Statsmodels ARIMA.fit方法返回的'u'参数含义及相关疑问

我在使用Statsmodels的ARIMA.fit方法估计ARMAX模型时，找不到结果中返回的'u'参数的相关说明。

示例代码 #

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

# 输入输出数据集样本
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())

运行输出 #

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).

我原本预期ARMAX(2,2)模型仅有4个参数（2个AR项、2个MA项），且已通过trend='n'关闭趋势项，因此该参数不应是趋势项。

我见过一个ARMAX(1,1)模型的示例，但结果中并未出现u参数。希望有人能指出Statsmodels文档中提及'u'参数含义的部分。

内容的提问来源于stack exchange，提问作者Bill