
Notes
-----
Tested against STATA xtscc package, which uses no small sample correction
This first averages relevant variables for each time period over all
individuals/groups, and then applies the same kernel weighted averaging
over time as in HAC.
Warning:
In the example with a short panel (few time periods and many individuals)
with mainly across individual variation this estimator did not produce
reasonable results.
Options might change when other kernels besides Bartlett and uniform are
available.
References
----------
Daniel Hoechle, xtscc paper
Driscoll and Kraay
'''
xu, hessian_inv = _get_sandwich_arrays(results)
#S_hac = S_nw_panel(xw, weights, groupidx)
S_hac = S_hac_groupsum(xu, time, nlags=nlags, weights_func=weights_func)
cov_hac = _HCCM2(hessian_inv, S_hac)
if use_correction:
nobs, k_params = xu.shape
if use_correction == 'hac':
cov_hac *= nobs / float(nobs - k_params)
elif use_correction in ['c', 'cluster']:
n_groups = len(np.unique(time))
cov_hac *= n_groups / (n_groups - 1.)
cov_hac *= ((nobs-1.) / float(nobs - k_params))
return cov_hac
