"""
Markov switching regression models
Author: Chad Fulton
License: BSD-3
"""
from __future__ import division, absolute_import, print_function
import numpy as np
import statsmodels.base.wrapper as wrap
from statsmodels.tsa.regime_switching import markov_switching
[docs]class MarkovRegression(markov_switching.MarkovSwitching):
r"""
First-order k-regime Markov switching regression model
Parameters
----------
endog : array_like
The endogenous variable.
k_regimes : integer
The number of regimes.
trend : {'nc', 'c', 't', 'ct'}
Whether or not to include a trend. To include an intercept, time trend,
or both, set `trend='c'`, `trend='t'`, or `trend='ct'`. For no trend,
set `trend='nc'`. Default is an intercept.
exog : array_like, optional
Array of exogenous regressors, shaped nobs x k.
order : integer, optional
The order of the model describes the dependence of the likelihood on
previous regimes. This depends on the model in question and should be
set appropriately by subclasses.
exog_tvtp : array_like, optional
Array of exogenous or lagged variables to use in calculating
time-varying transition probabilities (TVTP). TVTP is only used if this
variable is provided. If an intercept is desired, a column of ones must
be explicitly included in this array.
switching_trend : boolean or iterable, optional
If a boolean, sets whether or not all trend coefficients are
switching across regimes. If an iterable, should be of length equal
to the number of trend variables, where each element is
a boolean describing whether the corresponding coefficient is
switching. Default is True.
switching_exog : boolean or iterable, optional
If a boolean, sets whether or not all regression coefficients are
switching across regimes. If an iterable, should be of length equal
to the number of exogenous variables, where each element is
a boolean describing whether the corresponding coefficient is
switching. Default is True.
switching_variance : boolean, optional
Whether or not there is regime-specific heteroskedasticity, i.e.
whether or not the error term has a switching variance. Default is
False.
Notes
-----
This model is new and API stability is not guaranteed, although changes
will be made in a backwards compatible way if possible.
The model can be written as:
.. math::
y_t = a_{S_t} + x_t' \beta_{S_t} + \varepsilon_t \\
\varepsilon_t \sim N(0, \sigma_{S_t}^2)
i.e. the model is a dynamic linear regression where the coefficients and
the variance of the error term may be switching across regimes.
The `trend` is accomodated by prepending columns to the `exog` array. Thus
if `trend='c'`, the passed `exog` array should not already have a column of
ones.
References
----------
Kim, Chang-Jin, and Charles R. Nelson. 1999.
"State-Space Models with Regime Switching:
Classical and Gibbs-Sampling Approaches with Applications".
MIT Press Books. The MIT Press.
"""
def __init__(self, endog, k_regimes, trend='c', exog=None, order=0,
exog_tvtp=None, switching_trend=True, switching_exog=True,
switching_variance=False, dates=None, freq=None,
missing='none'):
# Properties
self.trend = trend
self.switching_trend = switching_trend
self.switching_exog = switching_exog
self.switching_variance = switching_variance
# Exogenous data
self.k_exog, exog = markov_switching.prepare_exog(exog)
# Trend
nobs = len(endog)
self.k_trend = 0
self._k_exog = self.k_exog
trend_exog = None
if trend == 'c':
trend_exog = np.ones((nobs, 1))
self.k_trend = 1
elif trend == 't':
trend_exog = (np.arange(nobs) + 1)[:, np.newaxis]
self.k_trend = 1
elif trend == 'ct':
trend_exog = np.c_[np.ones((nobs, 1)),
(np.arange(nobs) + 1)[:, np.newaxis]]
self.k_trend = 2
if trend_exog is not None:
exog = trend_exog if exog is None else np.c_[trend_exog, exog]
self._k_exog += self.k_trend
# Initialize the base model
super(MarkovRegression, self).__init__(
endog, k_regimes, order=order, exog_tvtp=exog_tvtp, exog=exog,
dates=dates, freq=freq, missing=missing)
# Switching options
if self.switching_trend is True or self.switching_trend is False:
self.switching_trend = [self.switching_trend] * self.k_trend
elif not len(self.switching_trend) == self.k_trend:
raise ValueError('Invalid iterable passed to `switching_trend`.')
