"""
Statistical tools for time series analysis
"""
from __future__ import division
from statsmodels.compat.python import (iteritems, range, lrange, string_types,
lzip, zip, long)
from statsmodels.compat.numpy import lstsq
from statsmodels.compat.scipy import _next_regular
import numpy as np
import pandas as pd
from numpy.linalg import LinAlgError
from scipy import stats
from statsmodels.regression.linear_model import OLS, yule_walker
from statsmodels.tools.sm_exceptions import (InterpolationWarning,
MissingDataError,
CollinearityWarning)
from statsmodels.tools.tools import add_constant, Bunch
from statsmodels.tsa._bds import bds
from statsmodels.tsa.adfvalues import mackinnonp, mackinnoncrit
from statsmodels.tsa.arima_model import ARMA
from statsmodels.tsa.tsatools import lagmat, lagmat2ds, add_trend
__all__ = ['acovf', 'acf', 'pacf', 'pacf_yw', 'pacf_ols', 'ccovf', 'ccf',
'periodogram', 'q_stat', 'coint', 'arma_order_select_ic',
'adfuller', 'kpss', 'bds', 'pacf_burg', 'innovations_algo',
'innovations_filter', 'levinson_durbin_pacf', 'levinson_durbin']
SQRTEPS = np.sqrt(np.finfo(np.double).eps)
#NOTE: now in two places to avoid circular import
#TODO: I like the bunch pattern for this too.
class ResultsStore(object):
def __str__(self):
return self._str # pylint: disable=E1101
def _autolag(mod, endog, exog, startlag, maxlag, method, modargs=(),
fitargs=(), regresults=False):
"""
Returns the results for the lag length that maximizes the info criterion.
Parameters
----------
mod : Model class
Model estimator class
endog : array-like
nobs array containing endogenous variable
exog : array-like
nobs by (startlag + maxlag) array containing lags and possibly other
variables
startlag : int
The first zero-indexed column to hold a lag. See Notes.
maxlag : int
The highest lag order for lag length selection.
method : {'aic', 'bic', 't-stat'}
aic - Akaike Information Criterion
bic - Bayes Information Criterion
t-stat - Based on last lag
modargs : tuple, optional
args to pass to model. See notes.
fitargs : tuple, optional
args to pass to fit. See notes.
regresults : bool, optional
Flag indicating to return optional return results
Returns
-------
icbest : float
Best information criteria.
bestlag : int
The lag length that maximizes the information criterion.
results : dict, optional
Dictionary containing all estimation results
Notes
-----
Does estimation like mod(endog, exog[:,:i], *modargs).fit(*fitargs)
where i goes from lagstart to lagstart+maxlag+1. Therefore, lags are
assumed to be in contiguous columns from low to high lag length with
the highest lag in the last column.
"""
#TODO: can tcol be replaced by maxlag + 2?
#TODO: This could be changed to laggedRHS and exog keyword arguments if
# this will be more general.
results = {}
method = method.lower()
for lag in range(startlag, startlag + maxlag + 1):
mod_instance = mod(endog, exog[:, :lag], *modargs)
results[lag] = mod_instance.fit()
if method == "aic":
icbest, bestlag = min((v.aic, k) for k, v in iteritems(results))
elif method == "bic":
icbest, bestlag = min((v.bic, k) for k, v in iteritems(results))
elif method == "t-stat":
#stop = stats.norm.ppf(.95)
stop = 1.6448536269514722
for lag in range(startlag + maxlag, startlag - 1, -1):
icbest = np.abs(results[lag].tvalues[-1])
if np.abs(icbest) >= stop:
bestlag = lag
icbest = icbest
break
else:
raise ValueError("Information Criterion %s not understood.") % method
if not regresults:
return icbest, bestlag
else:
return icbest, bestlag, results
#this needs to be converted to a class like HetGoldfeldQuandt,
# 3 different returns are a mess
# See:
#Ng and Perron(2001), Lag length selection and the construction of unit root
#tests with good size and power, Econometrica, Vol 69 (6) pp 1519-1554
#TODO: include drift keyword, only valid with regression == "c"
# just changes the distribution of the test statistic to a t distribution
#TODO: autolag is untested
[docs]def adfuller(x, maxlag=None, regression="c", autolag='AIC',
store=False, regresults=False):
"""
Augmented Dickey-Fuller unit root test
The Augmented Dickey-Fuller test can be used to test for a unit root in a
univariate process in the presence of serial correlation.
Parameters
----------
x : array_like, 1d
data series
maxlag : int
Maximum lag which is included in test, default 12*(nobs/100)^{1/4}
regression : {'c','ct','ctt','nc'}
Constant and trend order to include in regression
* 'c' : constant only (default)
* 'ct' : constant and trend
* 'ctt' : constant, and linear and quadratic trend
* 'nc' : no constant, no trend
autolag : {'AIC', 'BIC', 't-stat', None}
* if None, then maxlag lags are used
* if 'AIC' (default) or 'BIC', then the number of lags is chosen
to minimize the corresponding information criterion
* 't-stat' based choice of maxlag. Starts with maxlag and drops a
lag until the t-statistic on the last lag length is significant
using a 5%-sized test
store : bool
If True, then a result instance is returned additionally to
the adf statistic. Default is False
regresults : bool, optional
If True, the full regression results are returned. Default is False
Returns
-------
adf : float
Test statistic
pvalue : float
MacKinnon's approximate p-value based on MacKinnon (1994, 2010)
usedlag : int
Number of lags used
nobs : int
Number of observations used for the ADF regression and calculation of
the critical values
critical values : dict
Critical values for the test statistic at the 1 %, 5 %, and 10 %
levels. Based on MacKinnon (2010)
icbest : float
The maximized information criterion if autolag is not None.
resstore : ResultStore, optional
A dummy class with results attached as attributes
Notes
-----
The null hypothesis of the Augmented Dickey-Fuller is that there is a unit
root, with the alternative that there is no unit root. If the pvalue is
above a critical size, then we cannot reject that there is a unit root.
The p-values are obtained through regression surface approximation from
MacKinnon 1994, but using the updated 2010 tables. If the p-value is close
to significant, then the critical values should be used to judge whether
to reject the null.
The autolag option and maxlag for it are described in Greene.
Examples
--------
See example notebook
References
----------
.. [*] W. Green. "Econometric Analysis," 5th ed., Pearson, 2003.
.. [*] Hamilton, J.D. "Time Series Analysis". Princeton, 1994.
