code: 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
  
imports: 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 Union
  import warnings
  import numpy
  from numpy.linalg import LinAlgError
  import pandas
  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, ValueWarning
  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
label_desc: Use image node_id 3 for calling a global function with example usage: acovf(x, adjusted, demean, fft, missing, nlag) and returns: acov, acov, acov