def pacf_ols(x, nlags=40):
'''Calculate partial autocorrelations
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
Number of lags for which pacf is returned. Lag 0 is not returned.
Returns
-------
pacf : 1d array
partial autocorrelations, maxlag+1 elements
Notes
-----
This solves a separate OLS estimation for each desired lag.
'''
#TODO: add warnings for Yule-Walker
#NOTE: demeaning and not using a constant gave incorrect answers?
#JP: demeaning should have a better estimate of the constant
#maybe we can compare small sample properties with a MonteCarlo
xlags, x0 = lagmat(x, nlags, original='sep')
#xlags = sm.add_constant(lagmat(x, nlags), prepend=True)
xlags = add_constant(xlags, prepend=True)
pacf = [1.]
for k in range(1, nlags + 1):
res = OLS(x0[k:], xlags[k:, :k + 1]).fit()
#np.take(xlags[k:], range(1,k+1)+[-1],
pacf.append(res.params[-1])
return np.array(pacf)
def pacf_ols(x, nlags=40):
'''Calculate partial autocorrelations
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
Number of lags for which pacf is returned. Lag 0 is not returned.
Returns
-------
pacf : 1d array
partial autocorrelations, maxlag+1 elements
Notes
-----
This solves a separate OLS estimation for each desired lag.
'''
#TODO: add warnings for Yule-Walker
#NOTE: demeaning and not using a constant gave incorrect answers?
#JP: demeaning should have a better estimate of the constant
#maybe we can compare small sample properties with a MonteCarlo
xlags, x0 = lagmat(x, nlags, original='sep')
#xlags = sm.add_constant(lagmat(x, nlags), prepend=True)
xlags = add_constant(xlags, prepend=True)
pacf = [1.]
for k in range(1, nlags+1):
res = OLS(x0[k:], xlags[k:,:k+1]).fit()
#np.take(xlags[k:], range(1,k+1)+[-1],
pacf.append(res.params[-1])
return np.array(pacf)
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 : str {'c','ct','ctt','nc'}
Constant and trend order to include in regression
* 'c' : constant only
* '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' or 'BIC', then the number of lags is chosen to minimize the
corresponding information criterium
* 't-stat' based choice of maxlag. Starts with maxlag and drops a
lag until the t-statistic on the last lag length is significant at
the 95 % level.
store : bool
If True, then a result instance is returned additionally to
the adf statistic
regresults : bool
If True, the full regression results are returned.
Returns
-------
adf : float
Test statistic
pvalue : float
MacKinnon's approximate p-value based on MacKinnon (1994)
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.
regresults : RegressionResults instance
The
resstore : (optional) instance of ResultStore
an instance of 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 accept or reject the null.
The autolag option and maxlag for it are described in Greene.
Examples
--------
see example script
References
----------
Greene
Hamilton
P-Values (regression surface approximation)
MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
Critical values
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):
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]
xdall[:,0] = x[-nobs-1:-1] # replace 0 xdiff with level of x
xdshort = xdiff[-nobs:]
if store:
resstore = ResultsStore()
if autolag:
if regression != 'nc':
fullRHS = add_trend(xdall, regression, prepend=True)
else:
fullRHS = xdall
startlag = fullRHS.shape[1] - xdall.shape[1] + 1 # 1 for level
#search for lag length with smallest information criteria
#Note: use the same number of observations to have comparable IC
#aic and bic: smaller is better
if not regresults:
icbest, bestlag = _autolag(OLS, xdshort, fullRHS, startlag,
maxlag, autolag)
else:
icbest, bestlag, alres = _autolag(OLS, xdshort, fullRHS, startlag,
maxlag, autolag, regresults=regresults)
resstore.autolag_results = alres
bestlag -= startlag #convert to lag not column index
#rerun ols with best autolag
xdall = lagmat(xdiff[:,None], bestlag, trim='both', original='in')
nobs = xdall.shape[0]
xdall[:,0] = x[-nobs-1:-1] # replace 0 xdiff with level of x
xdshort = xdiff[-nobs:]
usedlag = bestlag
else:
usedlag = maxlag
icbest = None
if regression != 'nc':
resols = OLS(xdshort, add_trend(xdall[:,:usedlag+1], regression)).fit()
else:
resols = OLS(xdshort, xdall[:,:usedlag+1]).fit()
adfstat = resols.tvalues[0]
