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                 361.037
Date:                    Tue, 30 Nov 2021   AIC                           -696.075
Time:                            08:18:35   BIC                           -665.947
Sample:                        04-01-1960   HQIC                          -684.045
                             - 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):            0.05, 10.07   Jarque-Bera (JB):          11.05, 2.46
Prob(Q):                        0.82, 0.00   Prob(JB):                   0.00, 0.29
Heteroskedasticity (H):         0.45, 0.40   Skew:                      0.16, -0.38
Prob(H) (two-sided):            0.05, 0.03   Kurtosis:                   4.85, 3.44
                            Results for equation dln_inv
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv          -0.2399      0.093     -2.578      0.010      -0.422      -0.058
L1.dln_inc           0.2776      0.449      0.618      0.536      -0.602       1.157
L2.dln_inv          -0.1654      0.155     -1.066      0.286      -0.470       0.139
L2.dln_inc           0.0643      0.421      0.153      0.879      -0.761       0.889
beta.dln_consump     0.9840      0.637      1.545      0.122      -0.264       2.232
                            Results for equation dln_inc
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv           0.0633      0.036      1.770      0.077      -0.007       0.133
L1.dln_inc           0.0803      0.107      0.750      0.453      -0.129       0.290
L2.dln_inv           0.0111      0.033      0.337      0.736      -0.054       0.076
L2.dln_inc           0.0335      0.134      0.250      0.803      -0.229       0.296
beta.dln_consump     0.7756      0.113      6.893      0.000       0.555       0.996
                                  Error covariance matrix
============================================================================================
                               coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0434      0.004     12.295      0.000       0.036       0.050
sqrt.cov.dln_inv.dln_inc  6.006e-05      0.002      0.030      0.976      -0.004       0.004
sqrt.var.dln_inc             0.0109      0.001     11.212      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`');
../../../_images/examples_notebooks_generated_statespace_varmax_8_0.png

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                 353.883
                              + intercept   AIC                           -683.766
Date:                    Tue, 30 Nov 2021   BIC                           -655.956
Time:                            08:18:42   HQIC                          -672.661
Sample:                        04-01-1960
                             - 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):             0.02, 0.05   Jarque-Bera (JB):         11.85, 13.52
Prob(Q):                        0.88, 0.83   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.44, 0.81   Skew:                      0.05, -0.48
Prob(H) (two-sided):            0.05, 0.60   Kurtosis:                   4.95, 4.84
                           Results for equation dln_inv
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0182      0.005      3.815      0.000       0.009       0.028
L1.e(dln_inv)    -0.2710      0.105     -2.579      0.010      -0.477      -0.065
L1.e(dln_inc)     0.5424      0.631      0.859      0.390      -0.695       1.780
L2.e(dln_inv)     0.0397      0.146      0.271      0.786      -0.247       0.326
L2.e(dln_inc)     0.1665      0.478      0.348      0.728      -0.770       1.103
                           Results for equation dln_inc
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0207      0.002     12.977      0.000       0.018       0.024
L1.e(dln_inv)     0.0483      0.042      1.158      0.247      -0.033       0.130
L1.e(dln_inc)    -0.0742      0.140     -0.532      0.595      -0.348       0.199
L2.e(dln_inv)     0.0172      0.042      0.406      0.685      -0.066       0.100
L2.e(dln_inc)     0.1313      0.152      0.861      0.389      -0.168       0.430
                             Error covariance matrix
==================================================================================
                     coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------
sigma2.dln_inv     0.0020      0.000      7.384      0.000       0.001       0.003
sigma2.dln_inc     0.0001   2.34e-05      5.812      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())
/home/runner/work/statsmodels/statsmodels/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                 354.290
                              + intercept   AIC                           -682.580
Date:                    Tue, 30 Nov 2021   BIC                           -652.452
Time:                            08:18:48   HQIC                          -670.550
Sample:                        04-01-1960
                             - 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):             0.00, 0.05   Jarque-Bera (JB):         11.18, 13.96
Prob(Q):                        0.96, 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:                   4.89, 4.91
                           Results for equation dln_inv
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0104      0.066      0.159      0.874      -0.118       0.139
L1.dln_inv       -0.0051      0.704     -0.007      0.994      -1.385       1.375
L1.dln_inc        0.3827      2.766      0.138      0.890      -5.039       5.805
L1.e(dln_inv)    -0.2475      0.714     -0.347      0.729      -1.647       1.152
L1.e(dln_inc)     0.1232      3.017      0.041      0.967      -5.791       6.037
                           Results for equation dln_inc
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0165      0.027      0.600      0.548      -0.037       0.070
L1.dln_inv       -0.0328      0.282     -0.117      0.907      -0.585       0.519
L1.dln_inc        0.2351      1.114      0.211      0.833      -1.947       2.418
L1.e(dln_inv)     0.0887      0.288      0.308      0.758      -0.476       0.654
L1.e(dln_inc)    -0.2393      1.148     -0.208      0.835      -2.490       2.012
                                  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      0.652      0.514      -0.003       0.007
sqrt.var.dln_inc             0.0116      0.001     11.729      0.000       0.010       0.013
============================================================================================

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