import numpy as np
from statsmodels.tools.decorators import (cache_readonly,
cache_writable, resettable_cache)
from scipy import optimize
from numpy import dot, identity, kron, log, zeros, pi, exp, eye, abs, empty
from numpy.linalg import inv, pinv
import statsmodels.base.model as base
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.wrapper as wrap
from statsmodels.regression.linear_model import yule_walker, GLS
from statsmodels.tsa.tsatools import (lagmat, add_trend,
_ar_transparams, _ar_invtransparams, _ma_transparams,
_ma_invtransparams)
from statsmodels.tsa.vector_ar import util
from statsmodels.tsa.ar_model import AR
from statsmodels.tsa.arima_process import arma2ma
from statsmodels.sandbox.regression.numdiff import (approx_fprime,
approx_fprime_cs, approx_hess, approx_hess_cs)
from statsmodels.tsa.kalmanf import KalmanFilter
from scipy.stats import t, norm
from scipy.signal import lfilter
try:
from kalmanf import kalman_loglike
fast_kalman = 1
except:
fast_kalman = 0
def _unpack_params(params, order, k_trend, k_exog, reverse=False):
p, q = order
k = k_trend + k_exog
maparams = params[k+p:]
arparams = params[k:k+p]
trend = params[:k_trend]
exparams = params[k_trend:k]
if reverse:
return trend, exparams, arparams[::-1], maparams[::-1]
return trend, exparams, arparams, maparams
def _unpack_order(order):
k_ar, k_ma, k = order
k_lags = max(k_ar, k_ma+1)
return k_ar, k_ma, order, k_lags
def _make_arma_names(data, k_trend, order):
k_ar, k_ma = order
exog = data.exog
if exog is not None:
exog_names = data._get_names(data._orig_exog) or []
else:
exog_names = []
ar_lag_names = util.make_lag_names(data.ynames, k_ar, 0)
ar_lag_names = [''.join(('ar.', i))
for i in ar_lag_names]
ma_lag_names = util.make_lag_names(data.ynames, k_ma, 0)
ma_lag_names = [''.join(('ma.', i)) for i in ma_lag_names]
trend_name = util.make_lag_names('', 0, k_trend)
exog_names = trend_name + exog_names + ar_lag_names + ma_lag_names
return exog_names
def _make_arma_exog(endog, exog, trend):
k_trend = 1 # overwritten if no constant
if exog is None and trend == 'c': # constant only
exog = np.ones((len(endog),1))
elif exog is not None and trend == 'c': # constant plus exogenous
exog = add_trend(exog, trend='c', prepend=True)
elif exog is not None and trend == 'nc':
# make sure it's not holding constant from last run
if exog.var() == 0:
exog = None
k_trend = 0
if trend == 'nc':
k_trend = 0
return k_trend, exog
[docs]class ARMA(tsbase.TimeSeriesModel):
"""
Autoregressive Moving Average ARMA(p,q) Model
Parameters
----------
endog : array-like
The endogenous variable.
exog : array-like, optional
An optional arry of exogenous variables. This should *not* include a
constant or trend. You can specify this in the `fit` method.
"""
def __init__(self, endog, exog=None, dates=None, freq=None):
super(ARMA, self).__init__(endog, exog, dates, freq)
if exog is not None:
k_exog = exog.shape[1] # number of exog. variables excl. const
else:
k_exog = 0
self.k_exog = k_exog
def _fit_start_params_hr(self, order):
"""
Get starting parameters for fit.
Parameters
----------
order : iterable
(p,q,k) - AR lags, MA lags, and number of exogenous variables
including the constant.
Returns
-------
start_params : array
A first guess at the starting parameters.
Notes
-----
If necessary, fits an AR process with the laglength selected according
to best BIC. Obtain the residuals. Then fit an ARMA(p,q) model via
OLS using these residuals for a first approximation. Uses a separate
OLS regression to find the coefficients of exogenous variables.
References
----------
Hannan, E.J. and Rissanen, J. 1982. "Recursive estimation of mixed
autoregressive-moving average order." `Biometrika`. 69.1.
