text: """
  Statistical tools for time series analysis
  """
  from __future__ import annotations
  
  from statsmodels.compat.numpy import lstsq
  from statsmodels.compat.pandas import deprecate_kwarg
  from statsmodels.compat.python import Literal, lzip
  from statsmodels.compat.scipy import _next_regular
  
  from typing import Tuple
  import warnings
  
  import numpy as np
  from numpy.linalg import LinAlgError
  import pandas as pd
  from scipy import stats
  from scipy.interpolate import interp1d
  from scipy.signal import correlate
  
  from statsmodels.regression.linear_model import OLS, yule_walker
  from statsmodels.tools.sm_exceptions import (
      CollinearityWarning,
      InfeasibleTestError,
      InterpolationWarning,
      MissingDataError,
  )
  from statsmodels.tools.tools import Bunch, add_constant
  from statsmodels.tools.validation import (
      array_like,
      bool_like,
      dict_like,
      float_like,
      int_like,
      string_like,
  )
  from statsmodels.tsa._bds import bds
  from statsmodels.tsa._innovations import innovations_algo, innovations_filter
  from statsmodels.tsa.adfvalues import mackinnoncrit, mackinnonp
  from statsmodels.tsa.tsatools import add_trend, lagmat, lagmat2ds
  
  __all__ = [
      "acovf",
      "acf",
      "pacf",
      "pacf_yw",
      "pacf_ols",
      "ccovf",
      "ccf",
      "q_stat",
      "coint",
      "arma_order_select_ic",
      "adfuller",
      "kpss",
      "bds",
      "pacf_burg",
      "innovations_algo",
      "innovations_filter",
      "levinson_durbin_pacf",
      "levinson_durbin",
      "zivot_andrews",
      "range_unit_root_test",
  ]
  
  SQRTEPS = np.sqrt(np.finfo(np.double).eps)
  
  
  def _autolag(
      mod,
      endog,
      exog,
      startlag,
      maxlag,
      method,
      modargs=(),
      fitargs=(),
      regresults=False,
  ):
      """
      Returns the results for the lag length that maximizes the info criterion.
  
      Parameters
      ----------
      mod : Model class
          Model estimator class
      endog : array_like
          nobs array containing endogenous variable
      exog : array_like
          nobs by (startlag + maxlag) array containing lags and possibly other
          variables
      startlag : int
          The first zero-indexed column to hold a lag.  See Notes.
      maxlag : int
          The highest lag order for lag length selection.
      method : {"aic", "bic", "t-stat"}
          aic - Akaike Information Criterion
          bic - Bayes Information Criterion
          t-stat - Based on last lag
      modargs : tuple, optional
          args to pass to model.  See notes.
      fitargs : tuple, optional
          args to pass to fit.  See notes.
      regresults : bool, optional
          Flag indicating to return optional return results
  
      Returns
      -------
      icbest : float
          Best information criteria.
      bestlag : int
          The lag length that maximizes the information criterion.
      results : dict, optional
          Dictionary containing all estimation results
  
      Notes
      -----
      Does estimation like mod(endog, exog[:,:i], *modargs).fit(*fitargs)
      where i goes from lagstart to lagstart+maxlag+1.  Therefore, lags are
      assumed to be in contiguous columns from low to high lag length with
      the highest lag in the last column.
      """
      # TODO: can tcol be replaced by maxlag + 2?
      # TODO: This could be changed to laggedRHS and exog keyword arguments if
      #    this will be more general.
  
      results = {}
      method = method.lower()
      for lag in range(startlag, startlag + maxlag + 1):
          mod_instance = mod(endog, exog[:, :lag], *modargs)
          results[lag] = mod_instance.fit()
  
      if method == "aic":
          icbest, bestlag = min((v.aic, k) for k, v in results.items())
      elif method == "bic":
          icbest, bestlag = min((v.bic, k) for k, v in results.items())
      elif method == "t-stat":
          # stop = stats.norm.ppf(.95)
          stop = 1.6448536269514722
          # Default values to ensure that always set
          bestlag = startlag + maxlag
          icbest = 0.0
          for lag in range(startlag + maxlag, startlag - 1, -1):
              icbest = np.abs(results[lag].tvalues[-1])
              bestlag = lag
              if np.abs(icbest) >= stop:
                  # Break for first lag with a significant t-stat
                  break
      else:
          raise ValueError(f"Information Criterion {method} not understood.")
  
      if not regresults:
          return icbest, bestlag
      else:
          return icbest, bestlag, results
  
  
  # this needs to be converted to a class like HetGoldfeldQuandt,
  # 3 different returns are a mess
  # See:
  # Ng and Perron(2001), Lag length selection and the construction of unit root
  # tests with good size and power, Econometrica, Vol 69 (6) pp 1519-1554
  # TODO: include drift keyword, only valid with regression == "c"
  # just changes the distribution of the test statistic to a t distribution
  # TODO: autolag is untested
  def adfuller(
      x,
      maxlag: int | None = 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 : {None, int}
          Maximum lag which is included in test, default value of
          12*(nobs/100)^{1/4} is used when ``None``.
      regression : {"c","ct","ctt","n"}
          Constant and trend order to include in regression.
  
          * "c" : constant only (default).
          * "ct" : constant and trend.
          * "ctt" : constant, and linear and quadratic trend.
          * "n" : 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", "n")
      )
      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: "n", 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 != "n" 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 != "n":
              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 != "n":
          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
  
  
  @deprecate_kwarg("unbiased", "adjusted")
  def acovf(x, adjusted=False, demean=True, fft=True, missing="none", nlag=None):
      """
      Estimate autocovariances.
  
      Parameters
      ----------
      x : array_like
          Time series data. Must be 1d.
      adjusted : bool, default False
          If True, then denominators is n-k, otherwise n.
      demean : bool, default True
          If True, then subtract the mean x from each element of x.
      fft : bool, default True
          If True, use FFT convolution.  This method should be preferred
          for long time series.
      missing : str, default "none"
          A string in ["none", "raise", "conservative", "drop"] specifying how
          the NaNs are to be treated. "none" performs no checks. "raise" raises
          an exception if NaN values are found. "drop" removes the missing
          observations and then estimates the autocovariances treating the
          non-missing as contiguous. "conservative" computes the autocovariance
          using nan-ops so that nans are removed when computing the mean
          and cross-products that are used to estimate the autocovariance.
          When using "conservative", n is set to the number of non-missing
          observations.
      nlag : {int, None}, default None
          Limit the number of autocovariances returned.  Size of returned
          array is nlag + 1.  Setting nlag when fft is False uses a simple,
          direct estimator of the autocovariances that only computes the first
          nlag + 1 values. This can be much faster when the time series is long
          and only a small number of autocovariances are needed.
  
      Returns
      -------
      ndarray
          The estimated autocovariances.
  
      References
      -----------
      .. [1] Parzen, E., 1963. On spectral analysis with missing observations
             and amplitude modulation. Sankhya: The Indian Journal of
             Statistics, Series A, pp.383-392.
      """
      adjusted = bool_like(adjusted, "adjusted")
      demean = bool_like(demean, "demean")
      fft = bool_like(fft, "fft", optional=False)
      missing = string_like(
          missing, "missing", options=("none", "raise", "conservative", "drop")
      )
      nlag = int_like(nlag, "nlag", optional=True)
  
      x = array_like(x, "x", ndim=1)
  
      missing = missing.lower()
      if missing == "none":
          deal_with_masked = False
      else:
          deal_with_masked = has_missing(x)
      if deal_with_masked:
          if missing == "raise":
              raise MissingDataError("NaNs were encountered in the data")
          notmask_bool = ~np.isnan(x)  # bool
          if missing == "conservative":
              # Must copy for thread safety
              x = x.copy()
              x[~notmask_bool] = 0
          else:  # "drop"
              x = x[notmask_bool]  # copies non-missing
          notmask_int = notmask_bool.astype(int)  # int
  
      if demean and deal_with_masked:
          # whether "drop" or "conservative":
          xo = x - x.sum() / notmask_int.sum()
          if missing == "conservative":
              xo[~notmask_bool] = 0
      elif demean:
          xo = x - x.mean()
      else:
          xo = x
  
      n = len(x)
      lag_len = nlag
      if nlag is None:
          lag_len = n - 1
      elif nlag > n - 1:
          raise ValueError("nlag must be smaller than nobs - 1")
  
