"""
Statistical tools for time series analysis
"""
from statsmodels.compat.numpy import lstsq
from statsmodels.compat.pandas import deprecate_kwarg
from statsmodels.compat.python import lrange, lzip
from statsmodels.compat.scipy import _next_regular
import warnings
import numpy as np
from numpy.linalg import LinAlgError
import pandas as pd
from scipy import stats
from statsmodels.regression.linear_model import OLS, yule_walker
from statsmodels.tools.sm_exceptions import (
CollinearityWarning,
InfeasibleTestError,
InterpolationWarning,
MissingDataError,
)
from statsmodels.tools.tools import Bunch, add_constant
from statsmodels.tools.validation import (
array_like,
bool_like,
dict_like,
float_like,
int_like,
string_like,
)
from statsmodels.tsa._bds import bds
from statsmodels.tsa._innovations import innovations_algo, innovations_filter
from statsmodels.tsa.adfvalues import mackinnoncrit, mackinnonp
from statsmodels.tsa.arima_model import ARMA
from statsmodels.tsa.tsatools import add_trend, lagmat, lagmat2ds
__all__ = [
"acovf",
"acf",
"pacf",
"pacf_yw",
"pacf_ols",
"ccovf",
"ccf",
"q_stat",
"coint",
"arma_order_select_ic",
"adfuller",
"kpss",
"bds",
"pacf_burg",
"innovations_algo",
"innovations_filter",
"levinson_durbin_pacf",
"levinson_durbin",
"zivot_andrews",
]
SQRTEPS = np.sqrt(np.finfo(np.double).eps)
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 results.items())
elif method == "bic":
icbest, bestlag = min((v.bic, k) for k, v in results.items())
elif method == "t-stat":
# stop = stats.norm.ppf(.95)
stop = 1.6448536269514722
# Default values to ensure that always set
bestlag = startlag + maxlag
icbest = 0.0
for lag in range(startlag + maxlag, startlag - 1, -1):
icbest = np.abs(results[lag].tvalues[-1])
bestlag = lag
if np.abs(icbest) >= stop:
# Break for first lag with a significant t-stat
break
else:
raise ValueError(f"Information Criterion {method} not understood.")
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
The data series to test.
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}
Method to use when automatically determining the lag length among the
values 0, 1, ..., maxlag.
* 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.
* If None, then the number of included lags is set to maxlag.
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
The test statistic.
pvalue : float
MacKinnon"s approximate p-value based on MacKinnon (1994, 2010).
usedlag : int
The number of lags used.
nobs : int
The 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.
References
----------
.. [1] W. Green. "Econometric Analysis," 5th ed., Pearson, 2003.
.. [2] Hamilton, J.D. "Time Series Analysis". Princeton, 1994.
.. [3] MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
.. [4] 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
Examples
--------
See example notebook
"""
x = array_like(x, "x")
maxlag = int_like(maxlag, "maxlag", optional=True)
regression = string_like(
regression, "regression", options=("c", "ct", "ctt", "nc")
)
autolag = string_like(
autolag, "autolag", optional=True, options=("aic", "bic", "t-stat")
)
store = bool_like(store, "store")
regresults = bool_like(regresults, "regresults")
if regresults:
store = True
trenddict = {None: "nc", 0: "c", 1: "ct", 2: "ctt"}
if regression is None or isinstance(regression, int):
regression = trenddict[regression]
regression = regression.lower()
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(12.0 * np.power(nobs / 100.0, 1 / 4.0)))
# -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:
from statsmodels.stats.diagnostic import ResultsStore
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]@deprecate_kwarg("unbiased", "adjusted")
def acovf(x, adjusted=False, demean=True, fft=None, missing="none", nlag=None):
"""
Estimate autocovariances.
Parameters
----------
x : array_like
Time series data. Must be 1d.
adjusted : bool, default False
If True, then denominators is n-k, otherwise n.
demean : bool, default True
If True, then subtract the mean x from each element of x.
fft : bool, default None
If True, use FFT convolution. This method should be preferred
for long time series.
missing : str, default "none"
A string in ["none", "raise", "conservative", "drop"] specifying how
the NaNs are to be treated. "none" performs no checks. "raise" raises
an exception if NaN values are found. "drop" removes the missing
observations and then estimates the autocovariances treating the
non-missing as contiguous. "conservative" computes the autocovariance
using nan-ops so that nans are removed when computing the mean
and cross-products that are used to estimate the autocovariance.
When using "conservative", n is set to the number of non-missing
observations.
nlag : {int, None}, default 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
-------
ndarray
The estimated autocovariances.
References
-----------
.. [1] Parzen, E., 1963. On spectral analysis with missing observations
and amplitude modulation. Sankhya: The Indian Journal of
Statistics, Series A, pp.383-392.
"""
adjusted = bool_like(adjusted, "adjusted")
demean = bool_like(demean, "demean")
fft = bool_like(fft, "fft", optional=True)
missing = string_like(
missing, "missing", options=("none", "raise", "conservative", "drop")
)
nlag = int_like(nlag, "nlag", optional=True)
if fft is None:
msg = (
"fft=True will become the default after the release of the 0.12 "
"release of statsmodels. To suppress this warning, explicitly "
"set fft=False."
