"""
Recursive least squares model
Author: Chad Fulton
License: Simplified-BSD
"""
from __future__ import division, absolute_import, print_function
from warnings import warn
from statsmodels.compat.collections import OrderedDict
import numpy as np
import pandas as pd
from statsmodels.regression.linear_model import OLS
from statsmodels.tools.data import _is_using_pandas
from statsmodels.tsa.statespace.mlemodel import (
MLEModel, MLEResults, MLEResultsWrapper)
from statsmodels.tools.tools import Bunch
from statsmodels.tools.decorators import cache_readonly, resettable_cache
import statsmodels.base.wrapper as wrap
# Columns are alpha = 0.1, 0.05, 0.025, 0.01, 0.005
_cusum_squares_scalars = np.array([
[1.0729830, 1.2238734, 1.3581015, 1.5174271, 1.6276236],
[-0.6698868, -0.6700069, -0.6701218, -0.6702672, -0.6703724],
[-0.5816458, -0.7351697, -0.8858694, -1.0847745, -1.2365861]
])
[docs]class RecursiveLS(MLEModel):
r"""
Recursive least squares
Parameters
----------
endog : array_like
The observed time-series process :math:`y`
exog : array_like
Array of exogenous regressors, shaped nobs x k.
Notes
-----
Recursive least squares (RLS) corresponds to expanding window ordinary
least squares (OLS).
This model applies the Kalman filter to compute recursive estimates of the
coefficients and recursive residuals.
References
----------
.. [1] Durbin, James, and Siem Jan Koopman. 2012.
Time Series Analysis by State Space Methods: Second Edition.
Oxford University Press.
"""
def __init__(self, endog, exog, **kwargs):
# Standardize data
if not _is_using_pandas(endog, None):
endog = np.asanyarray(endog)
exog_is_using_pandas = _is_using_pandas(exog, None)
if not exog_is_using_pandas:
exog = np.asarray(exog)
# Make sure we have 2-dimensional array
if exog.ndim == 1:
if not exog_is_using_pandas:
exog = exog[:, None]
else:
exog = pd.DataFrame(exog)
self.k_exog = exog.shape[1]
# Handle coefficient initialization
# By default, do not calculate likelihood while it is controlled by
# diffuse initial conditions.
kwargs.setdefault('loglikelihood_burn', self.k_exog)
kwargs.setdefault('initialization', 'approximate_diffuse')
kwargs.setdefault('initial_variance', 1e9)
# Initialize the state space representation
super(RecursiveLS, self).__init__(
endog, k_states=self.k_exog, exog=exog, **kwargs
)
# Setup the state space representation
self['design'] = self.exog[:, :, None].T
self['transition'] = np.eye(self.k_states)
# Notice that the filter output does not depend on the measurement
# variance, so we set it here to 1
self['obs_cov', 0, 0] = 1.
@classmethod
[docs] def fit(self):
"""
Fits the model by application of the Kalman filter
Returns
-------
RecursiveLSResults
"""
# Get the smoother results with an arbitrary measurement variance
smoother_results = self.smooth(return_ssm=True)
# Compute the MLE of sigma2 (see Harvey, 1989 equation 4.2.5)
resid = smoother_results.standardized_forecasts_error[0]
sigma2 = (np.inner(resid, resid) /
(self.nobs - self.loglikelihood_burn))
# Now construct a results class, where the params are the final
# estimates of the regression coefficients
self['obs_cov', 0, 0] = sigma2
return self.smooth()
[docs] def filter(self, return_ssm=False, **kwargs):
# Get the state space output
result = super(RecursiveLS, self).filter([], transformed=True,
cov_type='none',
return_ssm=True, **kwargs)
# Wrap in a results object
if not return_ssm:
params = result.filtered_state[:, -1]
cov_kwds = {
'custom_cov_type': 'nonrobust',
'custom_cov_params': result.filtered_state_cov[:, :, -1],
'custom_description': ('Parameters and covariance matrix'
' estimates are RLS estimates'
' conditional on the entire sample.')