      if not fft and nlag is not None:
          acov = np.empty(lag_len + 1)
          acov[0] = xo.dot(xo)
          for i in range(lag_len):
              acov[i + 1] = xo[i + 1 :].dot(xo[: -(i + 1)])
          if not deal_with_masked or missing == "drop":
              if adjusted:
                  acov /= n - np.arange(lag_len + 1)
              else:
                  acov /= n
          else:
              if adjusted:
                  divisor = np.empty(lag_len + 1, dtype=np.int64)
                  divisor[0] = notmask_int.sum()
                  for i in range(lag_len):
                      divisor[i + 1] = notmask_int[i + 1 :].dot(
                          notmask_int[: -(i + 1)]
                      )
                  divisor[divisor == 0] = 1
                  acov /= divisor
              else:  # biased, missing data but npt "drop"
                  acov /= notmask_int.sum()
          return acov
  
      if adjusted and deal_with_masked and missing == "conservative":
          d = np.correlate(notmask_int, notmask_int, "full")
          d[d == 0] = 1
      elif adjusted:
          xi = np.arange(1, n + 1)
          d = np.hstack((xi, xi[:-1][::-1]))
      elif deal_with_masked:
          # biased and NaNs given and ("drop" or "conservative")
          d = notmask_int.sum() * np.ones(2 * n - 1)
      else:  # biased and no NaNs or missing=="none"
          d = n * np.ones(2 * n - 1)
  
      if fft:
          nobs = len(xo)
          n = _next_regular(2 * nobs + 1)
          Frf = np.fft.fft(xo, n=n)
          acov = np.fft.ifft(Frf * np.conjugate(Frf))[:nobs] / d[nobs - 1 :]
          acov = acov.real
      else:
          acov = np.correlate(xo, xo, "full")[n - 1 :] / d[n - 1 :]
  
      if nlag is not None:
          # Copy to allow gc of full array rather than view
          return acov[: lag_len + 1].copy()
      return acov
  
  
  def q_stat(x, nobs):
      """
      Compute Ljung-Box Q Statistic.
  
      Parameters
      ----------
      x : array_like
          Array of autocorrelation coefficients.  Can be obtained from acf.
      nobs : int, optional
          Number of observations in the entire sample (ie., not just the length
          of the autocorrelation function results.
  
      Returns
      -------
      q-stat : ndarray
          Ljung-Box Q-statistic for autocorrelation parameters.
      p-value : ndarray
          P-value of the Q statistic.
  
      Notes
      -----
      Designed to be used with acf.
      """
      x = array_like(x, "x")
      nobs = int_like(nobs, "nobs")
  
      ret = (
          nobs
          * (nobs + 2)
          * np.cumsum((1.0 / (nobs - np.arange(1, len(x) + 1))) * x ** 2)
      )
      chi2 = stats.chi2.sf(ret, np.arange(1, len(x) + 1))
      return ret, chi2
  
  
  # NOTE: Changed unbiased to False
  # see for example
  # http://www.itl.nist.gov/div898/handbook/eda/section3/autocopl.htm
  def acf(
      x,
      adjusted=False,
      nlags=None,
      qstat=False,
      fft=True,
      alpha=None,
      bartlett_confint=True,
      missing="none",
  ):
      """
      Calculate the autocorrelation function.
  
      Parameters
      ----------
      x : array_like
         The time series data.
      adjusted : bool, default False
         If True, then denominators for autocovariance are n-k, otherwise n.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs - 1). The returned value
          includes lag 0 (ie., 1) so size of the acf vector is (nlags + 1,).
      qstat : bool, default False
          If True, returns the Ljung-Box q statistic for each autocorrelation
          coefficient.  See q_stat for more information.
      fft : bool, default True
          If True, computes the ACF via FFT.
      alpha : scalar, default None
          If a number is given, the confidence intervals for the given level are
          returned. For instance if alpha=.05, 95 % confidence intervals are
          returned where the standard deviation is computed according to
          Bartlett"s formula.
      bartlett_confint : bool, default True
          Confidence intervals for ACF values are generally placed at 2
          standard errors around r_k. The formula used for standard error
          depends upon the situation. If the autocorrelations are being used
          to test for randomness of residuals as part of the ARIMA routine,
          the standard errors are determined assuming the residuals are white
          noise. The approximate formula for any lag is that standard error
          of each r_k = 1/sqrt(N). See section 9.4 of [2] for more details on
          the 1/sqrt(N) result. For more elementary discussion, see section 5.3.2
          in [3].
          For the ACF of raw data, the standard error at a lag k is
          found as if the right model was an MA(k-1). This allows the possible
          interpretation that if all autocorrelations past a certain lag are
          within the limits, the model might be an MA of order defined by the
          last significant autocorrelation. In this case, a moving average
          model is assumed for the data and the standard errors for the
          confidence intervals should be generated using Bartlett's formula.
          For more details on Bartlett formula result, see section 7.2 in [2].
      missing : str, default "none"
          A string in ["none", "raise", "conservative", "drop"] specifying how
          the NaNs are to be treated. "none" performs no checks. "raise" raises
          an exception if NaN values are found. "drop" removes the missing
          observations and then estimates the autocovariances treating the
          non-missing as contiguous. "conservative" computes the autocovariance
          using nan-ops so that nans are removed when computing the mean
          and cross-products that are used to estimate the autocovariance.
          When using "conservative", n is set to the number of non-missing
          observations.
  
      Returns
      -------
      acf : ndarray
          The autocorrelation function for lags 0, 1, ..., nlags. Shape
          (nlags+1,).
      confint : ndarray, optional
          Confidence intervals for the ACF at lags 0, 1, ..., nlags. Shape
          (nlags + 1, 2). Returned if alpha is not None.
      qstat : ndarray, optional
          The Ljung-Box Q-Statistic for lags 1, 2, ..., nlags (excludes lag
          zero). Returned if q_stat is True.
      pvalues : ndarray, optional
          The p-values associated with the Q-statistics for lags 1, 2, ...,
          nlags (excludes lag zero). Returned if q_stat is True.
  
      Notes
      -----
      The acf at lag 0 (ie., 1) is returned.
  
      For very long time series it is recommended to use fft convolution instead.
      When fft is False uses a simple, direct estimator of the autocovariances
      that only computes the first nlag + 1 values. This can be much faster when
      the time series is long and only a small number of autocovariances are
      needed.
  
      If adjusted is true, the denominator for the autocovariance is adjusted
      for the loss of data.
  
      References
      ----------
      .. [1] Parzen, E., 1963. On spectral analysis with missing observations
         and amplitude modulation. Sankhya: The Indian Journal of
         Statistics, Series A, pp.383-392.
      .. [2] Brockwell and Davis, 1987. Time Series Theory and Methods
      .. [3] Brockwell and Davis, 2010. Introduction to Time Series and
         Forecasting, 2nd edition.
      """
      adjusted = bool_like(adjusted, "adjusted")
      nlags = int_like(nlags, "nlags", optional=True)
      qstat = bool_like(qstat, "qstat")
      fft = bool_like(fft, "fft", optional=False)
      alpha = float_like(alpha, "alpha", optional=True)
      missing = string_like(
          missing, "missing", options=("none", "raise", "conservative", "drop")
      )
      x = array_like(x, "x")
      # TODO: should this shrink for missing="drop" and NaNs in x?
      nobs = x.shape[0]
      if nlags is None:
          nlags = min(int(10 * np.log10(nobs)), nobs - 1)
  
      avf = acovf(x, adjusted=adjusted, demean=True, fft=fft, missing=missing)
      acf = avf[: nlags + 1] / avf[0]
      if not (qstat or alpha):
          return acf
      if alpha is not None:
          if bartlett_confint:
              varacf = np.ones_like(acf) / nobs
              varacf[0] = 0
              varacf[1] = 1.0 / nobs
              varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1] ** 2)
          else:
              varacf = 1.0 / len(x)
          interval = stats.norm.ppf(1 - alpha / 2.0) * np.sqrt(varacf)
          confint = np.array(lzip(acf - interval, acf + interval))
          if not qstat:
              return acf, confint
      if qstat:
          qstat, pvalue = q_stat(acf[1:], nobs=nobs)  # drop lag 0
          if alpha is not None:
              return acf, confint, qstat, pvalue
          else:
              return acf, qstat, pvalue
  
  
  def pacf_yw(x, nlags=None, method="adjusted"):
      """
      Partial autocorrelation estimated with non-recursive yule_walker.
  
      Parameters
      ----------
      x : array_like
          The observations of time series for which pacf is calculated.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs - 1).
      method : {"adjusted", "mle"}, default "adjusted"
          The method for the autocovariance calculations in yule walker.
  
