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 : int
Indicates the number of lags to be used. If None (default),
lags is set to 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
----------
D. Kwiatkowski, P. C. B. Phillips, P. Schmidt, and Y. Shin (1992): Testing
the Null Hypothesis of Stationarity against the Alternative of a Unit Root.
`Journal of Econometrics` 54, 159-178.
"""
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:
# from Kwiatkowski et al. referencing Schwert (1989)
lags = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
pvals = [0.10, 0.05, 0.025, 0.01]
eta = 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 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 : int
Indicates the number of lags to be used. If None (default),
lags is set to 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
----------
D. Kwiatkowski, P. C. B. Phillips, P. Schmidt, and Y. Shin (1992): Testing
the Null Hypothesis of Stationarity against the Alternative of a Unit Root.
`Journal of Econometrics` 54, 159-178.
"""
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:
# from Kwiatkowski et al. referencing Schwert (1989)
lags = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
pvals = [0.10, 0.05, 0.025, 0.01]
eta = 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 run(self, x, arlags=1, regression='c', method='mle', varest='var94'):
"""
Leybourne-McCabe stationarity test
The Leybourne-McCabe test can be used to test for stationarity in a
univariate process.
Parameters
----------
x : array_like
data series
arlags : int
number of autoregressive terms to include, default=None
regression : {'c','ct'}
Constant and trend order to include in regression
* 'c' : constant only (default)
* 'ct' : constant and trend
method : {'mle','ols'}
Method used to estimate ARIMA(p, 1, 1) filter model
* 'mle' : condition sum of squares maximum likelihood (default)
* 'ols' : two-stage least squares
varest : {'var94','var99'}
Method used for residual variance estimation
* 'var94' : method used in original Leybourne-McCabe paper (1994)
(default)
* 'var99' : method used in follow-up paper (1999)
Returns
-------
lmstat : float
test statistic
pvalue : float
based on MC-derived critical values
arlags : int
AR(p) order used to create the filtered series
cvdict : dict
critical values for the test statistic at the 1%, 5%, and 10%
levels
Notes
-----
H0 = series is stationary
Basic process is to create a filtered series which removes the AR(p)
effects from the series under test followed by an auxiliary regression
similar to that of Kwiatkowski et al (1992). The AR(p) coefficients
are obtained by estimating an ARIMA(p, 1, 1) model. Two methods are
provided for ARIMA estimation: MLE and two-stage least squares.
Two methods are provided for residual variance estimation used in the
calculation of the test statistic. The first method ('var94') is the
mean of the squared residuals from the filtered regression. The second
method ('var99') is the MA(1) coefficient times the mean of the squared
residuals from the ARIMA(p, 1, 1) filtering model.
An empirical autolag procedure is provided. In this context, the number
of lags is equal to the number of AR(p) terms used in the filtering
step. The number of AR(p) terms is set equal to the to the first PACF
falling within the 95% confidence interval. Maximum nuber of AR lags is
limited to 1/2 series length.
References
----------
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.
Leybourne, S.J., & McCabe, B.P.M. (1994). A consistent test for a
unit root. Journal of Business and Economic Statistics, 12: 157–166.
Leybourne, S.J., & McCabe, B.P.M. (1999). Modified stationarity tests
with data-dependent model-selection rules. Journal of Business and
Economic Statistics, 17: 264-270.
Schwert, G W. (1987). Effects of model specification on tests for unit
roots in macroeconomic data. Journal of Monetary Economics, 20: 73–103.
"""
if regression not in ['c', 'ct']:
raise ValueError('LM: regression option \'%s\' not understood' %
regression)
if method not in ['mle', 'ols']:
raise ValueError('LM: method option \'%s\' not understood' %
method)
if varest not in ['var94', 'var99']:
raise ValueError('LM: varest option \'%s\' not understood' %
varest)
x = np.asarray(x)
if x.ndim > 2 or (x.ndim == 2 and x.shape[1] != 1):
raise ValueError(
'LM: x must be a 1d array or a 2d array with a single column')
x = np.reshape(x, (-1, 1))
# determine AR order if not specified
if arlags == None:
arlags = self._autolag(x)
elif not isinstance(arlags,
int) or arlags < 1 or arlags > int(len(x) / 2):
raise ValueError('LM: arlags must be an integer in range [1..%s]' %
str(int(len(x) / 2)))
# estimate the reduced ARIMA(p, 1, 1) model
if method == 'mle':
arfit = ARIMA(x, order=(arlags, 1, 1), trend=regression).fit()
resids = arfit.resid
arcoeffs = arfit.arparams
theta = arfit.maparams[0]
else:
arcoeffs, theta, resids = self._tsls_arima(x,
arlags,
model=regression)
# variance estimator from (1999) LM paper
var99 = abs(theta * np.sum(resids**2) / len(resids))
# create the filtered series:
# z(t) = x(t) - arcoeffs[0]*x(t-1) - ... - arcoeffs[p-1]*x(t-p)
z = np.full(len(x) - arlags, np.inf)
for i in range(len(z)):
z[i] = x[i + arlags]
for j in range(len(arcoeffs)):
z[i] -= arcoeffs[j] * x[i + arlags - j - 1]
# regress the filtered series against a constant and
# trend term (if requested)
if regression == 'c':
resids = z - z.mean()
else:
resids = OLS(z, add_constant(np.arange(1, len(z) + 1))).fit().resid
# variance estimator from (1994) LM paper
var94 = np.sum(resids**2) / len(resids)
# compute test statistic with specified variance estimator
eta = np.sum(resids.cumsum()**2) / (len(resids)**2)
if varest == 'var99':
lmstat = eta / var99
else:
lmstat = eta / var94
# calculate pval
crit = self.__leybourne_crit(lmstat, regression)
lmpval = crit[0]
cvdict = crit[1]
return lmstat, lmpval, arlags, cvdict