"""
p,q,k = order
start_params = zeros((p+q+k))
endog = self.endog.copy() # copy because overwritten
exog = self.exog
if k != 0:
ols_params = GLS(endog, exog).fit().params
start_params[:k] = ols_params
endog -= np.dot(exog, ols_params).squeeze()
if q != 0:
if p != 0:
armod = AR(endog).fit(ic='bic', trend='nc')
arcoefs_tmp = armod.params
p_tmp = armod.k_ar
resid = endog[p_tmp:] - np.dot(lagmat(endog, p_tmp,
trim='both'), arcoefs_tmp)
if p < p_tmp + q:
endog_start = p_tmp + q - p
resid_start = 0
else:
endog_start = 0
resid_start = p - p_tmp - q
lag_endog = lagmat(endog, p, 'both')[endog_start:]
lag_resid = lagmat(resid, q, 'both')[resid_start:]
# stack ar lags and resids
X = np.column_stack((lag_endog, lag_resid))
coefs = GLS(endog[max(p_tmp+q,p):], X).fit().params
start_params[k:k+p+q] = coefs
else:
start_params[k+p:k+p+q] = yule_walker(endog, order=q)[0]
if q==0 and p != 0:
arcoefs = yule_walker(endog, order=p)[0]
start_params[k:k+p] = arcoefs
return start_params
def _fit_start_params(self, order, method):
if method != 'css-mle': # use Hannan-Rissanen to get start params
start_params = self._fit_start_params_hr(order)
else: # use CSS to get start params
func = lambda params: -self.loglike_css(params)
#start_params = [.1]*(k_ar+k_ma+k_exog) # different one for k?
start_params = self._fit_start_params_hr(order)
if self.transparams:
start_params = self._invtransparams(start_params)
bounds = [(None,)*2]*sum(order)
mlefit = optimize.fmin_l_bfgs_b(func, start_params,
approx_grad=True, m=12, pgtol=1e-7, factr=1e3,
bounds = bounds, iprint=-1)
start_params = self._transparams(mlefit[0])
return start_params
[docs] def score(self, params):
"""
Compute the score function at params.
Notes
-----
This is a numerical approximation.
"""
loglike = self.loglike
#if self.transparams:
# params = self._invtransparams(params)
#return approx_fprime(params, loglike, epsilon=1e-5)
return approx_fprime_cs(params, loglike)
[docs] def hessian(self, params):
"""
Compute the Hessian at params,
Notes
-----
This is a numerical approximation.
"""
loglike = self.loglike
#if self.transparams:
# params = self._invtransparams(params)
if not fast_kalman or self.method == "css":
return approx_hess_cs(params, loglike, epsilon=1e-5)
else:
return approx_hess(params, self.loglike, epsilon=1e-3)[0]
def _transparams(self, params):
"""
Transforms params to induce stationarity/invertability.
Reference
---------
Jones(1980)
"""
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
newparams = np.zeros_like(params)
# just copy exogenous parameters
if k != 0:
newparams[:k] = params[:k]
# AR Coeffs
if k_ar != 0:
newparams[k:k+k_ar] = _ar_transparams(params[k:k+k_ar].copy())
# MA Coeffs
if k_ma != 0:
newparams[k+k_ar:] = _ma_transparams(params[k+k_ar:].copy())
return newparams
def _invtransparams(self, start_params):
"""
Inverse of the Jones reparameterization
"""
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
newparams = start_params.copy()
arcoefs = newparams[k:k+k_ar]
macoefs = newparams[k+k_ar:]
# AR coeffs
if k_ar != 0:
newparams[k:k+k_ar] = _ar_invtransparams(arcoefs)
# MA coeffs
if k_ma != 0:
newparams[k+k_ar:k+k_ar+k_ma] = _ma_invtransparams(macoefs)
return newparams
def _get_predict_start(self, start):
# do some defaults
if start is None:
if 'mle' in self.method:
start = 0
else:
start = self.k_ar
if 'mle' not in self.method:
if start < self.k_ar:
raise ValueError("Start must be >= k_ar")
return super(ARMA, self)._get_predict_start(start)
[docs] def geterrors(self, params):
"""
Get the errors of the ARMA process.
