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 = 0.0646438286283131
      expected_predictions = pd.DataFrame(
          {
              "pred": np.array(
                  [
                      0.55717779,
                      0.43355138,
                      0.54969767,
                      0.52945466,
                      0.39585199,
                      0.55935949,
                      0.45263533,
                      0.4578669,
                      0.36988237,
                      0.57912951,
                      0.48686057,
                      0.45709952,
                  ]
              ),
              "lower_bound": np.array(
                  [
                      0.19882822,
                      0.08272406,
                      0.18106389,
                      0.18777395,
                      0.03750242,
                      0.20853217,
                      0.08400155,
                      0.11618618,
                      0.01153279,
                      0.22830219,
                      0.11822679,
                      0.1154188,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      0.95368172,
                      0.81704742,
                      0.93685716,
                      0.9407976,
                      0.79235592,
                      0.94285553,
                      0.83979482,
                      0.86920984,
                      0.76638629,
                      0.96262555,
                      0.87402007,
                      0.86844245,
                  ]
              ),
          },
          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=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.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_no_remainder_with_mocked() without return types