)
warnings.warn(msg, FutureWarning)
fft = False
x = array_like(x, "x", ndim=1)
missing = missing.lower()
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 adjusted:
acov /= n - np.arange(lag_len + 1)
else:
acov /= n
else:
if adjusted:
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 adjusted and deal_with_masked and missing == "conservative":
d = np.correlate(notmask_int, notmask_int, "full")
d[d == 0] = 1
elif adjusted:
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=None):
"""
Compute Ljung-Box Q Statistic.
Parameters
----------
x : array_like
Array of autocorrelation coefficients. Can be obtained from acf.
nobs : int, optional
Number of observations in the entire sample (ie., not just the length
of the autocorrelation function results.
Returns
-------
q-stat : ndarray
Ljung-Box Q-statistic for autocorrelation parameters.
p-value : ndarray
P-value of the Q statistic.
Notes
-----
Designed to be used with acf.
"""
x = array_like(x, "x")
nobs = int_like(nobs, "nobs")
if type is not None:
warnings.warn(
"The `type` argument is deprecated and has no effect. This "
"argument will be removed after the 0.12 release.",
FutureWarning,
)
ret = (
nobs
* (nobs + 2)
* np.cumsum((1.0 / (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]@deprecate_kwarg("unbiased", "adjusted")
def acf(
x,
adjusted=False,
nlags=None,
qstat=False,
fft=None,
alpha=None,
missing="none",
):
"""
Calculate the autocorrelation function.
Parameters
----------
x : array_like
The time series data.
adjusted : bool, default False
If True, then denominators for autocovariance are n-k, otherwise n.
nlags : int, default 40
Number of lags to return autocorrelation for.
qstat : bool, default False
If True, returns the Ljung-Box q statistic for each autocorrelation
coefficient. See q_stat for more information.
fft : bool, default None
If True, computes the ACF via FFT.
alpha : scalar, default None
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, default "none"
A string in ["none", "raise", "conservative", "drop"] specifying how
the NaNs are to be treated. "none" performs no checks. "raise" raises
an exception if NaN values are found. "drop" removes the missing
observations and then estimates the autocovariances treating the
non-missing as contiguous. "conservative" computes the autocovariance
using nan-ops so that nans are removed when computing the mean
and cross-products that are used to estimate the autocovariance.
When using "conservative", n is set to the number of non-missing
observations.
Returns
-------
acf : ndarray
The autocorrelation function.
confint : ndarray, optional
Confidence intervals for the ACF. Returned if alpha is not None.
qstat : ndarray, optional
The Ljung-Box Q-Statistic. Returned if q_stat is True.
pvalues : ndarray, 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 adjusted is true, the denominator for the autocovariance is adjusted
for the loss of data.
References
----------
.. [1] Parzen, E., 1963. On spectral analysis with missing observations
and amplitude modulation. Sankhya: The Indian Journal of
Statistics, Series A, pp.383-392.
"""
adjusted = bool_like(adjusted, "adjusted")
nlags = int_like(nlags, "nlags", optional=True)
qstat = bool_like(qstat, "qstat")
fft = bool_like(fft, "fft", optional=True)
alpha = float_like(alpha, "alpha", optional=True)
missing = string_like(
missing, "missing", options=("none", "raise", "conservative", "drop")
)
if nlags is None:
warnings.warn(
"The default number of lags is changing from 40 to"
"min(int(10 * np.log10(nobs)), nobs - 1) after 0.12"
"is released. Set the number of lags to an integer to "
" silence this warning.",
FutureWarning,
)
nlags = 40
if fft is None:
warnings.warn(
"fft=True will become the default after the release of the 0.12 "
"release of statsmodels. To suppress this warning, explicitly "
"set fft=False.",
FutureWarning,
)
fft = False
x = array_like(x, "x")
nobs = len(x) # TODO: should this shrink for missing="drop" and NaNs in x?
avf = acovf(x, adjusted=adjusted, 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_like(acf) / nobs
varacf[0] = 0
varacf[1] = 1.0 / nobs
varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1] ** 2)
interval = stats.norm.ppf(1 - alpha / 2.0) * 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=None, method="adjusted"):
"""
Partial autocorrelation estimated with non-recursive yule_walker.
Parameters
----------
x : array_like
The observations of time series for which pacf is calculated.
nlags : int, default 40
The largest lag for which pacf is returned.
method : {"adjusted", "mle"}, default "adjusted"
The method for the autocovariance calculations in yule walker.
Returns
-------
ndarray
The partial autocorrelations, maxlag+1 elements.
See Also
--------
statsmodels.tsa.stattools.pacf
Partial autocorrelation estimation.
statsmodels.tsa.stattools.pacf_ols
Partial autocorrelation estimation using OLS.
statsmodels.tsa.stattools.pacf_burg
Partial autocorrelation estimation using Burg"s method.