      Returns
      -------
      ndarray
          The partial autocorrelations, maxlag+1 elements.
  
      See Also
      --------
      statsmodels.tsa.stattools.pacf
          Partial autocorrelation estimation.
      statsmodels.tsa.stattools.pacf_ols
          Partial autocorrelation estimation using OLS.
      statsmodels.tsa.stattools.pacf_burg
          Partial autocorrelation estimation using Burg"s method.
  
      Notes
      -----
      This solves yule_walker for each desired lag and contains
      currently duplicate calculations.
      """
      x = array_like(x, "x")
      nlags = int_like(nlags, "nlags", optional=True)
      nobs = x.shape[0]
      if nlags is None:
          nlags = min(int(10 * np.log10(nobs)), nobs - 1)
  
      method = string_like(method, "method", options=("adjusted", "mle"))
      pacf = [1.0]
      for k in range(1, nlags + 1):
          pacf.append(yule_walker(x, k, method=method)[0][-1])
      return np.array(pacf)
  
  
  def pacf_burg(x, nlags=None, demean=True):
      """
      Calculate Burg"s partial autocorrelation estimator.
  
      Parameters
      ----------
      x : array_like
          Observations of time series for which pacf is calculated.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs - 1).
      demean : bool, optional
          Flag indicating to demean that data. Set to False if x has been
          previously demeaned.
  
      Returns
      -------
      pacf : ndarray
          Partial autocorrelations for lags 0, 1, ..., nlag.
      sigma2 : ndarray
          Residual variance estimates where the value in position m is the
          residual variance in an AR model that includes m lags.
  
      See Also
      --------
      statsmodels.tsa.stattools.pacf
          Partial autocorrelation estimation.
      statsmodels.tsa.stattools.pacf_yw
           Partial autocorrelation estimation using Yule-Walker.
      statsmodels.tsa.stattools.pacf_ols
          Partial autocorrelation estimation using OLS.
  
      References
      ----------
      .. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
          and forecasting. Springer.
      """
      x = array_like(x, "x")
      if demean:
          x = x - x.mean()
      nobs = x.shape[0]
      p = nlags if nlags is not None else min(int(10 * np.log10(nobs)), nobs - 1)
      if p > nobs - 1:
          raise ValueError("nlags must be smaller than nobs - 1")
      d = np.zeros(p + 1)
      d[0] = 2 * x.dot(x)
      pacf = np.zeros(p + 1)
      u = x[::-1].copy()
      v = x[::-1].copy()
      d[1] = u[:-1].dot(u[:-1]) + v[1:].dot(v[1:])
      pacf[1] = 2 / d[1] * v[1:].dot(u[:-1])
      last_u = np.empty_like(u)
      last_v = np.empty_like(v)
      for i in range(1, p):
          last_u[:] = u
          last_v[:] = v
          u[1:] = last_u[:-1] - pacf[i] * last_v[1:]
          v[1:] = last_v[1:] - pacf[i] * last_u[:-1]
          d[i + 1] = (1 - pacf[i] ** 2) * d[i] - v[i] ** 2 - u[-1] ** 2
          pacf[i + 1] = 2 / d[i + 1] * v[i + 1 :].dot(u[i:-1])
      sigma2 = (1 - pacf ** 2) * d / (2.0 * (nobs - np.arange(0, p + 1)))
      pacf[0] = 1  # Insert the 0 lag partial autocorrel
  
      return pacf, sigma2
  
  
  @deprecate_kwarg("unbiased", "adjusted")
  def pacf_ols(x, nlags=None, efficient=True, adjusted=False):
      """
      Calculate partial autocorrelations via OLS.
  
      Parameters
      ----------
      x : array_like
          Observations of time series for which pacf is calculated.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs - 1).
      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.
      adjusted : bool, optional
          Adjust each partial autocorrelation by n / (n - lag).
  
      Returns
      -------
      ndarray
          The partial autocorrelations, (maxlag,) array corresponding to lags
          0, 1, ..., maxlag.
  
      See Also
      --------
      statsmodels.tsa.stattools.pacf
          Partial autocorrelation estimation.
      statsmodels.tsa.stattools.pacf_yw
           Partial autocorrelation estimation using Yule-Walker.
      statsmodels.tsa.stattools.pacf_burg
          Partial autocorrelation estimation using Burg"s method.
  
      Notes
      -----
      This solves a separate OLS estimation for each desired lag using method in
      [1]_. 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.
  
      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
      """
      x = array_like(x, "x")
      nlags = int_like(nlags, "nlags", optional=True)
      efficient = bool_like(efficient, "efficient")
      adjusted = bool_like(adjusted, "adjusted")
      nobs = x.shape[0]
      if nlags is None:
          nlags = min(int(10 * np.log10(nobs)), nobs - 1)
  
      pacf = np.empty(nlags + 1)
      pacf[0] = 1.0
      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 adjusted:
          pacf *= nobs / (nobs - np.arange(nlags + 1))
  
      return pacf
  
  
  def pacf(x, nlags=None, method="ywadjusted", alpha=None):
      """
      Partial autocorrelation estimate.
  
      Parameters
      ----------
      x : array_like
          Observations of time series for which pacf is calculated.
      nlags : int, optional
          Number of lags to return autocorrelation for. If not provided,
          uses min(10 * np.log10(nobs), nobs // 2 - 1). The returned value
          includes lag 0 (ie., 1) so size of the pacf vector is (nlags + 1,).
      method : str, default "ywunbiased"
          Specifies which method for the calculations to use.
  
          - "yw" or "ywadjusted" : Yule-Walker with sample-size adjustment in
            denominator for acovf. Default.
          - "ywm" or "ywmle" : Yule-Walker without adjustment.
          - "ols" : regression of time series on lags of it and on constant.
          - "ols-inefficient" : regression of time series on lags using a single
            common sample to estimate all pacf coefficients.
          - "ols-adjusted" : regression of time series on lags with a bias
            adjustment.
          - "ld" or "ldadjusted" : Levinson-Durbin recursion with bias
            correction.
          - "ldb" or "ldbiased" : Levinson-Durbin recursion without bias
            correction.
  
      alpha : float, optional
          If a number is given, the confidence intervals for the given level are
          returned. For instance if alpha=.05, 95 % confidence intervals are
          returned where the standard deviation is computed according to
          1/sqrt(len(x)).
  
      Returns
      -------
      pacf : ndarray
          The partial autocorrelations for lags 0, 1, ..., nlags. Shape
          (nlags+1,).
      confint : ndarray, optional
          Confidence intervals for the PACF at lags 0, 1, ..., nlags. Shape
          (nlags + 1, 2). Returned if alpha is not None.
  
      See Also
      --------
      statsmodels.tsa.stattools.acf
          Estimate the autocorrelation function.
      statsmodels.tsa.stattools.pacf
          Partial autocorrelation estimation.
      statsmodels.tsa.stattools.pacf_yw
           Partial autocorrelation estimation using Yule-Walker.
      statsmodels.tsa.stattools.pacf_ols
          Partial autocorrelation estimation using OLS.
      statsmodels.tsa.stattools.pacf_burg
          Partial autocorrelation estimation using Burg"s method.
  
      Notes
      -----
      Based on simulation evidence across a range of low-order ARMA models,
      the best methods based on root MSE are Yule-Walker (MLW), Levinson-Durbin
      (MLE) and Burg, respectively. The estimators with the lowest bias included
      included these three in addition to OLS and OLS-adjusted.
  
