code: def test_output_backtesting_forecaster_interval_yes_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, exog is mocked,
      12 observations to backtest, steps=5 (2 remainder), metric='mean_squared_error',
      'in_sample_residuals = True'
      """
      expected_metric = 0.061723961096013524
      expected_predictions = pd.DataFrame(
          {
              "pred": np.array(
                  [
                      0.59059622,
                      0.47257504,
                      0.53024098,
                      0.46163343,
                      0.50035119,
                      0.43595809,
                      0.4349167,
                      0.42381237,
                      0.55165332,
                      0.53442833,
                      0.65361802,
                      0.51297419,
                  ]
              ),
              "lower_bound": np.array(
                  [
                      0.24619375,
                      0.10545295,
                      0.13120713,
                      0.08044217,
                      0.13725077,
                      0.08041239,
                      0.05015513,
                      0.07677812,
                      0.17434611,
                      0.16051962,
                      0.29167326,
                      0.15775686,
                  ]
              ),
              "upper_bound": np.array(
                  [
                      0.95777604,
                      0.88685543,
                      0.90755063,
                      0.87811336,
                      0.86891022,
                      0.74808834,
                      0.80296989,
                      0.77919033,
                      0.97680126,
                      0.8877086,
                      1.07608747,
                      0.90555785,
                  ]
              ),
          },
          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=exog,
          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 11 for calling a global function with example usage: test_output_backtesting_forecaster_interval_yes_exog_yes_remainder_with_mocked() without return types