.. [*] MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
.. [*] MacKinnon, J.G. 2010. "Critical Values for Cointegration Tests." Queen's
University, Dept of Economics, Working Papers. Available at
http://ideas.repec.org/p/qed/wpaper/1227.html
"""
if regresults:
store = True
trenddict = {None: 'nc', 0: 'c', 1: 'ct', 2: 'ctt'}
if regression is None or isinstance(regression, (int, long)):
regression = trenddict[regression]
regression = regression.lower()
if regression not in ['c', 'nc', 'ct', 'ctt']:
raise ValueError("regression option %s not understood") % regression
x = np.asarray(x)
nobs = x.shape[0]
ntrend = len(regression) if regression != 'nc' else 0
if maxlag is None:
# from Greene referencing Schwert 1989
maxlag = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
# -1 for the diff
maxlag = min(nobs // 2 - ntrend - 1, maxlag)
if maxlag < 0:
raise ValueError('sample size is too short to use selected '
'regression component')
elif maxlag > nobs // 2 - ntrend - 1:
raise ValueError('maxlag must be less than (nobs/2 - 1 - ntrend) '
'where n trend is the number of included '
'deterministic regressors')
xdiff = np.diff(x)
xdall = lagmat(xdiff[:, None], maxlag, trim='both', original='in')
nobs = xdall.shape[0]
xdall[:, 0] = x[-nobs - 1:-1] # replace 0 xdiff with level of x
xdshort = xdiff[-nobs:]
if store:
resstore = ResultsStore()
if autolag:
if regression != 'nc':
fullRHS = add_trend(xdall, regression, prepend=True)
else:
fullRHS = xdall
startlag = fullRHS.shape[1] - xdall.shape[1] + 1
# 1 for level
# search for lag length with smallest information criteria
# Note: use the same number of observations to have comparable IC
# aic and bic: smaller is better
if not regresults:
icbest, bestlag = _autolag(OLS, xdshort, fullRHS, startlag,
maxlag, autolag)
else:
icbest, bestlag, alres = _autolag(OLS, xdshort, fullRHS, startlag,
maxlag, autolag,
regresults=regresults)
resstore.autolag_results = alres
bestlag -= startlag # convert to lag not column index
# rerun ols with best autolag
xdall = lagmat(xdiff[:, None], bestlag, trim='both', original='in')
nobs = xdall.shape[0]
xdall[:, 0] = x[-nobs - 1:-1] # replace 0 xdiff with level of x
xdshort = xdiff[-nobs:]
usedlag = bestlag
else:
usedlag = maxlag
icbest = None
if regression != 'nc':
resols = OLS(xdshort, add_trend(xdall[:, :usedlag + 1],
regression)).fit()
else:
resols = OLS(xdshort, xdall[:, :usedlag + 1]).fit()
adfstat = resols.tvalues[0]
# adfstat = (resols.params[0]-1.0)/resols.bse[0]
# the "asymptotically correct" z statistic is obtained as
# nobs/(1-np.sum(resols.params[1:-(trendorder+1)])) (resols.params[0] - 1)
# I think this is the statistic that is used for series that are integrated
# for orders higher than I(1), ie., not ADF but cointegration tests.
# Get approx p-value and critical values
pvalue = mackinnonp(adfstat, regression=regression, N=1)
critvalues = mackinnoncrit(N=1, regression=regression, nobs=nobs)
critvalues = {"1%" : critvalues[0], "5%" : critvalues[1],
"10%" : critvalues[2]}
if store:
resstore.resols = resols
resstore.maxlag = maxlag
resstore.usedlag = usedlag
resstore.adfstat = adfstat
resstore.critvalues = critvalues
resstore.nobs = nobs
resstore.H0 = ("The coefficient on the lagged level equals 1 - "
"unit root")
resstore.HA = "The coefficient on the lagged level < 1 - stationary"
resstore.icbest = icbest
resstore._str = 'Augmented Dickey-Fuller Test Results'
return adfstat, pvalue, critvalues, resstore
else:
if not autolag:
return adfstat, pvalue, usedlag, nobs, critvalues
else:
return adfstat, pvalue, usedlag, nobs, critvalues, icbest
[docs]def acovf(x, unbiased=False, demean=True, fft=None, missing='none', nlag=None):
"""
Autocovariance for 1D
Parameters
----------
x : array
Time series data. Must be 1d.
unbiased : bool
If True, then denominators is n-k, otherwise n
demean : bool
If True, then subtract the mean x from each element of x
fft : bool
If True, use FFT convolution. This method should be preferred
for long time series.
missing : str
A string in ['none', 'raise', 'conservative', 'drop'] specifying how
the NaNs are to be treated.
nlag : {int, None}
Limit the number of autocovariances returned. Size of returned
array is nlag + 1. Setting nlag when fft is False uses a simple,
direct estimator of the autocovariances that only computes the first
nlag + 1 values. This can be much faster when the time series is long
and only a small number of autocovariances are needed.
Returns
-------
acovf : array
autocovariance function
References
-----------
.. [*] Parzen, E., 1963. On spectral analysis with missing observations
and amplitude modulation. Sankhya: The Indian Journal of
Statistics, Series A, pp.383-392.
"""
if fft is None:
import warnings
msg = 'fft=True will become the default in a future version of ' \
'statsmodels. To suppress this warning, explicitly set ' \
'fft=False.'
warnings.warn(msg, FutureWarning)
fft = False
x = np.squeeze(np.asarray(x))
if x.ndim > 1:
raise ValueError("x must be 1d. Got %d dims." % x.ndim)
missing = missing.lower()
if missing not in ['none', 'raise', 'conservative', 'drop']:
raise ValueError("missing option %s not understood" % missing)
if missing == 'none':
deal_with_masked = False
else:
deal_with_masked = has_missing(x)
if deal_with_masked:
if missing == 'raise':
raise MissingDataError("NaNs were encountered in the data")
notmask_bool = ~np.isnan(x) # bool
if missing == 'conservative':
# Must copy for thread safety
x = x.copy()
x[~notmask_bool] = 0
else: # 'drop'
x = x[notmask_bool] # copies non-missing
notmask_int = notmask_bool.astype(int) # int
if demean and deal_with_masked:
# whether 'drop' or 'conservative':
xo = x - x.sum() / notmask_int.sum()
if missing == 'conservative':
xo[~notmask_bool] = 0
elif demean:
xo = x - x.mean()
else:
xo = x
n = len(x)
lag_len = nlag
if nlag is None:
lag_len = n - 1
elif nlag > n - 1:
raise ValueError('nlag must be smaller than nobs - 1')
if not fft and nlag is not None:
acov = np.empty(lag_len + 1)
acov[0] = xo.dot(xo)
for i in range(lag_len):
acov[i + 1] = xo[i + 1:].dot(xo[:-(i + 1)])
if not deal_with_masked or missing == 'drop':
if unbiased:
acov /= (n - np.arange(lag_len + 1))
else:
acov /= n
else:
if unbiased:
divisor = np.empty(lag_len + 1, dtype=np.int64)
divisor[0] = notmask_int.sum()
for i in range(lag_len):
divisor[i + 1] = notmask_int[i + 1:].dot(notmask_int[:-(i + 1)])
divisor[divisor == 0] = 1
acov /= divisor
else: # biased, missing data but npt 'drop'
acov /= notmask_int.sum()
return acov
if unbiased and deal_with_masked and missing == 'conservative':
d = np.correlate(notmask_int, notmask_int, 'full')
d[d == 0] = 1
elif unbiased:
xi = np.arange(1, n + 1)
d = np.hstack((xi, xi[:-1][::-1]))
elif deal_with_masked: # biased and NaNs given and ('drop' or 'conservative')
d = notmask_int.sum() * np.ones(2 * n - 1)
else: # biased and no NaNs or missing=='none'
d = n * np.ones(2 * n - 1)
if fft:
nobs = len(xo)
n = _next_regular(2 * nobs + 1)
Frf = np.fft.fft(xo, n=n)
acov = np.fft.ifft(Frf * np.conjugate(Frf))[:nobs] / d[nobs - 1:]
acov = acov.real
else:
acov = np.correlate(xo, xo, 'full')[n - 1:] / d[n - 1:]
if nlag is not None:
# Copy to allow gc of full array rather than view
return acov[:lag_len + 1].copy()
return acov
[docs]def q_stat(x, nobs, type="ljungbox"):
"""
Return's Ljung-Box Q Statistic
x : array-like
Array of autocorrelation coefficients. Can be obtained from acf.
nobs : int
Number of observations in the entire sample (ie., not just the length
of the autocorrelation function results.