Notes
-----
This solves yule_walker for each desired lag and contains
currently duplicate calculations.
"""
x = array_like(x, "x")
nlags = int_like(nlags, "nlags", optional=True)
if nlags is None:
warnings.warn(
"The default number of lags is changing from 40 to"
"min(int(10 * np.log10(nobs)), nobs - 1) after 0.12"
"is released. Set the number of lags to an integer to "
" silence this warning.",
FutureWarning,
)
nlags = 40
method = string_like(
method, "method", options=("adjusted", "unbiased", "mle")
)
if method == "unbiased":
warnings.warn(
"unbiased is deprecated in factor of adjusted to reflect that the "
"term is adjusting the sample size used in the autocovariance "
"calculation rather than estimating an unbiased autocovariance. "
"After release 0.13, using 'unbiased' will raise.",
FutureWarning,
)
method = "adjusted"
pacf = [1.0]
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):
"""
Calculate 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
Flag indicating to demean that data. Set to False if x has been
previously demeaned.
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
Partial autocorrelation estimation.
statsmodels.tsa.stattools.pacf_yw
Partial autocorrelation estimation using Yule-Walker.
statsmodels.tsa.stattools.pacf_ols
Partial autocorrelation estimation using OLS.
References
----------
.. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
x = array_like(x, "x")
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 / (2.0 * (nobs - np.arange(0, p + 1)))
pacf[0] = 1 # Insert the 0 lag partial autocorrel
return pacf, sigma2
[docs]@deprecate_kwarg("unbiased", "adjusted")
def pacf_ols(x, nlags=None, efficient=True, adjusted=False):
"""
Calculate partial autocorrelations via OLS.
Parameters
----------
x : array_like
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.
adjusted : bool, optional
Adjust each partial autocorrelation by n / (n - lag).
Returns
-------
ndarray
The partial autocorrelations, (maxlag,) array corresponding to lags
0, 1, ..., maxlag.
See Also
--------
statsmodels.tsa.stattools.pacf
Partial autocorrelation estimation.
statsmodels.tsa.stattools.pacf_yw
Partial autocorrelation estimation using Yule-Walker.
statsmodels.tsa.stattools.pacf_burg
Partial autocorrelation estimation using Burg"s method.
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.
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
"""
x = array_like(x, "x")
nlags = int_like(nlags, "nlags", optional=True)
efficient = bool_like(efficient, "efficient")
adjusted = bool_like(adjusted, "adjusted")
if nlags is None:
warnings.warn(
"The default number of lags is changing from 40 to"
"min(int(10 * np.log10(nobs)), nobs - 1) after 0.12"
"is released. Set the number of lags to an integer to "
" silence this warning.",
FutureWarning,
)
nlags = 40
pacf = np.empty(nlags + 1)
pacf[0] = 1.0
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 adjusted:
n = len(x)
pacf *= n / (n - np.arange(nlags + 1))
return pacf
[docs]def pacf(x, nlags=None, method="ywadjusted", alpha=None):
"""
Partial autocorrelation estimate.
Parameters
----------
x : array_like
Observations of time series for which pacf is calculated.
nlags : int
The largest lag for which the pacf is returned. The default
is currently 40, but will change to
min(int(10 * np.log10(nobs)), nobs // 2 - 1) in the future
method : str, default "ywunbiased"
Specifies which method for the calculations to use.
- "yw" or "ywadjusted" : Yule-Walker with sample-size adjustment in
denominator for acovf. Default.
- "ywm" or "ywmle" : Yule-Walker without adjustment.
- "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-adjusted" : regression of time series on lags with a bias
adjustment.
- "ld" or "ldadjusted" : 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 : ndarray
Partial autocorrelations, nlags elements, including lag zero.
confint : ndarray, optional
Confidence intervals for the PACF. Returned if confint is not None.
See Also
--------
statsmodels.tsa.stattools.acf
Estimate the autocorrelation function.
statsmodels.tsa.stattools.pacf
Partial autocorrelation estimation.
statsmodels.tsa.stattools.pacf_yw
Partial autocorrelation estimation using Yule-Walker.
statsmodels.tsa.stattools.pacf_ols
Partial autocorrelation estimation using OLS.
statsmodels.tsa.stattools.pacf_burg
Partial autocorrelation estimation using Burg"s method.
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-adjusted.
Yule-Walker (adjusted) and Levinson-Durbin (adjusted) performed
consistently worse than the other options.
