时间序列模型中的确定性项¶

[1]:

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

plt.rc("figure", figsize=(16, 9))
plt.rc("font", size=16)

基本用法¶

基本配置可以通过 DeterministicProcess 直接构建。这些可以包括一个常数、任意阶的时间趋势，以及季节性或傅里叶分量。

该过程需要一个索引，即全样本（或样本内）的索引。

首先，我们用一个常数、一个线性时间趋势和一个5周期季节项初始化一个确定性过程。in_sample方法返回与索引匹配的完整值集。

[2]:

from statsmodels.tsa.deterministic import DeterministicProcess

index = pd.RangeIndex(0, 100)
det_proc = DeterministicProcess(index, constant=True, order=1, seasonal=True, period=5)
det_proc.in_sample()

[2]:

	const	trend	s(2,5)	s(3,5)	s(4,5)	s(5,5)
0	1.0	1.0	0.0	0.0	0.0	0.0
1	1.0	2.0	1.0	0.0	0.0	0.0
2	1.0	3.0	0.0	1.0	0.0	0.0
3	1.0	4.0	0.0	0.0	1.0	0.0
4	1.0	5.0	0.0	0.0	0.0	1.0
...	...	...	...	...	...	...
95	1.0	96.0	0.0	0.0	0.0	0.0
96	1.0	97.0	1.0	0.0	0.0	0.0
97	1.0	98.0	0.0	1.0	0.0	0.0
98	1.0	99.0	0.0	0.0	1.0	0.0
99	1.0	100.0	0.0	0.0	0.0	1.0

100行 × 6列

The out_of_sample returns the next steps values after the end of the in-sample.

[3]:

det_proc.out_of_sample(15)

[3]:

	const	trend	s(2,5)	s(3,5)	s(4,5)	s(5,5)
100	1.0	101.0	0.0	0.0	0.0	0.0
101	1.0	102.0	1.0	0.0	0.0	0.0
102	1.0	103.0	0.0	1.0	0.0	0.0
103	1.0	104.0	0.0	0.0	1.0	0.0
104	1.0	105.0	0.0	0.0	0.0	1.0
105	1.0	106.0	0.0	0.0	0.0	0.0
106	1.0	107.0	1.0	0.0	0.0	0.0
107	1.0	108.0	0.0	1.0	0.0	0.0
108	1.0	109.0	0.0	0.0	1.0	0.0
109	1.0	110.0	0.0	0.0	0.0	1.0
110	1.0	111.0	0.0	0.0	0.0	0.0
111	1.0	112.0	1.0	0.0	0.0	0.0
112	1.0	113.0	0.0	1.0	0.0	0.0
113	1.0	114.0	0.0	0.0	1.0	0.0
114	1.0	115.0	0.0	0.0	0.0	1.0

range(start, stop) 也可以用于在任何范围内生成确定性项，包括样本内和样本外。

笔记¶

当索引是 pandas 的 DatetimeIndex 或 PeriodIndex 时，start 和 stop 可以是类似日期的（字符串，例如“2020-06-01”，或 Timestamp）或整数。
stop 总是包含在范围内。虽然这并不是非常Pythonic，但由于statsmodels和Pandas在处理类似日期的切片时都包含stop，因此这是必要的。

[4]:

det_proc.range(190, 210)

[4]:

	const	trend	s(2,5)	s(3,5)	s(4,5)	s(5,5)
190	1.0	191.0	0.0	0.0	0.0	0.0
191	1.0	192.0	1.0	0.0	0.0	0.0
192	1.0	193.0	0.0	1.0	0.0	0.0
193	1.0	194.0	0.0	0.0	1.0	0.0
194	1.0	195.0	0.0	0.0	0.0	1.0
195	1.0	196.0	0.0	0.0	0.0	0.0
196	1.0	197.0	1.0	0.0	0.0	0.0
197	1.0	198.0	0.0	1.0	0.0	0.0
198	1.0	199.0	0.0	0.0	1.0	0.0
199	1.0	200.0	0.0	0.0	0.0	1.0
200	1.0	201.0	0.0	0.0	0.0	0.0
201	1.0	202.0	1.0	0.0	0.0	0.0
202	1.0	203.0	0.0	1.0	0.0	0.0
203	1.0	204.0	0.0	0.0	1.0	0.0
204	1.0	205.0	0.0	0.0	0.0	1.0
205	1.0	206.0	0.0	0.0	0.0	0.0
206	1.0	207.0	1.0	0.0	0.0	0.0
207	1.0	208.0	0.0	1.0	0.0	0.0
208	1.0	209.0	0.0	0.0	1.0	0.0
209	1.0	210.0	0.0	0.0	0.0	1.0
210	1.0	211.0	0.0	0.0	0.0	0.0