}
result = RecursiveLSResultsWrapper(
RecursiveLSResults(self, params, result, cov_type='custom',
cov_kwds=cov_kwds)
)
return result
[docs] def smooth(self, return_ssm=False, **kwargs):
# Get the state space output
result = super(RecursiveLS, self).smooth([], transformed=True,
cov_type='none',
return_ssm=True, **kwargs)
# Wrap in a results object
if not return_ssm:
params = result.filtered_state[:, -1]
cov_kwds = {
'custom_cov_type': 'nonrobust',
'custom_cov_params': result.filtered_state_cov[:, :, -1],
'custom_description': ('Parameters and covariance matrix'
' estimates are RLS estimates'
' conditional on the entire sample.')
}
result = RecursiveLSResultsWrapper(
RecursiveLSResults(self, params, result, cov_type='custom',
cov_kwds=cov_kwds)
)
return result
@property
def param_names(self):
return self.exog_names
@property
def start_params(self):
# Only parameter is the measurement disturbance standard deviation
return np.zeros(0)
[docs] def update(self, params, **kwargs):
"""
Update the parameters of the model
Updates the representation matrices to fill in the new parameter
values.
Parameters
----------
params : array_like
Array of new parameters.
transformed : boolean, optional
Whether or not `params` is already transformed. If set to False,
`transform_params` is called. Default is True..
Returns
-------
params : array_like
Array of parameters.
"""
pass
[docs]class RecursiveLSResults(MLEResults):
"""
Class to hold results from fitting a recursive least squares model.
Parameters
----------
model : RecursiveLS instance
The fitted model instance
Attributes
----------
specification : dictionary
Dictionary including all attributes from the recursive least squares
model instance.
See Also
--------
statsmodels.tsa.statespace.kalman_filter.FilterResults
statsmodels.tsa.statespace.mlemodel.MLEResults
"""
def __init__(self, model, params, filter_results, cov_type='opg',
**kwargs):
super(RecursiveLSResults, self).__init__(
model, params, filter_results, cov_type, **kwargs)
self.df_resid = np.inf # attribute required for wald tests
# Save _init_kwds
self._init_kwds = self.model._get_init_kwds()
# Save the model specification
self.specification = Bunch(**{
'k_exog': self.model.k_exog})
@property
def recursive_coefficients(self):
"""
Estimates of regression coefficients, recursively estimated
Returns
-------
out: Bunch
Has the following attributes:
- `filtered`: a time series array with the filtered estimate of
the component
- `filtered_cov`: a time series array with the filtered estimate of
the variance/covariance of the component
- `smoothed`: a time series array with the smoothed estimate of
the component
- `smoothed_cov`: a time series array with the smoothed estimate of
the variance/covariance of the component
- `offset`: an integer giving the offset in the state vector where
this component begins
"""
out = None
spec = self.specification
start = offset = 0
end = offset + spec.k_exog
out = Bunch(
filtered=self.filtered_state[start:end],
filtered_cov=self.filtered_state_cov[start:end, start:end],
smoothed=None, smoothed_cov=None,
offset=offset
)
if self.smoothed_state is not None:
out.smoothed = self.smoothed_state[start:end]
if self.smoothed_state_cov is not None:
out.smoothed_cov = (
self.smoothed_state_cov[start:end, start:end])
return out
@cache_readonly
[docs] def resid_recursive(self):
"""
Recursive residuals
Returns
-------
resid_recursive : array_like
An array of length `nobs` holding the recursive
residuals.
Notes
-----
The first `k_exog` residuals are typically unreliable due to
initialization.
"""
# See Harvey (1989) section 5.4; he defines the standardized
# innovations in 5.4.1, but they have non-unit variance, whereas
# the standardized forecast errors assume unit variance. To convert
# to Harvey's definition, we need to multiply by the standard
# deviation.
return (self.filter_results.standardized_forecasts_error.squeeze() *
self.filter_results.obs_cov[0, 0]**0.5)
@cache_readonly
[docs] def cusum(self):
r"""
Cumulative sum of standardized recursive residuals statistics
Returns
-------
cusum : array_like
An array of length `nobs - k_exog` holding the
CUSUM statistics.
