def ad_fuller(series, maxlag=None):
"""Get series and return the p-value and the t-stat of the coefficient"""
if maxlag is None:
n = int((len(series) - 1)**(1. / 3))
elif maxlag < 1:
n = 1
else:
n = maxlag
# Putting the X values on a Tensor with Double as type
X = series
# Generating the lagged tensor to calculate the difference
X_1 = narrow(X, 0, 1, X.shape[0] - 1)
# Re-sizing the x values to get the difference
X = narrow(X, 0, 0, X.shape[0] - 1)
dX = X_1 - X
# Generating the lagged difference tensors
# and concatenating the lagged tensors into a single one
for i in range(1, n + 1):
lagged_n = narrow(dX, 0, n - i, (dX.shape[0] - n))
lagged_reshape = np.reshape(lagged_n, (lagged_n.shape[0], 1))
if i == 1:
lagged_tensors = lagged_reshape
else:
lagged_tensors = np.concatenate((lagged_tensors, lagged_reshape),
1)
# Reshaping the X and the difference tensor
# to match the dimension of the lagged ones
X = narrow(X, 0, 0, X.shape[0] - n)
dX = narrow(dX, 0, n, dX.shape[0] - n)
dX = np.reshape(dX, (dX.shape[0], 1))
# Concatenating the lagged tensors to the X one
# and adding a column full of ones for the Linear Regression
X = np.concatenate((np.reshape(X, (X.shape[0], 1)), lagged_tensors), 1)
ones_columns = np.ones((X.shape[0], 1))
X_ = np.concatenate((X, np.ones_like(ones_columns, dtype=np.float64)), 1)
nobs = X_.shape[0]
# Xb = y -> Xt.X.b = Xt.y -> b = (Xt.X)^-1.Xt.y
coeff = np.matmul(np.matmul(np.linalg.inv(np.matmul(X_.T, X_)), X_.T), dX)
std_error = get_std_error(X_, dX, coeff)
coeff_std_err = get_coeff_std_error(X_, std_error, coeff)[0]
t_stat = (coeff[0] / coeff_std_err).item()
p_value = mackinnonp(t_stat, regression="c", N=1)
critvalues = mackinnoncrit(N=1, regression="c", nobs=nobs)
critvalues = {
"1%": critvalues[0],
"5%": critvalues[1],
"10%": critvalues[2]
}
return t_stat, p_value, n, nobs, critvalues
def aeg_pca(X0, X1, trend):
__sqrteps = np.sqrt(np.finfo(np.double).eps)
# Compute residual
if trend == "nc":
# pseudo PCA from origin
rms0 = rms(X0)
rms1 = rms(X1)
residual = X0 / rms0 - X1 / rms1
collinearity = 1.0 - residual.std() / (X0 / rms1 + 1 / rms0).std()
if trend == "c":
X = np.array([X0, X1]).T
pca = PCA(n_components=2).fit(X)
residual = pca.transform(X)[:, 1] # PC1
collinearity = pca.explained_variance_ratio_[0]
# Get ADF statistics
if collinearity < 1.0 - __sqrteps:
adf_stat = adfuller(residual, regression="nc")[0]
else:
adf_stat = -np.inf
# Get pvalue
pvalue = mackinnonp(adf_stat, regression=trend, N=1)
# Get critical values
if trend == "nc":
crit = [np.nan, np.nan, np.nan]
if trend == "c":
crit = mackinnoncrit(N=1, regression=trend, nobs=X0.shape[0] - 1)
return adf_stat, pvalue, crit
def coint(y1, y2, regression="c"):
"""
This is a simple cointegration test. Uses unit-root test on residuals to
test for cointegrated relationship
See Hamilton (1994) 19.2
Parameters
----------
y1 : array_like, 1d
first element in cointegrating vector
y2 : array_like
remaining elements in cointegrating vector
c : str {'c'}
Included in regression
* 'c' : Constant
Returns
-------
coint_t : float
t-statistic of unit-root test on residuals
pvalue : float
MacKinnon's approximate p-value based on MacKinnon (1994)
crit_value : dict
Critical values for the test statistic at the 1 %, 5 %, and 10 %
levels.
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 are obtained through regression surface approximation from
MacKinnon 1994.
