code: def test_output_backtesting_forecaster_interval_out_sample_residuals_no_exog_no_remainder_with_mocked():
      """
      Test output of _backtesting_forecaster with backtesting mocked, interval yes.
      Regressor is LinearRegression with lags=3, Series y is mocked, no exog,
      12 observations to backtest, steps=4 (no remainder), metric='mean_squared_error',
      'in_sample_residuals = False'
      """
      expected_metric = 0.06598802629306816
      expected_predictions = pd.DataFrame(
          {
              "pred": np.array(
                  [
                      0.55717779,
                      0.43355138,
                      0.54969767,
                      0.52945466,
                      0.38969292,
                      0.52778339,
                      0.49152015,
                      0.4841678,
                      0.4076433,
                      0.50904672,
                      0.50249462,
                      0.49232817,
                  ]
              ),
              "lower_bound": np.array(
                  [
                      0.63654358,
                      0.62989756,
                      0.67156167,
                      0.70363265,
                      0.46905871,
                      0.73507276,
                      0.64554988,
                      0.64183541,
                      0.48700909,
                      0.68845988,
                      0.63865297,
                      0.60684242,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      1.54070487,
                      1.5131313,
                      1.56749058,
                      1.62564968,
                      1.37322,
                      1.61930035,
                      1.54870568,
                      1.55335041,
                      1.39117037,
                      1.56935123,
                      1.51973211,
                      1.50300901,
                  ]
              ),
          },
          index=pd.RangeIndex(start=38, stop=50, step=1),
      )
  
      forecaster = ForecasterAutoreg(
          regressor=LinearRegression(), lags=3
      )
      forecaster.set_out_sample_residuals(
          residuals=out_sample_residuals, append=False
      )
  
      n_backtest = 12
      y_train = y[:-n_backtest]
  
      metric, backtest_predictions = _backtesting_forecaster(
          forecaster=forecaster,
          y=y,
          exog=None,
          refit=True,
          initial_train_size=len(y_train),
          fixed_train_size=False,
          steps=4,
          metric="mean_squared_error",
          interval=[5, 95],
          n_boot=500,
          random_state=123,
          in_sample_residuals=False,
          verbose=False,
      )
  
      assert expected_metric == approx(metric)
      pd.testing.assert_frame_equal(
          expected_predictions, backtest_predictions
      )
  
imports: import pytest
  import numpy
  import pandas
  from pytest import approx
  from sklearn.linear_model import LinearRegression
  from sklearn.linear_model import Ridge
  from sklearn.metrics import mean_squared_error
  from skforecast.ForecasterAutoreg import ForecasterAutoreg
  from skforecast.ForecasterAutoregCustom import ForecasterAutoregCustom
  from skforecast.ForecasterAutoregDirect import ForecasterAutoregDirect
  from skforecast.model_selection.model_selection import _backtesting_forecaster
  from .fixtures_model_selection import y
  from .fixtures_model_selection import exog
  from .fixtures_model_selection import out_sample_residuals
label_desc: Use image node_id 12 for calling a global function with example usage: test_output_backtesting_forecaster_interval_out_sample_residuals_no_exog_no_remainder_with_mocked() without return types