      Yule-Walker (adjusted) and Levinson-Durbin (adjusted) performed
      consistently worse than the other options.
      """
      nlags = int_like(nlags, "nlags", optional=True)
      methods = (
          "ols",
          "ols-inefficient",
          "ols-adjusted",
          "yw",
          "ywa",
          "ld",
          "ywadjusted",
          "yw_adjusted",
          "ywm",
          "ywmle",
          "yw_mle",
          "lda",
          "ldadjusted",
          "ld_adjusted",
          "ldb",
          "ldbiased",
          "ld_biased",
      )
      x = array_like(x, "x", maxdim=2)
      method = string_like(method, "method", options=methods)
      alpha = float_like(alpha, "alpha", optional=True)
  
      nobs = x.shape[0]
      if nlags is None:
          nlags = min(int(10 * np.log10(nobs)), nobs // 2 - 1)
      if nlags >= x.shape[0] // 2:
          raise ValueError(
              "Can only compute partial correlations for lags up to 50% of the "
              f"sample size. The requested nlags {nlags} must be < "
              f"{x.shape[0] // 2}."
          )
  
      if method in ("ols", "ols-inefficient", "ols-adjusted"):
          efficient = "inefficient" not in method
          adjusted = "adjusted" in method
          ret = pacf_ols(x, nlags=nlags, efficient=efficient, adjusted=adjusted)
      elif method in ("yw", "ywa", "ywadjusted", "yw_adjusted"):
          ret = pacf_yw(x, nlags=nlags, method="adjusted")
      elif method in ("ywm", "ywmle", "yw_mle"):
          ret = pacf_yw(x, nlags=nlags, method="mle")
      elif method in ("ld", "lda", "ldadjusted", "ld_adjusted"):
          acv = acovf(x, adjusted=True, fft=False)
          ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
          ret = ld_[2]
      # inconsistent naming with ywmle
      else:  # method in ("ldb", "ldbiased", "ld_biased")
          acv = acovf(x, adjusted=False, fft=False)
          ld_ = levinson_durbin(acv, nlags=nlags, isacov=True)
          ret = ld_[2]
  
      if alpha is not None:
          varacf = 1.0 / len(x)  # for all lags >=1
          interval = stats.norm.ppf(1.0 - alpha / 2.0) * np.sqrt(varacf)
          confint = np.array(lzip(ret - interval, ret + interval))
          confint[0] = ret[0]  # fix confidence interval for lag 0 to varpacf=0
          return ret, confint
      else:
          return ret
  
  
  @deprecate_kwarg("unbiased", "adjusted")
  def ccovf(x, y, adjusted=True, demean=True, fft=True):
      """
      Calculate the crosscovariance between two series.
  
      Parameters
      ----------
      x, y : array_like
         The time series data to use in the calculation.
      adjusted : bool, optional
         If True, then denominators for crosscovariance is n-k, otherwise n.
      demean : bool, optional
          Flag indicating whether to demean x and y.
      fft : bool, default True
          If True, use FFT convolution.  This method should be preferred
          for long time series.
  
      Returns
      -------
      ndarray
          The estimated crosscovariance function.
      """
      x = array_like(x, "x")
      y = array_like(y, "y")
      adjusted = bool_like(adjusted, "adjusted")
      demean = bool_like(demean, "demean")
      fft = bool_like(fft, "fft", optional=False)
  
      n = len(x)
      if demean:
          xo = x - x.mean()
          yo = y - y.mean()
      else:
          xo = x
          yo = y
      if adjusted:
          d = np.arange(n, 0, -1)
      else:
          d = n
  
      method = "fft" if fft else "direct"
      return correlate(xo, yo, "full", method=method)[n - 1 :] / d
  
  
  @deprecate_kwarg("unbiased", "adjusted")
  def ccf(x, y, adjusted=True, fft=True):
      """
      The cross-correlation function.
  
      Parameters
      ----------
      x, y : array_like
          The time series data to use in the calculation.
      adjusted : bool
          If True, then denominators for cross-correlation is n-k, otherwise n.
      fft : bool, default True
          If True, use FFT convolution.  This method should be preferred
          for long time series.
  
      Returns
      -------
      ndarray
          The cross-correlation function of x and y.
  
      Notes
      -----
      If adjusted is true, the denominator for the autocovariance is adjusted.
      """
      x = array_like(x, "x")
      y = array_like(y, "y")
      adjusted = bool_like(adjusted, "adjusted")
      fft = bool_like(fft, "fft", optional=False)
  
      cvf = ccovf(x, y, adjusted=adjusted, demean=True, fft=fft)
      return cvf / (np.std(x) * np.std(y))
  
  
  # moved from sandbox.tsa.examples.try_ld_nitime, via nitime
  # TODO: check what to return, for testing and trying out returns everything
  def levinson_durbin(s, nlags=10, isacov=False):
      """
      Levinson-Durbin recursion for autoregressive processes.
  
      Parameters
      ----------
      s : array_like
          If isacov is False, then this is the time series. If iasacov is true
          then this is interpreted as autocovariance starting with lag 0.
      nlags : int, optional
          The largest lag to include in recursion or order of the autoregressive
          process.
      isacov : bool, optional
          Flag indicating whether the first argument, s, contains the
          autocovariances or the data series.
  
      Returns
      -------
      sigma_v : float
          The estimate of the error variance.
      arcoefs : ndarray
          The estimate of the autoregressive coefficients for a model including
          nlags.
      pacf : ndarray
          The partial autocorrelation function.
      sigma : ndarray
          The entire sigma array from intermediate result, last value is sigma_v.
      phi : ndarray
          The entire phi array from intermediate result, last column contains
          autoregressive coefficients for AR(nlags).
  
      Notes
      -----
      This function returns currently all results, but maybe we drop sigma and
      phi from the returns.
  
      If this function is called with the time series (isacov=False), then the
      sample autocovariance function is calculated with the default options
      (biased, no fft).
      """
      s = array_like(s, "s")
      nlags = int_like(nlags, "nlags")
      isacov = bool_like(isacov, "isacov")
  
      order = nlags
  
      if isacov:
          sxx_m = s
      else:
          sxx_m = acovf(s, fft=False)[: order + 1]  # not tested
  
      phi = np.zeros((order + 1, order + 1), "d")
      sig = np.zeros(order + 1)
      # initial points for the recursion
      phi[1, 1] = sxx_m[1] / sxx_m[0]
      sig[1] = sxx_m[0] - phi[1, 1] * sxx_m[1]
      for k in range(2, order + 1):
          phi[k, k] = (
              sxx_m[k] - np.dot(phi[1:k, k - 1], sxx_m[1:k][::-1])
          ) / sig[k - 1]
          for j in range(1, k):
              phi[j, k] = phi[j, k - 1] - phi[k, k] * phi[k - j, k - 1]
          sig[k] = sig[k - 1] * (1 - phi[k, k] ** 2)
  
      sigma_v = sig[-1]
      arcoefs = phi[1:, -1]
      pacf_ = np.diag(phi).copy()
      pacf_[0] = 1.0
      return sigma_v, arcoefs, pacf_, sig, phi  # return everything
  
  
  def levinson_durbin_pacf(pacf, nlags=None):
      """
      Levinson-Durbin algorithm that returns the acf and ar coefficients.
  
      Parameters
      ----------
      pacf : array_like
          Partial autocorrelation array for lags 0, 1, ... p.
      nlags : int, optional
          Number of lags in the AR model.  If omitted, returns coefficients from
          an AR(p) and the first p autocorrelations.
  
      Returns
      -------
      arcoefs : ndarray
          AR coefficients computed from the partial autocorrelations.
      acf : ndarray
          The acf computed from the partial autocorrelations. Array returned
          contains the autocorrelations corresponding to lags 0, 1, ..., p.
  
      References
      ----------
      .. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series
          and forecasting. Springer.
      """
      pacf = array_like(pacf, "pacf")
      nlags = int_like(nlags, "nlags", optional=True)
      pacf = np.squeeze(np.asarray(pacf))
  
      if pacf[0] != 1:
          raise ValueError(
              "The first entry of the pacf corresponds to lags 0 "
              "and so must be 1."
          )
      pacf = pacf[1:]
      n = pacf.shape[0]
      if nlags is not None:
          if nlags > n:
              raise ValueError(
                  "Must provide at least as many values from the "
                  "pacf as the number of lags."
              )
          pacf = pacf[:nlags]
          n = pacf.shape[0]
  
      acf = np.zeros(n + 1)
      acf[1] = pacf[0]
      nu = np.cumprod(1 - pacf ** 2)
      arcoefs = pacf.copy()
      for i in range(1, n):
          prev = arcoefs[: -(n - i)].copy()
          arcoefs[: -(n - i)] = prev - arcoefs[i] * prev[::-1]
          acf[i + 1] = arcoefs[i] * nu[i - 1] + prev.dot(acf[1 : -(n - i)][::-1])
      acf[0] = 1
      return arcoefs, acf
  
  
  def breakvar_heteroskedasticity_test(
      resid, subset_length=1 / 3, alternative="two-sided", use_f=True
  ):
      r"""
      Test for heteroskedasticity of residuals
  
      Tests whether the sum-of-squares in the first subset of the sample is
      significantly different than the sum-of-squares in the last subset
      of the sample. Analogous to a Goldfeld-Quandt test. The null hypothesis
      is of no heteroskedasticity.
  
