def test_series_both(self):
expected = pd.DataFrame(
index=self.series.index,
columns=['cpi', 'cpi.L.1', 'cpi.L.2', 'cpi.L.3'])
expected['cpi'] = self.series
for lag in range(1, 4):
expected['cpi.L.' + str(int(lag))] = self.series.shift(lag)
expected = expected.iloc[3:]
both = tsatools.lagmat(self.series,
3,
trim='both',
original='in',
use_pandas=True)
tm.assert_frame_equal(both, expected)
lags = tsatools.lagmat(self.series,
3,
trim='both',
original='ex',
use_pandas=True)
tm.assert_frame_equal(lags, expected.iloc[:, 1:])
lags, lead = tsatools.lagmat(self.series,
3,
trim='both',
original='sep',
use_pandas=True)
tm.assert_frame_equal(lead, expected.iloc[:, :1])
tm.assert_frame_equal(lags, expected.iloc[:, 1:])
def test_dataframe_forward(self):
data = self.macro_df
columns = list(data.columns)
n = data.shape[0]
values = np.zeros((n + 3, 16))
values[:n, :4] = data.values
for lag in range(1, 4):
new_cols = [col + '.L.' + str(lag) for col in data]
columns.extend(new_cols)
values[lag:n + lag, 4 * lag:4 * (lag + 1)] = data.values
index = data.index
values = values[:n]
expected = pd.DataFrame(values, columns=columns, index=index)
both = tsatools.lagmat(self.macro_df,
3,
trim='forward',
original='in',
use_pandas=True)
tm.assert_frame_equal(both, expected)
lags = tsatools.lagmat(self.macro_df,
3,
trim='forward',
original='ex',
use_pandas=True)
tm.assert_frame_equal(lags, expected.iloc[:, 4:])
lags, lead = tsatools.lagmat(self.macro_df,
3,
trim='forward',
original='sep',
use_pandas=True)
tm.assert_frame_equal(lags, expected.iloc[:, 4:])
tm.assert_frame_equal(lead, expected.iloc[:, :4])
def test_add_lag1d(self):
data = self.random_data
lagmat = tsatools.lagmat(data, 3, trim='Both')
results = np.column_stack((data[3:], lagmat))
lag_data = tsatools.add_lag(data, lags=3, insert=True)
assert_equal(results, lag_data)
# add index
data = data[:, None]
lagmat = tsatools.lagmat(data, 3, trim='Both') # test for lagmat too
results = np.column_stack((data[3:], lagmat))
lag_data = tsatools.add_lag(data, lags=3, insert=True)
assert_equal(results, lag_data)
def test_add_lag_ndarray(self):
data = self.macro_df.values
nddata = data.astype(float)
lagmat = tsatools.lagmat(nddata[:, 2], 3, trim='Both')
results = np.column_stack((nddata[3:, :3], lagmat, nddata[3:, -1]))
lag_data = tsatools.add_lag(nddata, 2, 3)
assert_equal(lag_data, results)
def test_add_lag_noinsert(self):
data = self.macro_df.values
nddata = data.astype(float)
lagmat = tsatools.lagmat(nddata[:, 2], 3, trim='Both')
results = np.column_stack((nddata[3:, :], lagmat))
lag_data = tsatools.add_lag(data, self.realgdp_loc, 3, insert=False)
assert_equal(lag_data, results)
def test_add_lag_1d_drop_struct(self):
data = np.zeros(100, dtype=[('variable', float)])
nddata = self.random_data
data['variable'] = nddata
lagmat = tsatools.lagmat(nddata, 3, trim='Both')
lag_data = tsatools.add_lag(data, lags=3, drop=True)
assert_equal(lagmat, lag_data.view((float, 3)))
def test_add_lag1d_drop(self):
data = self.random_data
lagmat = tsatools.lagmat(data, 3, trim='Both')
lag_data = tsatools.add_lag(data, lags=3, drop=True, insert=True)
assert_equal(lagmat, lag_data)
# no insert, should be the same
lag_data = tsatools.add_lag(data, lags=3, drop=True, insert=False)
assert_equal(lagmat, lag_data)
def test_add_lag_noinsertatend_ndarray(self):
data = self.macro_df.values
nddata = data.astype(float)
lagmat = tsatools.lagmat(nddata[:, -1], 3, trim='Both')
results = np.column_stack((nddata[3:, :], lagmat))
lag_data = tsatools.add_lag(nddata, 3, 3, insert=False)
assert_equal(lag_data, results)
# should be the same as insert also check negative col number
lag_data2 = tsatools.add_lag(nddata, -1, 3, insert=True)
assert_equal(lag_data2, results)
def test_sep_return(self):
data = self.random_data
n = data.shape[0]
lagmat, leads = tsatools.lagmat(data, 3, trim='none', original='sep')
expected = np.zeros((n + 3, 4))
for i in range(4):
expected[i:i + n, i] = data
expected_leads = expected[:, :1]
expected_lags = expected[:, 1:]
assert_equal(expected_lags, lagmat)
assert_equal(expected_leads, leads)
def test_add_lag_noinsert_atend(self):
data = self.macro_df.values
nddata = data.astype(float)
lagmat = tsatools.lagmat(nddata[:, -1], 3, trim='Both')
results = np.column_stack((nddata[3:, :], lagmat))
lag_data = tsatools.add_lag(data, self.cpi_loc, 3, insert=False)
assert_equal(lag_data, results)
# should be the same as insert
lag_data2 = tsatools.add_lag(data, self.cpi_loc, 3, insert=True)
assert_equal(lag_data2, results)
def _stackX(self, k_ar, trend):
"""
Private method to build the RHS matrix for estimation.
