code: def test_ar1_forecast(self):
      am = arch_model(self.ar1, mean="AR", vol="Constant", lags=[1])
      res = am.fit()
  
      fcast = res.forecast(horizon=5, start=0)
      params = np.asarray(res.params)
      direct = self.ar1.values
  
      for i in range(5):
          direct = params[0] + params[1] * direct
          assert_allclose(direct, fcast.mean.iloc[:, i])
          scale = np.sum((params[1] ** np.arange(i + 1)) ** 2.0)
          var = fcast.variance.iloc[1:, i]
          assert_allclose(var, scale * params[2] * np.ones_like(var))
  
      assert np.all(
          np.asarray(fcast.residual_variance[1:] == params[2])
      )
  
      fcast = res.forecast(horizon=5)
      params = np.asarray(res.params)
      assert np.all(np.asarray(np.isnan(fcast.mean[:-1])))
      assert np.all(np.asarray(np.isnan(fcast.variance[:-1])))
      assert np.all(np.asarray(np.isnan(fcast.residual_variance[:-1])))
      assert np.all(
          np.asarray(fcast.residual_variance.iloc[-1] == params[-1])
      )
      means = np.zeros(5)
      means[0] = params[0] + params[1] * self.ar1.iloc[-1]
      for i in range(1, 5):
          means[i] = params[0] + params[1] * means[i - 1]
      assert_allclose(means, fcast.mean.iloc[-1].values)
  
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 6 for calling the TestForecasting obj's underlying member method code with example usage: obj.test_ar1_forecast() without return types