Returns
-------
q-stat : array
Ljung-Box Q-statistic for autocorrelation parameters
p-value : array
P-value of the Q statistic
Notes
-----
Written to be used with acf.
"""
x = np.asarray(x)
if type == "ljungbox":
ret = (nobs * (nobs + 2) *
np.cumsum((1. / (nobs - np.arange(1, len(x) + 1))) * x**2))
chi2 = stats.chi2.sf(ret, np.arange(1, len(x) + 1))
return ret, chi2
#NOTE: Changed unbiased to False
#see for example
# http://www.itl.nist.gov/div898/handbook/eda/section3/autocopl.htm
[docs]def acf(x, unbiased=False, nlags=40, qstat=False, fft=None, alpha=None,
missing='none'):
"""
Autocorrelation function for 1d arrays.
Parameters
----------
x : array
Time series data
unbiased : bool
If True, then denominators for autocovariance are n-k, otherwise n
nlags: int, optional
Number of lags to return autocorrelation for.
qstat : bool, optional
If True, returns the Ljung-Box q statistic for each autocorrelation
coefficient. See q_stat for more information.
fft : bool, optional
If True, computes the ACF via FFT.
alpha : scalar, optional
If a number is given, the confidence intervals for the given level are
returned. For instance if alpha=.05, 95 % confidence intervals are
returned where the standard deviation is computed according to
Bartlett\'s formula.
missing : str, optional
A string in ['none', 'raise', 'conservative', 'drop'] specifying how the NaNs
are to be treated.
Returns
-------
acf : array
autocorrelation function
confint : array, optional
Confidence intervals for the ACF. Returned if alpha is not None.
qstat : array, optional
The Ljung-Box Q-Statistic. Returned if q_stat is True.
pvalues : array, optional
The p-values associated with the Q-statistics. Returned if q_stat is
True.
Notes
-----
The acf at lag 0 (ie., 1) is returned.
For very long time series it is recommended to use fft convolution instead.
When fft is False uses a simple, direct estimator of the autocovariances
that only computes the first nlag + 1 values. This can be much faster when
the time series is long and only a small number of autocovariances are
needed.
If unbiased is true, the denominator for the autocovariance is adjusted
but the autocorrelation is not an unbiased estimator.
References
----------
.. [*] Parzen, E., 1963. On spectral analysis with missing observations
and amplitude modulation. Sankhya: The Indian Journal of
Statistics, Series A, pp.383-392.
"""
if fft is None:
import warnings
msg = 'fft=True will become the default in a future version of ' \
'statsmodels. To suppress this warning, explicitly set ' \
'fft=False.'
warnings.warn(msg, FutureWarning)
fft = False
nobs = len(x) # should this shrink for missing='drop' and NaNs in x?
avf = acovf(x, unbiased=unbiased, demean=True, fft=fft, missing=missing)
acf = avf[:nlags + 1] / avf[0]
if not (qstat or alpha):
return acf
if alpha is not None:
varacf = np.ones(nlags + 1) / nobs
varacf[0] = 0
varacf[1] = 1. / nobs
varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1]**2)
interval = stats.norm.ppf(1 - alpha / 2.) * np.sqrt(varacf)
confint = np.array(lzip(acf - interval, acf + interval))
if not qstat:
return acf, confint
if qstat:
qstat, pvalue = q_stat(acf[1:], nobs=nobs) # drop lag 0
if alpha is not None:
return acf, confint, qstat, pvalue
else:
return acf, qstat, pvalue
[docs]def pacf_yw(x, nlags=40, method='unbiased'):
'''Partial autocorrelation estimated with non-recursive yule_walker
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
largest lag for which pacf is returned
method : 'unbiased' (default) or 'mle'
method for the autocovariance calculations in yule walker
Returns
-------
pacf : 1d array
partial autocorrelations, maxlag+1 elements
See Also
--------
statsmodels.tsa.stattools.pacf
statsmodels.tsa.stattools.pacf_burg
statsmodels.tsa.stattools.pacf_ols
Notes
-----
This solves yule_walker for each desired lag and contains
currently duplicate calculations.
'''
pacf = [1.]
for k in range(1, nlags + 1):
pacf.append(yule_walker(x, k, method=method)[0][-1])
return np.array(pacf)
[docs]def pacf_burg(x, nlags=None, demean=True):
"""
Burg's partial autocorrelation estimator
Parameters
----------
x : array-like
Observations of time series for which pacf is calculated
nlags : int, optional
Number of lags to compute the partial autocorrelations. If omitted,
uses the smaller of 10(log10(nobs)) or nobs - 1
demean : bool, optional
Returns
-------
pacf : ndarray
Partial autocorrelations for lags 0, 1, ..., nlag
sigma2 : ndarray
Residual variance estimates where the value in position m is the
residual variance in an AR model that includes m lags
See Also
--------
statsmodels.tsa.stattools.pacf
statsmodels.tsa.stattools.pacf_yw
statsmodels.tsa.stattools.pacf_ols
References
----------
.. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
x = np.squeeze(np.asarray(x))
if x.ndim != 1:
raise ValueError('x must be 1-d or squeezable to 1-d.')
if demean:
x = x - x.mean()
nobs = x.shape[0]
p = nlags if nlags is not None else min(int(10 * np.log10(nobs)), nobs - 1)
if p > nobs - 1:
raise ValueError('nlags must be smaller than nobs - 1')
d = np.zeros(p + 1)
d[0] = 2 * x.dot(x)
pacf = np.zeros(p + 1)
u = x[::-1].copy()
v = x[::-1].copy()
d[1] = u[:-1].dot(u[:-1]) + v[1:].dot(v[1:])
pacf[1] = 2 / d[1] * v[1:].dot(u[:-1])
last_u = np.empty_like(u)
last_v = np.empty_like(v)
for i in range(1, p):
last_u[:] = u
last_v[:] = v
u[1:] = last_u[:-1] - pacf[i] * last_v[1:]
v[1:] = last_v[1:] - pacf[i] * last_u[:-1]
d[i + 1] = (1 - pacf[i] ** 2) * d[i] - v[i] ** 2 - u[-1] ** 2
pacf[i + 1] = 2 / d[i + 1] * v[i + 1:].dot(u[i:-1])
sigma2 = (1 - pacf ** 2) * d / (2. * (nobs - np.arange(0, p + 1)))
pacf[0] = 1 # Insert the 0 lag partial autocorrel
return pacf, sigma2
[docs]def pacf_ols(x, nlags=40, efficient=True, unbiased=False):
"""
Calculate partial autocorrelations via OLS
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
Number of lags for which pacf is returned. Lag 0 is not returned.