      Parameters
      ----------
      resid : array_like
          Residuals of a time series model.
          The shape is 1d (nobs,) or 2d (nobs, nvars).
      subset_length : {int, float}
          Length of the subsets to test (h in Notes below).
          If a float in 0 < subset_length < 1, it is interpreted as fraction.
          Default is 1/3.
      alternative : str, 'increasing', 'decreasing' or 'two-sided'
          This specifies the alternative for the p-value calculation. Default
          is two-sided.
      use_f : bool, optional
          Whether or not to compare against the asymptotic distribution
          (chi-squared) or the approximate small-sample distribution (F).
          Default is True (i.e. default is to compare against an F
          distribution).
  
      Returns
      -------
      test_statistic : {float, ndarray}
          Test statistic(s) H(h).
      p_value : {float, ndarray}
          p-value(s) of test statistic(s).
  
      Notes
      -----
      The null hypothesis is of no heteroskedasticity. That means different
      things depending on which alternative is selected:
  
      - Increasing: Null hypothesis is that the variance is not increasing
          throughout the sample; that the sum-of-squares in the later
          subsample is *not* greater than the sum-of-squares in the earlier
          subsample.
      - Decreasing: Null hypothesis is that the variance is not decreasing
          throughout the sample; that the sum-of-squares in the earlier
          subsample is *not* greater than the sum-of-squares in the later
          subsample.
      - Two-sided: Null hypothesis is that the variance is not changing
          throughout the sample. Both that the sum-of-squares in the earlier
          subsample is not greater than the sum-of-squares in the later
          subsample *and* that the sum-of-squares in the later subsample is
          not greater than the sum-of-squares in the earlier subsample.
  
      For :math:`h = [T/3]`, the test statistic is:
  
      .. math::
  
          H(h) = \sum_{t=T-h+1}^T  \tilde v_t^2
          \Bigg / \sum_{t=1}^{h} \tilde v_t^2
  
      This statistic can be tested against an :math:`F(h,h)` distribution.
      Alternatively, :math:`h H(h)` is asymptotically distributed according
      to :math:`\chi_h^2`; this second test can be applied by passing
      `use_f=False` as an argument.
  
      See section 5.4 of [1]_ for the above formula and discussion, as well
      as additional details.
  
      References
      ----------
      .. [1] Harvey, Andrew C. 1990. *Forecasting, Structural Time Series*
              *Models and the Kalman Filter.* Cambridge University Press.
      """
      squared_resid = np.asarray(resid, dtype=float) ** 2
      if squared_resid.ndim == 1:
          squared_resid = squared_resid.reshape(-1, 1)
      nobs = len(resid)
  
      if 0 < subset_length < 1:
          h = int(np.round(nobs * subset_length))
      elif type(subset_length) is int and subset_length >= 1:
          h = subset_length
  
      numer_resid = squared_resid[-h:]
      numer_dof = (~np.isnan(numer_resid)).sum(axis=0)
      numer_squared_sum = np.nansum(numer_resid, axis=0)
      for i, dof in enumerate(numer_dof):
          if dof < 2:
              warnings.warn(
                  "Early subset of data for variable %d"
                  " has too few non-missing observations to"
                  " calculate test statistic." % i,
                  stacklevel=2,
              )
              numer_squared_sum[i] = np.nan
  
      denom_resid = squared_resid[:h]
      denom_dof = (~np.isnan(denom_resid)).sum(axis=0)
      denom_squared_sum = np.nansum(denom_resid, axis=0)
      for i, dof in enumerate(denom_dof):
          if dof < 2:
              warnings.warn(
                  "Later subset of data for variable %d"
                  " has too few non-missing observations to"
                  " calculate test statistic." % i,
                  stacklevel=2,
              )
              denom_squared_sum[i] = np.nan
  
      test_statistic = numer_squared_sum / denom_squared_sum
  
      # Setup functions to calculate the p-values
      if use_f:
          from scipy.stats import f
  
          pval_lower = lambda test_statistics: f.cdf(  # noqa:E731
              test_statistics, numer_dof, denom_dof
          )
          pval_upper = lambda test_statistics: f.sf(  # noqa:E731
              test_statistics, numer_dof, denom_dof
          )
      else:
          from scipy.stats import chi2
  
          pval_lower = lambda test_statistics: chi2.cdf(  # noqa:E731
              numer_dof * test_statistics, denom_dof
          )
          pval_upper = lambda test_statistics: chi2.sf(  # noqa:E731
              numer_dof * test_statistics, denom_dof
          )
  
      # Calculate the one- or two-sided p-values
      alternative = alternative.lower()
      if alternative in ["i", "inc", "increasing"]:
          p_value = pval_upper(test_statistic)
      elif alternative in ["d", "dec", "decreasing"]:
          test_statistic = 1.0 / test_statistic
          p_value = pval_upper(test_statistic)
      elif alternative in ["2", "2-sided", "two-sided"]:
          p_value = 2 * np.minimum(
              pval_lower(test_statistic), pval_upper(test_statistic)
          )
      else:
          raise ValueError("Invalid alternative.")
  
      if len(test_statistic) == 1:
          return test_statistic[0], p_value[0]
  
      return test_statistic, p_value
  
  
  def grangercausalitytests(x, maxlag, addconst=True, verbose=None):
      """
      Four tests for granger non causality of 2 time series.
  
      All four tests give similar results. `params_ftest` and `ssr_ftest` are
      equivalent based on F test which is identical to lmtest:grangertest in R.
  
      Parameters
      ----------
      x : array_like
          The data for testing whether the time series in the second column Granger
          causes the time series in the first column. Missing values are not
          supported.
      maxlag : {int, Iterable[int]}
          If an integer, computes the test for all lags up to maxlag. If an
          iterable, computes the tests only for the lags in maxlag.
      addconst : bool
          Include a constant in the model.
      verbose : bool
          Print results. Deprecated
  
          .. deprecated: 0.14
  
             verbose is deprecated and will be removed after 0.15 is released
  
  
  
      Returns
      -------
      dict
          All test results, dictionary keys are the number of lags. For each
          lag the values are a tuple, with the first element a dictionary with
          test statistic, pvalues, degrees of freedom, the second element are
          the OLS estimation results for the restricted model, the unrestricted
          model and the restriction (contrast) matrix for the parameter f_test.
  
      Notes
      -----
      TODO: convert to class and attach results properly
  
      The Null hypothesis for grangercausalitytests is that the time series in
      the second column, x2, does NOT Granger cause the time series in the first
      column, x1. Grange causality means that past values of x2 have a
      statistically significant effect on the current value of x1, taking past
      values of x1 into account as regressors. We reject the null hypothesis
      that x2 does not Granger cause x1 if the pvalues are below a desired size
      of the test.
  
      The null hypothesis for all four test is that the coefficients
      corresponding to past values of the second time series are zero.
  
      `params_ftest`, `ssr_ftest` are based on F distribution
  
      `ssr_chi2test`, `lrtest` are based on chi-square distribution
  
      References
      ----------
      .. [1] https://en.wikipedia.org/wiki/Granger_causality
  
      .. [2] Greene: Econometric Analysis
  
      Examples
      --------
      >>> import statsmodels.api as sm
      >>> from statsmodels.tsa.stattools import grangercausalitytests
      >>> import numpy as np
      >>> data = sm.datasets.macrodata.load_pandas()
      >>> data = data.data[["realgdp", "realcons"]].pct_change().dropna()
  
      All lags up to 4
      >>> gc_res = grangercausalitytests(data, 4)
  
      Only lag 4
      >>> gc_res = grangercausalitytests(data, [4])
      """
      x = array_like(x, "x", ndim=2)
      if not np.isfinite(x).all():
          raise ValueError("x contains NaN or inf values.")
      addconst = bool_like(addconst, "addconst")
      if verbose is not None:
          verbose = bool_like(verbose, "verbose")
          warnings.warn(
              "verbose is deprecated since functions should not print results",
              FutureWarning,
          )
      else:
          verbose = True  # old default
  
      try:
          maxlag = int_like(maxlag, "maxlag")
          if maxlag <= 0:
              raise ValueError("maxlag must be a positive integer")
          lags = np.arange(1, maxlag + 1)
      except TypeError:
          lags = np.array([int(lag) for lag in maxlag])
          maxlag = lags.max()
          if lags.min() <= 0 or lags.size == 0:
              raise ValueError(
                  "maxlag must be a non-empty list containing only "
                  "positive integers"
              )
  
      if x.shape[0] <= 3 * maxlag + int(addconst):
          raise ValueError(
              "Insufficient observations. Maximum allowable "
              "lag is {0}".format(int((x.shape[0] - int(addconst)) / 3) - 1)
          )
  
      resli = {}
  
      for mlg in lags:
          result = {}
          if verbose:
              print("\nGranger Causality")
              print("number of lags (no zero)", mlg)
          mxlg = mlg
  