# adfstat = (resols.params[0]-1.0)/resols.bse[0]
# the "asymptotically correct" z statistic is obtained as
# nobs/(1-np.sum(resols.params[1:-(trendorder+1)])) (resols.params[0] - 1)
# I think this is the statistic that is used for series that are integrated
# for orders higher than I(1), ie., not ADF but cointegration tests.
# Get approx p-value and critical values
pvalue = mackinnonp(adfstat, regression=regression, N=1)
critvalues = mackinnoncrit(N=1, regression=regression, nobs=nobs)
critvalues = {"1%" : critvalues[0], "5%" : critvalues[1],
"10%" : critvalues[2]}
if store:
resstore.resols = resols
resstore.maxlag = maxlag
resstore.usedlag = usedlag
resstore.adfstat = adfstat
resstore.critvalues = critvalues
resstore.nobs = nobs
resstore.H0 = "The coefficient on the lagged level equals 1 - unit root"
resstore.HA = "The coefficient on the lagged level < 1 - stationary"
resstore.icbest = icbest
return adfstat, pvalue, critvalues, resstore
else:
if not autolag:
return adfstat, pvalue, usedlag, nobs, critvalues
else:
return adfstat, pvalue, usedlag, nobs, critvalues, icbest
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 : str {'c','ct','ctt','nc'}
Constant and trend order to include in regression
* 'c' : constant only
* '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' or 'BIC', then the number of lags is chosen to minimize the
corresponding information criterium
* 't-stat' based choice of maxlag. Starts with maxlag and drops a
lag until the t-statistic on the last lag length is significant at
the 95 % level.
store : bool
If True, then a result instance is returned additionally to
the adf statistic
regresults : bool
If True, the full regression results are returned.
Returns
-------
adf : float
Test statistic
pvalue : float
MacKinnon's approximate p-value based on MacKinnon (1994)
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.
regresults : RegressionResults instance
The
resstore : (optional) instance of ResultStore
an instance of 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 accept or reject the null.
Examples
--------
see example script
References
----------
Greene
Hamilton
P-Values (regression surface approximation)
MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for
unit-root and cointegration tests. `Journal of Business and Economic
Statistics` 12, 167-76.
Critical values
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
'''
trenddict = {None: 'nc', 0: 'c', 1: 'ct', 2: 'ctt'}
if regression is None or isinstance(regression, int):
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(round(12. * np.power(nobs / 100., 1 / 4.)))
xdiff = np.diff(x)
xdall = lagmat(xdiff[:, None], maxlag, trim='both', original='in')
nobs = xdall.shape[0]
xdall[:, 0] = x[-nobs - 1:-1] # replace 0 xdiff with level of x
xdshort = xdiff[-nobs:]
if store:
resstore = ResultsStore()
if autolag:
if regression != 'nc':
fullRHS = add_trend(xdall, regression, prepend=True)
else:
fullRHS = xdall
startlag = fullRHS.shape[1] - xdall.shape[1] + 1 # 1 for level
#search for lag length with highest information criteria
#Note: use the same number of observations to have comparable IC
icbest, bestlag = _autolag(OLS, xdshort, fullRHS, startlag, maxlag,
autolag)
#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.usedlag = usedlag
resstore.adfstat = adfstat
resstore.critvalues = critvalues
resstore.nobs = nobs
resstore.H0 = "The coefficient on the lagged level equals 1"
resstore.HA = "The coefficient on the lagged level < 1"
resstore.icbest = icbest
return adfstat, pvalue, critvalues, resstore
else:
if not autolag:
return adfstat, pvalue, usedlag, nobs, critvalues
else:
return adfstat, pvalue, usedlag, nobs, critvalues, icbest