efficient : bool, optional
If true, uses the maximum number of available observations to compute
each partial autocorrelation. If not, uses the same number of
observations to compute all pacf values.
unbiased : bool, optional
Adjust each partial autocorrelation by n / (n - lag)
Returns
-------
pacf : 1d array
partial autocorrelations, (maxlag,) array corresponding to lags
0, 1, ..., maxlag
Notes
-----
This solves a separate OLS estimation for each desired lag using method in
[1]_. Setting efficient to True has two effects. First, it uses
`nobs - lag` observations of estimate each pacf. Second, it re-estimates
the mean in each regression. If efficient is False, then the data are first
demeaned, and then `nobs - maxlag` observations are used to estimate each
partial autocorrelation.
The inefficient estimator appears to have better finite sample properties.
This option should only be used in time series that are covariance
stationary.
OLS estimation of the pacf does not guarantee that all pacf values are
between -1 and 1.
See Also
--------
statsmodels.tsa.stattools.pacf
statsmodels.tsa.stattools.pacf_yw
statsmodels.tsa.stattools.pacf_burg
References
----------
.. [1] Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015).
Time series analysis: forecasting and control. John Wiley & Sons, p. 66
"""
pacf = np.empty(nlags + 1)
pacf[0] = 1.0
x = np.squeeze(np.asarray(x))
if x.ndim != 1:
raise ValueError('x must be squeezable to a 1-d array')
if efficient:
xlags, x0 = lagmat(x, nlags, original='sep')
xlags = add_constant(xlags)
for k in range(1, nlags + 1):
params = lstsq(xlags[k:, :k + 1], x0[k:], rcond=None)[0]
pacf[k] = params[-1]
else:
x = x - np.mean(x)
# Create a single set of lags for multivariate OLS
xlags, x0 = lagmat(x, nlags, original='sep', trim='both')
for k in range(1, nlags + 1):
params = lstsq(xlags[:, :k], x0, rcond=None)[0]
# Last coefficient corresponds to PACF value (see [1])
pacf[k] = params[-1]
if unbiased:
n = len(x)
pacf *= n / (n - np.arange(nlags + 1))
return pacf
[docs]def pacf(x, nlags=40, method='ywunbiased', alpha=None):
"""
Partial autocorrelation estimated
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
largest lag for which the pacf is returned
method : str
specifies which method for the calculations to use:
- 'yw' or 'ywunbiased' : Yule-Walker with bias correction in
denominator for acovf. Default.
- 'ywm' or 'ywmle' : Yule-Walker without bias correction
- 'ols' : regression of time series on lags of it and on constant
- 'ols-inefficient' : regression of time series on lags using a single
common sample to estimate all pacf coefficients
- 'ols-unbiased' : regression of time series on lags with a bias
adjustment
- 'ld' or 'ldunbiased' : Levinson-Durbin recursion with bias correction
- 'ldb' or 'ldbiased' : Levinson-Durbin recursion without bias
correction
alpha : float, optional
If a number is given, the confidence intervals for the given level are
returned. For instance if alpha=.05, 95 % confidence intervals are
returned where the standard deviation is computed according to
1/sqrt(len(x))
Returns
-------
pacf : 1d array
partial autocorrelations, nlags elements, including lag zero
confint : array, optional
Confidence intervals for the PACF. Returned if confint is not None.
See Also
--------
statsmodels.tsa.stattools.acf
statsmodels.tsa.stattools.pacf_yw
statsmodels.tsa.stattools.pacf_burg
statsmodels.tsa.stattools.pacf_ols
Notes
-----
Based on simulation evidence across a range of low-order ARMA models,
the best methods based on root MSE are Yule-Walker (MLW), Levinson-Durbin
(MLE) and Burg, respectively. The estimators with the lowest bias included
included these three in addition to OLS and OLS-unbiased.
Yule-Walker (unbiased) and Levinson-Durbin (unbiased) performed
consistently worse than the other options.
"""
if method in ('ols', 'ols-inefficient', 'ols-unbiased'):
efficient = 'inefficient' not in method
unbiased = 'unbiased' in method
ret = pacf_ols(x, nlags=nlags, efficient=efficient, unbiased=unbiased)
elif method in ('yw', 'ywu', 'ywunbiased', 'yw_unbiased'):
ret = pacf_yw(x, nlags=nlags, method='unbiased')
elif method in ('ywm', 'ywmle', 'yw_mle'):
ret = pacf_yw(x, nlags=nlags, method='mle')
elif method in ('ld', 'ldu', 'ldunbiased', 'ld_unbiased'):
acv = acovf(x, unbiased=True, fft=False)
ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
ret = ld_[2]
# inconsistent naming with ywmle
elif method in ('ldb', 'ldbiased', 'ld_biased'):
acv = acovf(x, unbiased=False, fft=False)
ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
ret = ld_[2]
else:
raise ValueError('method not available')
if alpha is not None:
varacf = 1. / len(x) # for all lags >=1
interval = stats.norm.ppf(1. - alpha / 2.) * np.sqrt(varacf)
confint = np.array(lzip(ret - interval, ret + interval))
confint[0] = ret[0] # fix confidence interval for lag 0 to varpacf=0
return ret, confint
else:
return ret
[docs]def ccovf(x, y, unbiased=True, demean=True):
''' crosscovariance for 1D
Parameters
----------
x, y : arrays
time series data
unbiased : boolean
if True, then denominators is n-k, otherwise n
Returns
-------
ccovf : array
autocovariance function
Notes
-----
This uses np.correlate which does full convolution. For very long time
series it is recommended to use fft convolution instead.
'''
n = len(x)
if demean:
xo = x - x.mean()
yo = y - y.mean()
else:
xo = x
yo = y
if unbiased:
xi = np.ones(n)
d = np.correlate(xi, xi, 'full')
else:
d = n
return (np.correlate(xo, yo, 'full') / d)[n - 1:]
[docs]def ccf(x, y, unbiased=True):
'''cross-correlation function for 1d
Parameters
----------
x, y : arrays
time series data
unbiased : boolean
if True, then denominators for autocovariance is n-k, otherwise n
Returns
-------
ccf : array
cross-correlation function of x and y
Notes
-----
This is based np.correlate which does full convolution. For very long time
series it is recommended to use fft convolution instead.
If unbiased is true, the denominator for the autocovariance is adjusted
but the autocorrelation is not an unbiased estimtor.
'''
cvf = ccovf(x, y, unbiased=unbiased, demean=True)
return cvf / (np.std(x) * np.std(y))
[docs]def periodogram(X):
"""
Returns the periodogram for the natural frequency of X
Parameters
----------
X : array-like
Array for which the periodogram is desired.