          # create lagmat of both time series
          dta = lagmat2ds(x, mxlg, trim="both", dropex=1)
  
          # add constant
          if addconst:
              dtaown = add_constant(dta[:, 1 : (mxlg + 1)], prepend=False)
              dtajoint = add_constant(dta[:, 1:], prepend=False)
              if (
                  dtajoint.shape[1] == (dta.shape[1] - 1)
                  or (dtajoint.max(0) == dtajoint.min(0)).sum() != 1
              ):
                  raise InfeasibleTestError(
                      "The x values include a column with constant values and so"
                      " the test statistic cannot be computed."
                  )
          else:
              raise NotImplementedError("Not Implemented")
              # dtaown = dta[:, 1:mxlg]
              # dtajoint = dta[:, 1:]
  
          # Run ols on both models without and with lags of second variable
          res2down = OLS(dta[:, 0], dtaown).fit()
          res2djoint = OLS(dta[:, 0], dtajoint).fit()
  
          # print results
          # for ssr based tests see:
          # http://support.sas.com/rnd/app/examples/ets/granger/index.htm
          # the other tests are made-up
  
          # Granger Causality test using ssr (F statistic)
          if res2djoint.model.k_constant:
              tss = res2djoint.centered_tss
          else:
              tss = res2djoint.uncentered_tss
          if (
              tss == 0
              or res2djoint.ssr == 0
              or np.isnan(res2djoint.rsquared)
              or (res2djoint.ssr / tss) < np.finfo(float).eps
              or res2djoint.params.shape[0] != dtajoint.shape[1]
          ):
              raise InfeasibleTestError(
                  "The Granger causality test statistic cannot be compute "
                  "because the VAR has a perfect fit of the data."
              )
          fgc1 = (
              (res2down.ssr - res2djoint.ssr)
              / res2djoint.ssr
              / mxlg
              * res2djoint.df_resid
          )
          if verbose:
              print(
                  "ssr based F test:         F=%-8.4f, p=%-8.4f, df_denom=%d,"
                  " df_num=%d"
                  % (
                      fgc1,
                      stats.f.sf(fgc1, mxlg, res2djoint.df_resid),
                      res2djoint.df_resid,
                      mxlg,
                  )
              )
          result["ssr_ftest"] = (
              fgc1,
              stats.f.sf(fgc1, mxlg, res2djoint.df_resid),
              res2djoint.df_resid,
              mxlg,
          )
  
          # Granger Causality test using ssr (ch2 statistic)
          fgc2 = res2down.nobs * (res2down.ssr - res2djoint.ssr) / res2djoint.ssr
          if verbose:
              print(
                  "ssr based chi2 test:   chi2=%-8.4f, p=%-8.4f, "
                  "df=%d" % (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg)
              )
          result["ssr_chi2test"] = (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg)
  
          # likelihood ratio test pvalue:
          lr = -2 * (res2down.llf - res2djoint.llf)
          if verbose:
              print(
                  "likelihood ratio test: chi2=%-8.4f, p=%-8.4f, df=%d"
                  % (lr, stats.chi2.sf(lr, mxlg), mxlg)
              )
          result["lrtest"] = (lr, stats.chi2.sf(lr, mxlg), mxlg)
  
          # F test that all lag coefficients of exog are zero
          rconstr = np.column_stack(
              (np.zeros((mxlg, mxlg)), np.eye(mxlg, mxlg), np.zeros((mxlg, 1)))
          )
          ftres = res2djoint.f_test(rconstr)
          if verbose:
              print(
                  "parameter F test:         F=%-8.4f, p=%-8.4f, df_denom=%d,"
                  " df_num=%d"
                  % (ftres.fvalue, ftres.pvalue, ftres.df_denom, ftres.df_num)
              )
          result["params_ftest"] = (
              np.squeeze(ftres.fvalue)[()],
              np.squeeze(ftres.pvalue)[()],
              ftres.df_denom,
              ftres.df_num,
          )
  
          resli[mxlg] = (result, [res2down, res2djoint, rconstr])
  
      return resli
  
  
  def coint(
      y0,
      y1,
      trend="c",
      method="aeg",
      maxlag=None,
      autolag: str | None = "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.
  
      **Warning:** The autolag default has changed compared to statsmodels 0.8.
      In 0.8 autolag was always None, no the keyword is used and defaults to
      "aic". Use `autolag=None` to avoid the lag search.
  
      Parameters
      ----------
      y0 : array_like
          The first element in cointegrated system. Must be 1-d.
      y1 : array_like
          The remaining elements in cointegrated system.
      trend : str {"c", "ct"}
          The trend term included in regression for cointegrating equation.
  
          * "c" : constant.
          * "ct" : constant and linear trend.
          * also available quadratic trend "ctt", and no constant "n".
  
      method : {"aeg"}
          Only "aeg" (augmented Engle-Granger) is available.
      maxlag : None or int
          Argument for `adfuller`, largest or given number of lags.
      autolag : str
          Argument for `adfuller`, lag selection criterion.
  
          * If None, then maxlag lags are used without lag search.
          * 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.
      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
          The 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.
  
      If the two series are almost perfectly collinear, then computing the
      test is numerically unstable. However, the two series will be cointegrated
      under the maintained assumption that they are integrated. In this case
      the t-statistic will be set to -inf and the pvalue to zero.
  
      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
      ----------
      .. [1] MacKinnon, J.G. 1994  "Approximate Asymptotic Distribution Functions
         for Unit-Root and Cointegration Tests." Journal of Business & Economics
         Statistics, 12.2, 167-76.
      .. [2] 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
      """
      y0 = array_like(y0, "y0")
      y1 = array_like(y1, "y1", ndim=2)
      trend = string_like(trend, "trend", options=("c", "n", "ct", "ctt"))
      string_like(method, "method", options=("aeg",))
      maxlag = int_like(maxlag, "maxlag", optional=True)
      autolag = string_like(
          autolag, "autolag", optional=True, options=("aic", "bic", "t-stat")
      )
      return_results = bool_like(return_results, "return_results", optional=True)
  
      nobs, k_vars = y1.shape
      k_vars += 1  # add 1 for y0
  
      if trend == "n":
          xx = y1
      else:
          xx = add_trend(y1, trend=trend, prepend=False)
  
      res_co = OLS(y0, xx).fit()
  
      if res_co.rsquared < 1 - 100 * SQRTEPS:
          res_adf = adfuller(
              res_co.resid, maxlag=maxlag, autolag=autolag, regression="n"
          )
      else:
          warnings.warn(
              "y0 and y1 are (almost) perfectly colinear."
              "Cointegration test is not reliable in this case.",
              CollinearityWarning,
              stacklevel=2,
          )
          # Edge case where series are too similar
          res_adf = (-np.inf,)
  
      # no constant or trend, see egranger in Stata and MacKinnon
      if trend == "n":
          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 do not 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 _safe_arma_fit(y, order, model_kw, trend, fit_kw, start_params=None):
      from statsmodels.tsa.arima.model import ARIMA
  
      try:
          return ARIMA(y, order=order, **model_kw, trend=trend).fit(
              start_params=start_params, **fit_kw
          )
      except LinAlgError:
          # SVD convergence failure on badly misspecified models
          return
  
      except ValueError as error:
          if start_params is not None:  # do not recurse again
              # user supplied start_params only get one chance
              return
          # try a little harder, should be handled in fit really
          elif "initial" not in error.args[0] or "initial" in str(error):
              start_params = [0.1] * sum(order)
              if trend == "c":
                  start_params = [0.1] + start_params
              return _safe_arma_fit(
                  y, order, model_kw, trend, fit_kw, start_params
              )
          else:
              return
      except:  # no idea what happened
          return
  
  
  def arma_order_select_ic(
      y, max_ar=4, max_ma=2, ic="bic", trend="c", model_kw=None, fit_kw=None
  ):
      """
      Compute information criteria for many ARMA models.
  