Notes
-----
The CUSUM statistic takes the form:
.. math::
W_t = \frac{1}{\hat \sigma} \sum_{j=k+1}^t w_j
where :math:`w_j` is the recursive residual at time :math:`j` and
:math:`\hat \sigma` is the estimate of the standard deviation
from the full sample.
Excludes the first `k_exog` datapoints.
Due to differences in the way :math:`\hat \sigma` is calculated, the
output of this function differs slightly from the output in the
R package strucchange and the Stata contributed .ado file cusum6. The
calculation in this package is consistent with the description of
Brown et al. (1975)
References
----------
.. [1] Brown, R. L., J. Durbin, and J. M. Evans. 1975.
"Techniques for Testing the Constancy of
Regression Relationships over Time."
Journal of the Royal Statistical Society.
Series B (Methodological) 37 (2): 149-92.
"""
llb = self.loglikelihood_burn
return (np.cumsum(self.resid_recursive[self.loglikelihood_burn:]) /
np.std(self.resid_recursive[llb:], ddof=1))
@cache_readonly
[docs] def cusum_squares(self):
r"""
Cumulative sum of squares of standardized recursive residuals
statistics
Returns
-------
cusum_squares : array_like
An array of length `nobs - k_exog` holding the
CUSUM of squares statistics.
Notes
-----
The CUSUM of squares statistic takes the form:
.. math::
s_t = \left ( \sum_{j=k+1}^t w_j^2 \right ) \Bigg /
\left ( \sum_{j=k+1}^T w_j^2 \right )
where :math:`w_j` is the recursive residual at time :math:`j`.
Excludes the first `k_exog` datapoints.
References
----------
.. [1] Brown, R. L., J. Durbin, and J. M. Evans. 1975.
"Techniques for Testing the Constancy of
Regression Relationships over Time."
Journal of the Royal Statistical Society.
Series B (Methodological) 37 (2): 149-92.
"""
numer = np.cumsum(self.resid_recursive[self.loglikelihood_burn:]**2)
denom = numer[-1]
return numer / denom
[docs] def plot_recursive_coefficient(self, variables=0, alpha=0.05,
legend_loc='upper left', fig=None,
figsize=None):
r"""
Plot the recursively estimated coefficients on a given variable
Parameters
----------
variables : int or str or iterable of int or string, optional
Integer index or string name of the variable whose coefficient will
be plotted. Can also be an iterable of integers or strings. Default
is the first variable.
alpha : float, optional
The confidence intervals for the coefficient are (1 - alpha) %
legend_loc : string, optional
The location of the legend in the plot. Default is upper left.
fig : Matplotlib Figure instance, optional
If given, subplots are created in this figure instead of in a new
figure. Note that the grid will be created in the provided
figure using `fig.add_subplot()`.
figsize : tuple, optional
If a figure is created, this argument allows specifying a size.
The tuple is (width, height).
Notes
-----
All plots contain (1 - `alpha`) % confidence intervals.
"""
# Get variables
if isinstance(variables, (int, str)):
variables = [variables]
k_variables = len(variables)
# If a string was given for `variable`, try to get it from exog names
exog_names = self.model.exog_names
for i in range(k_variables):
variable = variables[i]
if isinstance(variable, str):
variables[i] = exog_names.index(variable)
# Create the plot
from scipy.stats import norm
from statsmodels.graphics.utils import _import_mpl, create_mpl_fig
plt = _import_mpl()
fig = create_mpl_fig(fig, figsize)
for i in range(k_variables):
variable = variables[i]
ax = fig.add_subplot(k_variables, 1, i + 1)
# Get dates, if applicable
if hasattr(self.data, 'dates') and self.data.dates is not None:
dates = self.data.dates._mpl_repr()
else:
dates = np.arange(self.nobs)
llb = self.loglikelihood_burn
# Plot the coefficient
coef = self.recursive_coefficients
ax.plot(dates[llb:], coef.filtered[variable, llb:],
label='Recursive estimates: %s' % exog_names[variable])
# Legend
handles, labels = ax.get_legend_handles_labels()
# Get the critical value for confidence intervals
if alpha is not None:
critical_value = norm.ppf(1 - alpha / 2.)