Returns
-------
pgram : array
1./len(X) * np.abs(np.fft.fft(X))**2
References
----------
.. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
X = np.asarray(X)
# if kernel == "bartlett":
# w = 1 - np.arange(M+1.)/M #JP removed integer division
pergr = 1. / len(X) * np.abs(np.fft.fft(X))**2
pergr[0] = 0. # what are the implications of this?
return pergr
# moved from sandbox.tsa.examples.try_ld_nitime, via nitime
# TODO: check what to return, for testing and trying out returns everything
[docs]def levinson_durbin(s, nlags=10, isacov=False):
"""
Levinson-Durbin recursion for autoregressive processes
Parameters
----------
s : array_like
If isacov is False, then this is the time series. If iasacov is true
then this is interpreted as autocovariance starting with lag 0
nlags : integer
largest lag to include in recursion or order of the autoregressive
process
isacov : boolean
flag to indicate whether the first argument, s, contains the
autocovariances or the data series.
Returns
-------
sigma_v : float
estimate of the error variance ?
arcoefs : ndarray
estimate of the autoregressive coefficients for a model including nlags
pacf : ndarray
partial autocorrelation function
sigma : ndarray
entire sigma array from intermediate result, last value is sigma_v
phi : ndarray
entire phi array from intermediate result, last column contains
autoregressive coefficients for AR(nlags)
Notes
-----
This function returns currently all results, but maybe we drop sigma and
phi from the returns.
If this function is called with the time series (isacov=False), then the
sample autocovariance function is calculated with the default options
(biased, no fft).
"""
s = np.asarray(s)
order = nlags
if isacov:
sxx_m = s
else:
sxx_m = acovf(s, fft=False)[:order + 1] # not tested
phi = np.zeros((order + 1, order + 1), 'd')
sig = np.zeros(order + 1)
# initial points for the recursion
phi[1, 1] = sxx_m[1] / sxx_m[0]
sig[1] = sxx_m[0] - phi[1, 1] * sxx_m[1]
for k in range(2, order + 1):
phi[k, k] = (sxx_m[k] - np.dot(phi[1:k, k-1],
sxx_m[1:k][::-1])) / sig[k-1]
for j in range(1, k):
phi[j, k] = phi[j, k-1] - phi[k, k] * phi[k-j, k-1]
sig[k] = sig[k-1] * (1 - phi[k, k]**2)
sigma_v = sig[-1]
arcoefs = phi[1:, -1]
pacf_ = np.diag(phi).copy()
pacf_[0] = 1.
return sigma_v, arcoefs, pacf_, sig, phi # return everything
[docs]def levinson_durbin_pacf(pacf, nlags=None):
"""
Levinson-Durbin algorithm that returns the acf and ar coefficients
Parameters
----------
pacf : array-like
Partial autocorrelation array for lags 0, 1, ... p
nlags : int, optional
Number of lags in the AR model. If omitted, returns coefficients from
an AR(p) and the first p autocorrelations
Returns
-------
arcoefs : ndarray
AR coefficients computed from the partial autocorrelations
acf : ndarray
acf computed from the partial autocorrelations. Array returned contains
the autocorelations corresponding to lags 0, 1, ..., p
References
----------
.. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
pacf = np.squeeze(np.asarray(pacf))
if pacf.ndim != 1:
raise ValueError('pacf must be 1-d or squeezable to 1-d.')
if pacf[0] != 1:
raise ValueError('The first entry of the pacf corresponds to lags 0 '
'and so must be 1.')
pacf = pacf[1:]
n = pacf.shape[0]
if nlags is not None:
if nlags > n:
raise ValueError('Must provide at least as many values from the '
'pacf as the number of lags.')
pacf = pacf[:nlags]
n = pacf.shape[0]
acf = np.zeros(n + 1)
acf[1] = pacf[0]
nu = np.cumprod(1 - pacf ** 2)
arcoefs = pacf.copy()
for i in range(1, n):
prev = arcoefs[:-(n - i)].copy()
arcoefs[:-(n - i)] = prev - arcoefs[i] * prev[::-1]
acf[i + 1] = arcoefs[i] * nu[i-1] + prev.dot(acf[1:-(n - i)][::-1])
acf[0] = 1
return arcoefs, acf
[docs]def innovations_algo(acov, nobs=None, rtol=None):
"""
Innovations algorithm to convert autocovariances to MA parameters
Parameters
----------
acov : array-like
Array containing autocovariances including lag 0
nobs : int, optional
Number of periods to run the algorithm. If not provided, nobs is
equal to the length of acovf
rtol : float, optional
Tolerance used to check for convergence. Default value is 0 which will
never prematurely end the algorithm. Checks after 10 iterations and
stops if sigma2[i] - sigma2[i - 10] < rtol * sigma2[0]. When the
stopping condition is met, the remaining values in theta and sigma2
are forward filled using the value of the final iteration.
Returns
-------
theta : ndarray
Innovation coefficients of MA representation. Array is (nobs, q) where
q is the largest index of a non-zero autocovariance. theta
corresponds to the first q columns of the coefficient matrix in the
common description of the innovation algorithm.
sigma2 : ndarray
The prediction error variance (nobs,).
Examples
--------
>>> import statsmodels.api as sm
>>> data = sm.datasets.macrodata.load_pandas()
>>> rgdpg = data.data['realgdp'].pct_change().dropna()
>>> acov = sm.tsa.acovf(rgdpg)
>>> nobs = activity.shape[0]
>>> theta, sigma2 = innovations_algo(acov[:4], nobs=nobs)
See Also
--------
innovations_filter
References
----------
.. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
acov = np.squeeze(np.asarray(acov))
if acov.ndim != 1:
raise ValueError('acov must be 1-d or squeezable to 1-d.')
rtol = 0.0 if rtol is None else rtol
if not isinstance(rtol, float):
raise ValueError('rtol must be a non-negative float or None.')
if nobs is not None and (nobs != int(nobs) or nobs < 1):
raise ValueError('nobs must be a positive integer')
n = acov.shape[0] if nobs is None else int(nobs)
max_lag = int(np.max(np.argwhere(acov != 0)))
v = np.zeros(n + 1)
v[0] = acov[0]
# Retain only the relevant columns of theta
theta = np.zeros((n + 1, max_lag + 1))
for i in range(1, n):
for k in range(max(i - max_lag, 0), i):
sub = 0
for j in range(max(i - max_lag, 0), k):
sub += theta[k, k - j] * theta[i, i - j] * v[j]
theta[i, i - k] = 1. / v[k] * (acov[i - k] - sub)
v[i] = acov[0]
for j in range(max(i - max_lag, 0), i):
v[i] -= theta[i, i - j] ** 2 * v[j]
# Break if v has converged
if i >= 10:
if v[i - 10] - v[i] < v[0] * rtol:
# Forward fill all remaining values
v[i + 1:] = v[i]
theta[i + 1:] = theta[i]
break
theta = theta[:-1, 1:]
v = v[:-1]
return theta, v
[docs]def innovations_filter(endog, theta):
"""
Filter observations using the innovations algorithm
Parameters
----------
endog : array-like
The time series to filter (nobs,). Should be demeaned if not mean 0.
theta : ndarray
Innovation coefficients of MA representation. Array must be (nobs, q)
where q order of the MA.
