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 = 0.07085869503962372
      expected_predictions = pd.DataFrame(
          {
              "pred": np.array(
                  [
                      0.55717779,
                      0.43355138,
                      0.54969767,
                      0.52945466,
                      0.48308861,
                      0.54252612,
                      0.46139434,
                      0.44730047,
                      0.50031862,
                      0.49831103,
                      0.55613172,
                      0.42970914,
                  ]
              ),
              "lower_bound": np.array(
                  [
                      0.19882822,
                      0.08272406,
                      0.18106389,
                      0.18777395,
                      0.1238825,
                      0.18417655,
                      0.11056702,
                      0.07866669,
                      0.1586379,
                      0.13910492,
                      0.19778215,
                      0.07888182,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      0.95368172,
                      0.81704742,
                      0.93685716,
                      0.9407976,
                      0.85396419,
                      0.93903005,
                      0.84489038,
                      0.83445996,
                      0.91166155,
                      0.86918661,
                      0.95263565,
                      0.81320518,
                  ]
              ),
          },
          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=False,
          initial_train_size=len(y_train),
          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.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 10 for calling a global function with example usage: test_output_backtesting_forecaster_interval_no_exog_yes_remainder_with_mocked() without return types