code: def test_model_forecast_recursive():
      vol = GARCH()
      base = ConstantMean(SP500.iloc[:-10], volatility=vol).fit(
          disp=False
      )
      params = base.params
      fcasts = {}
      for i in range(11):
          end = SP500.shape[0] - 10 + i
          mod = ConstantMean(SP500.iloc[:end], volatility=vol)
          fcasts[i] = mod.forecast(params, reindex=True)
  
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 12 for calling a global function with example usage: test_model_forecast_recursive() without return types