Returns
-------
resid : ndarray
Array of filtered innovations
Examples
--------
>>> import statsmodels.api as sm
>>> data = sm.datasets.macrodata.load_pandas()
>>> rgdpg = data.data['realgdp'].pct_change().dropna()
>>> acov = sm.tsa.acovf(rgdpg)
>>> nobs = activity.shape[0]
>>> theta, sigma2 = innovations_algo(acov[:4], nobs=nobs)
>>> resid = innovations_filter(rgdpg, theta)
See Also
--------
innovations_algo
References
----------
.. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
orig_endog = endog
endog = np.squeeze(np.asarray(endog))
if endog.ndim != 1:
raise ValueError('endog must be 1-d or squeezable to 1-d.')
nobs = endog.shape[0]
n_theta, k = theta.shape
if nobs != n_theta:
raise ValueError('theta must be (nobs, q) where q is the moder order')
is_pandas = isinstance(orig_endog, (pd.DataFrame, pd.Series))
if is_pandas:
if len(orig_endog.index) != nobs:
msg = 'If endog is a Series or DataFrame, the index must ' \
'correspond to the number of time series observations.'
raise ValueError(msg)
u = np.empty(nobs)
u[0] = endog[0]
for i in range(1, nobs):
if i < k:
hat = (theta[i, :i] * u[:i][::-1]).sum()
else:
hat = (theta[i] * u[i - k:i][::-1]).sum()
u[i] = endog[i] - hat
if is_pandas:
u = pd.Series(u, index=orig_endog.index.copy())
return u
[docs]def grangercausalitytests(x, maxlag, addconst=True, verbose=True):
"""four tests for granger non causality of 2 timeseries
all four tests give similar results
`params_ftest` and `ssr_ftest` are equivalent based on F test which is
identical to lmtest:grangertest in R
Parameters
----------
x : array, 2d
data for test whether the time series in the second column Granger
causes the time series in the first column
maxlag : integer
the Granger causality test results are calculated for all lags up to
maxlag
verbose : bool
print results if true
Returns
-------
results : dictionary
all test results, dictionary keys are the number of lags. For each
lag the values are a tuple, with the first element a dictionary with
teststatistic, pvalues, degrees of freedom, the second element are
the OLS estimation results for the restricted model, the unrestricted
model and the restriction (contrast) matrix for the parameter f_test.
Notes
-----
TODO: convert to class and attach results properly
The Null hypothesis for grangercausalitytests is that the time series in
the second column, x2, does NOT Granger cause the time series in the first
column, x1. Grange causality means that past values of x2 have a
statistically significant effect on the current value of x1, taking past
values of x1 into account as regressors. We reject the null hypothesis
that x2 does not Granger cause x1 if the pvalues are below a desired size
of the test.
The null hypothesis for all four test is that the coefficients
corresponding to past values of the second time series are zero.
'params_ftest', 'ssr_ftest' are based on F distribution
'ssr_chi2test', 'lrtest' are based on chi-square distribution
References
----------
http://en.wikipedia.org/wiki/Granger_causality
Greene: Econometric Analysis
"""
from scipy import stats
x = np.asarray(x)
if x.shape[0] <= 3 * maxlag + int(addconst):
raise ValueError("Insufficient observations. Maximum allowable "
"lag is {0}".format(int((x.shape[0] - int(addconst)) /
3) - 1))
resli = {}
for mlg in range(1, maxlag + 1):
result = {}
if verbose:
print('\nGranger Causality')
print('number of lags (no zero)', mlg)
mxlg = mlg
# create lagmat of both time series
dta = lagmat2ds(x, mxlg, trim='both', dropex=1)
#add constant
if addconst:
dtaown = add_constant(dta[:, 1:(mxlg + 1)], prepend=False)
dtajoint = add_constant(dta[:, 1:], prepend=False)
else:
raise NotImplementedError('Not Implemented')
#dtaown = dta[:, 1:mxlg]
#dtajoint = dta[:, 1:]
# Run ols on both models without and with lags of second variable
res2down = OLS(dta[:, 0], dtaown).fit()
res2djoint = OLS(dta[:, 0], dtajoint).fit()
#print results
#for ssr based tests see:
#http://support.sas.com/rnd/app/examples/ets/granger/index.htm
#the other tests are made-up
# Granger Causality test using ssr (F statistic)
fgc1 = ((res2down.ssr - res2djoint.ssr) /
res2djoint.ssr / mxlg * res2djoint.df_resid)
if verbose:
print('ssr based F test: F=%-8.4f, p=%-8.4f, df_denom=%d,'
' df_num=%d' % (fgc1,
stats.f.sf(fgc1, mxlg,
res2djoint.df_resid),
res2djoint.df_resid, mxlg))
result['ssr_ftest'] = (fgc1,
stats.f.sf(fgc1, mxlg, res2djoint.df_resid),
res2djoint.df_resid, mxlg)
# Granger Causality test using ssr (ch2 statistic)
fgc2 = res2down.nobs * (res2down.ssr - res2djoint.ssr) / res2djoint.ssr
if verbose:
print('ssr based chi2 test: chi2=%-8.4f, p=%-8.4f, '
'df=%d' % (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg))
result['ssr_chi2test'] = (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg)
#likelihood ratio test pvalue:
lr = -2 * (res2down.llf - res2djoint.llf)
if verbose:
print('likelihood ratio test: chi2=%-8.4f, p=%-8.4f, df=%d' %
(lr, stats.chi2.sf(lr, mxlg), mxlg))
result['lrtest'] = (lr, stats.chi2.sf(lr, mxlg), mxlg)
# F test that all lag coefficients of exog are zero
rconstr = np.column_stack((np.zeros((mxlg, mxlg)),
np.eye(mxlg, mxlg),
np.zeros((mxlg, 1))))
ftres = res2djoint.f_test(rconstr)
if verbose:
print('parameter F test: F=%-8.4f, p=%-8.4f, df_denom=%d,'
' df_num=%d' % (ftres.fvalue, ftres.pvalue, ftres.df_denom,
ftres.df_num))
result['params_ftest'] = (np.squeeze(ftres.fvalue)[()],
np.squeeze(ftres.pvalue)[()],
ftres.df_denom, ftres.df_num)
resli[mxlg] = (result, [res2down, res2djoint, rconstr])
return resli
[docs]def coint(y0, y1, trend='c', method='aeg', maxlag=None, autolag='aic',
return_results=None):
"""Test for no-cointegration of a univariate equation
The null hypothesis is no cointegration. Variables in y0 and y1 are
assumed to be integrated of order 1, I(1).
This uses the augmented Engle-Granger two-step cointegration test.
Constant or trend is included in 1st stage regression, i.e. in
cointegrating equation.
**Warning:** The autolag default has changed compared to statsmodels 0.8.
In 0.8 autolag was always None, no the keyword is used and defaults to
'aic'. Use `autolag=None` to avoid the lag search.