"""
nlags = int_like(nlags, "nlags", optional=True)
renames = {
"ydu": "yda",
"ywu": "ywa",
"ywunbiased": "ywadjusted",
"ldunbiased": "ldadjusted",
"ld_unbiased": "ld_adjusted",
"ldu": "lda",
"ols-unbiased": "ols-adjusted",
}
if method in renames:
warnings.warn(
f"{method} has been renamed {renames[method]}. After release 0.13, "
"using the old name will raise.",
FutureWarning,
)
method = renames[method]
methods = (
"ols",
"ols-inefficient",
"ols-adjusted",
"yw",
"ywa",
"ld",
"ywadjusted",
"yw_adjusted",
"ywm",
"ywmle",
"yw_mle",
"lda",
"ldadjusted",
"ld_adjusted",
"ldb",
"ldbiased",
"ld_biased",
)
x = array_like(x, "x", maxdim=2)
method = string_like(method, "method", options=methods)
alpha = float_like(alpha, "alpha", optional=True)
if nlags is None:
warnings.warn(
"The default number of lags is changing from 40 to"
"min(int(10 * np.log10(nobs)), nobs // 2 - 1) after 0.12"
"is released. Set the number of lags to an integer to "
" silence this warning.",
FutureWarning,
)
nlags = 40
if nlags >= x.shape[0] // 2:
raise ValueError(
"Can only compute partial correlations for lags up to 50% of the "
f"sample size. The requested nlags {nlags} must be < "
f"{x.shape[0] // 2}."
)
if method in ("ols", "ols-inefficient", "ols-adjusted"):
efficient = "inefficient" not in method
adjusted = "adjusted" in method
ret = pacf_ols(x, nlags=nlags, efficient=efficient, adjusted=adjusted)
elif method in ("yw", "ywa", "ywadjusted", "yw_adjusted"):
ret = pacf_yw(x, nlags=nlags, method="adjusted")
elif method in ("ywm", "ywmle", "yw_mle"):
ret = pacf_yw(x, nlags=nlags, method="mle")
elif method in ("ld", "lda", "ldadjusted", "ld_adjusted"):
acv = acovf(x, adjusted=True, fft=False)
ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
ret = ld_[2]
# inconsistent naming with ywmle
else: # method in ("ldb", "ldbiased", "ld_biased")
acv = acovf(x, adjusted=False, fft=False)
ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
ret = ld_[2]
if alpha is not None:
varacf = 1.0 / len(x) # for all lags >=1
interval = stats.norm.ppf(1.0 - alpha / 2.0) * 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]@deprecate_kwarg("unbiased", "adjusted")
def ccovf(x, y, adjusted=True, demean=True):
"""
Calculate the crosscovariance between two series.
Parameters
----------
x, y : array_like
The time series data to use in the calculation.
adjusted : bool, optional
If True, then denominators for autocovariance is n-k, otherwise n.
demean : bool, optional
Flag indicating whether to demean x and y.
Returns
-------
ndarray
The estimated crosscovariance function.
Notes
-----
This uses np.correlate which does full convolution. For very long time
series it is recommended to use fft convolution instead.
"""
x = array_like(x, "x")
y = array_like(y, "y")
adjusted = bool_like(adjusted, "adjusted")
demean = bool_like(demean, "demean")
n = len(x)
if demean:
xo = x - x.mean()
yo = y - y.mean()
else:
xo = x
yo = y
if adjusted:
xi = np.ones(n)
d = np.correlate(xi, xi, "full")
else:
d = n
return (np.correlate(xo, yo, "full") / d)[n - 1 :]
[docs]@deprecate_kwarg("unbiased", "adjusted")
def ccf(x, y, adjusted=True):
"""
The cross-correlation function.
Parameters
----------
x, y : array_like
The time series data to use in the calculation.
adjusted : bool
If True, then denominators for autocovariance is n-k, otherwise n.
Returns
-------
ndarray
The 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 adjusted is true, the denominator for the autocovariance is adjusted.
"""
x = array_like(x, "x")
y = array_like(y, "y")
adjusted = bool_like(adjusted, "adjusted")
cvf = ccovf(x, y, adjusted=adjusted, demean=True)
return cvf / (np.std(x) * np.std(y))
# 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 : int, optional
The largest lag to include in recursion or order of the autoregressive
process.
isacov : bool, optional
Flag indicating whether the first argument, s, contains the
autocovariances or the data series.
Returns
-------
sigma_v : float
The estimate of the error variance.
arcoefs : ndarray
The estimate of the autoregressive coefficients for a model including
nlags.
pacf : ndarray
The partial autocorrelation function.
sigma : ndarray
The entire sigma array from intermediate result, last value is sigma_v.
phi : ndarray
The 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 = array_like(s, "s")
nlags = int_like(nlags, "nlags")
isacov = bool_like(isacov, "isacov")
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] = 1.0
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
The acf computed from the partial autocorrelations. Array returned
contains the autocorrelations corresponding to lags 0, 1, ..., p.
References
----------
.. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
and forecasting. Springer.
"""
pacf = array_like(pacf, "pacf")
nlags = int_like(nlags, "nlags", optional=True)
pacf = np.squeeze(np.asarray(pacf))
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 grangercausalitytests(x, maxlag, addconst=True, verbose=True):
"""
Four tests for granger non causality of 2 time series.
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_like
The data for test whether the time series in the second column Granger
causes the time series in the first column. Missing values are not
supported.
maxlag : {int, Iterable[int]}
If an integer, computes the test for all lags up to maxlag. If an
iterable, computes the tests only for the lags in maxlag.
addconst : bool
Include a constant in the model.
verbose : bool
Print results.
