我在某些时间序列上使用statsmodel-特别是手动生成ADF结果以更好地了解过程。但是，我不确定如果手动将exog变量包括在内，为什么趋势/漂移项没有出现在ARIMA（1,1,0）结果中，而是出现在ARIMA（0,0,0）结果中。下面的两个模型应该相等。
res_arima = sm.tsa.arima.ARIMA(df.log_biz_machine_investment.loc[start:end], order=(1,1,0), trend='ct', freq='QS', trend_offset=df.index.get_loc(start)+1)
print(res_arima.fit().summary())
SARIMAX Results
======================================================================================
Dep. Variable: log_biz_machine_investment No. Observations: 123
Model: ARIMA(1, 1, 0) Log Likelihood 119.336
Date: Fri, 22 May 2020 AIC -230.672
Time: 17:33:22 BIC -219.456
Sample: 10-01-1972 HQIC -226.116
- 04-01-2003
Covariance Type: opg
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
const 1.155e-13 9.14e-13 0.126 0.900 -1.68e-12 1.91e-12
x1 0.0101 0.007 1.468 0.142 -0.003 0.024
ar.L1 -0.2216 0.081 -2.723 0.006 -0.381 -0.062
sigma2 0.0083 0.001 11.192 0.000 0.007 0.010
===================================================================================
Ljung-Box (Q): 43.43 Jarque-Bera (JB): 32.63
Prob(Q): 0.33 Prob(JB): 0.00
Heteroskedasticity (H): 0.33 Skew: -0.38
Prob(H) (two-sided): 0.00 Kurtosis: 5.42
===================================================================================
exog_data_without_lag = df.loc[:, ['log_biz_machine_investment', 'dlog_biz_machine_investment']]
endog = df.loc[:, 'dlog_biz_machine_investment']
maxlag = 1
exog_data = sm.tsa.tsatools.lagmat(df.dlog_biz_machine_investment, maxlag=maxlag, use_pandas=True)
exog_data = pd.concat([exog_data, df.log_biz_machine_investment.shift(1)], axis=1)
exog_data = exog_data.loc[start:end]
mod = sm.tsa.arima.ARIMA(df.dlog_biz_machine_investment.loc[start:end], exog=exog_data, order=(0,0,0), freq='QS', trend='ct', trend_offset=df.index.get_loc(start)+1)
results = mod.fit(method='innovations_mle')
print(results.summary())
SARIMAX Results
=======================================================================================
Dep. Variable: dlog_biz_machine_investment No. Observations: 123
Model: ARIMA Log Likelihood 123.957
Date: Fri, 22 May 2020 AIC -237.913
Time: 17:31:35 BIC -223.852
Sample: 10-01-1972 HQIC -232.202
- 04-01-2003
Covariance Type: opg
===================================================================================================
coef std err z P>|z| [0.025 0.975]
---------------------------------------------------------------------------------------------------
const 0.4019 0.178 2.264 0.024 0.054 0.750
drift 0.0013 0.001 2.482 0.013 0.000 0.002
dlog_biz_machine_investment.L.1 -0.1960 0.082 -2.403 0.016 -0.356 -0.036
log_biz_machine_investment -0.1146 0.050 -2.271 0.023 -0.213 -0.016
sigma2 0.0078 0.001 11.727 0.000 0.006 0.009
===================================================================================
Ljung-Box (Q): 45.13 Jarque-Bera (JB): 59.25
Prob(Q): 0.27 Prob(JB): 0.00
Heteroskedasticity (H): 0.32 Skew: -0.53
Prob(H) (two-sided): 0.00 Kurtosis: 6.23
===================================================================================
这导致回归结果有所不同。漂移项应同时出现在两个结果中。