Parameters
----------
y1 : array_like, 1d
first element in cointegrating vector
y2 : array_like
remaining elements in cointegrating vector
trend : str {'c', 'ct'}
trend term included in regression for cointegrating equation
* 'c' : constant
* 'ct' : constant and linear trend
* also available quadratic trend 'ctt', and no constant 'nc'
method : string
currently only 'aeg' for augmented Engle-Granger test is available.
default might change.
maxlag : None or int
keyword for `adfuller`, largest or given number of lags
autolag : string
keyword for `adfuller`, lag selection criterion.
* if None, then maxlag lags are used without lag search
* if 'AIC' (default) or 'BIC', then the number of lags is chosen
to minimize the corresponding information criterion
* 't-stat' based choice of maxlag. Starts with maxlag and drops a
lag until the t-statistic on the last lag length is significant
using a 5%-sized test
return_results : bool
for future compatibility, currently only tuple available.
If True, then a results instance is returned. Otherwise, a tuple
with the test outcome is returned.
Set `return_results=False` to avoid future changes in return.
Returns
-------
coint_t : float
t-statistic of unit-root test on residuals
pvalue : float
MacKinnon's approximate, asymptotic p-value based on MacKinnon (1994)
crit_value : dict
Critical values for the test statistic at the 1 %, 5 %, and 10 %
levels based on regression curve. This depends on the number of
observations.
Notes
-----
The Null hypothesis is that there is no cointegration, the alternative
hypothesis is that there is cointegrating relationship. If the pvalue is
small, below a critical size, then we can reject the hypothesis that there
is no cointegrating relationship.
P-values and critical values are obtained through regression surface
approximation from MacKinnon 1994 and 2010.
If the two series are almost perfectly collinear, then computing the
test is numerically unstable. However, the two series will be cointegrated
under the maintained assumption that they are integrated. In this case
the t-statistic will be set to -inf and the pvalue to zero.
TODO: We could handle gaps in data by dropping rows with nans in the
auxiliary regressions. Not implemented yet, currently assumes no nans
and no gaps in time series.
References
----------
MacKinnon, J.G. 1994 "Approximate Asymptotic Distribution Functions for
Unit-Root and Cointegration Tests." Journal of Business & Economics
Statistics, 12.2, 167-76.
MacKinnon, J.G. 2010. "Critical Values for Cointegration Tests."
Queen's University, Dept of Economics Working Papers 1227.
http://ideas.repec.org/p/qed/wpaper/1227.html
"""
trend = trend.lower()
if trend not in ['c', 'nc', 'ct', 'ctt']:
raise ValueError("trend option %s not understood" % trend)
y0 = np.asarray(y0)
y1 = np.asarray(y1)
if y1.ndim < 2:
y1 = y1[:, None]
nobs, k_vars = y1.shape
k_vars += 1 # add 1 for y0
if trend == 'nc':
xx = y1
else:
xx = add_trend(y1, trend=trend, prepend=False)
res_co = OLS(y0, xx).fit()
if res_co.rsquared < 1 - 100 * SQRTEPS:
res_adf = adfuller(res_co.resid, maxlag=maxlag, autolag=autolag,
regression='nc')
else:
import warnings
warnings.warn("y0 and y1 are (almost) perfectly colinear."
"Cointegration test is not reliable in this case.",
CollinearityWarning)
# Edge case where series are too similar
res_adf = (-np.inf,)
# no constant or trend, see egranger in Stata and MacKinnon
if trend == 'nc':
crit = [np.nan] * 3 # 2010 critical values not available
else:
crit = mackinnoncrit(N=k_vars, regression=trend, nobs=nobs - 1)
# nobs - 1, the -1 is to match egranger in Stata, I don't know why.
# TODO: check nobs or df = nobs - k
pval_asy = mackinnonp(res_adf[0], regression=trend, N=k_vars)
return res_adf[0], pval_asy, crit
def _safe_arma_fit(y, order, model_kw, trend, fit_kw, start_params=None):
try:
return ARMA(y, order=order, **model_kw).fit(disp=0, trend=trend,
start_params=start_params,
**fit_kw)
except LinAlgError:
# SVD convergence failure on badly misspecified models
return
except ValueError as error:
if start_params is not None: # don't recurse again
# user supplied start_params only get one chance
return
# try a little harder, should be handled in fit really
elif ('initial' not in error.args[0] or 'initial' in str(error)):
start_params = [.1] * sum(order)
if trend == 'c':
start_params = [.1] + start_params
return _safe_arma_fit(y, order, model_kw, trend, fit_kw,
start_params)
else:
return
except: # no idea what happened
return
[docs]def arma_order_select_ic(y, max_ar=4, max_ma=2, ic='bic', trend='c',
model_kw=None, fit_kw=None):
"""
Returns information criteria for many ARMA models
Parameters
----------
y : array-like
Time-series data
max_ar : int
Maximum number of AR lags to use. Default 4.
max_ma : int
Maximum number of MA lags to use. Default 2.
ic : str, list
Information criteria to report. Either a single string or a list
of different criteria is possible.
trend : str
The trend to use when fitting the ARMA models.
model_kw : dict
Keyword arguments to be passed to the ``ARMA`` model
fit_kw : dict
Keyword arguments to be passed to ``ARMA.fit``.
Returns
-------
obj : Results object
Each ic is an attribute with a DataFrame for the results. The AR order
used is the row index. The ma order used is the column index. The
minimum orders are available as ``ic_min_order``.
Examples
--------
>>> from statsmodels.tsa.arima_process import arma_generate_sample
>>> import statsmodels.api as sm
>>> import numpy as np
>>> arparams = np.array([.75, -.25])
>>> maparams = np.array([.65, .35])
>>> arparams = np.r_[1, -arparams]
>>> maparam = np.r_[1, maparams]
>>> nobs = 250
>>> np.random.seed(2014)
>>> y = arma_generate_sample(arparams, maparams, nobs)
>>> res = sm.tsa.arma_order_select_ic(y, ic=['aic', 'bic'], trend='nc')
>>> res.aic_min_order
>>> res.bic_min_order
Notes
-----
This method can be used to tentatively identify the order of an ARMA
process, provided that the time series is stationary and invertible. This
function computes the full exact MLE estimate of each model and can be,
therefore a little slow. An implementation using approximate estimates
will be provided in the future. In the meantime, consider passing
{method : 'css'} to fit_kw.
"""
from pandas import DataFrame
ar_range = lrange(0, max_ar + 1)
ma_range = lrange(0, max_ma + 1)
if isinstance(ic, string_types):
ic = [ic]
elif not isinstance(ic, (list, tuple)):
raise ValueError("Need a list or a tuple for ic if not a string.")
results = np.zeros((len(ic), max_ar + 1, max_ma + 1))
model_kw = {} if model_kw is None else model_kw
fit_kw = {} if fit_kw is None else fit_kw
y_arr = np.asarray(y)
for ar in ar_range:
for ma in ma_range:
if ar == 0 and ma == 0 and trend == 'nc':
results[:, ar, ma] = np.nan
continue
mod = _safe_arma_fit(y_arr, (ar, ma), model_kw, trend, fit_kw)
if mod is None:
results[:, ar, ma] = np.nan
continue
for i, criteria in enumerate(ic):
results[i, ar, ma] = getattr(mod, criteria)
dfs = [DataFrame(res, columns=ma_range, index=ar_range) for res in results]
res = dict(zip(ic, dfs))
# add the minimums to the results dict
min_res = {}
for i, result in iteritems(res):
mins = np.where(result.min().min() == result)
min_res.update({i + '_min_order': (mins[0][0], mins[1][0])})
res.update(min_res)
return Bunch(**res)
def has_missing(data):
"""
Returns True if 'data' contains missing entries, otherwise False
"""
return np.isnan(np.sum(data))
[docs]def kpss(x, regression='c', lags=None, store=False):
"""
Kwiatkowski-Phillips-Schmidt-Shin test for stationarity.
