[ENH] Add exogenous variable support to ARIMA

[satwiksps]

[ENH] Add exogenous variable support to ARIMA

Reference Issues

• Implements [ENH] ARIMA support for exogenous variables #3077.

What does this implement ? Explain your changes.

This PR adds full support for exogenous variables (exog) to ARIMA and AutoARIMA.

Summary of enhancements:

• Added "capability:exogenous": True for both ARIMA and AutoARIMA.
• Modified _fit, _predict, and iterative_forecast in ARIMA to:
  • accept exogenous variables,
  • perform an OLS regression of y on exog,
  • run ARIMA on the residual series,
  • include regression contribution back into predictions,
  • support multi-step forecasting with future exog.
• Updated AutoARIMA so that if exog is provided, it is correctly forwarded to the wrapped ARIMA model.

Tests added

• test_arima_with_exog_basic_fit_predict
  Verifies that ARIMA fits and predicts correctly when exogenous variables are supplied.
• test_arima_exog_shape_mismatch_raises
  Ensures shape mismatch errors are raised for incorrect exog dimensions in fit and predict.
• test_arima_iterative_forecast_with_exog
  Tests multi-step forecasting with future exogenous values.
• test_arima_no_exog_backward_compatibility
  Ensures ARIMA without exog behaves exactly as before (backwards compatibility).

All existing ARIMA tests continue to pass, and the new tests confirm correctness and robustness of the exogenous variable support.

Does your contribution introduce a new dependency? If yes, which one?

No new dependencies.

Any other comments?

• All changes were validated through both the existing ARIMA test suite and dedicated exog tests.
• Closes - [ENH] ARIMA support for exogenous variables #3077

PR checklist

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• I've added myself to the list of contributors.
• The PR title starts with [ENH].

For new estimators and functions

• Not applicable

For developers with write access

• Not applicable.

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I have added the following labels to this PR based on the title: [ enhancement ].
I have added the following labels to this PR based on the changes made: [ forecasting ]. Feel free to change these if they do not properly represent the PR.

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