Parameters
----------
params : array-like
The fitted ARMA parameters
order : array-like
3 item iterable, with the number of AR, MA, and exogenous
parameters, including the trend
"""
#start = self._get_predict_start(start) # will be an index of a date
#end, out_of_sample = self._get_predict_end(end)
params = np.asarray(params)
k_ar, k_ma = self.k_ar, self.k_ma
k = self.k_exog + self.k_trend
if 'mle' in self.method: # use KalmanFilter to get errors
(y, k, nobs, k_ar, k_ma, k_lags, newparams, Z_mat, m, R_mat,
T_mat, paramsdtype) = KalmanFilter._init_kalman_state(params, self)
errors = KalmanFilter.geterrors(y,k,k_ar,k_ma, k_lags, nobs,
Z_mat, m, R_mat, T_mat, paramsdtype)
if isinstance(errors, tuple):
errors = errors[0] # non-cython version returns a tuple
else: # use scipy.signal.lfilter
y = self.endog.copy()
k = self.k_exog + self.k_trend
if k > 0:
y -= dot(self.exog, params[:k])
k_ar = self.k_ar
k_ma = self.k_ma
(trendparams, exparams,
arparams, maparams) = _unpack_params(params, (k_ar, k_ma),
self.k_trend, self.k_exog,
reverse=False)
b,a = np.r_[1,-arparams], np.r_[1,maparams]
zi = zeros((max(k_ar, k_ma)))
for i in range(k_ar):
zi[i] = sum(-b[:i+1][::-1]*y[:i+1])
e = lfilter(b,a,y,zi=zi)
errors = e[0][k_ar:]
return errors.squeeze()
def _predict_out_of_sample(self, params, steps, errors, exog=None):
p = self.k_ar
q = self.k_ma
k_exog = self.k_exog
k_trend = self.k_trend
(trendparam, exparams,
arparams, maparams) = _unpack_params(params, (p,q), k_trend,
k_exog, reverse=True)
if exog is None and k_exog > 0:
raise ValueError("You must provide exog for ARMAX")
if q:
i = 0 # in case q == steps == 1
resid = np.zeros(2*q)
resid[:q] = errors[-q:] #only need last q
else:
i = -1 # since we don't run first loop below
y = self.endog
if k_trend == 1:
mu = trendparam * (1-arparams.sum()) # use expectation
# not constant
mu = np.array([mu]*steps) # repeat it so you can slice if exog
else:
mu = np.zeros(steps)
if k_exog > 0: # add exogenous process to constant
mu += np.dot(exparams, exog)
endog = np.zeros(p+steps-1)
if p:
endog[:p] = y[-p:] #only need p
forecast = np.zeros(steps)
for i in range(min(q,steps-1)):
fcast = mu[i] + np.dot(arparams,endog[i:i+p]) + \
np.dot(maparams,resid[i:i+q])
forecast[i] = fcast
endog[i+p] = fcast
for i in range(i+1,steps-1):
fcast = mu[i] + np.dot(arparams,endog[i:i+p])
forecast[i] = fcast
endog[i+p] = fcast
#need to do one more without updating endog
forecast[-1] = mu[-1] + np.dot(arparams,endog[steps-1:])
return forecast
[docs] def predict(self, params, start=None, end=None, exog=None):
"""
In-sample and out-of-sample prediction.
Parameters
----------
params : array-like
The fitted parameters of the model.
start : int, str, or datetime
Zero-indexed observation number at which to start forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type.
end : int, str, or datetime
Zero-indexed observation number at which to end forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type.
exog : array-like, optional
If the model is an ARMAX and out-of-sample forecasting is
requestion, exog must be given.
Notes
------
Consider using the results prediction.
"""
method = self.method
#params = np.asarray(params)
start = self._get_predict_start(start) # will be an index of a date
end, out_of_sample = self._get_predict_end(end)
if end < start:
raise ValueError("end is before start")
if end == start + out_of_sample:
return np.array([])
k_ar = self.k_ar
predictedvalues = np.zeros(end+1-start + out_of_sample)
endog = self.endog
resid = self.geterrors(params)
# this does pre- and in-sample fitting
fittedvalues = endog - resid #get them all then trim
fv_start = start
if 'mle' not in method:
fv_start -= k_ar # start is in terms of endog index
pv_end = min(len(predictedvalues), len(fittedvalues) - fv_start)
fv_end = min(len(fittedvalues), end+1)
predictedvalues[:pv_end] = fittedvalues[fv_start:fv_end]
if out_of_sample == 0:
return predictedvalues
# do out of sample fitting
predictedvalues[pv_end:] = self._predict_out_of_sample(params,
out_of_sample, resid, exog)
return predictedvalues
[docs] def loglike(self, params):
"""
Compute the log-likelihood for ARMA(p,q) model
Notes
-----
Likelihood used depends on the method set in fit
"""
method = self.method
if method in ['mle', 'css-mle']:
return self.loglike_kalman(params)
elif method == 'css':
return self.loglike_css(params)
else:
raise ValueError("Method %s not understood" % method)
[docs] def loglike_kalman(self, params):
"""
Compute exact loglikelihood for ARMA(p,q) model using the Kalman Filter.
