code: def test_ar2_forecast(self):
      am = arch_model(self.ar2, mean="AR", vol="Constant", lags=[1, 2])
      res = am.fit()
  
      fcast = res.forecast(horizon=5)
      params = np.asarray(res.params)
      expected = np.zeros(7)
      expected[:2] = self.ar2.iloc[-2:]
      for i in range(2, 7):
          expected[i] = (
              params[0]
              + params[1] * expected[i - 1]
              + params[2] * expected[i - 2]
          )
  
      expected = expected[2:]
      assert np.all(np.asarray(np.isnan(fcast.mean.iloc[:-1])))
      assert_allclose(fcast.mean.iloc[-1], expected)
  
      expected = np.zeros(5)
      comp = np.array([res.params.iloc[1:3], [1, 0]])
      a = np.eye(2)
      for i in range(5):
          expected[i] = a[0, 0]
          a = a.dot(comp)
      expected = res.params.iloc[-1] * np.cumsum(expected**2)
      assert_allclose(fcast.variance.iloc[-1], expected)
  
      expected = np.empty((1000, 5))
      expected[:2] = np.nan
      expected[2:] = res.params.iloc[-1]
  
      fcast = res.forecast(horizon=5, start=1)
      expected = np.zeros((999, 7))
      expected[:, 0] = self.ar2.iloc[0:-1]
      expected[:, 1] = self.ar2.iloc[1:]
      for i in range(2, 7):
          expected[:, i] = (
              params[0]
              + params[1] * expected[:, i - 1]
              + params[2] * expected[:, i - 2]
          )
      fill = np.empty((1, 5))
      fill.fill(np.nan)
      expected = np.concatenate((fill, expected[:, 2:]))
      assert_allclose(np.asarray(fcast.mean), expected[1:])
  
      expected = np.empty((1000, 5))
      expected[:2] = np.nan
      expected[2:] = res.params.iloc[-1]
      assert_allclose(np.asarray(fcast.residual_variance), expected[1:])
  
      with pytest.raises(ValueError):
          res.forecast(horizon=5, start=0)
  
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 8 for calling the TestForecasting obj's underlying member method code with example usage: obj.test_ar2_forecast() without return types