# Plot confidence intervals
std_errors = np.sqrt(coef.filtered_cov[variable, variable, :])
ci_lower = (
coef.filtered[variable] - critical_value * std_errors)
ci_upper = (
coef.filtered[variable] + critical_value * std_errors)
ci_poly = ax.fill_between(
dates[llb:], ci_lower[llb:], ci_upper[llb:], alpha=0.2
)
ci_label = ('$%.3g \\%%$ confidence interval'
% ((1 - alpha)*100))
# Only add CI to legend for the first plot
if i == 0:
# Proxy artist for fill_between legend entry
# See http://matplotlib.org/1.3.1/users/legend_guide.html
p = plt.Rectangle((0, 0), 1, 1,
fc=ci_poly.get_facecolor()[0])
handles.append(p)
labels.append(ci_label)
ax.legend(handles, labels, loc=legend_loc)
# Remove xticks for all but the last plot
if i < k_variables - 1:
ax.xaxis.set_ticklabels([])
fig.tight_layout()
return fig
def _cusum_significance_bounds(self, alpha, ddof=0, points=None):
"""
Parameters
----------
alpha : float, optional
The significance bound is alpha %.
ddof : int, optional
The number of periods additional to `k_exog` to exclude in
constructing the bounds. Default is zero. This is usually used
only for testing purposes.
points : iterable, optional
The points at which to evaluate the significance bounds. Default is
two points, beginning and end of the sample.
Notes
-----
Comparing against the cusum6 package for Stata, this does not produce
exactly the same confidence bands (which are produced in cusum6 by
lw, uw) because they burn the first k_exog + 1 periods instead of the
first k_exog. If this change is performed
(so that `tmp = (self.nobs - llb - 1)**0.5`), then the output here
matches cusum6.
The cusum6 behavior does not seem to be consistent with
Brown et al. (1975); it is likely they did that because they needed
three initial observations to get the initial OLS estimates, whereas
we do not need to do that.
"""
# Get the constant associated with the significance level
if alpha == 0.01:
scalar = 1.143
elif alpha == 0.05:
scalar = 0.948
elif alpha == 0.10:
scalar = 0.950
else:
raise ValueError('Invalid significance level.')
# Get the points for the significance bound lines
llb = self.loglikelihood_burn
tmp = (self.nobs - llb - ddof)**0.5
upper_line = lambda x: scalar * tmp + 2 * scalar * (x - llb) / tmp
if points is None:
points = np.array([llb, self.nobs])
return -upper_line(points), upper_line(points)
[docs] def plot_cusum(self, alpha=0.05, legend_loc='upper left',
fig=None, figsize=None):
r"""
Plot the CUSUM statistic and significance bounds.
Parameters
----------
alpha : float, optional
The plotted significance bounds are alpha %.
legend_loc : string, optional
The location of the legend in the plot. Default is upper left.
fig : Matplotlib Figure instance, optional
If given, subplots are created in this figure instead of in a new
figure. Note that the grid will be created in the provided
figure using `fig.add_subplot()`.
figsize : tuple, optional
If a figure is created, this argument allows specifying a size.
The tuple is (width, height).
Notes
-----
Evidence of parameter instability may be found if the CUSUM statistic
moves out of the significance bounds.
References
----------
.. [1] Brown, R. L., J. Durbin, and J. M. Evans. 1975.
"Techniques for Testing the Constancy of
Regression Relationships over Time."
Journal of the Royal Statistical Society.
Series B (Methodological) 37 (2): 149-92.