"""
return KalmanFilter.loglike(params, self)
[docs] def loglike_css(self, params):
"""
Conditional Sum of Squares likelihood function.
"""
k_ar = self.k_ar
k_ma = self.k_ma
k = self.k_exog + self.k_trend
y = self.endog.copy().astype(params.dtype)
nobs = self.nobs
# how to handle if empty?
if self.transparams:
newparams = self._transparams(params)
else:
newparams = params
if k > 0:
y -= dot(self.exog, newparams[:k])
# the order of p determines how many zeros errors to set for lfilter
b,a = np.r_[1,-newparams[k:k+k_ar]], np.r_[1,newparams[k+k_ar:]]
zi = np.zeros((max(k_ar,k_ma)), dtype=params.dtype)
for i in range(k_ar):
zi[i] = sum(-b[:i+1][::-1] * y[:i+1])
errors = lfilter(b,a, y, zi=zi)[0][k_ar:]
ssr = np.dot(errors,errors)
sigma2 = ssr/nobs
self.sigma2 = sigma2
llf = -nobs/2.*(log(2*pi) + log(sigma2)) - ssr/(2*sigma2)
return llf
[docs] def fit(self, order, start_params=None, trend='c', method = "css-mle",
transparams=True, solver=None, maxiter=35, full_output=1,
disp=5, callback=None, **kwargs):
"""
Fits ARMA(p,q) model using exact maximum likelihood via Kalman filter.
Parameters
----------
start_params : array-like, optional
Starting parameters for ARMA(p,q). If None, the default is given
by ARMA._fit_start_params. See there for more information.
transparams : bool, optional
Whehter or not to transform the parameters to ensure stationarity.
Uses the transformation suggested in Jones (1980). If False,
no checking for stationarity or invertibility is done.
method : str {'css-mle','mle','css'}
This is the loglikelihood to maximize. If "css-mle", the
conditional sum of squares likelihood is maximized and its values
are used as starting values for the computation of the exact
likelihood via the Kalman filter. If "mle", the exact likelihood
is maximized via the Kalman Filter. If "css" the conditional sum
of squares likelihood is maximized. All three methods use
`start_params` as starting parameters. See above for more
information.
trend : str {'c','nc'}
Whehter to include a constant or not. 'c' includes constant,
'nc' no constant.
solver : str or None, optional
Solver to be used. The default is 'l_bfgs' (limited memory Broyden-
Fletcher-Goldfarb-Shanno). Other choices are 'bfgs', 'newton'
(Newton-Raphson), 'nm' (Nelder-Mead), 'cg' - (conjugate gradient),
'ncg' (non-conjugate gradient), and 'powell'.
The limited memory BFGS uses m=30 to approximate the Hessian,
projected gradient tolerance of 1e-7 and factr = 1e3. These
cannot currently be changed for l_bfgs. See notes for more
information.
maxiter : int, optional
The maximum number of function evaluations. Default is 35.
tol : float
The convergence tolerance. Default is 1e-08.
full_output : bool, optional
If True, all output from solver will be available in
the Results object's mle_retvals attribute. Output is dependent
on the solver. See Notes for more information.
disp : bool, optional
If True, convergence information is printed. For the default
l_bfgs_b solver, disp controls the frequency of the output during
the iterations. disp < 0 means no output in this case.
callback : function, optional
Called after each iteration as callback(xk) where xk is the current
parameter vector.
kwargs
See Notes for keyword arguments that can be passed to fit.
Returns
-------
`statsmodels.tsa.arima.ARMAResults` class
See also
--------
statsmodels.model.LikelihoodModel.fit for more information
on using the solvers.
