code: def test_output_backtesting_forecaster_interval_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 = True'
      """
      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(
                  [
                      0.19882822,
                      0.08272406,
                      0.18106389,
                      0.18777395,
                      0.03520425,
                      0.12926384,
                      0.0495347,
                      0.04527341,
                      0.0113795,
                      0.13676538,
                      0.12478441,
                      0.06814153,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      0.95368172,
                      0.81704742,
                      0.93685716,
                      0.9407976,
                      0.78486946,
                      0.93084605,
                      0.84533191,
                      0.90255909,
                      0.80099612,
                      0.88747244,
                      0.88292664,
                      0.88718366,
                  ]
              ),
          },
          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=4,
          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 8 for calling a global function with example usage: test_output_backtesting_forecaster_interval_no_exog_no_remainder_with_mocked() without return types