Computes the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for the null
hypothesis that x is level or trend stationary.
Parameters
----------
x : array_like, 1d
Data series
regression : str{'c', 'ct'}
Indicates the null hypothesis for the KPSS test
* 'c' : The data is stationary around a constant (default)
* 'ct' : The data is stationary around a trend
lags : {None, str, int}, optional
Indicates the number of lags to be used. If None (default), lags is
calculated using the legacy method. If 'auto', lags is calculated
using the data-dependent method of Hobijn et al. (1998). See also
Andrews (1991), Newey & West (1994), and Schwert (1989). If set to
'legacy', uses int(12 * (n / 100)**(1 / 4)) , as outlined in
Schwert (1989).
store : bool
If True, then a result instance is returned additionally to
the KPSS statistic (default is False).
Returns
-------
kpss_stat : float
The KPSS test statistic
p_value : float
The p-value of the test. The p-value is interpolated from
Table 1 in Kwiatkowski et al. (1992), and a boundary point
is returned if the test statistic is outside the table of
critical values, that is, if the p-value is outside the
interval (0.01, 0.1).
lags : int
The truncation lag parameter
crit : dict
The critical values at 10%, 5%, 2.5% and 1%. Based on
Kwiatkowski et al. (1992).
resstore : (optional) instance of ResultStore
An instance of a dummy class with results attached as attributes
Notes
-----
To estimate sigma^2 the Newey-West estimator is used. If lags is None,
the truncation lag parameter is set to int(12 * (n / 100) ** (1 / 4)),
as outlined in Schwert (1989). The p-values are interpolated from
Table 1 of Kwiatkowski et al. (1992). If the computed statistic is
outside the table of critical values, then a warning message is
generated.
Missing values are not handled.
References
----------
Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent
covariance matrix estimation. Econometrica, 59: 817-858.
Hobijn, B., Frances, B.H., & Ooms, M. (2004). Generalizations of the
KPSS-test for stationarity. Statistica Neerlandica, 52: 483-502.
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992). Testing
the null hypothesis of stationarity against the alternative of a unit root.
Journal of Econometrics, 54: 159-178.
Newey, W.K., & West, K.D. (1994). Automatic lag selection in covariance
matrix estimation. Review of Economic Studies, 61: 631-653.
Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation.
Journal of Business and Economic Statistics, 7 (2): 147-159.
"""
from warnings import warn
nobs = len(x)
x = np.asarray(x)
hypo = regression.lower()
# if m is not one, n != m * n
if nobs != x.size:
raise ValueError("x of shape {0} not understood".format(x.shape))
if hypo == 'ct':
# p. 162 Kwiatkowski et al. (1992): y_t = beta * t + r_t + e_t,
# where beta is the trend, r_t a random walk and e_t a stationary
# error term.
resids = OLS(x, add_constant(np.arange(1, nobs + 1))).fit().resid
crit = [0.119, 0.146, 0.176, 0.216]
elif hypo == 'c':
# special case of the model above, where beta = 0 (so the null
# hypothesis is that the data is stationary around r_0).
resids = x - x.mean()
crit = [0.347, 0.463, 0.574, 0.739]
else:
raise ValueError("hypothesis '{0}' not understood".format(hypo))
if lags is None:
lags = 'legacy'
msg = 'The behavior of using lags=None will change in the next ' \
'release. Currently lags=None is the same as ' \
'lags=\'legacy\', and so a sample-size lag length is used. ' \
'After the next release, the default will change to be the ' \
'same as lags=\'auto\' which uses an automatic lag length ' \
'selection method. To silence this warning, either use ' \
'\'auto\' or \'legacy\''
warn(msg, FutureWarning)
if lags == 'legacy':
lags = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
lags = min(lags, nobs - 1)
elif lags == 'auto':
# autolag method of Hobijn et al. (1998)
lags = _kpss_autolag(resids, nobs)
lags = min(lags, nobs - 1)
else:
lags = int(lags)
if lags >= nobs:
raise ValueError("lags ({}) must be < number of observations ({})"
.format(lags, nobs))
pvals = [0.10, 0.05, 0.025, 0.01]
eta = np.sum(resids.cumsum()**2) / (nobs**2) # eq. 11, p. 165
s_hat = _sigma_est_kpss(resids, nobs, lags)
kpss_stat = eta / s_hat
p_value = np.interp(kpss_stat, crit, pvals)
if p_value == pvals[-1]:
warn("p-value is smaller than the indicated p-value", InterpolationWarning)
elif p_value == pvals[0]:
warn("p-value is greater than the indicated p-value", InterpolationWarning)
crit_dict = {'10%': crit[0], '5%': crit[1], '2.5%': crit[2], '1%': crit[3]}
if store:
rstore = ResultsStore()
rstore.lags = lags
rstore.nobs = nobs
stationary_type = "level" if hypo == 'c' else "trend"
rstore.H0 = "The series is {0} stationary".format(stationary_type)
rstore.HA = "The series is not {0} stationary".format(stationary_type)
return kpss_stat, p_value, crit_dict, rstore
else:
return kpss_stat, p_value, lags, crit_dict
def _sigma_est_kpss(resids, nobs, lags):
"""
Computes equation 10, p. 164 of Kwiatkowski et al. (1992). This is the
consistent estimator for the variance.
"""
s_hat = np.sum(resids**2)
for i in range(1, lags + 1):
resids_prod = np.dot(resids[i:], resids[:nobs - i])
s_hat += 2 * resids_prod * (1. - (i / (lags + 1.)))
return s_hat / nobs
def _kpss_autolag(resids, nobs):
"""
Computes the number of lags for covariance matrix estimation in KPSS test
using method of Hobijn et al (1998). See also Andrews (1991), Newey & West
(1994), and Schwert (1989). Assumes Bartlett / Newey-West kernel.
"""
covlags = int(np.power(nobs, 2. / 9.))
s0 = np.sum(resids**2) / nobs
s1 = 0
for i in range(1, covlags + 1):
resids_prod = np.dot(resids[i:], resids[:nobs - i])
resids_prod /= (nobs / 2.)
s0 += resids_prod
s1 += i * resids_prod
s_hat = s1 / s0
pwr = 1. / 3.
gamma_hat = 1.1447 * np.power(s_hat * s_hat, pwr)
autolags = int(gamma_hat * np.power(nobs, pwr))
return autolags
