code: def pacf_ols(
      x: ArrayLike1D,
      nlags: int | None = None,
      efficient: bool = True,
      adjusted: bool = False,
  ) -> np.ndarray:
      """
      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 = max(min(int(10 * np.log10(nobs)), nobs // 2), 1)
      if nlags > nobs // 2:
          raise ValueError(
              f"nlags must be smaller than nobs // 2 ({nobs//2})"
          )
      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] = np.squeeze(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] = np.squeeze(params[-1])
      if adjusted:
          pacf *= nobs / (nobs - np.arange(nlags + 1))
      return pacf
  
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 8 for calling a global function with example usage: pacf_ols(x, nlags, efficient, adjusted) and returns: pacf