code: def test_output_backtesting_forecaster_interval_no_exog_yes_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=5 (2 remainder), metric='mean_squared_error',
      'in_sample_residuals = True'
      """
      expected_metric = 0.06916732087926723
      expected_predictions = pd.DataFrame(
          {
              "pred": np.array(
                  [
                      0.55717779,
                      0.43355138,
                      0.54969767,
                      0.52945466,
                      0.48308861,
                      0.5096801,
                      0.49519677,
                      0.47997916,
                      0.49177914,
                      0.495797,
                      0.57738724,
                      0.44370472,
                  ]
              ),
              "lower_bound": np.array(
                  [
                      0.19882822,
                      0.08272406,
                      0.18106389,
                      0.18777395,
                      0.1238825,
                      0.06681772,
                      0.09795868,
                      0.08383945,
                      0.10160946,
                      0.08917676,
                      0.23321023,
                      0.08685352,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      0.95368172,
                      0.81704742,
                      0.93685716,
                      0.9407976,
                      0.85396419,
                      0.86172991,
                      0.88313129,
                      0.82354636,
                      0.93875053,
                      0.86176335,
                      0.96037185,
                      0.84205069,
                  ]
              ),
          },
          index=pd.RangeIndex(start=38, stop=50, step=1),
      )
  
      forecaster = ForecasterAutoreg(
          regressor=LinearRegression(), lags=3
      )
  
      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=5,
          metric="mean_squared_error",
          interval=[5, 95],
          n_boot=500,
          random_state=123,
          in_sample_residuals=True,
          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 9 for calling a global function with example usage: test_output_backtesting_forecaster_interval_no_exog_yes_remainder_with_mocked() without return types