Returns
-------
dict
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
test statistic, 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
----------
.. [1] https://en.wikipedia.org/wiki/Granger_causality
.. [2] Greene: Econometric Analysis
Examples
--------
>>> import statsmodels.api as sm
>>> from statsmodels.tsa.stattools import grangercausalitytests
>>> import numpy as np
>>> data = sm.datasets.macrodata.load_pandas()
>>> data = data.data[["realgdp", "realcons"]].pct_change().dropna()
All lags up to 4
>>> gc_res = grangercausalitytests(data, 4)
Only lag 4
>>> gc_res = grangercausalitytests(data, [4])
"""
x = array_like(x, "x", ndim=2)
if not np.isfinite(x).all():
raise ValueError("x contains NaN or inf values.")
addconst = bool_like(addconst, "addconst")
verbose = bool_like(verbose, "verbose")
try:
maxlag = int_like(maxlag, "maxlag")
if maxlag <= 0:
raise ValueError("maxlag must a a positive integer")
lags = np.arange(1, maxlag + 1)
except TypeError:
lags = np.array([int(lag) for lag in maxlag])
maxlag = lags.max()
if lags.min() <= 0 or lags.size == 0:
raise ValueError(
"maxlag must be a non-empty list containing only "
"positive integers"
)
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 lags:
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)
if (
dtajoint.shape[1] == (dta.shape[1] - 1)
or (dtajoint.max(0) == dtajoint.min(0)).sum() != 1
):
raise InfeasibleTestError(
"The x values include a column with constant values and so"
" the test statistic cannot be computed."
)
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)
if res2djoint.model.k_constant:
tss = res2djoint.centered_tss
else:
tss =res2djoint.centered_tss
if (
tss == 0
or res2djoint.ssr == 0
or np.isnan(res2djoint.rsquared)
or (res2djoint.ssr / tss) < np.finfo(float).eps
or res2djoint.params.shape[0] != dtajoint.shape[1]
):
raise InfeasibleTestError(
"The Granger causality test statistic cannot be compute "
"because the VAR has a perfect fit of the data."
)
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
----------
y0 : array_like
The first element in cointegrated system. Must be 1-d.
y1 : array_like
The remaining elements in cointegrated system.
trend : str {"c", "ct"}
The 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 : {"aeg"}
Only "aeg" (augmented Engle-Granger) is available.
maxlag : None or int
Argument for `adfuller`, largest or given number of lags.
autolag : str
Argument 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
The 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
----------
.. [1] MacKinnon, J.G. 1994 "Approximate Asymptotic Distribution Functions
for Unit-Root and Cointegration Tests." Journal of Business & Economics
Statistics, 12.2, 167-76.
.. [2] 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
"""
y0 = array_like(y0, "y0")
y1 = array_like(y1, "y1", ndim=2)
trend = string_like(trend, "trend", options=("c", "nc", "ct", "ctt"))
method = string_like(method, "method", options=("aeg",))
maxlag = int_like(maxlag, "maxlag", optional=True)
autolag = string_like(
autolag, "autolag", optional=True, options=("aic", "bic", "t-stat")
)
return_results = bool_like(return_results, "return_results", optional=True)
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:
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 do not 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: # do not 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 = [0.1] * sum(order)
if trend == "c":
start_params = [0.1] + 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
):
"""
Compute information criteria for many ARMA models.
Parameters
----------
y : array_like
Array of 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
-------
Bunch
Dict-like object with attribute access. 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``.
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.
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
"""
max_ar = int_like(max_ar, "max_ar")
max_ma = int_like(max_ma, "max_ma")
trend = string_like(trend, "trend", options=("nc", "c"))
model_kw = dict_like(model_kw, "model_kw", optional=True)
fit_kw = dict_like(fit_kw, "fit_kw", optional=True)
ar_range = lrange(0, max_ar + 1)
ma_range = lrange(0, max_ma + 1)
if isinstance(ic, str):
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 = array_like(y, "y", contiguous=True)
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 = [
pd.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 res.items():
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]@deprecate_kwarg("lags", "nlags")
def kpss(x, regression="c", nlags=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
The data series to test.
regression : str{"c", "ct"}
The null hypothesis for the KPSS test.
* "c" : The data is stationary around a constant (default).
* "ct" : The data is stationary around a trend.
nlags : {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
----------
.. [1] Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation
consistent covariance matrix estimation. Econometrica, 59: 817-858.
.. [2] Hobijn, B., Frances, B.H., & Ooms, M. (2004). Generalizations of the
KPSS-test for stationarity. Statistica Neerlandica, 52: 483-502.
.. [3] 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.
.. [4] Newey, W.K., & West, K.D. (1994). Automatic lag selection in
covariance matrix estimation. Review of Economic Studies, 61: 631-653.
.. [5] Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo
investigation. Journal of Business and Economic Statistics, 7 (2):
147-159.