Notes
------
If fit by 'mle', it is assumed for the Kalman Filter that the initial
unkown state is zero, and that the inital variance is
P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
r, order = 'F')
The below is the docstring from
`statsmodels.LikelihoodModel.fit`
"""
# enforce invertibility
self.transparams = transparams
self.method = method.lower()
# get model order and constants
self.k_ar = k_ar = int(order[0])
self.k_ma = k_ma = int(order[1])
self.k_lags = k_lags = max(k_ar,k_ma+1)
endog, exog = self.endog, self.exog
k_exog = self.k_exog
self.nobs = len(endog) # this is overwritten if method is 'css'
# (re)set trend and handle exogenous variables
# always pass original exog
k_trend, exog = _make_arma_exog(endog, self._data.exog, trend)
self.k_trend = k_trend
self.exog = exog # overwrites original exog from __init__
# (re)set names for this model
self.exog_names = _make_arma_names(self._data, k_trend, order)
k = k_trend + k_exog
# choose objective function
method = method.lower()
# adjust nobs for css
if method == 'css':
self.nobs = len(self.endog) - self.k_ar
loglike = lambda params: -self.loglike(params)
if start_params is not None:
start_params = np.asarray(start_params)
else: # estimate starting parameters
start_params = self._fit_start_params((k_ar,k_ma,k), method)
if transparams: # transform initial parameters to ensure invertibility
start_params = self._invtransparams(start_params)
if solver is None: # use default limited memory bfgs
bounds = [(None,)*2]*(k_ar+k_ma+k)
mlefit = optimize.fmin_l_bfgs_b(loglike, start_params,
approx_grad=True, m=12, pgtol=1e-8, factr=1e2,
bounds=bounds, iprint=disp)
self.mlefit = mlefit
params = mlefit[0]
else: # call the solver from LikelihoodModel
mlefit = super(ARMA, self).fit(start_params, method=solver,
maxiter=maxiter, full_output=full_output, disp=disp,
callback = callback, **kwargs)
self.mlefit = mlefit
params = mlefit.params
if transparams: # transform parameters back
params = self._transparams(params)
self.transparams = False # set to false so methods don't expect transf.
normalized_cov_params = None #TODO: fix this
armafit = ARMAResults(self, params, normalized_cov_params)
return ARMAResultsWrapper(armafit)
fit.__doc__ += base.LikelihoodModel.fit.__doc__
[docs]class ARMAResults(tsbase.TimeSeriesModelResults):
"""
Class to hold results from fitting an ARMA model.
Parameters
----------
model : ARMA instance
The fitted model instance
params : array
Fitted parameters
normalized_cov_params : array, optional
The normalized variance covariance matrix
scale : float, optional
Optional argument to scale the variance covariance matrix.
Returns
--------
**Attributes**
aic : float
Akaikie Information Criterion
:math:`-2*llf+2*(df_model+1)`
arparams : array
The parameters associated with the AR coefficients in the model.
arroots : array
The roots of the AR coefficients are the solution to
(1 - arparams[0]*z - arparams[1]*z**2 -...- arparams[p-1]*z**k_ar) = 0
Stability requires that the roots in modulus lie outside the unit
circle.
bic : float
Bayes Information Criterion
-2*llf + log(nobs)*(df_model+1)
Where if the model is fit using conditional sum of squares, the
number of observations `nobs` does not include the `p` pre-sample
observations.
bse : array
The standard errors of the parameters. These are computed using the
numerical Hessian.
df_model : array
The model degrees of freedom = `k_exog` + `k_trend` + `k_ar` + `k_ma`
df_resid : array
The residual degrees of freedom = `nobs` - `df_model`
fittedvalues : array
The predicted values of the model.
hqic : float
Hannan-Quinn Information Criterion
-2*llf + 2*(`df_model`)*log(log(nobs))
Like `bic` if the model is fit using conditional sum of squares then
the `k_ar` pre-sample observations are not counted in `nobs`.
k_ar : int
The number of AR coefficients in the model.
k_exog : int
The number of exogenous variables included in the model. Does not
include the constant.
k_ma : int
The number of MA coefficients.
k_trend : int
This is 0 for no constant or 1 if a constant is included.
llf : float
The value of the log-likelihood function evaluated at `params`.
maparams : array
The value of the moving average coefficients.