if self.switching_exog is True or self.switching_exog is False:
self.switching_exog = [self.switching_exog] * self.k_exog
elif not len(self.switching_exog) == self.k_exog:
raise ValueError('Invalid iterable passed to `switching_exog`.')
self.switching_coeffs = (
np.r_[self.switching_trend,
self.switching_exog].astype(bool).tolist())
# Parameters
self.parameters['exog'] = self.switching_coeffs
self.parameters['variance'] = [1] if self.switching_variance else [0]
[docs] def predict_conditional(self, params):
"""
In-sample prediction, conditional on the current regime
Parameters
----------
params : array_like
Array of parameters at which to perform prediction.
Returns
-------
predict : array_like
Array of predictions conditional on current, and possibly past,
regimes
"""
params = np.array(params, ndmin=1)
# Since in the base model the values are the same across columns, we
# only compute a single column, and then expand it below.
predict = np.zeros((self.k_regimes, self.nobs), dtype=params.dtype)
for i in range(self.k_regimes):
# Predict
if self._k_exog > 0:
coeffs = params[self.parameters[i, 'exog']]
predict[i] = np.dot(self.exog, coeffs)
return predict[:, None, :]
def _resid(self, params):
predict = np.repeat(self.predict_conditional(params),
self.k_regimes, axis=1)
return self.endog - predict
def _conditional_likelihoods(self, params):
"""
Compute likelihoods conditional on the current period's regime
"""
# Get residuals
resid = self._resid(params)
# Compute the conditional likelihoods
variance = params[self.parameters['variance']].squeeze()
if self.switching_variance:
variance = np.reshape(variance, (self.k_regimes, 1, 1))
conditional_likelihoods = (
np.exp(-0.5 * resid**2 / variance) / np.sqrt(2 * np.pi * variance))
return conditional_likelihoods
@property
def _res_classes(self):
return {'fit': (MarkovRegressionResults,
MarkovRegressionResultsWrapper)}
def _em_iteration(self, params0):
"""
EM iteration
Notes
-----
This uses the inherited _em_iteration method for computing the
non-TVTP transition probabilities and then performs the EM step for
regression coefficients and variances.
"""
# Inherited parameters
result, params1 = super(MarkovRegression, self)._em_iteration(params0)
tmp = np.sqrt(result.smoothed_marginal_probabilities)
# Regression coefficients
coeffs = None
if self._k_exog > 0:
coeffs = self._em_exog(result, self.endog, self.exog,
self.parameters.switching['exog'], tmp)
for i in range(self.k_regimes):
params1[self.parameters[i, 'exog']] = coeffs[i]
# Variances
params1[self.parameters['variance']] = self._em_variance(
result, self.endog, self.exog, coeffs, tmp)
# params1[self.parameters['variance']] = 0.33282116
return result, params1
def _em_exog(self, result, endog, exog, switching, tmp=None):
"""
EM step for regression coefficients
"""
k_exog = exog.shape[1]
coeffs = np.zeros((self.k_regimes, k_exog))
# First, estimate non-switching coefficients
if not np.all(switching):
nonswitching_exog = exog[:, ~switching]
nonswitching_coeffs = (
np.dot(np.linalg.pinv(nonswitching_exog), endog))
coeffs[:, ~switching] = nonswitching_coeffs
endog = endog - np.dot(nonswitching_exog, nonswitching_coeffs)
# Next, get switching coefficients
if np.any(switching):
switching_exog = exog[:, switching]
if tmp is None:
tmp = np.sqrt(result.smoothed_marginal_probabilities)
for i in range(self.k_regimes):
tmp_endog = tmp[i] * endog
tmp_exog = tmp[i][:, np.newaxis] * switching_exog
coeffs[i, switching] = (
np.dot(np.linalg.pinv(tmp_exog), tmp_endog))
return coeffs
def _em_variance(self, result, endog, exog, betas, tmp=None):
"""
EM step for variances
"""
k_exog = 0 if exog is None else exog.shape[1]
if self.switching_variance:
variance = np.zeros(self.k_regimes)
for i in range(self.k_regimes):
if k_exog > 0:
resid = endog - np.dot(exog, betas[i])
else:
resid = endog
variance[i] = (
np.sum(resid**2 *
result.smoothed_marginal_probabilities[i]) /
np.sum(result.smoothed_marginal_probabilities[i]))
else:
variance = 0
if tmp is None:
tmp = np.sqrt(result.smoothed_marginal_probabilities)
for i in range(self.k_regimes):
tmp_endog = tmp[i] * endog
if k_exog > 0:
tmp_exog = tmp[i][:, np.newaxis] * exog
resid = tmp_endog - np.dot(tmp_exog, betas[i])
else:
resid = tmp_endog
variance += np.sum(resid**2)
variance /= self.nobs
return variance
@property
def start_params(self):
"""
(array) Starting parameters for maximum likelihood estimation.