"""
x = array_like(x, "x")
regression = string_like(regression, "regression", options=("c", "ct"))
store = bool_like(store, "store")
nobs = x.shape[0]
hypo = regression
# 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]
if nlags is None:
nlags = "legacy"
msg = (
"The behavior of using nlags=None will change in release 0.13."
"Currently nlags=None is the same as "
'nlags="legacy", and so a sample-size lag length is used. '
"After the next release, the default will change to be the "
'same as nlags="auto" which uses an automatic lag length '
"selection method. To silence this warning, either use "
'"auto" or "legacy"'
)
warnings.warn(msg, FutureWarning)
if nlags == "legacy":
nlags = int(np.ceil(12.0 * np.power(nobs / 100.0, 1 / 4.0)))
nlags = min(nlags, nobs - 1)
elif nlags == "auto":
# autolag method of Hobijn et al. (1998)
nlags = _kpss_autolag(resids, nobs)
nlags = min(nlags, nobs - 1)
else:
nlags = int(nlags)
if nlags >= nobs:
raise ValueError(
"lags ({}) must be < number of observations ({})".format(
nlags, 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, nlags)
kpss_stat = eta / s_hat
p_value = np.interp(kpss_stat, crit, pvals)
warn_msg = """\
The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is {direction} than the p-value returned.
"""
if p_value == pvals[-1]:
warnings.warn(
warn_msg.format(direction="smaller"), InterpolationWarning
)
elif p_value == pvals[0]:
warnings.warn(
warn_msg.format(direction="greater"), InterpolationWarning
)
crit_dict = {"10%": crit[0], "5%": crit[1], "2.5%": crit[2], "1%": crit[3]}
if store:
from statsmodels.stats.diagnostic import ResultsStore
rstore = ResultsStore()
rstore.lags = nlags
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, nlags, 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 * (1.0 - (i / (lags + 1.0)))
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, 2.0 / 9.0))
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 / 2.0
s0 += resids_prod
s1 += i * resids_prod
s_hat = s1 / s0
pwr = 1.0 / 3.0
gamma_hat = 1.1447 * np.power(s_hat * s_hat, pwr)
autolags = int(gamma_hat * np.power(nobs, pwr))
return autolags
class ZivotAndrewsUnitRoot(object):
"""
Class wrapper for Zivot-Andrews structural-break unit-root test
"""
def __init__(self):
"""
Critical values for the three different models specified for the
Zivot-Andrews unit-root test.
Notes
-----
The p-values are generated through Monte Carlo simulation using
100,000 replications and 2000 data points.
"""
self._za_critical_values = {}
# constant-only model
self._c = (
(0.001, -6.78442),
(0.100, -5.83192),
(0.200, -5.68139),
(0.300, -5.58461),
(0.400, -5.51308),
(0.500, -5.45043),
(0.600, -5.39924),
(0.700, -5.36023),
(0.800, -5.33219),
(0.900, -5.30294),
(1.000, -5.27644),
(2.500, -5.03340),
(5.000, -4.81067),
(7.500, -4.67636),
(10.000, -4.56618),
(12.500, -4.48130),
(15.000, -4.40507),
(17.500, -4.33947),
(20.000, -4.28155),
(22.500, -4.22683),
(25.000, -4.17830),
(27.500, -4.13101),
(30.000, -4.08586),
(32.500, -4.04455),
(35.000, -4.00380),
(37.500, -3.96144),
(40.000, -3.92078),
(42.500, -3.88178),
(45.000, -3.84503),
(47.500, -3.80549),
(50.000, -3.77031),
(52.500, -3.73209),
(55.000, -3.69600),
(57.500, -3.65985),
(60.000, -3.62126),
(65.000, -3.54580),
(70.000, -3.46848),
(75.000, -3.38533),
(80.000, -3.29112),
(85.000, -3.17832),
(90.000, -3.04165),
(92.500, -2.95146),
(95.000, -2.83179),
(96.000, -2.76465),
(97.000, -2.68624),
(98.000, -2.57884),
(99.000, -2.40044),
(99.900, -1.88932),
)
self._za_critical_values["c"] = np.asarray(self._c)
# trend-only model
self._t = (
(0.001, -83.9094),
(0.100, -13.8837),
(0.200, -9.13205),
(0.300, -6.32564),
(0.400, -5.60803),
(0.500, -5.38794),
(0.600, -5.26585),
(0.700, -5.18734),
(0.800, -5.12756),
(0.900, -5.07984),
(1.000, -5.03421),
(2.500, -4.65634),
(5.000, -4.40580),
(7.500, -4.25214),
(10.000, -4.13678),
(12.500, -4.03765),
(15.000, -3.95185),
(17.500, -3.87945),
(20.000, -3.81295),
(22.500, -3.75273),
(25.000, -3.69836),
(27.500, -3.64785),
(30.000, -3.59819),