maroots : array
The roots of the MA coefficients are the solution to
(1 + maparams[0]*z + maparams[1]*z**2 + ... + maparams[q-1]*z**q) = 0
Stability requires that the roots in modules lie outside the unit
circle.
model : ARMA instance
A reference to the model that was fit.
nobs : float
The number of observations used to fit the model. If the model is fit
using exact maximum likelihood this is equal to the total number of
observations, `n_totobs`. If the model is fit using conditional
maximum likelihood this is equal to `n_totobs` - `k_ar`.
n_totobs : float
The total number of observations for `endog`. This includes all
observations, even pre-sample values if the model is fit using `css`.
params : array
The parameters of the model. The order of variables is the trend
coefficients and the `k_exog` exognous coefficients, then the
`k_ar` AR coefficients, and finally the `k_ma` MA coefficients.
pvalues : array
The p-values associated with the t-values of the coefficients. Note
that the coefficients are assumed to have a Student's T distribution.
resid : array
The model residuals. If the model is fit using 'mle' then the
residuals are created via the Kalman Filter. If the model is fit
using 'css' then the residuals are obtained via `scipy.signal.lfilter`
adjusted such that the first `k_ma` residuals are zero. These zero
residuals are not returned.
scale : float
This is currently set to 1.0 and not used by the model or its results.
sigma2 : float
The variance of the residuals. If the model is fit by 'css',
sigma2 = ssr/nobs, where ssr is the sum of squared residuals. If
the model is fit by 'mle', then sigma2 = 1/nobs * sum(v**2 / F)
where v is the one-step forecast error and F is the forecast error
variance. See `nobs` for the difference in definitions depending on the
fit.
"""
_cache = {}
#TODO: use this for docstring when we fix nobs issue
def __init__(self, model, params, normalized_cov_params=None, scale=1.):
super(ARMAResults, self).__init__(model, params, normalized_cov_params,
scale)
self.sigma2 = model.sigma2
nobs = model.nobs
self.nobs = nobs
k_exog = model.k_exog
self.k_exog = k_exog
k_trend = model.k_trend
self.k_trend = k_trend
k_ar = model.k_ar
self.k_ar = k_ar
self.n_totobs = len(model.endog)
k_ma = model.k_ma
self.k_ma = k_ma
df_model = k_exog + k_trend + k_ar + k_ma
self.df_model = df_model
self.df_resid = self.nobs - df_model
self._cache = resettable_cache()
@cache_readonly
def arroots(self):
return np.roots(np.r_[1,-self.arparams])**-1
@cache_readonly
def maroots(self):
return np.roots(np.r_[1,self.maparams])**-1
#@cache_readonly
#def arfreq(self):
# return (np.log(arroots/abs(arroots))/(2j*pi)).real
#NOTE: why don't root finding functions work well?
#@cache_readonly
#def mafreq(eslf):
# return
@cache_readonly
def arparams(self):
k = self.k_exog + self.k_trend
return self.params[k:k+self.k_ar]
@cache_readonly
def maparams(self):
k = self.k_exog + self.k_trend
k_ar = self.k_ar
return self.params[k+k_ar:]
@cache_readonly
def llf(self):
return self.model.loglike(self.params)
@cache_readonly
def bse(self):
params = self.params
hess = self.model.hessian(params)
if len(params) == 1: # can't take an inverse
return np.sqrt(-1./hess)
return np.sqrt(np.diag(-inv(hess)))
[docs] def cov_params(self): # add scale argument?
params = self.params
hess = self.model.hessian(params)
return -inv(hess)
@cache_readonly
def aic(self):
return -2*self.llf + 2*(self.df_model+1)
@cache_readonly
def bic(self):
nobs = self.nobs
return -2*self.llf + np.log(nobs)*(self.df_model+1)
@cache_readonly
def hqic(self):
nobs = self.nobs
return -2*self.llf + 2*(self.df_model+1)*np.log(np.log(nobs))
@cache_readonly
def fittedvalues(self):
model = self.model
endog = model.endog.copy()
k_ar = self.k_ar
exog = model.exog # this is a copy
if exog is not None:
if model.method == "css" and k_ar > 0:
exog = exog[k_ar:]
if model.method == "css" and k_ar > 0:
endog = endog[k_ar:]
fv = endog - self.resid
# add deterministic part back in
k = self.k_exog + self.k_trend
#TODO: this needs to be commented out for MLE with constant
# if k != 0:
# fv += dot(exog, self.params[:k])
return fv
@cache_readonly
def resid(self):
return self.model.geterrors(self.params)
@cache_readonly
def pvalues(self):
#TODO: same for conditional and unconditional?
df_resid = self.df_resid
return t.sf(np.abs(self.tvalues), df_resid) * 2
[docs] def predict(self, start=None, end=None, exog=None):
"""
In-sample and out-of-sample prediction.
