Repository: statsmodels/statsmodels

Issue:
ENH: Zivot-Andrews unit-root test (issue 4746)
Initial commit for Zivot-Andrews structural-break unit-root test (issue 4746).


Discussion:

Resolving patch:
diff --git a/statsmodels/tools/validation/validation.py b/statsmodels/tools/validation/validation.py
--- a/statsmodels/tools/validation/validation.py
+++ b/statsmodels/tools/validation/validation.py
@@ -354,7 +354,7 @@ def string_like(value, name, optional=False, options=None, lower=True):
     if lower:
         value = value.lower()
     if options is not None and value not in options:
-        extra_text = 'If not None,' if optional else ''
+        extra_text = 'If not None, ' if optional else ''
         options_text = "'" + '\', \''.join(options) + "'"
         msg = '{0}{1} must be one of: {2}'.format(extra_text,
                                                   name, options_text)
@@ -386,7 +386,7 @@ def dict_like(value, name, optional=False, strict=True):
         return None
     if (not isinstance(value, Mapping) or
             (strict and not(isinstance(value, dict)))):
-        extra_text = 'If not None,' if optional else ''
+        extra_text = 'If not None, ' if optional else ''
         strict_text = ' or dict_like (i.e., a Mapping)' if strict else ''
         msg = '{0}{1} must be a dict{2}'.format(extra_text, name, strict_text)
         raise TypeError(msg)
diff --git a/statsmodels/tsa/stattools.py b/statsmodels/tsa/stattools.py
--- a/statsmodels/tsa/stattools.py
+++ b/statsmodels/tsa/stattools.py
@@ -27,7 +27,8 @@
 __all__ = ['acovf', 'acf', 'pacf', 'pacf_yw', 'pacf_ols', 'ccovf', 'ccf',
            'periodogram', 'q_stat', 'coint', 'arma_order_select_ic',
            'adfuller', 'kpss', 'bds', 'pacf_burg', 'innovations_algo',
-           'innovations_filter', 'levinson_durbin_pacf', 'levinson_durbin']
+           'innovations_filter', 'levinson_durbin_pacf', 'levinson_durbin',
+           'zivot_andrews']
 
 SQRTEPS = np.sqrt(np.finfo(np.double).eps)
 
@@ -1700,3 +1701,306 @@ def _kpss_autolag(resids, nobs):
     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
+            data series
+        trim : float
+            percentage of series at begin/end to exclude from break-period
+            calculation in range [0, 0.333] (default=0.15)
+        maxlag : int
+            maximum lag which is included in test, default=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}
+            * 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
+            test statistic
+        pvalue : float
+            based on MC-derived critical values
+        cvdict : dict
+            critical values for the test statistic at the 1%, 5%, and 10%
+            levels
+        bpidx : int
+            index of x corresponding to endogenously calculated break period
+            with values in the range [0..nobs-1]
+        baselag : int
+            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 tradeoff.
+
+        References
+        ----------
+        .. [*] 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.
+
+        .. [*] Schwert, G.W. (1989). Tests for unit roots: A Monte Carlo
+           investigation. Journal of Business & Economic Statistics, 7:
+           147-159.
+
+        .. [*] 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. / 3.):
+            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. * np.power(nobs / 100., 1 / 4.))
+        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__


Test patch:
diff --git a/statsmodels/tsa/tests/results/gnpdef.csv b/statsmodels/tsa/tests/results/gnpdef.csv
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/gnpdef.csv
@@ -0,0 +1,83 @@
+GNP_DEFLATOR
+3.254242969
+3.234749174
+3.214867803
+3.17805383
+3.198673118
+3.135494216
+3.122364924
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diff --git a/statsmodels/tsa/tests/results/rand10000.csv b/statsmodels/tsa/tests/results/rand10000.csv
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/rand10000.csv
@@ -0,0 +1,10001 @@
+RAND10000
+0
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+1.103351908
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+-0.256822186
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diff --git a/statsmodels/tsa/tests/results/rgnp.csv b/statsmodels/tsa/tests/results/rgnp.csv
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/rgnp.csv
@@ -0,0 +1,63 @@
+REAL_GNP
+4.76046307
+4.788324729
+4.813809051
+4.86907173
+4.878246106
+4.833102254
+4.824305716
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diff --git a/statsmodels/tsa/tests/results/rgnpq.csv b/statsmodels/tsa/tests/results/rgnpq.csv
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/rgnpq.csv
@@ -0,0 +1,160 @@
+REAL_GNP_QTR
+7.574046546
+7.57317714
+7.572137692
+7.587975262
+7.603009239
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+7.624913865
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diff --git a/statsmodels/tsa/tests/results/stkprc.csv b/statsmodels/tsa/tests/results/stkprc.csv
new file mode 100644
--- /dev/null
+++ b/statsmodels/tsa/tests/results/stkprc.csv
@@ -0,0 +1,101 @@
+STOCK_PRICES
+1.545432582
+1.615419984
+1.568615918
+1.519513205
+1.492904096
+1.401182974
+1.1442228
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diff --git a/statsmodels/tsa/tests/test_stattools.py b/statsmodels/tsa/tests/test_stattools.py
--- a/statsmodels/tsa/tests/test_stattools.py
+++ b/statsmodels/tsa/tests/test_stattools.py
@@ -24,7 +24,7 @@
                                        arma_order_select_ic, levinson_durbin,
                                        levinson_durbin_pacf, pacf_burg,
                                        innovations_algo, innovations_filter,
-                                       periodogram)
+                                       periodogram, zivot_andrews)
 