Columns are trend terms then lags.
"""
endog = self.endog
X = lagmat(endog, maxlag=k_ar, trim='both')
k_trend = util.get_trendorder(trend)
if k_trend:
X = add_trend(X, prepend=True, trend=trend)
self.k_trend = k_trend # TODO: Don't set this here
return X
def test_add_lag1d_struct(self):
data = np.zeros(100, dtype=[('variable', float)])
nddata = self.random_data
data['variable'] = nddata
lagmat = tsatools.lagmat(nddata, 3, trim='Both', original='in')
lag_data = tsatools.add_lag(data, 'variable', lags=3, insert=True)
assert_equal(lagmat, lag_data.view((float, 4)))
lag_data = tsatools.add_lag(data, 'variable', lags=3, insert=False)
assert_equal(lagmat, lag_data.view((float, 4)))
lag_data = tsatools.add_lag(data, lags=3, insert=True)
assert_equal(lagmat, lag_data.view((float, 4)))
def _em_autoregressive(self, result, betas, tmp=None):
"""
EM step for autoregressive coefficients and variances
"""
if tmp is None:
tmp = np.sqrt(result.smoothed_marginal_probabilities)
resid = np.zeros((self.k_regimes, self.nobs + self.order))
resid[:] = self.orig_endog
if self._k_exog > 0:
for i in range(self.k_regimes):
resid[i] -= np.dot(self.orig_exog, betas[i])
# The difference between this and `_em_exog` is that here we have a
# different endog and exog for each regime
coeffs = np.zeros((self.k_regimes, ) + (self.order, ))
variance = np.zeros((self.k_regimes, ))
exog = np.zeros((self.nobs, self.order))
for i in range(self.k_regimes):
endog = resid[i, self.order:]
exog = lagmat(resid[i], self.order)[self.order:]
tmp_endog = tmp[i] * endog
tmp_exog = tmp[i][:, None] * exog
coeffs[i] = np.dot(np.linalg.pinv(tmp_exog), tmp_endog)
if self.switching_variance:
tmp_resid = endog - np.dot(exog, coeffs[i])
variance[i] = (
np.sum(tmp_resid**2 *
result.smoothed_marginal_probabilities[i]) /
np.sum(result.smoothed_marginal_probabilities[i]))
else:
tmp_resid = tmp_endog - np.dot(tmp_exog, coeffs[i])
variance[i] = np.sum(tmp_resid**2)
# Variances
if not self.switching_variance:
variance = variance.sum() / self.nobs
return coeffs, variance
def test_dataframe_without_pandas(self):
data = self.macro_df
both = tsatools.lagmat(data, 3, trim='both', original='in')
both_np = tsatools.lagmat(data.values, 3, trim='both', original='in')
assert_equal(both, both_np)
lags = tsatools.lagmat(data, 3, trim='none', original='ex')
lags_np = tsatools.lagmat(data.values, 3, trim='none', original='ex')
assert_equal(lags, lags_np)
lags, lead = tsatools.lagmat(data, 3, trim='forward', original='sep')
lags_np, lead_np = tsatools.lagmat(data.values,
3,
trim='forward',
original='sep')
assert_equal(lags, lags_np)
assert_equal(lead, lead_np)
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, integer_types):
regression = trenddict[regression]
regression = regression.lower()
if regression not in ['c', 'nc', 'ct', 'ctt']: # pragma: no cover
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
# add +1 for level
startlag = fullRHS.shape[1] - xdall.shape[1] + 1
# 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
icbest, bestlag, alres = _autolag(OLS,
xdshort,
fullRHS,
startlag,
maxlag,
autolag,
regresults=True)
if 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 __init__(self,
endog,
k_regimes,
order,
trend='c',
exog=None,
exog_tvtp=None,
switching_ar=True,
switching_trend=True,
switching_exog=False,
switching_variance=False,
dates=None,
freq=None,
missing='none'):
# Properties
self.order = order # this also gets set in MarkovSwitching.__init__
self.switching_ar = switching_ar
# Switching options
if self.switching_ar is True or self.switching_ar is False:
self.switching_ar = [self.switching_ar] * order
elif len(self.switching_ar) != order:
raise ValueError('Invalid iterable passed to `switching_ar`.')
