SARIMAX

import pandas as pd
uc = pd.read_excel('uschange.xlsx')
uc.head()

	Date	Consumption	Income	Production	Savings	Unemployment
0	1970-01-01	0.615986	0.972261	-2.452700	4.810312	0.9
1	1970-04-01	0.460376	1.169085	-0.551525	7.287992	0.5
2	1970-07-01	0.876791	1.553271	-0.358708	7.289013	0.5
3	1970-10-01	-0.274245	-0.255272	-2.185455	0.985230	0.7
4	1971-01-01	1.897371	1.987154	1.909734	3.657771	-0.1

from statsmodels.formula.api import ols
ols('Consumption ~ Income', uc).fit().summary()

OLS Regression Results

Dep. Variable:	Consumption	R-squared:	0.159
Model:	OLS	Adj. R-squared:	0.154
Method:	Least Squares	F-statistic:	34.98
Date:	Sat, 01 Jul 2023	Prob (F-statistic):	1.58e-08
Time:	19:29:57	Log-Likelihood:	-169.62
No. Observations:	187	AIC:	343.2
Df Residuals:	185	BIC:	349.7
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	0.5451	0.056	9.789	0.000	0.435	0.655
Income	0.2806	0.047	5.915	0.000	0.187	0.374

Omnibus:	16.528	Durbin-Watson:	1.696
Prob(Omnibus):	0.000	Jarque-Bera (JB):	29.145
Skew:	-0.454	Prob(JB):	4.69e-07
Kurtosis:	4.707	Cond. No.	2.08

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

from patsy import dmatrices
y, x = dmatrices('Consumption ~ Income', uc)

from statsmodels.regression.linear_model import OLS
OLS(y, x).fit().summary()

OLS Regression Results

Dep. Variable:	Consumption	R-squared:	0.159
Model:	OLS	Adj. R-squared:	0.154
Method:	Least Squares	F-statistic:	34.98
Date:	Sat, 01 Jul 2023	Prob (F-statistic):	1.58e-08
Time:	19:30:13	Log-Likelihood:	-169.62
No. Observations:	187	AIC:	343.2
Df Residuals:	185	BIC:	349.7
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	0.5451	0.056	9.789	0.000	0.435	0.655
Income	0.2806	0.047	5.915	0.000	0.187	0.374

Omnibus:	16.528	Durbin-Watson:	1.696
Prob(Omnibus):	0.000	Jarque-Bera (JB):	29.145
Skew:	-0.454	Prob(JB):	4.69e-07
Kurtosis:	4.707	Cond. No.	2.08

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

from  statsmodels.tsa.api import SARIMAX
m = SARIMAX(y, order=(1, 0, 0)).fit()
m.summary()

SARIMAX Results

Dep. Variable:	Consumption	No. Observations:	187
Model:	SARIMAX(1, 0, 0)	Log Likelihood	-196.683
Date:	Sat, 01 Jul 2023	AIC	397.367
Time:	19:39:49	BIC	403.829
Sample:	0	HQIC	399.985
	- 187
Covariance Type:	opg

	coef	std err	z	P>\|z\|	[0.025	0.975]
ar.L1	0.7152	0.051	14.133	0.000	0.616	0.814
sigma2	0.4780	0.040	11.973	0.000	0.400	0.556

Ljung-Box (L1) (Q):	31.48	Jarque-Bera (JB):	24.93
Prob(Q):	0.00	Prob(JB):	0.00
Heteroskedasticity (H):	0.22	Skew:	0.15
Prob(H) (two-sided):	0.00	Kurtosis:	4.76

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

import matplotlib.pyplot as plt
uc.Consumption.plot()
plt.plot(m.predict(start=0, end=186))

[<matplotlib.lines.Line2D at 0x2d376654ca0>]

m = SARIMAX(y, order=(0, 0, 1), trend='c').fit()
m.summary()

SARIMAX Results

Dep. Variable:	Consumption	No. Observations:	187
Model:	SARIMAX(0, 0, 1)	Log Likelihood	-177.185
Date:	Sat, 01 Jul 2023	AIC	360.371
Time:	19:39:50	BIC	370.064
Sample:	0	HQIC	364.298
	- 187
Covariance Type:	opg

	coef	std err	z	P>\|z\|	[0.025	0.975]
intercept	0.7463	0.063	11.844	0.000	0.623	0.870
ma.L1	0.2666	0.069	3.878	0.000	0.132	0.401
sigma2	0.3894	0.029	13.466	0.000	0.333	0.446

Ljung-Box (L1) (Q):	0.45	Jarque-Bera (JB):	52.67
Prob(Q):	0.50	Prob(JB):	0.00
Heteroskedasticity (H):	0.31	Skew:	-0.59
Prob(H) (two-sided):	0.00	Kurtosis:	5.31

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

uc.Consumption.plot()
plt.plot(m.predict(start=0, end=186))

[<matplotlib.lines.Line2D at 0x2d3766d3e50>]

m = SARIMAX(y, x, order=(1, 0, 0)).fit()
m.summary()

SARIMAX Results

Dep. Variable:	y	No. Observations:	187
Model:	SARIMAX(1, 0, 0)	Log Likelihood	-166.064
Date:	Sat, 01 Jul 2023	AIC	340.129
Time:	19:39:51	BIC	353.053
Sample:	0	HQIC	345.366
	- 187
Covariance Type:	opg

	coef	std err	z	P>\|z\|	[0.025	0.975]
const	0.6093	0.062	9.898	0.000	0.489	0.730
x1	0.1907	0.034	5.633	0.000	0.124	0.257
ar.L1	0.2353	0.067	3.514	0.000	0.104	0.367
sigma2	0.3457	0.028	12.308	0.000	0.291	0.401

Ljung-Box (L1) (Q):	0.72	Jarque-Bera (JB):	25.86
Prob(Q):	0.40	Prob(JB):	0.00
Heteroskedasticity (H):	0.42	Skew:	-0.42
Prob(H) (two-sided):	0.00	Kurtosis:	4.62

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

uc.Consumption.plot()
plt.plot(m.predict(start=0, end=186, exog=x))

[<matplotlib.lines.Line2D at 0x2d3777a0220>]

<statsmodels.tsa.statespace.sarimax.SARIMAXResultsWrapper at 0x2d373963fd0>

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