VARMAX models¶
This is a brief introduction notebook to VARMAX models in statsmodels. The VARMAX model is generically specified as:
\[y_t = \nu + A_1 y_{t-1} + \dots + A_p y_{t-p} + B x_t + \epsilon_t +
M_1 \epsilon_{t-1} + \dots M_q \epsilon_{t-q}\]
where \(y_t\) is a \(\text{k_endog} \times 1\) vector.
[1]:
%matplotlib inline
[2]:
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
[3]:
dta = sm.datasets.webuse('lutkepohl2', 'https://www.stata-press.com/data/r12/')
dta.index = dta.qtr
dta.index.freq = dta.index.inferred_freq
endog = dta.loc['1960-04-01':'1978-10-01', ['dln_inv', 'dln_inc', 'dln_consump']]
Model specification¶
The VARMAX class in statsmodels allows estimation of VAR, VMA, and VARMA models (through the order argument), optionally with a constant term (via the trend argument). Exogenous regressors may also be included (as usual in statsmodels, by the exog argument), and in this way a time trend may be added. Finally, the class allows measurement error (via the measurement_error argument) and allows specifying either a diagonal or unstructured innovation covariance matrix (via the
error_cov_type argument).
Example 1: VAR¶
Below is a simple VARX(2) model in two endogenous variables and an exogenous series, but no constant term. Notice that we needed to allow for more iterations than the default (which is maxiter=50) in order for the likelihood estimation to converge. This is not unusual in VAR models which have to estimate a large number of parameters, often on a relatively small number of time series: this model, for example, estimates 27 parameters off of 75 observations of 3 variables.
[4]:
exog = endog['dln_consump']
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(2,0), trend='n', exog=exog)
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
Statespace Model Results
==================================================================================
Dep. Variable: ['dln_inv', 'dln_inc'] No. Observations: 75
Model: VARX(2) Log Likelihood 361.034
Date: Fri, 01 Sep 2023 AIC -696.068
Time: 11:17:11 BIC -665.941
Sample: 04-01-1960 HQIC -684.038
- 10-01-1978
Covariance Type: opg
===================================================================================
Ljung-Box (L1) (Q): 0.04, 9.99 Jarque-Bera (JB): 11.10, 2.43
Prob(Q): 0.84, 0.00 Prob(JB): 0.00, 0.30
Heteroskedasticity (H): 0.45, 0.40 Skew: 0.15, -0.38
Prob(H) (two-sided): 0.05, 0.03 Kurtosis: 4.86, 3.44
Results for equation dln_inv
====================================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------------
L1.dln_inv -0.2419 0.093 -2.598 0.009 -0.424 -0.059
L1.dln_inc 0.2741 0.448 0.613 0.540 -0.603 1.151
L2.dln_inv -0.1632 0.155 -1.052 0.293 -0.467 0.141
L2.dln_inc 0.0788 0.423 0.187 0.852 -0.750 0.907
beta.dln_consump 0.9662 0.638 1.514 0.130 -0.285 2.217
Results for equation dln_inc
====================================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------------
L1.dln_inv 0.0634 0.036 1.773 0.076 -0.007 0.133
L1.dln_inc 0.0857 0.107 0.804 0.421 -0.123 0.295
L2.dln_inv 0.0113 0.033 0.341 0.733 -0.053 0.076
L2.dln_inc 0.0388 0.135 0.289 0.773 -0.225 0.302
beta.dln_consump 0.7638 0.113 6.789 0.000 0.543 0.984
Error covariance matrix
============================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv 0.0434 0.004 12.267 0.000 0.036 0.050
sqrt.cov.dln_inv.dln_inc 5.933e-05 0.002 0.029 0.977 -0.004 0.004
sqrt.var.dln_inc 0.0109 0.001 11.228 0.000 0.009 0.013
============================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
From the estimated VAR model, we can plot the impulse response functions of the endogenous variables.
[5]:
ax = res.impulse_responses(10, orthogonalized=True, impulse=[1, 0]).plot(figsize=(13,3))
ax.set(xlabel='t', title='Responses to a shock to `dln_inv`');
Example 2: VMA¶
A vector moving average model can also be formulated. Below we show a VMA(2) on the same data, but where the innovations to the process are uncorrelated. In this example we leave out the exogenous regressor but now include the constant term.