使用类似日期的索引¶

接下来，我们使用 PeriodIndex 展示相同的步骤。

[5]:

index = pd.period_range("2020-03-01", freq="M", periods=60)
det_proc = DeterministicProcess(index, constant=True, fourier=2)
det_proc.in_sample().head(12)

[5]:

	const	sin(1,12)	cos(1,12)	sin(2,12)	cos(2,12)
2020-03	1.0	0.000000e+00	1.000000e+00	0.000000e+00	1.0
2020-04	1.0	5.000000e-01	8.660254e-01	8.660254e-01	0.5
2020-05	1.0	8.660254e-01	5.000000e-01	8.660254e-01	-0.5
2020-06	1.0	1.000000e+00	6.123234e-17	1.224647e-16	-1.0
2020-07	1.0	8.660254e-01	-5.000000e-01	-8.660254e-01	-0.5
2020-08	1.0	5.000000e-01	-8.660254e-01	-8.660254e-01	0.5
2020-09	1.0	1.224647e-16	-1.000000e+00	-2.449294e-16	1.0
2020-10	1.0	-5.000000e-01	-8.660254e-01	8.660254e-01	0.5
2020-11	1.0	-8.660254e-01	-5.000000e-01	8.660254e-01	-0.5
2020-12	1.0	-1.000000e+00	-1.836970e-16	3.673940e-16	-1.0
2021-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5
2021-02	1.0	-5.000000e-01	8.660254e-01	-8.660254e-01	0.5

[6]:

det_proc.out_of_sample(12)

[6]:

	const	sin(1,12)	cos(1,12)	sin(2,12)	cos(2,12)
2025-03	1.0	-1.224647e-15	1.000000e+00	-2.449294e-15	1.0
2025-04	1.0	5.000000e-01	8.660254e-01	8.660254e-01	0.5
2025-05	1.0	8.660254e-01	5.000000e-01	8.660254e-01	-0.5
2025-06	1.0	1.000000e+00	-4.904777e-16	-9.809554e-16	-1.0
2025-07	1.0	8.660254e-01	-5.000000e-01	-8.660254e-01	-0.5
2025-08	1.0	5.000000e-01	-8.660254e-01	-8.660254e-01	0.5
2025-09	1.0	4.899825e-15	-1.000000e+00	-9.799650e-15	1.0
2025-10	1.0	-5.000000e-01	-8.660254e-01	8.660254e-01	0.5
2025-11	1.0	-8.660254e-01	-5.000000e-01	8.660254e-01	-0.5
2025-12	1.0	-1.000000e+00	-3.184701e-15	6.369401e-15	-1.0
2026-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5
2026-02	1.0	-5.000000e-01	8.660254e-01	-8.660254e-01	0.5

range 接受类似日期的参数，这些参数通常以字符串形式给出。

[7]:

det_proc.range("2025-01", "2026-01")

[7]:

	const	sin(1,12)	cos(1,12)	sin(2,12)	cos(2,12)
2025-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5
2025-02	1.0	-5.000000e-01	8.660254e-01	-8.660254e-01	0.5
2025-03	1.0	-1.224647e-15	1.000000e+00	-2.449294e-15	1.0
2025-04	1.0	5.000000e-01	8.660254e-01	8.660254e-01	0.5
2025-05	1.0	8.660254e-01	5.000000e-01	8.660254e-01	-0.5
2025-06	1.0	1.000000e+00	-4.904777e-16	-9.809554e-16	-1.0
2025-07	1.0	8.660254e-01	-5.000000e-01	-8.660254e-01	-0.5
2025-08	1.0	5.000000e-01	-8.660254e-01	-8.660254e-01	0.5
2025-09	1.0	4.899825e-15	-1.000000e+00	-9.799650e-15	1.0
2025-10	1.0	-5.000000e-01	-8.660254e-01	8.660254e-01	0.5
2025-11	1.0	-8.660254e-01	-5.000000e-01	8.660254e-01	-0.5
2025-12	1.0	-1.000000e+00	-3.184701e-15	6.369401e-15	-1.0
2026-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5