References
----------
MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
"""
regression = regression.lower()
if regression not in ['c', 'nc', 'ct', 'ctt']:
raise ValueError("regression option %s not understood" % regression)
y1 = np.asarray(y1)
y2 = np.asarray(y2)
if regression == 'c':
y2 = add_constant(y2, prepend=False)
st1_resid = OLS(y1, y2).fit().resid # stage one residuals
lgresid_cons = add_constant(st1_resid[0:-1], prepend=False)
uroot_reg = OLS(st1_resid[1:], lgresid_cons).fit()
coint_t = (uroot_reg.params[0] - 1) / uroot_reg.bse[0]
pvalue = mackinnonp(coint_t, regression="c", N=2, lags=None)
crit_value = mackinnoncrit(N=1, regression="c", nobs=len(y1))
return coint_t, pvalue, crit_value
def coint(y1, y2, regression="c"):
"""
This is a simple cointegration test. Uses unit-root test on residuals to
test for cointegrated relationship
See Hamilton (1994) 19.2
Parameters
----------
y1 : array_like, 1d
first element in cointegrating vector
y2 : array_like
remaining elements in cointegrating vector
c : str {'c'}
Included in regression
* 'c' : Constant
Returns
-------
coint_t : float
t-statistic of unit-root test on residuals
pvalue : float
MacKinnon's approximate p-value based on MacKinnon (1994)
crit_value : dict
Critical values for the test statistic at the 1 %, 5 %, and 10 %
levels.
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 are obtained through regression surface approximation from
MacKinnon 1994.
References
----------
MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
"""
regression = regression.lower()
if regression not in ['c', 'nc', 'ct', 'ctt']:
raise ValueError("regression option %s not understood" % regression)
y1 = np.asarray(y1)
y2 = np.asarray(y2)
if regression == 'c':
y2 = add_constant(y2, prepend=False)
st1_resid = OLS(y1, y2).fit().resid # stage one residuals
lgresid_cons = add_constant(st1_resid[0:-1], prepend=False)
uroot_reg = OLS(st1_resid[1:], lgresid_cons).fit()
coint_t = (uroot_reg.params[0] - 1) / uroot_reg.bse[0]
pvalue = mackinnonp(coint_t, regression="c", N=2, lags=None)
crit_value = mackinnoncrit(N=1, regression="c", nobs=len(y1))
return coint_t, pvalue, crit_value
def df_test(df):
df_vect = df
df_size = len(df_vect)
autolag = 'AIC'
max_lag = None
regression = 'c'
trend_size = len(regression) # размер тренда
# Максимальное запаздывание, вычисляется как ТВГ соотвествующего выражения
max_lag = int(np.ceil(12. * np.power(df_size / 100., 1 / 2)))
max_lag = min(df_size // 2 - trend_size, max_lag)
if max_lag < 0:
raise ValueError('Dataset is too short')
# массив с первыми разностями: элем_i = a[i+1] - a[i]
df_diff = np.diff(df_vect)
# массив с лагами, где max_lag - число "сдвигов" вниз
df_diff_all = sm.tsa.lagmat(df_diff[:, None],
max_lag,
trim='both',
original='in')
df_size = df_diff_all.shape[0] # количество столбцов в массиве лагов
# заменяем первый столбец df_diff_all на df_vect
df_diff_all[:, 0] = df_vect[-df_size - 1:-1]
df_diff_short = df_diff[-df_size:] # оставляем последние df_size элементов
df_diff_full = df_diff_all
start_lag = df_diff_full.shape[1] - \
df_diff_all.shape[1] + 1 # начальный лаг
best_inf_crit, best_lag = get_lag(sm.OLS, df_diff_short, df_diff_full,
start_lag, max_lag, autolag)
best_lag -= start_lag # оптимальное значение лага
# массив с лагами, но уже при оптимальном значении лага
df_diff_all = sm.tsa.lagmat(df_diff[:, None],
best_lag,
trim='both',
original='in')
df_size = df_diff_all.shape[0]
# заменяем первый столбец df_diff_all на df_vect
df_diff_all[:, 0] = df_vect[-df_size - 1:-1]
df_diff_short = df_diff[-df_size:]
use_lag = best_lag
# аппроксимация ряда методом наименьших квадратов
resols = sm.OLS(df_diff_short,
sm.tsa.add_trend(df_diff_all[:, :use_lag + 1],
regression)).fit()
adfstat = resols.tvalues[0] # получение необходимой статистики
pvalue = mackinnonp(adfstat, regression=regression, N=1)
critvalues = mackinnoncrit(N=1, regression=regression, nobs=df_size)
if adfstat < critvalues[1]:
print("Time series is stationary with crit value ", adfstat)
return True
else:
print("Time series is not stationary with crit value ", adfstat)
return False
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.
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.
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.
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 - np.sqrt(np.finfo(np.double).eps):
res_adf = adfuller(res_co.resid, maxlag=maxlag, autolag=None,
regression='nc')
else:
import warnings
warnings.warn("y0 and y1 are perfectly colinear. Cointegration test "
"is not reliable in this case.")
