马尔可夫切换动态回归模型¶

本笔记本提供了一个使用statsmodels中的马尔可夫切换模型来估计具有状态变化的动态回归模型的示例。它遵循Stata马尔可夫切换文档中的示例，该文档可以在http://www.stata.com/manuals14/tsmswitch.pdf找到。

[1]:

%matplotlib inline

import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt

# NBER recessions
from pandas_datareader.data import DataReader
from datetime import datetime

usrec = DataReader(
    "USREC", "fred", start=datetime(1947, 1, 1), end=datetime(2013, 4, 1)
)

带有切换截距的联邦基金利率¶

第一个示例将联邦基金利率建模为围绕一个常数截距的噪声，但在不同制度期间截距会发生变化。模型非常简单：

\[r_t = \mu_{S_t} + \varepsilon_t \qquad \varepsilon_t \sim N(0, \sigma^2)\]

其中 \(S_t \in \{0, 1\}\)，并且状态根据

\[\begin{split} P(S_t = s_t | S_{t-1} = s_{t-1}) = \begin{bmatrix} p_{00} & p_{10} \\ 1 - p_{00} & 1 - p_{10} \end{bmatrix}\end{split}\]

我们将通过最大似然估计这个模型的参数：\(p_{00}, p_{10}, \mu_0, \mu_1, \sigma^2\)。

本示例中使用的数据可以在 https://www.stata-press.com/data/r14/usmacro 找到。

[2]:

# Get the federal funds rate data
from statsmodels.tsa.regime_switching.tests.test_markov_regression import fedfunds

dta_fedfunds = pd.Series(
    fedfunds, index=pd.date_range("1954-07-01", "2010-10-01", freq="QS")
)

# Plot the data
dta_fedfunds.plot(title="Federal funds rate", figsize=(12, 3))

# Fit the model
# (a switching mean is the default of the MarkovRegession model)
mod_fedfunds = sm.tsa.MarkovRegression(dta_fedfunds, k_regimes=2)
res_fedfunds = mod_fedfunds.fit()

../../../_images/examples_notebooks_generated_markov_regression_4_0.png

[3]:

res_fedfunds.summary()

[3]:

Markov Switching Model Results
Dep. Variable:	y	No. Observations:	226
Model:	MarkovRegression	Log Likelihood	-508.636
Date:	Wed, 16 Oct 2024	AIC	1027.272
Time:	18:35:45	BIC	1044.375
Sample:	07-01-1954	HQIC	1034.174
	- 10-01-2010
Covariance Type:	approx

Regime 0 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	3.7088	0.177	20.988	0.000	3.362	4.055

Regime 1 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	9.5568	0.300	31.857	0.000	8.969	10.145

Non-switching parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
sigma2	4.4418	0.425	10.447	0.000	3.608	5.275

Regime transition parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
p[0->0]	0.9821	0.010	94.443	0.000	0.962	1.002
p[1->0]	0.0504	0.027	1.876	0.061	-0.002	0.103

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

从摘要输出中，第一个制度（“低制度”）的平均联邦基金利率估计为\(3.7\)，而在“高制度”中，它为\(9.6\)。下面我们绘制了处于高制度的概率平滑图。该模型表明，20世纪80年代是一个存在高联邦基金利率的时期。

[4]:

res_fedfunds.smoothed_marginal_probabilities[1].plot(
    title="Probability of being in the high regime", figsize=(12, 3)
)

[4]:

<Axes: title={'center': 'Probability of being in the high regime'}>

../../../_images/examples_notebooks_generated_markov_regression_7_1.png

从估计的转移矩阵中，我们可以计算出低制度与高制度的预期持续时间。

[5]:

print(res_fedfunds.expected_durations)

[55.85400626 19.85506546]

预计低制度将持续约十四年，而高制度预计仅持续约五年。

带有切换截距和滞后因变量的联邦基金利率¶

第二个示例在前一个模型的基础上增加了联邦基金利率的滞后值。

\[r_t = \mu_{S_t} + r_{t-1} \beta_{S_t} + \varepsilon_t \qquad \varepsilon_t \sim N(0, \sigma^2)\]

其中 \(S_t \in \{0, 1\}\)，并且状态根据

\[\begin{split} P(S_t = s_t | S_{t-1} = s_{t-1}) = \begin{bmatrix} p_{00} & p_{10} \\ 1 - p_{00} & 1 - p_{10} \end{bmatrix}\end{split}\]

我们将通过最大似然估计这个模型的参数：\(p_{00}, p_{10}, \mu_0, \mu_1, \beta_0, \beta_1, \sigma^2\)。

[6]:

# Fit the model
mod_fedfunds2 = sm.tsa.MarkovRegression(
    dta_fedfunds.iloc[1:], k_regimes=2, exog=dta_fedfunds.iloc[:-1]
)
res_fedfunds2 = mod_fedfunds2.fit()

[7]:

res_fedfunds2.summary()

[7]:

Markov Switching Model Results
Dep. Variable:	y	No. Observations:	225
Model:	MarkovRegression	Log Likelihood	-264.711
Date:	Wed, 16 Oct 2024	AIC	543.421
Time:	18:35:46	BIC	567.334
Sample:	10-01-1954	HQIC	553.073
	- 10-01-2010
Covariance Type:	approx

Regime 0 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	0.7245	0.289	2.510	0.012	0.159	1.290
x1	0.7631	0.034	22.629	0.000	0.697	0.829

Regime 1 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	-0.0989	0.118	-0.835	0.404	-0.331	0.133
x1	1.0612	0.019	57.351	0.000	1.025	1.097

Non-switching parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
sigma2	0.4783	0.050	9.642	0.000	0.381	0.576

Regime transition parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
p[0->0]	0.6378	0.120	5.304	0.000	0.402	0.874
p[1->0]	0.1306	0.050	2.634	0.008	0.033	0.228

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

从摘要输出中可以注意到以下几点：

信息准则显著下降，表明该模型比之前的模型拟合得更好。
在截距方面，对制度的理解已经发生了变化。现在第一个制度具有较高的截距，而第二个制度具有较低的截距。

检查高机制状态的平滑概率，我们现在看到了更多的变化性。

[8]:

res_fedfunds2.smoothed_marginal_probabilities[0].plot(
    title="Probability of being in the high regime", figsize=(12, 3)
)

[8]:

<Axes: title={'center': 'Probability of being in the high regime'}>

../../../_images/examples_notebooks_generated_markov_regression_15_1.png

最后，每个制度预期的持续时间已经大大减少了。

[9]:

print(res_fedfunds2.expected_durations)

[2.76105188 7.65529154]

带有2或3个体制的泰勒规则¶

我们现在引入两个额外的外生变量——产出缺口的衡量指标和通货膨胀的衡量指标——来估计一个具有2和3种体制的切换型泰勒规则，以观察哪种更适合数据。

由于模型通常难以估计，对于3-状态模型，我们采用对初始参数进行搜索以改进结果，指定了20次随机搜索重复。

[10]:

# Get the additional data
from statsmodels.tsa.regime_switching.tests.test_markov_regression import ogap, inf

dta_ogap = pd.Series(ogap, index=pd.date_range("1954-07-01", "2010-10-01", freq="QS"))
dta_inf = pd.Series(inf, index=pd.date_range("1954-07-01", "2010-10-01", freq="QS"))

exog = pd.concat((dta_fedfunds.shift(), dta_ogap, dta_inf), axis=1).iloc[4:]

# Fit the 2-regime model
mod_fedfunds3 = sm.tsa.MarkovRegression(dta_fedfunds.iloc[4:], k_regimes=2, exog=exog)
res_fedfunds3 = mod_fedfunds3.fit()

# Fit the 3-regime model
np.random.seed(12345)
mod_fedfunds4 = sm.tsa.MarkovRegression(dta_fedfunds.iloc[4:], k_regimes=3, exog=exog)
res_fedfunds4 = mod_fedfunds4.fit(search_reps=20)

[11]:

res_fedfunds3.summary()

[11]:

Markov Switching Model Results
Dep. Variable:	y	No. Observations:	222
Model:	MarkovRegression	Log Likelihood	-229.256
Date:	Wed, 16 Oct 2024	AIC	480.512
Time:	18:35:46	BIC	517.942
Sample:	07-01-1955	HQIC	495.624
	- 10-01-2010
Covariance Type:	approx

Regime 0 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	0.6555	0.137	4.771	0.000	0.386	0.925
x1	0.8314	0.033	24.951	0.000	0.766	0.897
x2	0.1355	0.029	4.609	0.000	0.078	0.193
x3	-0.0274	0.041	-0.671	0.502	-0.107	0.053

Regime 1 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	-0.0945	0.128	-0.739	0.460	-0.345	0.156
x1	0.9293	0.027	34.309	0.000	0.876	0.982
x2	0.0343	0.024	1.429	0.153	-0.013	0.081
x3	0.2125	0.030	7.147	0.000	0.154	0.271

Non-switching parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
sigma2	0.3323	0.035	9.526	0.000	0.264	0.401

Regime transition parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
p[0->0]	0.7279	0.093	7.828	0.000	0.546	0.910
p[1->0]	0.2115	0.064	3.298	0.001	0.086	0.337

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

[12]:

res_fedfunds4.summary()

[12]:

Markov Switching Model Results
Dep. Variable:	y	No. Observations:	222
Model:	MarkovRegression	Log Likelihood	-180.806
Date:	Wed, 16 Oct 2024	AIC	399.611
Time:	18:35:47	BIC	464.262
Sample:	07-01-1955	HQIC	425.713
	- 10-01-2010
Covariance Type:	approx

Regime 0 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	-1.0250	0.290	-3.531	0.000	-1.594	-0.456
x1	0.3277	0.086	3.812	0.000	0.159	0.496
x2	0.2036	0.049	4.152	0.000	0.107	0.300
x3	1.1381	0.081	13.977	0.000	0.978	1.298