"""
# Create the plot
from statsmodels.graphics.utils import _import_mpl, create_mpl_fig
plt = _import_mpl()
fig = create_mpl_fig(fig, figsize)
ax = fig.add_subplot(1, 1, 1)
# Get dates, if applicable
if hasattr(self.data, 'dates') and self.data.dates is not None:
dates = self.data.dates._mpl_repr()
else:
dates = np.arange(self.nobs)
llb = self.loglikelihood_burn
# Plot cusum series and reference line
ax.plot(dates[llb:], self.cusum, label='CUSUM')
ax.hlines(0, dates[llb], dates[-1], color='k', alpha=0.3)
# Plot significance bounds
lower_line, upper_line = self._cusum_significance_bounds(alpha)
ax.plot([dates[llb], dates[-1]], upper_line, 'k--',
label='%d%% significance' % (alpha * 100))
ax.plot([dates[llb], dates[-1]], lower_line, 'k--')
ax.legend(loc=legend_loc)
return fig
def _cusum_squares_significance_bounds(self, alpha, points=None):
"""
Notes
-----
Comparing against the cusum6 package for Stata, this does not produce
exactly the same confidence bands (which are produced in cusum6 by
lww, uww) because they use a different method for computing the
critical value; in particular, they use tabled values from
Table C, pp. 364-365 of "The Econometric Analysis of Time Series"
Harvey, (1990), and use the value given to 99 observations for any
larger number of observations. In contrast, we use the approximating
critical values suggested in Edgerton and Wells (1994) which allows
computing relatively good approximations for any number of
observations.
"""
# Get the approximate critical value associated with the significance
# level
llb = self.loglikelihood_burn
n = 0.5 * (self.nobs - llb) - 1
try:
ix = [0.1, 0.05, 0.025, 0.01, 0.005].index(alpha / 2)
except ValueError:
raise ValueError('Invalid significance level.')
scalars = _cusum_squares_scalars[:, ix]
crit = scalars[0] / n**0.5 + scalars[1] / n + scalars[2] / n**1.5
# Get the points for the significance bound lines
if points is None:
points = np.array([llb, self.nobs])
line = (points - llb) / (self.nobs - llb)
return line - crit, line + crit
[docs] def plot_cusum_squares(self, alpha=0.05, legend_loc='upper left',
fig=None, figsize=None):
r"""
Plot the CUSUM of squares statistic and significance bounds.
Parameters
----------
alpha : float, optional
The plotted significance bounds are alpha %.
legend_loc : string, optional
The location of the legend in the plot. Default is upper left.
fig : Matplotlib Figure instance, optional
If given, subplots are created in this figure instead of in a new
figure. Note that the grid will be created in the provided
figure using `fig.add_subplot()`.
figsize : tuple, optional
If a figure is created, this argument allows specifying a size.
The tuple is (width, height).
Notes
-----
Evidence of parameter instability may be found if the CUSUM of squares
statistic moves out of the significance bounds.
Critical values used in creating the significance bounds are computed
using the approximate formula of [2]_.
References
----------
.. [1] Brown, R. L., J. Durbin, and J. M. Evans. 1975.
"Techniques for Testing the Constancy of
Regression Relationships over Time."
Journal of the Royal Statistical Society.
Series B (Methodological) 37 (2): 149-92.
.. [2] Edgerton, David, and Curt Wells. 1994.
"Critical Values for the Cusumsq Statistic
in Medium and Large Sized Samples."
Oxford Bulletin of Economics and Statistics 56 (3): 355-65.
"""
# Create the plot
from statsmodels.graphics.utils import _import_mpl, create_mpl_fig
plt = _import_mpl()
fig = create_mpl_fig(fig, figsize)
ax = fig.add_subplot(1, 1, 1)
# Get dates, if applicable
if hasattr(self.data, 'dates') and self.data.dates is not None:
dates = self.data.dates._mpl_repr()
else:
dates = np.arange(self.nobs)
llb = self.loglikelihood_burn
# Plot cusum series and reference line
ax.plot(dates[llb:], self.cusum_squares, label='CUSUM of squares')
ref_line = (np.arange(llb, self.nobs) - llb) / (self.nobs - llb)
ax.plot(dates[llb:], ref_line, 'k', alpha=0.3)
# Plot significance bounds
lower_line, upper_line = self._cusum_squares_significance_bounds(alpha)
ax.plot([dates[llb], dates[-1]], upper_line, 'k--',
label='%d%% significance' % (alpha * 100))
ax.plot([dates[llb], dates[-1]], lower_line, 'k--')
ax.legend(loc=legend_loc)
return fig
class RecursiveLSResultsWrapper(MLEResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(RecursiveLSResultsWrapper, RecursiveLSResults)
