code: def test_append_multistep(data):
      # Test that `MLEResults.append` works when called repeatedly
      endog = [1.0, 0.5, 1.5, 0.9, 0.2, 0.34]
      if data == "numpy":
          endog = np.array(endog)
      elif data == "range":
          endog = pd.Series(endog)
      elif data == "date":
          index = pd.date_range(
              start="2000-01-01", periods=6, freq="MS"
          )
          endog = pd.Series(endog, index=index)
      elif data == "period":
          index = pd.period_range(start="2000-01", periods=6, freq="M")
          endog = pd.Series(endog, index=index)
  
      # Base model fitting
      mod = sarimax.SARIMAX(endog[:2], order=(1, 0, 0))
      res = mod.smooth([0.5, 1.0])
      assert_allclose(res.model.endog[:, 0], [1.0, 0.5])
      assert_allclose(res.forecast(1), 0.25)
  
      # First append
      res1 = res.append(endog[2:3])
      assert_allclose(res1.model.endog[:, 0], [1.0, 0.5, 1.5])
      assert_allclose(res1.forecast(1), 0.75)
  
      # Second append
      res2 = res1.append(endog[3:5])
      assert_allclose(res2.model.endog[:, 0], [1.0, 0.5, 1.5, 0.9, 0.2])
      assert_allclose(res2.forecast(1), 0.1)
  
      # Third append
      res3 = res2.append(endog[5:6])
      print(res3.model.endog)
      assert_allclose(
          res3.model.endog[:, 0], [1.0, 0.5, 1.5, 0.9, 0.2, 0.34]
      )
      assert_allclose(res3.forecast(1), 0.17)
  
imports: import numpy
  import pandas
  import pytest
  from numpy.testing import assert_allclose
  from statsmodels.tsa.statespace import sarimax
label_desc: Use image node_id 1 for calling a global function with example usage: test_append_multistep(data) without return types