(32.500, -3.55146),
(35.000, -3.50522),
(37.500, -3.45987),
(40.000, -3.41672),
(42.500, -3.37465),
(45.000, -3.33394),
(47.500, -3.29393),
(50.000, -3.25316),
(52.500, -3.21244),
(55.000, -3.17124),
(57.500, -3.13211),
(60.000, -3.09204),
(65.000, -3.01135),
(70.000, -2.92897),
(75.000, -2.83614),
(80.000, -2.73893),
(85.000, -2.62840),
(90.000, -2.49611),
(92.500, -2.41337),
(95.000, -2.30820),
(96.000, -2.25797),
(97.000, -2.19648),
(98.000, -2.11320),
(99.000, -1.99138),
(99.900, -1.67466),
)
self._za_critical_values["t"] = np.asarray(self._t)
# constant + trend model
self._ct = (
(0.001, -38.17800),
(0.100, -6.43107),
(0.200, -6.07279),
(0.300, -5.95496),
(0.400, -5.86254),
(0.500, -5.77081),
(0.600, -5.72541),
(0.700, -5.68406),
(0.800, -5.65163),
(0.900, -5.60419),
(1.000, -5.57556),
(2.500, -5.29704),
(5.000, -5.07332),
(7.500, -4.93003),
(10.000, -4.82668),
(12.500, -4.73711),
(15.000, -4.66020),
(17.500, -4.58970),
(20.000, -4.52855),
(22.500, -4.47100),
(25.000, -4.42011),
(27.500, -4.37387),
(30.000, -4.32705),
(32.500, -4.28126),
(35.000, -4.23793),
(37.500, -4.19822),
(40.000, -4.15800),
(42.500, -4.11946),
(45.000, -4.08064),
(47.500, -4.04286),
(50.000, -4.00489),
(52.500, -3.96837),
(55.000, -3.93200),
(57.500, -3.89496),
(60.000, -3.85577),
(65.000, -3.77795),
(70.000, -3.69794),
(75.000, -3.61852),
(80.000, -3.52485),
(85.000, -3.41665),
(90.000, -3.28527),
(92.500, -3.19724),
(95.000, -3.08769),
(96.000, -3.03088),
(97.000, -2.96091),
(98.000, -2.85581),
(99.000, -2.71015),
(99.900, -2.28767),
)
self._za_critical_values["ct"] = np.asarray(self._ct)
def _za_crit(self, stat, model="c"):
"""
Linear interpolation for Zivot-Andrews p-values and critical values
Parameters
----------
stat : float
The ZA test statistic
model : {"c","t","ct"}
The model used when computing the ZA statistic. "c" is default.
Returns
-------
pvalue : float
The interpolated p-value
cvdict : dict
Critical values for the test statistic at the 1%, 5%, and 10%
levels
Notes
-----
The p-values are linear interpolated from the quantiles of the
simulated ZA test statistic distribution
"""
table = self._za_critical_values[model]
pcnts = table[:, 0]
stats = table[:, 1]
# ZA cv table contains quantiles multiplied by 100
pvalue = np.interp(stat, stats, pcnts) / 100.0
cv = [1.0, 5.0, 10.0]
crit_value = np.interp(cv, pcnts, stats)
cvdict = {
"1%": crit_value[0],
"5%": crit_value[1],
"10%": crit_value[2],
}
return pvalue, cvdict
def _quick_ols(self, endog, exog):
"""
Minimal implementation of LS estimator for internal use
"""
xpxi = np.linalg.inv(exog.T.dot(exog))
xpy = exog.T.dot(endog)
nobs, k_exog = exog.shape
b = xpxi.dot(xpy)
e = endog - exog.dot(b)
sigma2 = e.T.dot(e) / (nobs - k_exog)
return b / np.sqrt(np.diag(sigma2 * xpxi))
def _format_regression_data(self, series, nobs, const, trend, cols, lags):
"""
Create the endog/exog data for the auxiliary regressions
from the original (standardized) series under test.
"""
# first-diff y and standardize for numerical stability
endog = np.diff(series, axis=0)
endog /= np.sqrt(endog.T.dot(endog))
series /= np.sqrt(series.T.dot(series))
# reserve exog space
exog = np.zeros((endog[lags:].shape[0], cols + lags))
exog[:, 0] = const
# lagged y and dy
exog[:, cols - 1] = series[lags : (nobs - 1)]
exog[:, cols:] = lagmat(endog, lags, trim="none")[
lags : exog.shape[0] + lags
]
return endog, exog
def _update_regression_exog(
self, exog, regression, period, nobs, const, trend, cols, lags
):
"""
Update the exog array for the next regression.
"""
cutoff = period - (lags + 1)
if regression != "t":
exog[:cutoff, 1] = 0
exog[cutoff:, 1] = const
exog[:, 2] = trend[(lags + 2) : (nobs + 1)]
if regression == "ct":
exog[:cutoff, 3] = 0
exog[cutoff:, 3] = trend[1 : (nobs - period + 1)]
else:
exog[:, 1] = trend[(lags + 2) : (nobs + 1)]
exog[: (cutoff - 1), 2] = 0
exog[(cutoff - 1) :, 2] = trend[0 : (nobs - period + 1)]
return exog
def run(self, x, trim=0.15, maxlag=None, regression="c", autolag="AIC"):
"""
Zivot-Andrews structural-break unit-root test.