[6]:
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(0,2), error_cov_type='diagonal')
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
Statespace Model Results
==================================================================================
Dep. Variable: ['dln_inv', 'dln_inc'] No. Observations: 75
Model: VMA(2) Log Likelihood 353.888
+ intercept AIC -683.775
Date: Fri, 01 Sep 2023 BIC -655.965
Time: 11:17:19 HQIC -672.671
Sample: 04-01-1960
- 10-01-1978
Covariance Type: opg
===================================================================================
Ljung-Box (L1) (Q): 0.00, 0.05 Jarque-Bera (JB): 12.75, 13.71
Prob(Q): 0.95, 0.82 Prob(JB): 0.00, 0.00
Heteroskedasticity (H): 0.44, 0.81 Skew: 0.06, -0.49
Prob(H) (two-sided): 0.04, 0.60 Kurtosis: 5.02, 4.86
Results for equation dln_inv
=================================================================================
coef std err z P>|z| [0.025 0.975]
---------------------------------------------------------------------------------
intercept 0.0182 0.005 3.789 0.000 0.009 0.028
L1.e(dln_inv) -0.2579 0.106 -2.438 0.015 -0.465 -0.051
L1.e(dln_inc) 0.5084 0.630 0.807 0.420 -0.726 1.743
L2.e(dln_inv) 0.0313 0.149 0.210 0.834 -0.261 0.324
L2.e(dln_inc) 0.1934 0.476 0.406 0.685 -0.740 1.127
Results for equation dln_inc
=================================================================================
coef std err z P>|z| [0.025 0.975]
---------------------------------------------------------------------------------
intercept 0.0207 0.002 13.088 0.000 0.018 0.024
L1.e(dln_inv) 0.0478 0.042 1.148 0.251 -0.034 0.129
L1.e(dln_inc) -0.0737 0.140 -0.525 0.599 -0.349 0.201
L2.e(dln_inv) 0.0184 0.043 0.433 0.665 -0.065 0.102
L2.e(dln_inc) 0.1219 0.153 0.795 0.426 -0.178 0.422
Error covariance matrix
==================================================================================
coef std err z P>|z| [0.025 0.975]
----------------------------------------------------------------------------------
sigma2.dln_inv 0.0020 0.000 7.347 0.000 0.001 0.003
sigma2.dln_inc 0.0001 2.33e-05 5.831 0.000 9e-05 0.000
==================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
Caution: VARMA(p,q) specifications¶
Although the model allows estimating VARMA(p,q) specifications, these models are not identified without additional restrictions on the representation matrices, which are not built-in. For this reason, it is recommended that the user proceed with error (and indeed a warning is issued when these models are specified). Nonetheless, they may in some circumstances provide useful information.
[7]:
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
/opt/hostedtoolcache/Python/3.10.12/x64/lib/python3.10/site-packages/statsmodels/tsa/statespace/varmax.py:161: EstimationWarning: Estimation of VARMA(p,q) models is not generically robust, due especially to identification issues.
warn('Estimation of VARMA(p,q) models is not generically robust,'
Statespace Model Results
==================================================================================
Dep. Variable: ['dln_inv', 'dln_inc'] No. Observations: 75
Model: VARMA(1,1) Log Likelihood 354.288
+ intercept AIC -682.575
Date: Fri, 01 Sep 2023 BIC -652.448
Time: 11:17:22 HQIC -670.546
Sample: 04-01-1960
- 10-01-1978
Covariance Type: opg
===================================================================================
Ljung-Box (L1) (Q): 0.01, 0.05 Jarque-Bera (JB): 11.01, 13.99
Prob(Q): 0.93, 0.82 Prob(JB): 0.00, 0.00
Heteroskedasticity (H): 0.43, 0.91 Skew: 0.01, -0.45
Prob(H) (two-sided): 0.04, 0.81 Kurtosis: 4.88, 4.91
Results for equation dln_inv
=================================================================================
coef std err z P>|z| [0.025 0.975]
---------------------------------------------------------------------------------
intercept 0.0106 0.066 0.160 0.873 -0.119 0.141
L1.dln_inv -0.0079 0.707 -0.011 0.991 -1.394 1.378
L1.dln_inc 0.3757 2.795 0.134 0.893 -5.102 5.853
L1.e(dln_inv) -0.2476 0.718 -0.345 0.730 -1.655 1.160
L1.e(dln_inc) 0.1249 3.043 0.041 0.967 -5.839 6.089
Results for equation dln_inc
=================================================================================
coef std err z P>|z| [0.025 0.975]
---------------------------------------------------------------------------------
intercept 0.0165 0.028 0.595 0.552 -0.038 0.071
L1.dln_inv -0.0328 0.282 -0.116 0.907 -0.586 0.521
L1.dln_inc 0.2324 1.129 0.206 0.837 -1.980 2.445
L1.e(dln_inv) 0.0885 0.289 0.306 0.759 -0.478 0.655
L1.e(dln_inc) -0.2364 1.163 -0.203 0.839 -2.516 2.043
Error covariance matrix
============================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv 0.0449 0.003 14.527 0.000 0.039 0.051
sqrt.cov.dln_inv.dln_inc 0.0017 0.003 0.651 0.515 -0.003 0.007
sqrt.var.dln_inc 0.0116 0.001 11.708 0.000 0.010 0.013
============================================================================================
Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
Last update:
Sep 01, 2023