# Cache an array for holding slices
self._predict_slices = [slice(None, None, None)] * (self.order + 1)
# Autoregressive exog
self.exog_ar = lagmat(endog, self.order)[self.order:]
# Initialize the base model
super(MarkovAutoregression,
self).__init__(endog,
k_regimes,
trend=trend,
exog=exog,
order=order,
exog_tvtp=exog_tvtp,
switching_trend=switching_trend,
switching_exog=switching_exog,
switching_variance=switching_variance,
dates=dates,
freq=freq,
missing=missing)
# TODO: It looks like all existing tests have zero nulls in endog
# Sanity checks
if self.nobs <= self.order:
# in tests (as of 2018-03-31) self.nobs always matches len(endog)
# at this point, but it isn't obvious if this MUST hold
raise ValueError('Must have more observations than the order of'
' the autoregression.')
# Reshape other datasets
self.nobs -= self.order
# TODO: Do we have test cases where nulls have been dropped
# from self.endog by this point? can we write this in
# terms of anything fixed?
self.orig_endog = self.endog
# TODO: Does this necessarily match self.data.orig_endog?
self.endog = self.endog[self.order:]
if self._k_exog > 0:
self.orig_exog = self.exog
self.exog = self.exog[self.order:]
# Reset the ModelData datasets
# TODO: I'm not wild about altering self.data in-place...
self.data.endog, self.data.exog = (self.data._convert_endog_exog(
self.endog, self.exog))
# Reset indexes, if provided
if self.data.row_labels is not None:
self.data._cache['row_labels'] = self.data.row_labels[self.order:]
if self._index is not None:
if self._index_generated:
self._index = self._index[:-self.order]
else:
self._index = self._index[self.order:]
# Parameters
self.parameters['autoregressive'] = self.switching_ar
def test_pandas_errors(self):
# TODO: parametrize?
for trim in ['none', 'backward']:
for data in [self.macro_df, self.series]:
with pytest.raises(ValueError):
tsatools.lagmat(data, 3, trim=trim, use_pandas=True)
def pacf_ols(x, nlags=40, efficient=True, unbiased=False):
"""
Calculate partial autocorrelations via OLS
Parameters
----------
x : 1d array
observations of time series for which pacf is calculated
nlags : int
Number of lags for which pacf is returned. Lag 0 is not returned.
efficient : bool, optional
If true, uses the maximum number of available observations to compute
each partial autocorrelation. If not, uses the same number of
observations to compute all pacf values.
unbiased : bool, optional
Adjust each partial autocorrelation by n / (n - lag)
Returns
-------
pacf : 1d array
partial autocorrelations, (maxlag,) array corresponding to lags
0, 1, ..., maxlag
Notes
-----
This solves a separate OLS estimation for each desired lag. Setting
efficient to True has two effects. First, it uses `nobs - lag`
observations of estimate each pacf. Second, it re-estimates the mean in
each regression. If efficient is False, then the data are first demeaned,
and then `nobs - maxlag` observations are used to estimate each partial
autocorrelation.
The inefficient estimator appears to have better finite sample properties.
This option should only be used in time series that are covariance
stationary.
OLS estimation of the pacf does not guarantee that all pacf values are
between -1 and 1.
See also
--------
sm2.tsa.stattools.pacf
sm2.tsa.autocov.pacf_yw
sm2.tsa.autocov.pacf_burg
References
----------
.. [1] Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015).
Time series analysis: forecasting and control. John Wiley & Sons, p. 66
"""
pacf = np.empty(nlags + 1)
pacf[0] = 1.0
x = np.squeeze(np.asarray(x))
if x.ndim != 1:
raise ValueError('x must be squeezable to a 1-d array')
if efficient:
xlags, x0 = lagmat(x, nlags, original='sep')
xlags = add_constant(xlags)
for k in range(1, nlags + 1):
params = lstsq(xlags[k:, :k + 1], x0[k:], rcond=None)[0]
pacf[k] = params[-1]
else:
x = x - np.mean(x)
# Create a single set of lags for multivariate OLS
xlags, x0 = lagmat(x, nlags, original='sep', trim='both')
for k in range(1, nlags + 1):
params = lstsq(xlags[:, :k], x0, rcond=None)[0]
# Last coefficient corresponds to PACF value (see [1])
pacf[k] = params[-1]
if unbiased:
n = len(x)
pacf *= n / (n - np.arange(nlags + 1))
return pacf
def test_unknown_trim(self):
with pytest.raises(ValueError):
tsatools.lagmat(self.macro_df, 3, trim='unknown', use_pandas=True)
with pytest.raises(ValueError):
tsatools.lagmat(self.macro_df.values, 3, trim='unknown')
def test_too_few_observations(self):
with pytest.raises(ValueError):
tsatools.lagmat(self.macro_df, 300, use_pandas=True)
with pytest.raises(ValueError):
tsatools.lagmat(self.macro_df.values, 300)
