code: def test_basic():
      endog = lake.copy()
      exog = np.arange(1, len(endog) + 1) * 1.0
  
      # Test default options (include_constant=True, concentrate_scale=False)
      p, res = statespace(
          endog,
          exog=exog,
          order=(1, 0, 0),
          include_constant=True,
          concentrate_scale=False,
      )
  
      mod_ss = sarimax.SARIMAX(
          endog, exog=add_constant(exog), order=(1, 0, 0)
      )
      res_ss = mod_ss.filter(p.params)
  
      assert_allclose(res.statespace_results.llf, res_ss.llf)
  
      # Test include_constant=False
      p, res = statespace(
          endog,
          exog=exog,
          order=(1, 0, 0),
          include_constant=False,
          concentrate_scale=False,
      )
  
      mod_ss = sarimax.SARIMAX(endog, exog=exog, order=(1, 0, 0))
      res_ss = mod_ss.filter(p.params)
  
      assert_allclose(res.statespace_results.llf, res_ss.llf)
  
      # Test concentrate_scale=True
      p, res = statespace(
          endog,
          exog=exog,
          order=(1, 0, 0),
          include_constant=True,
          concentrate_scale=True,
      )
  
      mod_ss = sarimax.SARIMAX(
          endog,
          exog=add_constant(exog),
          order=(1, 0, 0),
          concentrate_scale=True,
      )
      res_ss = mod_ss.filter(p.params)
  
      assert_allclose(res.statespace_results.llf, res_ss.llf)
  
file_path: statsmodels/statsmodels/tsa/arima/estimators/tests/test_statespace.py
imports: import numpy
  from numpy.testing import assert_allclose, assert_raises
  from statsmodels.tools.tools import add_constant
  from statsmodels.tsa.statespace import sarimax
  from statsmodels.tsa.arima.datasets.brockwell_davis_2002 import lake
  from statsmodels.tsa.arima.estimators.statespace import statespace
label_desc: Use image node_id 1 for calling a global function with example usage: test_basic() without return types