 DECIMAL_8 = 8
 DECIMAL_6 = 6
@@ -34,6 +34,8 @@
 DECIMAL_2 = 2
 DECIMAL_1 = 1
 
+CURR_DIR = os.path.dirname(os.path.abspath(__file__))
+
 
 @pytest.fixture('module')
 def acovf_data():
@@ -158,13 +160,9 @@ class CheckCorrGram(object):
     """
     data = macrodata.load_pandas()
     x = data.data['realgdp']
-    filename = os.path.dirname(os.path.abspath(__file__))+\
-            "/results/results_corrgram.csv"
+    filename = os.path.join(CURR_DIR, 'results', 'results_corrgram.csv')
     results = pd.read_csv(filename, delimiter=',')
 
-    #not needed: add 1. for lag zero
-    #self.results['acvar'] = np.concatenate(([1.], self.results['acvar']))
-
 
 class TestACF(CheckCorrGram):
     """
@@ -953,3 +951,67 @@ def test_adfuller_maxlag_too_large(reset_randomstate):
 def test_periodogram_future_warning(reset_randomstate):
     with pytest.warns(FutureWarning):
         periodogram(np.random.standard_normal(100))
+
+
+class SetupZivotAndrews(object):
+    # test directory
+    cur_dir = CURR_DIR
+    run_dir = os.path.join(cur_dir, 'results')
+    # use same file for testing failure modes
+    fail_file = os.path.join(run_dir, 'rgnp.csv')
+    fail_mdl = np.asarray(pd.read_csv(fail_file))
+
+
+class TestZivotAndrews(SetupZivotAndrews):
+
+    # failure mode tests
+    def test_fail_regression_type(self):
+        with pytest.raises(ValueError):
+            zivot_andrews(self.fail_mdl, regression='x')
+
+    def test_fail_trim_value(self):
+        with pytest.raises(ValueError):
+            zivot_andrews(self.fail_mdl, trim=0.5)
+
+    def test_fail_array_shape(self):
+        with pytest.raises(ValueError):
+            zivot_andrews(np.random.rand(50, 2))
+
+    def test_fail_autolag_type(self):
+        with pytest.raises(ValueError):
+            zivot_andrews(self.fail_mdl, autolag='None')
+
+    # following tests compare results to R package urca.ur.za (1.13-0)
+    def test_rgnp_case(self):
+        res = zivot_andrews(self.fail_mdl, maxlag=8, regression='c',
+                            autolag=None)
+        assert_allclose([res[0], res[1], res[4]],
+                        [-5.57615, 0.00312, 20], rtol=1e-3)
+
+    def test_gnpdef_case(self):
+        mdlfile = os.path.join(self.run_dir, 'gnpdef.csv')
+        mdl = np.asarray(pd.read_csv(mdlfile))
+        res = zivot_andrews(mdl, maxlag=8, regression='c', autolag='t-stat')
+        assert_allclose([res[0], res[1], res[3], res[4]],
+                        [-4.12155, 0.28024, 5, 40], rtol=1e-3)
+
+    def test_stkprc_case(self):
+        mdlfile = os.path.join(self.run_dir, 'stkprc.csv')
+        mdl = np.asarray(pd.read_csv(mdlfile))
+        res = zivot_andrews(mdl, maxlag=8, regression='ct', autolag='t-stat')
+        assert_allclose([res[0], res[1], res[3], res[4]],
+                        [-5.60689, 0.00894, 1, 65], rtol=1e-3)
+
+    def test_rgnpq_case(self):
+        mdlfile = os.path.join(self.run_dir, 'rgnpq.csv')
+        mdl = np.asarray(pd.read_csv(mdlfile))
+        res = zivot_andrews(mdl, maxlag=12, regression='t', autolag='t-stat')
+        assert_allclose([res[0], res[1], res[3], res[4]],
+                        [-3.02761, 0.63993, 12, 102], rtol=1e-3)
+
+    def test_rand10000_case(self):
+        mdlfile = os.path.join(self.run_dir, 'rand10000.csv')
+        mdl = np.asarray(pd.read_csv(mdlfile))
+        res = zivot_andrews(mdl, regression='c', autolag='t-stat')
+        assert_allclose([res[0], res[1], res[3], res[4]],
+                        [-3.48223, 0.69111, 25, 7071], rtol=1e-3)