Notes
-----
These are not very sophisticated and / or good. We set equal transition
probabilities and interpolate regression coefficients between zero and
the OLS estimates, where the interpolation is based on the regime
number. We rely heavily on the EM algorithm to quickly find much better
starting parameters, which are then used by the typical scoring
approach.
"""
# Inherited parameters
params = markov_switching.MarkovSwitching.start_params.fget(self)
# Regression coefficients
if self._k_exog > 0:
beta = np.dot(np.linalg.pinv(self.exog), self.endog)
variance = np.var(self.endog - np.dot(self.exog, beta))
if np.any(self.switching_coeffs):
for i in range(self.k_regimes):
params[self.parameters[i, 'exog']] = (
beta * (i / self.k_regimes))
else:
params[self.parameters['exog']] = beta
else:
variance = np.var(self.endog)
# Variances
if self.switching_variance:
params[self.parameters['variance']] = (
np.linspace(variance / 10., variance, num=self.k_regimes))
else:
params[self.parameters['variance']] = variance
return params
@property
def param_names(self):
"""
(list of str) List of human readable parameter names (for parameters
actually included in the model).
"""
# Inherited parameters
param_names = np.array(
markov_switching.MarkovSwitching.param_names.fget(self),
dtype=object)
# Regression coefficients
if np.any(self.switching_coeffs):
for i in range(self.k_regimes):
param_names[self.parameters[i, 'exog']] = [
'%s[%d]' % (exog_name, i) for exog_name in self.exog_names]
else:
param_names[self.parameters['exog']] = self.exog_names
# Variances
if self.switching_variance:
for i in range(self.k_regimes):
param_names[self.parameters[i, 'variance']] = 'sigma2[%d]' % i
else:
param_names[self.parameters['variance']] = 'sigma2'
return param_names.tolist()
class MarkovRegressionResults(markov_switching.MarkovSwitchingResults):
r"""
Class to hold results from fitting a Markov switching regression model
Parameters
----------
model : MarkovRegression instance
The fitted model instance
params : array
Fitted parameters
filter_results : HamiltonFilterResults or KimSmootherResults instance
The underlying filter and, optionally, smoother output
cov_type : string
The type of covariance matrix estimator to use. Can be one of 'approx',
'opg', 'robust', or 'none'.
Attributes
----------
model : Model instance
A reference to the model that was fit.
filter_results : HamiltonFilterResults or KimSmootherResults instance
The underlying filter and, optionally, smoother output
nobs : float
The number of observations used to fit the model.
params : array
The parameters of the model.
scale : float
This is currently set to 1.0 and not used by the model or its results.
"""
pass
class MarkovRegressionResultsWrapper(
markov_switching.MarkovSwitchingResultsWrapper):
pass
wrap.populate_wrapper(MarkovRegressionResultsWrapper, # noqa:E305
MarkovRegressionResults)