这相当于使用整数值58和70。

[8]:

det_proc.range(58, 70)

[8]:

	const	sin(1,12)	cos(1,12)	sin(2,12)	cos(2,12)
2025-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5
2025-02	1.0	-5.000000e-01	8.660254e-01	-8.660254e-01	0.5
2025-03	1.0	-1.224647e-15	1.000000e+00	-2.449294e-15	1.0
2025-04	1.0	5.000000e-01	8.660254e-01	8.660254e-01	0.5
2025-05	1.0	8.660254e-01	5.000000e-01	8.660254e-01	-0.5
2025-06	1.0	1.000000e+00	-4.904777e-16	-9.809554e-16	-1.0
2025-07	1.0	8.660254e-01	-5.000000e-01	-8.660254e-01	-0.5
2025-08	1.0	5.000000e-01	-8.660254e-01	-8.660254e-01	0.5
2025-09	1.0	4.899825e-15	-1.000000e+00	-9.799650e-15	1.0
2025-10	1.0	-5.000000e-01	-8.660254e-01	8.660254e-01	0.5
2025-11	1.0	-8.660254e-01	-5.000000e-01	8.660254e-01	-0.5
2025-12	1.0	-1.000000e+00	-3.184701e-15	6.369401e-15	-1.0
2026-01	1.0	-8.660254e-01	5.000000e-01	-8.660254e-01	-0.5

高级构造¶

具有构造函数不直接支持的特征的确定性过程可以使用 additional_terms 创建，该函数接受一个 DetermisticTerm 列表。这里我们创建了一个具有两个季节性成分的确定性过程：周内日（5天周期）和通过傅里叶成分捕捉的年周期（365.25天周期）。

[9]:

from statsmodels.tsa.deterministic import Fourier, Seasonality, TimeTrend

index = pd.period_range("2020-03-01", freq="D", periods=2 * 365)
tt = TimeTrend(constant=True)
four = Fourier(period=365.25, order=2)
seas = Seasonality(period=7)
det_proc = DeterministicProcess(index, additional_terms=[tt, seas, four])
det_proc.in_sample().head(28)

[9]:

	const	s(2,7)	s(3,7)	s(4,7)	s(5,7)	s(6,7)	s(7,7)	sin(1,365.25)	cos(1,365.25)	sin(2,365.25)	cos(2,365.25)
2020-03-01	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.000000	1.000000	0.000000	1.000000
2020-03-02	1.0	1.0	0.0	0.0	0.0	0.0	0.0	0.017202	0.999852	0.034398	0.999408
2020-03-03	1.0	0.0	1.0	0.0	0.0	0.0	0.0	0.034398	0.999408	0.068755	0.997634
2020-03-04	1.0	0.0	0.0	1.0	0.0	0.0	0.0	0.051584	0.998669	0.103031	0.994678
2020-03-05	1.0	0.0	0.0	0.0	1.0	0.0	0.0	0.068755	0.997634	0.137185	0.990545
2020-03-06	1.0	0.0	0.0	0.0	0.0	1.0	0.0	0.085906	0.996303	0.171177	0.985240
2020-03-07	1.0	0.0	0.0	0.0	0.0	0.0	1.0	0.103031	0.994678	0.204966	0.978769
2020-03-08	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.120126	0.992759	0.238513	0.971139
2020-03-09	1.0	1.0	0.0	0.0	0.0	0.0	0.0	0.137185	0.990545	0.271777	0.962360
2020-03-10	1.0	0.0	1.0	0.0	0.0	0.0	0.0	0.154204	0.988039	0.304719	0.952442
2020-03-11	1.0	0.0	0.0	1.0	0.0	0.0	0.0	0.171177	0.985240	0.337301	0.941397
2020-03-12	1.0	0.0	0.0	0.0	1.0	0.0	0.0	0.188099	0.982150	0.369484	0.929237
2020-03-13	1.0	0.0	0.0	0.0	0.0	1.0	0.0	0.204966	0.978769	0.401229	0.915978
2020-03-14	1.0	0.0	0.0	0.0	0.0	0.0	1.0	0.221772	0.975099	0.432499	0.901634
2020-03-15	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.238513	0.971139	0.463258	0.886224
2020-03-16	1.0	1.0	0.0	0.0	0.0	0.0	0.0	0.255182	0.966893	0.493468	0.869764
2020-03-17	1.0	0.0	1.0	0.0	0.0	0.0	0.0	0.271777	0.962360	0.523094	0.852275
2020-03-18	1.0	0.0	0.0	1.0	0.0	0.0	0.0	0.288291	0.957543	0.552101	0.833777
2020-03-19	1.0	0.0	0.0	0.0	1.0	0.0	0.0	0.304719	0.952442	0.580455	0.814292
2020-03-20	1.0	0.0	0.0	0.0	0.0	1.0	0.0	0.321058	0.947060	0.608121	0.793844
2020-03-21	1.0	0.0	0.0	0.0	0.0	0.0	1.0	0.337301	0.941397	0.635068	0.772456
2020-03-22	1.0	0.0	0.0	0.0	0.0	0.0	0.0	0.353445	0.935455	0.661263	0.750154
2020-03-23	1.0	1.0	0.0	0.0	0.0	0.0	0.0	0.369484	0.929237	0.686676	0.726964
2020-03-24	1.0	0.0	1.0	0.0	0.0	0.0	0.0	0.385413	0.922744	0.711276	0.702913
2020-03-25	1.0	0.0	0.0	1.0	0.0	0.0	0.0	0.401229	0.915978	0.735034	0.678031
2020-03-26	1.0	0.0	0.0	0.0	1.0	0.0	0.0	0.416926	0.908940	0.757922	0.652346
2020-03-27	1.0	0.0	0.0	0.0	0.0	1.0	0.0	0.432499	0.901634	0.779913	0.625889
2020-03-28	1.0	0.0	0.0	0.0	0.0	0.0	1.0	0.447945	0.894061	0.800980	0.598691

自定义确定性术语¶

The DetermisticTerm 抽象基类旨在被继承，以帮助用户编写自定义的确定性项。接下来我们展示两个例子。第一个是允许在固定周期数后中断的时间趋势。第二个是一个“技巧”确定性项，它允许将外生数据（实际上不是确定性过程）视为确定性过程。这让我们可以简化收集预测所需项的过程。

这些旨在展示自定义术语的构建。在输入验证方面，它们肯定可以得到改进。

[10]:

from statsmodels.tsa.deterministic import DeterministicTerm


class BrokenTimeTrend(DeterministicTerm):
    def __init__(self, break_period: int):
        self._break_period = break_period

    def __str__(self):
        return "Broken Time Trend"

    def _eq_attr(self):
        return (self._break_period,)

    def in_sample(self, index: pd.Index):
        nobs = index.shape[0]
        terms = np.zeros((nobs, 2))
        terms[self._break_period :, 0] = 1
        terms[self._break_period :, 1] = np.arange(self._break_period + 1, nobs + 1)
        return pd.DataFrame(terms, columns=["const_break", "trend_break"], index=index)

    def out_of_sample(
        self, steps: int, index: pd.Index, forecast_index: pd.Index = None
    ):
        # Always call extend index first
        fcast_index = self._extend_index(index, steps, forecast_index)
        nobs = index.shape[0]
        terms = np.zeros((steps, 2))
        # Assume break period is in-sample
        terms[:, 0] = 1
        terms[:, 1] = np.arange(nobs + 1, nobs + steps + 1)
        return pd.DataFrame(
            terms, columns=["const_break", "trend_break"], index=fcast_index
        )

[11]:

btt = BrokenTimeTrend(60)
tt = TimeTrend(constant=True, order=1)
index = pd.RangeIndex(100)
det_proc = DeterministicProcess(index, additional_terms=[tt, btt])
det_proc.range(55, 65)

[11]:

	const	trend	const_break	trend_break
55	1.0	56.0	0.0	0.0
56	1.0	57.0	0.0	0.0
57	1.0	58.0	0.0	0.0
58	1.0	59.0	0.0	0.0
59	1.0	60.0	0.0	0.0
60	1.0	61.0	1.0	61.0
61	1.0	62.0	1.0	62.0
62	1.0	63.0	1.0	63.0
63	1.0	64.0	1.0	64.0
64	1.0	65.0	1.0	65.0
65	1.0	66.0	1.0	66.0

