BUG: SARIMAX Extend with constant exog doesn't work

[UgurKap]

Describe the bug

Similar to this issue: #7019

It seems the above bug is solved, but when you try to extend a SARIMAX model with constant exog values, it throws an error

Code Sample, a copy-pastable example if possible

from statsmodels.tsa.arima.model import ARIMA
import numpy as np

# A normal training demo

# Old data
old_data = np.random.rand(12*10, 1)
old_exo = np.random.rand(10, 1)
old_exo = np.repeat(old_exo, 12).reshape(-1, 1)

# Model fit with the available data
m = ARIMA(old_data, exog=old_exo)
res = m.fit()

# Some new data arrives
new_data = np.random.rand(12, 1)
new_exo = np.repeat(np.random.rand(1, 1), 12).reshape(-1, 1)

# The following line should not fail, but it does
res.extend(new_data, exog=new_exo)

We get ValueError: A constant trend was included in the model specification, but the exog data already contains a column of constants.

Expected Output

It should just extend the model without checking if exog is constant or not, as exog changes every "h" periods. When I try to extend the model after doing "h" periods where exog doesn't change, we get the error above.

Output of import statsmodels.api as sm; sm.show_versions()

INSTALLED VERSIONS

Python: 3.10.11.final.0
OS: Linux 5.14.0-70.30.1.el9_0.x86_64 #1 SMP PREEMPT Thu Nov 3 20:29:04 UTC 2022 x86_64
byteorder: little
LC_ALL: None
LANG: en_US.UTF-8

statsmodels

Installed: 0.14.0 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/statsmodels)

Required Dependencies

cython: Not installed
numpy: 1.23.5 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/numpy)
scipy: 1.11.1 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/scipy)
pandas: 2.0.3 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/pandas)
dateutil: 2.8.2 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/dateutil)
patsy: 0.5.3 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/patsy)

Optional Dependencies

matplotlib: 3.7.1 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/matplotlib)
backend: module://matplotlib_inline.backend_inline
cvxopt: Not installed
joblib: 1.3.1 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/joblib)

Developer Tools

IPython: 8.14.0 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/IPython)
jinja2: 3.1.2 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/jinja2)
sphinx: Not installed
pygments: 2.15.1 (/mnt/lustre/ludwig/lqb955/conda_envs/air_forecast/lib/python3.10/site-packages/pygments)
pytest: Not installed
virtualenv: Not installed

[ChadFulton]

Thanks very much for posting this, definitely a bug with ARIMA.extend.