code: def test_output_backtesting_forecaster_multiseries_ForecasterAutoregMultiVariate_refit_int_interval_yes_exog_not_allow_remainder_gap_with_mocked():
      """
      Test output of backtesting_forecaster_multiseries in ForecasterAutoregMultiVariate
      with refit int, interval, gap, with mocked using exog and intervals
      (mocked done in Skforecast v0.9.0).
      """
      series_with_index = series.copy()
      series_with_index.index = pd.date_range(
          start="2022-01-01", periods=50, freq="D"
      )
      exog_with_index = series["l1"].rename("exog_1").copy()
      exog_with_index.index = pd.date_range(
          start="2022-01-01", periods=50, freq="D"
      )
  
      forecaster = ForecasterAutoregMultiVariate(
          regressor=Ridge(random_state=123),
          level="l1",
          lags=2,
          steps=7,
          transformer_series=None,
      )
  
      refit = 3
      n_validation = 30
      steps = 4
      gap = 3
  
      (
          metrics_levels,
          backtest_predictions,
      ) = backtesting_forecaster_multiseries(
          forecaster=forecaster,
          series=series_with_index,
          steps=steps,
          levels=["l1"],
          metric="mean_absolute_error",
          initial_train_size=len(series_with_index) - n_validation,
          gap=gap,
          allow_incomplete_fold=False,
          refit=refit,
          fixed_train_size=False,
          exog=exog_with_index,
          interval=[5, 95],
          n_boot=100,
          random_state=123,
          in_sample_residuals=True,
          verbose=False,
      )
  
      expected_metric = pd.DataFrame(
          {
              "levels": ["l1"],
              "mean_absolute_error": [0.10066454067329249],
          }
      )
      expected_predictions = pd.DataFrame(
          data=np.array(
              [
                  [0.53203113, 0.27318916, 0.69931226],
                  [0.62129188, 0.46220068, 0.85870486],
                  [0.40145922, 0.17189084, 0.59797159],
                  [0.38821275, 0.17098498, 0.61456487],
                  [0.37636995, 0.11752798, 0.54365108],
                  [0.36599243, 0.20690123, 0.60340541],
                  [0.47654862, 0.24698023, 0.67306099],
                  [0.33725755, 0.12002978, 0.56360967],
                  [0.46122163, 0.20237966, 0.62850276],
                  [0.47541934, 0.31632814, 0.71283232],
                  [0.50601999, 0.27645161, 0.70253236],
                  [0.39293452, 0.17570675, 0.61928664],
                  [0.43793405, 0.27695885, 0.5990429],
                  [0.52600886, 0.38167775, 0.72224797],
                  [0.70882239, 0.55319058, 0.83377003],
                  [0.70063755, 0.51962777, 0.85415566],
                  [0.48631217, 0.32533698, 0.64742102],
                  [0.52248417, 0.37815306, 0.71872328],
                  [0.30010035, 0.14446853, 0.42504798],
                  [0.43342987, 0.25242008, 0.58694797],
                  [0.42479569, 0.2638205, 0.58590454],
                  [0.68566159, 0.54133048, 0.8819007],
                  [0.34632377, 0.19069195, 0.4712714],
                  [0.44695116, 0.26594138, 0.60046927],
              ]
          ),
          columns=["l1", "lower_bound", "upper_bound"],
          index=pd.date_range(start="2022-01-24", periods=24, freq="D"),
      )
      backtest_predictions = backtest_predictions.asfreq("D")
  
      pd.testing.assert_frame_equal(expected_metric, metrics_levels)
      pd.testing.assert_frame_equal(
          expected_predictions, backtest_predictions
      )
  
imports: import re
  import pytest
  import numpy
  import pandas
  from sklearn.linear_model import Ridge
  from sklearn.metrics import mean_absolute_error
  from skforecast.exceptions import IgnoredArgumentWarning
  from skforecast.ForecasterAutoreg import ForecasterAutoreg
  from skforecast.ForecasterAutoregMultiSeries import ForecasterAutoregMultiSeries
  from skforecast.ForecasterAutoregMultiSeriesCustom import ForecasterAutoregMultiSeriesCustom
  from skforecast.ForecasterAutoregMultiVariate import ForecasterAutoregMultiVariate
  from skforecast.model_selection_multiseries import backtesting_forecaster_multiseries
  from skforecast.model_selection_multiseries import backtesting_forecaster_multivariate
  from .fixtures_model_selection_multiseries import series
label_desc: Use image node_id 31 for calling a global function with example usage: test_output_backtesting_forecaster_multiseries_ForecasterAutoregMultiVariate_refit_int_interval_yes_exog_not_allow_remainder_gap_with_mocked() without return types