Regime 1 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	-0.0259	0.087	-0.298	0.765	-0.196	0.144
x1	0.9737	0.019	50.265	0.000	0.936	1.012
x2	0.0341	0.017	2.030	0.042	0.001	0.067
x3	0.1215	0.022	5.606	0.000	0.079	0.164

Regime 2 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	0.7346	0.130	5.632	0.000	0.479	0.990
x1	0.8436	0.024	35.198	0.000	0.797	0.891
x2	0.1633	0.025	6.515	0.000	0.114	0.212
x3	-0.0499	0.027	-1.835	0.067	-0.103	0.003

Non-switching parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
sigma2	0.1660	0.018	9.240	0.000	0.131	0.201

Regime transition parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
p[0->0]	0.7214	0.117	6.177	0.000	0.493	0.950
p[1->0]	4.001e-08	nan	nan	nan	nan	nan
p[2->0]	0.0783	0.038	2.079	0.038	0.004	0.152
p[0->1]	0.1044	0.095	1.103	0.270	-0.081	0.290
p[1->1]	0.8259	0.054	15.208	0.000	0.719	0.932
p[2->1]	0.2288	0.073	3.150	0.002	0.086	0.371

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

由于较低的信息准则，我们可能更倾向于选择3状态模型，其解释为低、中和高利率制度。每个制度的平滑概率如下图所示。

[13]:

fig, axes = plt.subplots(3, figsize=(10, 7))

ax = axes[0]
ax.plot(res_fedfunds4.smoothed_marginal_probabilities[0])
ax.set(title="Smoothed probability of a low-interest rate regime")

ax = axes[1]
ax.plot(res_fedfunds4.smoothed_marginal_probabilities[1])
ax.set(title="Smoothed probability of a medium-interest rate regime")

ax = axes[2]
ax.plot(res_fedfunds4.smoothed_marginal_probabilities[2])
ax.set(title="Smoothed probability of a high-interest rate regime")

fig.tight_layout()

../../../_images/examples_notebooks_generated_markov_regression_23_0.png

切换方差¶

我们也可以适应切换方差。特别是，我们考虑模型

\[y_t = \mu_{S_t} + y_{t-1} \beta_{S_t} + \varepsilon_t \quad \varepsilon_t \sim N(0, \sigma_{S_t}^2)\]

我们使用最大似然法来估计该模型的参数：\(p_{00}, p_{10}, \mu_0, \mu_1, \beta_0, \beta_1, \sigma_0^2, \sigma_1^2\)。

该应用程序针对股票的绝对回报，数据可以在https://www.stata-press.com/data/r14/snp500找到。

[14]:

# Get the federal funds rate data
from statsmodels.tsa.regime_switching.tests.test_markov_regression import areturns

dta_areturns = pd.Series(
    areturns, index=pd.date_range("2004-05-04", "2014-5-03", freq="W")
)

# Plot the data
dta_areturns.plot(title="Absolute returns, S&P500", figsize=(12, 3))

# Fit the model
mod_areturns = sm.tsa.MarkovRegression(
    dta_areturns.iloc[1:],
    k_regimes=2,
    exog=dta_areturns.iloc[:-1],
    switching_variance=True,
)
res_areturns = mod_areturns.fit()

../../../_images/examples_notebooks_generated_markov_regression_25_0.png

[15]:

res_areturns.summary()

[15]:

Markov Switching Model Results
Dep. Variable:	y	No. Observations:	520
Model:	MarkovRegression	Log Likelihood	-745.798
Date:	Wed, 16 Oct 2024	AIC	1507.595
Time:	18:35:47	BIC	1541.626
Sample:	05-16-2004	HQIC	1520.926
	- 04-27-2014
Covariance Type:	approx

Regime 0 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	0.7641	0.078	9.761	0.000	0.611	0.918
x1	0.0791	0.030	2.620	0.009	0.020	0.138
sigma2	0.3476	0.061	5.694	0.000	0.228	0.467

Regime 1 parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
const	1.9728	0.278	7.086	0.000	1.427	2.518
x1	0.5280	0.086	6.155	0.000	0.360	0.696
sigma2	2.5771	0.405	6.357	0.000	1.783	3.372

Regime transition parameters
	coef	std err	z	P>\|z\|	[0.025	0.975]
p[0->0]	0.7531	0.063	11.871	0.000	0.629	0.877
p[1->0]	0.6825	0.066	10.301	0.000	0.553	0.812

Warnings:
[1] Covariance matrix calculated using numerical (complex-step) differentiation.

第一个状态是低方差状态，第二个状态是高方差状态。下面我们绘制了处于低方差状态的概率。在2008年到2012年之间，似乎没有明确的迹象表明某一种状态在引导经济。

[16]:

res_areturns.smoothed_marginal_probabilities[0].plot(
    title="Probability of being in a low-variance regime", figsize=(12, 3)
)

[16]:

<Axes: title={'center': 'Probability of being in a low-variance regime'}>

../../../_images/examples_notebooks_generated_markov_regression_28_1.png

Last update: Oct 16, 2024