Parameters
----------
start : int, str, or datetime
Zero-indexed observation number at which to start forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type.
end : int, str, or datetime
Zero-indexed observation number at which to end forecasting, ie.,
the first forecast is start. Can also be a date string to
parse or a datetime type.
exog : array-like, optional
If the model is an ARMAX and out-of-sample forecasting is
requestion, exog must be given.
"""
return self.model.predict(self.params, start, end, exog)
[docs] def forecast(self, steps=1, exog=None, alpha=.05):
"""
Out-of-sample forecasts
Parameters
----------
steps : int
The number of out of sample forecasts from the end of the
sample.
exog : array
If the model is an ARMAX, you must provide out of sample
values for the exogenous variables. This should not include
the constant.
alpha : float
The confidence intervals for the forecasts are (1 - alpha) %
Returns
-------
forecast : array
Array of out of sample forecasts
stderr : array
Array of the standard error of the forecasts.
conf_int : array
2d array of the confidence interval for the forecast
"""
arparams = self.arparams
maparams = self.maparams
forecast = self.model._predict_out_of_sample(self.params,
steps, self.resid, exog)
# compute the standard errors
sigma2 = self.sigma2
ma_rep = arma2ma(np.r_[1,-arparams],
np.r_[1, maparams], nobs=steps)
fcasterr = np.sqrt(sigma2 * np.cumsum(ma_rep**2))
const = norm.ppf(1 - alpha/2.)
conf_int = np.c_[forecast - const*fcasterr, forecast + const*fcasterr]
return forecast, fcasterr, conf_int
class ARMAResultsWrapper(wrap.ResultsWrapper):
_attrs = {}
_wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
_attrs)
_methods = {}
_wrap_methods = wrap.union_dicts(
tsbase.TimeSeriesResultsWrapper._wrap_methods,
_methods)
wrap.populate_wrapper(ARMAResultsWrapper, ARMAResults)
if __name__ == "__main__":
import numpy as np
import statsmodels.api as sm
# simulate arma process
from statsmodels.tsa.arima_process import arma_generate_sample
y = arma_generate_sample([1., -.75],[1.,.25], nsample=1000)
arma = ARMA(y)
res = arma.fit(trend='nc', order=(1,1))
np.random.seed(12345)
y_arma22 = arma_generate_sample([1.,-.85,.35],[1,.25,-.9], nsample=1000)
arma22 = ARMA(y_arma22)
res22 = arma22.fit(trend = 'nc', order=(2,2))
# test CSS
arma22_css = ARMA(y_arma22)
res22css = arma22_css.fit(trend='nc', order=(2,2), method='css')
data = sm.datasets.sunspots.load()
ar = ARMA(data.endog)
resar = ar.fit(trend='nc', order=(9,0))
y_arma31 = arma_generate_sample([1,-.75,-.35,.25],[.1], nsample=1000)
arma31css = ARMA(y_arma31)
res31css = arma31css.fit(order=(3,1), method="css", trend="nc",
transparams=True)
y_arma13 = arma_generate_sample([1., -.75],[1,.25,-.5,.8], nsample=1000)
arma13css = ARMA(y_arma13)
res13css = arma13css.fit(order=(1,3), method='css', trend='nc')
# check css for p < q and q < p
y_arma41 = arma_generate_sample([1., -.75, .35, .25, -.3],[1,-.35],
nsample=1000)
arma41css = ARMA(y_arma41)
res41css = arma41css.fit(order=(4,1), trend='nc', method='css')
y_arma14 = arma_generate_sample([1, -.25], [1., -.75, .35, .25, -.3],
nsample=1000)
arma14css = ARMA(y_arma14)
res14css = arma14css.fit(order=(4,1), trend='nc', method='css')