接下来，我们为一些实际的外生数据编写一个简单的“包装器”，以便简化构建用于预测的样本外生数组。

[12]:

class ExogenousProcess(DeterministicTerm):
    def __init__(self, data):
        self._data = data

    def __str__(self):
        return "Custom Exog Process"

    def _eq_attr(self):
        return (id(self._data),)

    def in_sample(self, index: pd.Index):
        return self._data.loc[index]

    def out_of_sample(
        self, steps: int, index: pd.Index, forecast_index: pd.Index = None
    ):
        forecast_index = self._extend_index(index, steps, forecast_index)
        return self._data.loc[forecast_index]

[13]:

import numpy as np

gen = np.random.default_rng(98765432101234567890)
exog = pd.DataFrame(gen.integers(100, size=(300, 2)), columns=["exog1", "exog2"])
exog.head()

[13]:

	exog1	exog2
0	6	99
1	64	28
2	15	81
3	54	8
4	12	8

[14]:

ep = ExogenousProcess(exog)
tt = TimeTrend(constant=True, order=1)
# The in-sample index
idx = exog.index[:200]
det_proc = DeterministicProcess(idx, additional_terms=[tt, ep])

[15]:

det_proc.in_sample().head()

[15]:

	const	trend	exog1	exog2
0	1.0	1.0	6	99
1	1.0	2.0	64	28
2	1.0	3.0	15	81
3	1.0	4.0	54	8
4	1.0	5.0	12	8

[16]:

det_proc.out_of_sample(10)

[16]:

	const	trend	exog1	exog2
200	1.0	201.0	56	88
201	1.0	202.0	48	84
202	1.0	203.0	44	5
203	1.0	204.0	65	63
204	1.0	205.0	63	39
205	1.0	206.0	89	39
206	1.0	207.0	41	54
207	1.0	208.0	71	5
208	1.0	209.0	89	6
209	1.0	210.0	58	63

模型支持¶

唯一直接支持 DeterministicProcess 的模型是 AutoReg。可以使用 deterministic 关键字参数设置自定义项。

注意：使用自定义项需要设置 trend="n" 和 seasonal=False，以便所有确定性成分必须来自自定义确定性项。

模拟一些数据¶

这里我们模拟一些数据，这些数据具有由傅里叶级数捕捉到的每周季节性。

[17]:

gen = np.random.default_rng(98765432101234567890)
idx = pd.RangeIndex(200)
det_proc = DeterministicProcess(idx, constant=True, period=52, fourier=2)
det_terms = det_proc.in_sample().to_numpy()
params = np.array([1.0, 3, -1, 4, -2])
exog = det_terms @ params
y = np.empty(200)
y[0] = det_terms[0] @ params + gen.standard_normal()
for i in range(1, 200):
    y[i] = 0.9 * y[i - 1] + det_terms[i] @ params + gen.standard_normal()
y = pd.Series(y, index=idx)
ax = y.plot()

../../../_images/examples_notebooks_generated_deterministics_28_0.png

然后使用 deterministic 关键字参数来拟合模型。seasonal 默认为 False，但 trend 默认为 "c"，因此需要进行更改。

[18]:

from statsmodels.tsa.api import AutoReg

mod = AutoReg(y, 1, trend="n", deterministic=det_proc)
res = mod.fit()
print(res.summary())

                            AutoReg Model Results
==============================================================================
Dep. Variable:                      y   No. Observations:                  200
Model:                     AutoReg(1)   Log Likelihood                -270.964
Method:               Conditional MLE   S.D. of innovations              0.944
Date:                Wed, 16 Oct 2024   AIC                            555.927
Time:                        18:27:06   BIC                            578.980
Sample:                             1   HQIC                           565.258
                                  200
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.8436      0.172      4.916      0.000       0.507       1.180
sin(1,52)      2.9738      0.160     18.587      0.000       2.660       3.287
cos(1,52)     -0.6771      0.284     -2.380      0.017      -1.235      -0.120
sin(2,52)      3.9951      0.099     40.336      0.000       3.801       4.189
cos(2,52)     -1.7206      0.264     -6.519      0.000      -2.238      -1.203
y.L1           0.9116      0.014     63.264      0.000       0.883       0.940
                                    Roots
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.0970           +0.0000j            1.0970            0.0000
-----------------------------------------------------------------------------