      Parameters
      ----------
      y : array_like
          Array of time-series data.
      max_ar : int
          Maximum number of AR lags to use. Default 4.
      max_ma : int
          Maximum number of MA lags to use. Default 2.
      ic : str, list
          Information criteria to report. Either a single string or a list
          of different criteria is possible.
      trend : str
          The trend to use when fitting the ARMA models.
      model_kw : dict
          Keyword arguments to be passed to the ``ARMA`` model.
      fit_kw : dict
          Keyword arguments to be passed to ``ARMA.fit``.
  
      Returns
      -------
      Bunch
          Dict-like object with attribute access. Each ic is an attribute with a
          DataFrame for the results. The AR order used is the row index. The ma
          order used is the column index. The minimum orders are available as
          ``ic_min_order``.
  
      Notes
      -----
      This method can be used to tentatively identify the order of an ARMA
      process, provided that the time series is stationary and invertible. This
      function computes the full exact MLE estimate of each model and can be,
      therefore a little slow. An implementation using approximate estimates
      will be provided in the future. In the meantime, consider passing
      {method : "css"} to fit_kw.
  
      Examples
      --------
  
      >>> from statsmodels.tsa.arima_process import arma_generate_sample
      >>> import statsmodels.api as sm
      >>> import numpy as np
  
      >>> arparams = np.array([.75, -.25])
      >>> maparams = np.array([.65, .35])
      >>> arparams = np.r_[1, -arparams]
      >>> maparam = np.r_[1, maparams]
      >>> nobs = 250
      >>> np.random.seed(2014)
      >>> y = arma_generate_sample(arparams, maparams, nobs)
      >>> res = sm.tsa.arma_order_select_ic(y, ic=["aic", "bic"], trend="n")
      >>> res.aic_min_order
      >>> res.bic_min_order
      """
      max_ar = int_like(max_ar, "max_ar")
      max_ma = int_like(max_ma, "max_ma")
      trend = string_like(trend, "trend", options=("n", "c"))
      model_kw = dict_like(model_kw, "model_kw", optional=True)
      fit_kw = dict_like(fit_kw, "fit_kw", optional=True)
  
      ar_range = [i for i in range(max_ar + 1)]
      ma_range = [i for i in range(max_ma + 1)]
      if isinstance(ic, str):
          ic = [ic]
      elif not isinstance(ic, (list, tuple)):
          raise ValueError("Need a list or a tuple for ic if not a string.")
  
      results = np.zeros((len(ic), max_ar + 1, max_ma + 1))
      model_kw = {} if model_kw is None else model_kw
      fit_kw = {} if fit_kw is None else fit_kw
      y_arr = array_like(y, "y", contiguous=True)
      for ar in ar_range:
          for ma in ma_range:
              mod = _safe_arma_fit(y_arr, (ar, 0, ma), model_kw, trend, fit_kw)
              if mod is None:
                  results[:, ar, ma] = np.nan
                  continue
  
              for i, criteria in enumerate(ic):
                  results[i, ar, ma] = getattr(mod, criteria)
  
      dfs = [
          pd.DataFrame(res, columns=ma_range, index=ar_range) for res in results
      ]
  
      res = dict(zip(ic, dfs))
  
      # add the minimums to the results dict
      min_res = {}
      for i, result in res.items():
          mins = np.where(result.min().min() == result)
          min_res.update({i + "_min_order": (mins[0][0], mins[1][0])})
      res.update(min_res)
  
      return Bunch(**res)
  
  
  def has_missing(data):
      """
      Returns True if "data" contains missing entries, otherwise False
      """
      return np.isnan(np.sum(data))
  
  
  def kpss(
      x,
      regression: Literal["c", "ct"] = "c",
      nlags: Literal["auto", "legacy"] | int = "auto",
      store: bool = False,
  ) -> Tuple[float, float, int, dict[str, float]]:
      """
      Kwiatkowski-Phillips-Schmidt-Shin test for stationarity.
  
      Computes the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for the null
      hypothesis that x is level or trend stationary.
  
      Parameters
      ----------
      x : array_like, 1d
          The data series to test.
      regression : str{"c", "ct"}
          The null hypothesis for the KPSS test.
  
          * "c" : The data is stationary around a constant (default).
          * "ct" : The data is stationary around a trend.
      nlags : {str, int}, optional
          Indicates the number of lags to be used. If "auto" (default), lags
          is calculated using the data-dependent method of Hobijn et al. (1998).
          See also Andrews (1991), Newey & West (1994), and Schwert (1989). If
          set to "legacy",  uses int(12 * (n / 100)**(1 / 4)) , as outlined in
          Schwert (1989).
      store : bool
          If True, then a result instance is returned additionally to
          the KPSS statistic (default is False).
  
      Returns
      -------
      kpss_stat : float
          The KPSS test statistic.
      p_value : float
          The p-value of the test. The p-value is interpolated from
          Table 1 in Kwiatkowski et al. (1992), and a boundary point
          is returned if the test statistic is outside the table of
          critical values, that is, if the p-value is outside the
          interval (0.01, 0.1).
      lags : int
          The truncation lag parameter.
      crit : dict
          The critical values at 10%, 5%, 2.5% and 1%. Based on
          Kwiatkowski et al. (1992).
      resstore : (optional) instance of ResultStore
          An instance of a dummy class with results attached as attributes.
  
      Notes
      -----
      To estimate sigma^2 the Newey-West estimator is used. If lags is "legacy",
      the truncation lag parameter is set to int(12 * (n / 100) ** (1 / 4)),
      as outlined in Schwert (1989). The p-values are interpolated from
      Table 1 of Kwiatkowski et al. (1992). If the computed statistic is
      outside the table of critical values, then a warning message is
      generated.
  
      Missing values are not handled.
  
      References
      ----------
      .. [1] Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation
         consistent covariance matrix estimation. Econometrica, 59: 817-858.
  
      .. [2] Hobijn, B., Frances, B.H., & Ooms, M. (2004). Generalizations of the
         KPSS-test for stationarity. Statistica Neerlandica, 52: 483-502.
  
      .. [3] Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992).
         Testing the null hypothesis of stationarity against the alternative of a
         unit root. Journal of Econometrics, 54: 159-178.
  
      .. [4] Newey, W.K., & West, K.D. (1994). Automatic lag selection in
         covariance matrix estimation. Review of Economic Studies, 61: 631-653.
  
      .. [5] Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo
         investigation. Journal of Business and Economic Statistics, 7 (2):
         147-159.
      """
      x = array_like(x, "x")
      regression = string_like(regression, "regression", options=("c", "ct"))
      store = bool_like(store, "store")
  
      nobs = x.shape[0]
      hypo = regression
  
      # if m is not one, n != m * n
      if nobs != x.size:
          raise ValueError("x of shape {0} not understood".format(x.shape))
  
      if hypo == "ct":
          # p. 162 Kwiatkowski et al. (1992): y_t = beta * t + r_t + e_t,
          # where beta is the trend, r_t a random walk and e_t a stationary
          # error term.
          resids = OLS(x, add_constant(np.arange(1, nobs + 1))).fit().resid
          crit = [0.119, 0.146, 0.176, 0.216]
      else:  # hypo == "c"
          # special case of the model above, where beta = 0 (so the null
          # hypothesis is that the data is stationary around r_0).
          resids = x - x.mean()
          crit = [0.347, 0.463, 0.574, 0.739]
  
      if nlags == "legacy":
          nlags = int(np.ceil(12.0 * np.power(nobs / 100.0, 1 / 4.0)))
          nlags = min(nlags, nobs - 1)
      elif nlags == "auto" or nlags is None:
          if nlags is None:
              # TODO: Remove before 0.14 is released
              warnings.warn(
                  "None is not a valid value for nlags. It must be an integer, "
                  "'auto' or 'legacy'. None will raise starting in 0.14",
                  FutureWarning,
                  stacklevel=2,
              )
          # autolag method of Hobijn et al. (1998)
          nlags = _kpss_autolag(resids, nobs)
          nlags = min(nlags, nobs - 1)
      elif isinstance(nlags, str):
          raise ValueError("nvals must be 'auto' or 'legacy' when not an int")
      else:
          nlags = int_like(nlags, "nlags", optional=False)
  
          if nlags >= nobs:
              raise ValueError(
                  f"lags ({nlags}) must be < number of observations ({nobs})"
              )
  
      pvals = [0.10, 0.05, 0.025, 0.01]
  
      eta = np.sum(resids.cumsum() ** 2) / (nobs ** 2)  # eq. 11, p. 165
      s_hat = _sigma_est_kpss(resids, nobs, nlags)
  
