code: def test_first_last_obs(self):
      y = self.ar2_garch
      mod = arch_model(y)
      res = mod.fit(
          disp="off", first_obs=y.index[100], last_obs=y.index[900]
      )
      res_2 = mod.fit(disp="off", first_obs=100, last_obs=900)
      assert_allclose(res.params, res_2.params)
      mod = arch_model(y.iloc[100:900])
      res_3 = mod.fit(disp="off")
      assert_allclose(res.params, res_3.params)
  
      mod = arch_model(y)
      res_4 = mod.fit(disp="off", first_obs=100, last_obs=y.index[900])
      assert_allclose(res.params, res_4.params)
  
imports: import datetime
  from itertools import product
  import numpy
  from numpy.random import RandomState
  from numpy.testing import assert_allclose
  import pandas
  from pandas.testing import assert_frame_equal
  import pytest
  from arch.data import sp500
  from arch.tests.univariate.test_variance_forecasting import preserved_state
  from arch.univariate import APARCH, ARX, EGARCH, FIGARCH, GARCH, HARCH, HARX, ConstantMean, ConstantVariance, EWMAVariance, MIDASHyperbolic, RiskMetrics2006, ZeroMean, arch_model
  from arch.univariate.mean import _ar_forecast, _ar_to_impulse
label_desc: Use image node_id 18 for calling the TestForecasting obj's underlying member method code with example usage: obj.test_first_last_obs() without return types