我们可以使用 plot_predict 来显示预测值及其预测区间。样本外的确定性值由传递给 AutoReg 的确定性过程自动生成。

[19]:

fig = res.plot_predict(200, 200 + 2 * 52, True)

../../../_images/examples_notebooks_generated_deterministics_32_0.png

[20]:

auto_reg_forecast = res.predict(200, 211)
auto_reg_forecast

[20]:

200    -3.253482
201    -8.555660
202   -13.607557
203   -18.152622
204   -21.950370
205   -24.790116
206   -26.503171
207   -26.972781
208   -26.141244
209   -24.013773
210   -20.658891
211   -16.205310
dtype: float64

与其他模型一起使用¶

其他模型不直接支持 DeterministicProcess。我们可以改为手动将任何确定性项作为 exog 传递给支持外生值的模型。

请注意，带有外生变量的 SARIMAX 是带有SARIMA误差的OLS，因此模型为

\[\begin{split}\begin{align*} \nu_t & = y_t - x_t \beta \\ (1-\phi(L))\nu_t & = (1+\theta(L))\epsilon_t. \end{align*}\end{split}\]

确定性项上的参数不能直接与AutoReg进行比较，后者根据方程演化

\[(1-\phi(L)) y_t = x_t \beta + \epsilon_t.\]

当 \(x_t\) 仅包含确定性项时，这两种表示是等价的（假设 \(\theta(L)=0\) 以便没有MA）。

[21]:

from statsmodels.tsa.api import SARIMAX

det_proc = DeterministicProcess(idx, period=52, fourier=2)
det_terms = det_proc.in_sample()

mod = SARIMAX(y, order=(1, 0, 0), trend="c", exog=det_terms)
res = mod.fit(disp=False)
print(res.summary())

                               SARIMAX Results
==============================================================================
Dep. Variable:                      y   No. Observations:                  200
Model:               SARIMAX(1, 0, 0)   Log Likelihood                -293.381
Date:                Wed, 16 Oct 2024   AIC                            600.763
Time:                        18:27:07   BIC                            623.851
Sample:                             0   HQIC                           610.106
                                - 200
Covariance Type:                  opg
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
intercept      0.0796      0.140      0.567      0.570      -0.196       0.355
sin(1,52)      9.1916      0.876     10.492      0.000       7.475      10.909
cos(1,52)    -17.4348      0.891    -19.576      0.000     -19.180     -15.689
sin(2,52)      1.2512      0.466      2.683      0.007       0.337       2.165
cos(2,52)    -17.1863      0.434    -39.583      0.000     -18.037     -16.335
ar.L1          0.9957      0.007    150.762      0.000       0.983       1.009
sigma2         1.0748      0.119      9.068      0.000       0.842       1.307
===================================================================================
Ljung-Box (L1) (Q):                   2.16   Jarque-Bera (JB):                 1.03
Prob(Q):                              0.14   Prob(JB):                         0.60
Heteroskedasticity (H):               0.71   Skew:                            -0.14
Prob(H) (two-sided):                  0.16   Kurtosis:                         2.78
===================================================================================

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

预测结果相似但有所不同，因为SARIMAX的参数是使用MLE估计的，而AutoReg使用OLS。

[22]:

sarimax_forecast = res.forecast(12, exog=det_proc.out_of_sample(12))
df = pd.concat([auto_reg_forecast, sarimax_forecast], axis=1)
df.columns = columns = ["AutoReg", "SARIMAX"]
df

[22]:

	AutoReg	SARIMAX
200	-3.253482	-2.956562
201	-8.555660	-7.985589
202	-13.607557	-12.794072
203	-18.152622	-17.130963
204	-21.950370	-20.760473
205	-24.790116	-23.475511
206	-26.503171	-25.109629
207	-26.972781	-25.546790
208	-26.141244	-24.728385
209	-24.013773	-22.657094
210	-20.658891	-19.397352
211	-16.205310	-15.072385

Last update: Oct 16, 2024