What is the 'u' parameter estimated by Statsmodel's ARIMA fit method?

Question

I'm struggling to find mention of the 'u' parameter that is returned by the statsmodels ARIMA.fit method in the following ARMAX model parameter estimation example:

import pandas as pd
from statsmodels.tsa.arima.model import ARIMA

# Sample of the full input-output dataset
id_data = pd.DataFrame({
    'u': [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 
          1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
    'y': [-1.4369, -0.999, -0.0325, 0.8435, 0.4339, -0.2925, 
          -0.8885, -2.3191, -4.004, -5.4779, -7.053, -7.5489, 
          -8.779, -8.9262, -8.5207, -8.3915, -8.5699, -8.2192, 
          -8.284, -7.6011]
})

arma22 = ARIMA(id_data.y, exog=id_data.u, order=(2, 0, 2), trend='n').fit()
print(arma22.params)
print(arma22.summary())

Output:

u        -0.564553
ar.L1     1.798081
ar.L2    -0.829465
ma.L1    -0.859744
ma.L2     0.998002
sigma2    0.220407
dtype: float64
                               SARIMAX Results                                
==============================================================================
Dep. Variable:                      y   No. Observations:                   20
Model:                 ARIMA(2, 0, 2)   Log Likelihood                 -18.559
Date:                Sun, 25 Feb 2024   AIC                             49.119
Time:                        14:26:11   BIC                             55.093
Sample:                             0   HQIC                            50.285
                                 - 20                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
u             -0.5646      0.303     -1.861      0.063      -1.159       0.030
ar.L1          1.7981      0.158     11.407      0.000       1.489       2.107
ar.L2         -0.8295      0.164     -5.059      0.000      -1.151      -0.508
ma.L1         -0.8597     21.365     -0.040      0.968     -42.735      41.016
ma.L2          0.9980     49.501      0.020      0.984     -96.022      98.018
sigma2         0.2204     10.881      0.020      0.984     -21.106      21.547
===================================================================================
Ljung-Box (L1) (Q):                   1.05   Jarque-Bera (JB):                 0.81
Prob(Q):                              0.31   Prob(JB):                         0.67
Heteroskedasticity (H):               0.78   Skew:                             0.27
Prob(H) (two-sided):                  0.76   Kurtosis:                         2.18
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

I was expecting an ARMAX(2, 2) model to have four parameters (four degrees of freedom). I've turned off the trend (const) parameter so it can't be that.

There is a worked-example of an ARMAX(1, 1) model here but no u parameter appears in the results.

If someone can point me to part of the documentation which mentions what 'u' is I'd be grateful.

Answer 1

I think I figured out the answer to my own question.

If you pass raw data instead of a Pandas series the 'u' parameter turns to 'x1' as in the examples in the documentation.

arma22 = ARIMA(id_data.y.values, exog=id_data.u.values, order=(2, 0, 2), trend='n').fit()
print(arma22.params)
print(arma22.summary())

Output:

[-0.56455335  1.79808108 -0.82946546 -0.85974377  0.99800247  0.22040651]
                               SARIMAX Results                                
==============================================================================
Dep. Variable:                      y   No. Observations:                   20
Model:                 ARIMA(2, 0, 2)   Log Likelihood                 -18.559
Date:                Sun, 25 Feb 2024   AIC                             49.119
Time:                        14:58:40   BIC                             55.093
Sample:                             0   HQIC                            50.285
                                 - 20                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
x1            -0.5646      0.303     -1.861      0.063      -1.159       0.030
ar.L1          1.7981      0.158     11.407      0.000       1.489       2.107
ar.L2         -0.8295      0.164     -5.059      0.000      -1.151      -0.508
ma.L1         -0.8597     21.365     -0.040      0.968     -42.735      41.016
ma.L2          0.9980     49.501      0.020      0.984     -96.022      98.018
sigma2         0.2204     10.881      0.020      0.984     -21.106      21.547
===================================================================================
Ljung-Box (L1) (Q):                   1.05   Jarque-Bera (JB):                 0.81
Prob(Q):                              0.31   Prob(JB):                         0.67
Heteroskedasticity (H):               0.78   Skew:                             0.27
Prob(H) (two-sided):                  0.76   Kurtosis:                         2.18
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

So the 'u' in fact comes from the Pandas series name of my exogenous inputs.

So I'm guessing 'u' (or 'x1') is the beta parameter in this ARMAX model definition, which I found on the FAQ page:

If someone can confirm this that would be great. Or point me to the part of the document that explains the parameters of the ARIMA model and the model.summary() and model.params outputs.