# Edge case where series are too similar
res_adf = (0,)
# 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 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]
if maxlag is None:
#from Greene referencing Schwert 1989
maxlag = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
xdiff = np.diff(x)
xdall = lagmat(xdiff[:, None], maxlag, trim='both', original='in')
nobs = xdall.shape[0] # pylint: disable=E1103
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 # pylint: disable=E1103
#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] # pylint: disable=E1103
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
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.
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.
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.
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()
res_adf = adfuller(res_co.resid, maxlag=maxlag, autolag=None,
regression='nc')
# 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 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
----------
.. [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
"""
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]
if maxlag is None:
#from Greene referencing Schwert 1989
maxlag = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.)))
xdiff = np.diff(x)
xdall = lagmat(xdiff[:, None], maxlag, trim='both', original='in')
nobs = xdall.shape[0] # pylint: disable=E1103
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 # pylint: disable=E1103
#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] # pylint: disable=E1103
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
def run(self, x, regression='c', trunclag=None):
"""
Phillips-Perron unit-root test
The Phillips-Perron test (1998) can be used to test for a unit root in
a univariate process.
Parameters
----------
x : array_like, 1d
data series
regression : {'nc', 'c', 'ct'}
constant and trend order to include in regression
* 'nc' : no constant, no trend
* 'c' : constant only (default)
* 'ct' : constant and trend
trunclag : {int, None}
number of truncation lags, default=int(sqrt(nobs)/5) (SAS, 2015)
Returns
-------
tau : float
Z-tau test statistic
tau_pv : float
Z-tau p-value based on MacKinnon (1994) regression surface model
lags : int
number of truncation lags used for covariance estimation
nobs : int
number of observations used in regression
tau_cvdict : dict
critical values for the Z-tau test statistic at 1%, 5%, 10%
rho : float
Z-rho test statistic
rho_pv : float
Z-rho p-value based on interpolated simulation-derived critical
values
rho_cvdict : dict
critical values for the Z-rho test statistic at 1%, 5%, 10%
Notes
-----
H0 = series has a unit root (i.e., non-stationary)
Basic process is to fit the time series under test with an AR(1) model
using heteroscedasticity- and autocorrelation-consistent residual
covariance estimation in order to generate the Phillips-Perron Z-rho
and Z-tau statistics (1988) which are asymptotically equivalent to the
Dickey-Fuller (1979, 1981) rho/tau statistics. Z-tau p-values are
calculated using the statsmodel implementation of MacKinnon's (1994,
2010) regression surface model. Z-rho p-values are interpolated from
Monte-Carlo derived critical values.
References
----------
Dickey, D.A., and Fuller, W.A. (1979). Distribution of the estimators
for autoregressive time series with a unit root. Journal of the
American Statistical Association, 74: 427-431.
Dickey, D.A., and Fuller, W.A. (1981). Likelihood ratio statistics for
autoregressive time series with a unit root. Econometrica, 49:
1057-1072.
MacKinnon, J.G. (1994). Approximate asymptotic distribution functions
for unit-root and cointegration tests. Journal of Business and Economic
Statistics, 12: 167-176.
MacKinnon, J.G. (2010). Critical values for cointegration tests.
Working Paper 1227, Queen's University, Department of Economics.
Retrieved from
URL: https://www.econ.queensu.ca/research/working-papers.
Newey, W.K., and West, K.D. (1994). A simple, positive semi-definite,
heteroscedasticity- and autocorrelation-consistent covariance matrix.
Econometrica, 20: 73-103.
Phillips, P.C.B, and Perron, P. (1988). Testing for a unit root in time
series regression. Biometrika, 75: 335-346.
SAS Institute Inc. (2015). SAS/ETS 14.1 User's Guide. Cary, NC: SAS
Institute Inc.
Schwert, G.W. (1987). Effects of model specification on tests for unit
roots in macroeconomic data. Journal of Monetary Economics, 20: 73-103.
Seabold, S., and Perktold, J. (2010). Statsmodels: econometric and
statistical modeling with python. In S. van der Walt and J. Millman
(Eds.), Proceedings of the 9th Python in Science Conference (pp.
57-61).