      kpss_stat = eta / s_hat
      p_value = np.interp(kpss_stat, crit, pvals)
  
      warn_msg = """\
  The test statistic is outside of the range of p-values available in the
  look-up table. The actual p-value is {direction} than the p-value returned.
  """
      if p_value == pvals[-1]:
          warnings.warn(
              warn_msg.format(direction="smaller"),
              InterpolationWarning,
              stacklevel=2,
          )
      elif p_value == pvals[0]:
          warnings.warn(
              warn_msg.format(direction="greater"),
              InterpolationWarning,
              stacklevel=2,
          )
  
      crit_dict = {"10%": crit[0], "5%": crit[1], "2.5%": crit[2], "1%": crit[3]}
  
      if store:
          from statsmodels.stats.diagnostic import ResultsStore
  
          rstore = ResultsStore()
          rstore.lags = nlags
          rstore.nobs = nobs
  
          stationary_type = "level" if hypo == "c" else "trend"
          rstore.H0 = "The series is {0} stationary".format(stationary_type)
          rstore.HA = "The series is not {0} stationary".format(stationary_type)
  
          return kpss_stat, p_value, crit_dict, rstore
      else:
          return kpss_stat, p_value, nlags, crit_dict
  
  
  def _sigma_est_kpss(resids, nobs, lags):
      """
      Computes equation 10, p. 164 of Kwiatkowski et al. (1992). This is the
      consistent estimator for the variance.
      """
      s_hat = np.sum(resids ** 2)
      for i in range(1, lags + 1):
          resids_prod = np.dot(resids[i:], resids[: nobs - i])
          s_hat += 2 * resids_prod * (1.0 - (i / (lags + 1.0)))
      return s_hat / nobs
  
  
  def _kpss_autolag(resids, nobs):
      """
      Computes the number of lags for covariance matrix estimation in KPSS test
      using method of Hobijn et al (1998). See also Andrews (1991), Newey & West
      (1994), and Schwert (1989). Assumes Bartlett / Newey-West kernel.
      """
      covlags = int(np.power(nobs, 2.0 / 9.0))
      s0 = np.sum(resids ** 2) / nobs
      s1 = 0
      for i in range(1, covlags + 1):
          resids_prod = np.dot(resids[i:], resids[: nobs - i])
          resids_prod /= nobs / 2.0
          s0 += resids_prod
          s1 += i * resids_prod
      s_hat = s1 / s0
      pwr = 1.0 / 3.0
      gamma_hat = 1.1447 * np.power(s_hat * s_hat, pwr)
      autolags = int(gamma_hat * np.power(nobs, pwr))
      return autolags
  
  
  def range_unit_root_test(x, store=False):
      """
      Range unit-root test for stationarity.
  
      Computes the Range Unit-Root (RUR) test for the null
      hypothesis that x is stationary.
  
      Parameters
      ----------
      x : array_like, 1d
          The data series to test.
      store : bool
          If True, then a result instance is returned additionally to
          the RUR statistic (default is False).
  
      Returns
      -------
      rur_stat : float
          The RUR test statistic.
      p_value : float
          The p-value of the test. The p-value is interpolated from
          Table 1 in Aparicio et al. (2006), and a boundary point
          is returned if the test statistic is outside the table of
          critical values, that is, if the p-value is outside the
          interval (0.01, 0.1).
      crit : dict
          The critical values at 10%, 5%, 2.5% and 1%. Based on
          Aparicio et al. (2006).
      resstore : (optional) instance of ResultStore
          An instance of a dummy class with results attached as attributes.
  
      Notes
      -----
      The p-values are interpolated from
      Table 1 of Aparicio et al. (2006). If the computed statistic is
      outside the table of critical values, then a warning message is
      generated.
  
      Missing values are not handled.
  
      References
      ----------
      .. [1] Aparicio, F., Escribano A., Sipols, A.E. (2006). Range Unit-Root (RUR)
          tests: robust against nonlinearities, error distributions, structural breaks
          and outliers. Journal of Time Series Analysis, 27 (4): 545-576.
      """
      x = array_like(x, "x")
      store = bool_like(store, "store")
  
      nobs = x.shape[0]
  
      # if m is not one, n != m * n
      if nobs != x.size:
          raise ValueError("x of shape {0} not understood".format(x.shape))
  
      # Table from [1] has been replicated using 200,000 samples
      # Critical values for new n_obs values have been identified
      pvals = [0.01, 0.025, 0.05, 0.10, 0.90, 0.95]
      n = np.array(
          [25, 50, 100, 150, 200, 250, 500, 1000, 2000, 3000, 4000, 5000]
      )
      crit = np.array(
          [
              [0.6626, 0.8126, 0.9192, 1.0712, 2.4863, 2.7312],
              [0.7977, 0.9274, 1.0478, 1.1964, 2.6821, 2.9613],
              [0.9070, 1.0243, 1.1412, 1.2888, 2.8317, 3.1393],
              [0.9543, 1.0768, 1.1869, 1.3294, 2.8915, 3.2049],
              [0.9833, 1.0984, 1.2101, 1.3494, 2.9308, 3.2482],
              [0.9982, 1.1137, 1.2242, 1.3632, 2.9571, 3.2842],
              [1.0494, 1.1643, 1.2712, 1.4076, 3.0207, 3.3584],
              [1.0846, 1.1959, 1.2988, 1.4344, 3.0653, 3.4073],
              [1.1121, 1.2200, 1.3230, 1.4556, 3.0948, 3.4439],
              [1.1204, 1.2295, 1.3303, 1.4656, 3.1054, 3.4632],
              [1.1309, 1.2347, 1.3378, 1.4693, 3.1165, 3.4717],
              [1.1377, 1.2402, 1.3408, 1.4729, 3.1252, 3.4807],
          ]
      )
  
      # Interpolation for nobs
      inter_crit = np.zeros((1, crit.shape[1]))
      for i in range(crit.shape[1]):
          f = interp1d(n, crit[:, i])
          inter_crit[0, i] = f(nobs)
  
      # Calculate RUR stat
      xs = pd.Series(x)
      exp_max = xs.expanding(1).max().shift(1)
      exp_min = xs.expanding(1).min().shift(1)
      count = (xs > exp_max).sum() + (xs < exp_min).sum()
  
      rur_stat = count / np.sqrt(len(x))
  
      k = len(pvals) - 1
      for i in range(len(pvals) - 1, -1, -1):
          if rur_stat < inter_crit[0, i]:
              k = i
          else:
              break
  
      p_value = pvals[k]
  
      warn_msg = """\
  The test statistic is outside of the range of p-values available in the
  look-up table. The actual p-value is {direction} than the p-value returned.
  """
      direction = ""
      if p_value == pvals[-1]:
          direction = "smaller"
      elif p_value == pvals[0]:
          direction = "larger"
  
      if direction:
          warnings.warn(
              warn_msg.format(direction=direction),
              InterpolationWarning,
              stacklevel=2,
          )
  
      crit_dict = {
          "10%": inter_crit[0, 3],
          "5%": inter_crit[0, 2],
          "2.5%": inter_crit[0, 1],
          "1%": inter_crit[0, 0],
      }
  
      if store:
          from statsmodels.stats.diagnostic import ResultsStore
  
          rstore = ResultsStore()
          rstore.nobs = nobs
  
          rstore.H0 = "The series is not stationary"
          rstore.HA = "The series is stationary"
  
          return rur_stat, p_value, crit_dict, rstore
      else:
          return rur_stat, p_value, crit_dict
  
  
  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
              The data series to test.
          trim : float
              The percentage of series at begin/end to exclude from break-period
              calculation in range [0, 0.333] (default=0.15).
          maxlag : int
              The maximum lag which is included in test, default is
              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}
              The method to select the lag length when using automatic selection.
  
              * 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
              The test statistic.
          pvalue : float
              The pvalue based on MC-derived critical values.
          cvdict : dict
              The critical values for the test statistic at the 1%, 5%, and 10%
              levels.
          baselag : int
              The number of lags used for period regressions.
          bpidx : int
              The index of x corresponding to endogenously calculated break period
              with values in the range [0..nobs-1].
  
          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 trade-off.
  
          References
          ----------
          .. [1] 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.
  
          .. [2] Schwert, G.W. (1989). Tests for unit roots: A Monte Carlo
             investigation. Journal of Business & Economic Statistics, 7:
             147-159.
  
          .. [3] 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 > (1.0 / 3.0):
              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(12.0 * np.power(nobs / 100.0, 1 / 4.0))
          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__
  