The Zivot-Andrews test tests for a unit root in a univariate process
in the presence of serial correlation and a single structural break.
Parameters
----------
x : array_like
The data series to test.
trim : float
The percentage of series at begin/end to exclude from break-period
calculation in range [0, 0.333] (default=0.15).
maxlag : int
The maximum lag which is included in test, default is
12*(nobs/100)^{1/4} (Schwert, 1989).
regression : {"c","t","ct"}
Constant and trend order to include in regression.
* "c" : constant only (default).
* "t" : trend only.
* "ct" : constant and trend.
autolag : {"AIC", "BIC", "t-stat", None}
The method to select the lag length when using automatic selection.
* 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.
Returns
-------
zastat : float
The test statistic.
pvalue : float
The pvalue based on MC-derived critical values.
cvdict : dict
The critical values for the test statistic at the 1%, 5%, and 10%
levels.
bpidx : int
The index of x corresponding to endogenously calculated break period
with values in the range [0..nobs-1].
baselag : int
The number of lags used for period regressions.
Notes
-----
H0 = unit root with a single structural break
Algorithm follows Baum (2004/2015) approximation to original
Zivot-Andrews method. Rather than performing an autolag regression at
each candidate break period (as per the original paper), a single
autolag regression is run up-front on the base model (constant + trend
with no dummies) to determine the best lag length. This lag length is
then used for all subsequent break-period regressions. This results in
significant run time reduction but also slightly more pessimistic test
statistics than the original Zivot-Andrews method, although no attempt
has been made to characterize the size/power trade-off.
References
----------
.. [1] Baum, C.F. (2004). ZANDREWS: Stata module to calculate
Zivot-Andrews unit root test in presence of structural break,"
Statistical Software Components S437301, Boston College Department
of Economics, revised 2015.
.. [2] Schwert, G.W. (1989). Tests for unit roots: A Monte Carlo
investigation. Journal of Business & Economic Statistics, 7:
147-159.
.. [3] Zivot, E., and Andrews, D.W.K. (1992). Further evidence on the
great crash, the oil-price shock, and the unit-root hypothesis.
Journal of Business & Economic Studies, 10: 251-270.
"""
x = array_like(x, "x")
trim = float_like(trim, "trim")
maxlag = int_like(maxlag, "maxlag", optional=True)
regression = string_like(
regression, "regression", options=("c", "t", "ct")
)
autolag = string_like(
autolag, "autolag", options=("aic", "bic", "t-stat"), optional=True
)
if trim < 0 or trim > (1.0 / 3.0):
raise ValueError("trim value must be a float in range [0, 1/3)")
nobs = x.shape[0]
if autolag:
adf_res = adfuller(
x, maxlag=maxlag, regression="ct", autolag=autolag
)
baselags = adf_res[2]
elif maxlag:
baselags = maxlag
else:
baselags = int(12.0 * np.power(nobs / 100.0, 1 / 4.0))
trimcnt = int(nobs * trim)
start_period = trimcnt
end_period = nobs - trimcnt
if regression == "ct":
basecols = 5
else:
basecols = 4
# normalize constant and trend terms for stability
c_const = 1 / np.sqrt(nobs)
t_const = np.arange(1.0, nobs + 2)
t_const *= np.sqrt(3) / nobs ** (3 / 2)
# format the auxiliary regression data
endog, exog = self._format_regression_data(
x, nobs, c_const, t_const, basecols, baselags
)
# iterate through the time periods
stats = np.full(end_period + 1, np.inf)
for bp in range(start_period + 1, end_period + 1):
# update intercept dummy / trend / trend dummy
exog = self._update_regression_exog(
exog,
regression,
bp,
nobs,
c_const,
t_const,
basecols,
baselags,
)
# check exog rank on first iteration
if bp == start_period + 1:
o = OLS(endog[baselags:], exog, hasconst=1).fit()
if o.df_model < exog.shape[1] - 1:
raise ValueError(
"ZA: auxiliary exog matrix is not full rank.\n"
" cols (exc intercept) = {} rank = {}".format(
exog.shape[1] - 1, o.df_model
)
)
stats[bp] = o.tvalues[basecols - 1]
else:
stats[bp] = self._quick_ols(endog[baselags:], exog)[
basecols - 1
]
# return best seen
zastat = np.min(stats)
bpidx = np.argmin(stats) - 1
crit = self._za_crit(zastat, regression)
pval = crit[0]
cvdict = crit[1]
return zastat, pval, cvdict, baselags, bpidx
def __call__(
self, x, trim=0.15, maxlag=None, regression="c", autolag="AIC"
):
return self.run(
x, trim=trim, maxlag=maxlag, regression=regression, autolag=autolag
)
zivot_andrews = ZivotAndrewsUnitRoot()
zivot_andrews.__doc__ = zivot_andrews.run.__doc__