"""
if regression not in ['nc', 'c', 'ct']:
raise ValueError(
'PP: regression option \'{}\' not understood'.format(
regression))
if x.ndim > 2 or (x.ndim == 2 and x.shape[1] != 1):
raise ValueError(
'PP: x must be a 1d array or a 2d array with a single column')
nobs = x.shape[0] - 1
if trunclag is not None and (trunclag < 0
or trunclag > x.shape[0] - 2):
raise ValueError(
'PP: trunclags must be in range [0, {}]'.format(x.shape - 2))
# set up exog matrix
x = np.reshape(x, (-1, 1))
if regression == 'nc':
exog = np.ones(shape=(nobs, 1))
elif regression == 'c':
exog = np.ones(shape=(nobs, 2))
else:
exog = np.ones(shape=(x.shape[0] - 1, 3))
exog[:, 1] = np.arange(1, nobs + 1).reshape(nobs, )
exog[:, -1] = x[:-1].reshape(nobs, )
# set up endog vector
endog = x[1:].reshape(nobs, 1)
# run auxiliary regression
ols = OLS(endog, exog).fit()
# save the coefficient of the AR(1) term
rho_hat = ols.params[-1]
# save the standard error of the AR(1) term
se = ols.bse[-1]
# calculate bandwidth for Bartlett kernel. if trunclag
# is not provided, calculate according to SAS 9.4 (2015)
# methodology.
if trunclag is None:
lags = np.amax([1, int(np.sqrt(nobs) / 5)])
else:
lags = trunclag
bw = lags + 1
# get the residual self variances and covariances
var, cov = self._sigma_est_pp(ols.resid, nobs, bw)
s_hat = var + cov
mse = var * nobs / (nobs - exog.shape[1])
tau1 = np.sqrt(var / s_hat) * (rho_hat - 1) / se
tau2 = 0.5 * cov * nobs * se / (np.sqrt(s_hat) * np.sqrt(mse))
tau = tau1 - tau2
rho = nobs * (rho_hat - 1) - \
tau2 * np.sqrt(s_hat) * nobs * se / np.sqrt(mse)
# get Z-tau p-value and critical values using
# MacKinnon (1994, 2010) response surface model
# from statsmodels (2010)
tau_pv = mackinnonp(tau, regression=regression)
tau_cvs = mackinnoncrit(regression=regression, nobs=nobs)
tau_cvdict = {'1%': tau_cvs[0], '5%': tau_cvs[1], '10%': tau_cvs[2]}
# get Z-rho p-value and critical values using
# simulation-derived critical values
rho_crit = self.__pp_crit(rho, regression)
rho_pv = rho_crit[0]
rho_cvdict = rho_crit[1]
return tau, tau_pv, tau_cvdict, rho, rho_pv, rho_cvdict, lags, nobs
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
def test_shape(series, maxlag=None):
"""Get series and return the p-value and the t-stat of the coefficient"""
if maxlag is None:
n = int(12 * ((len(series) / 100)**(1. / 4)))
elif maxlag < 1:
n = 1
else:
n = maxlag
# Putting the X values on a Tensor with Double as type
X = torch.tensor(series)
X = X.type(torch.DoubleTensor)
# Generating the lagged tensor to calculate the difference
X_1 = X.narrow(0, 1, X.shape[0] - 1)
# Re-sizing the x values to get the difference
X = X.narrow(0, 0, X.shape[0] - 1)
dX = X_1 - X
expanded_dX = toeplitz_like(dX, n)
X = torch.cat(
(X.narrow(0, 0, expanded_dX.shape[0]).unsqueeze(1), expanded_dX),
dim=1)
'''# Generating the lagged difference tensors
# and concatenating the lagged tensors into a single one
for i in range(1, n + 1):
lagged_n = dX.narrow(0, n - i, (dX.shape[0] - n))
lagged_reshape = torch.reshape(lagged_n, (lagged_n.shape[0], 1))
if i == 1:
lagged_tensors = lagged_reshape
else:
lagged_tensors = torch.cat((lagged_tensors, lagged_reshape), 1)
# Reshaping the X and the difference tensor
# to match the dimension of the lagged ones
X = X.narrow(0, 0, X.shape[0] - n)
X = torch.cat((torch.reshape(X, (X.shape[0], 1)), lagged_tensors), 1)'''
dX = dX.narrow(0, n, dX.shape[0] - n).unsqueeze(0).t()
print(dX.shape)
ones_columns = torch.ones((X.shape[0], 1))
X_ = torch.cat((X, torch.ones_like(ones_columns, dtype=torch.float64)), 1)
nobs = X_.shape[0]
# Xb = y -> Xt.X.b = Xt.y -> b = (Xt.X)^-1.Xt.y
coeff = torch.mm(
torch.mm(torch.inverse(torch.mm(torch.t(X_), X_)), torch.t(X_)), dX)
std_error = get_std_error(X_, dX, coeff)
coeff_std_err = get_coeff_std_error(X_, std_error, coeff)[0]
t_stat = coeff[0] / coeff_std_err
p_value = mackinnonp(t_stat.item(), regression="c", N=1)
critvalues = mackinnoncrit(N=1, regression="c", nobs=nobs)
critvalues = {
"1%": critvalues[0],
"5%": critvalues[1],
"10%": critvalues[2]
}
return t_stat.item(), p_value, n, nobs, critvalues