# 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:                    Thu, 14 Dec 2023   AIC                           -696.074
Time:                            14:41:53   BIC                           -665.946
Sample:                        04-01-1960   HQIC                          -684.044
- 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):            0.04, 10.22   Jarque-Bera (JB):          11.10, 2.44
Prob(Q):                        0.84, 0.00   Prob(JB):                   0.00, 0.30
Heteroskedasticity (H):         0.45, 0.40   Skew:                      0.16, -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          -0.2424      0.093     -2.618      0.009      -0.424      -0.061
L1.dln_inc           0.2827      0.448      0.631      0.528      -0.595       1.161
L2.dln_inv          -0.1621      0.155     -1.045      0.296      -0.466       0.142
L2.dln_inc           0.0816      0.421      0.194      0.846      -0.744       0.907
beta.dln_consump     0.9634      0.637      1.513      0.130      -0.285       2.212
Results for equation dln_inc
====================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv           0.0622      0.036      1.732      0.083      -0.008       0.133
L1.dln_inc           0.0824      0.107      0.768      0.443      -0.128       0.293
L2.dln_inv           0.0093      0.033      0.282      0.778      -0.055       0.074
L2.dln_inc           0.0328      0.134      0.244      0.807      -0.230       0.296
beta.dln_consump     0.7754      0.112      6.905      0.000       0.555       0.995
Error covariance matrix
============================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0433      0.004     12.327      0.000       0.036       0.050
sqrt.cov.dln_inv.dln_inc  3.564e-05      0.002      0.018      0.986      -0.004       0.004
sqrt.var.dln_inc             0.0109      0.001     11.202      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                           -683.776
Date:                    Thu, 14 Dec 2023   BIC                           -655.966
Time:                            14:42:10   HQIC                          -672.672
Sample:                        04-01-1960
- 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):             0.00, 0.05   Jarque-Bera (JB):         12.58, 13.55
Prob(Q):                        0.95, 0.82   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.44, 0.81   Skew:                      0.05, -0.48
Prob(H) (two-sided):            0.04, 0.60   Kurtosis:                   5.00, 4.84
Results for equation dln_inv
=================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0182      0.005      3.802      0.000       0.009       0.028
L1.e(dln_inv)    -0.2593      0.106     -2.454      0.014      -0.466      -0.052
L1.e(dln_inc)     0.5277      0.631      0.836      0.403      -0.709       1.764
L2.e(dln_inv)     0.0311      0.149      0.209      0.834      -0.261       0.323
L2.e(dln_inc)     0.1759      0.476      0.369      0.712      -0.757       1.109
Results for equation dln_inc
=================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0207      0.002     13.058      0.000       0.018       0.024
L1.e(dln_inv)     0.0482      0.042      1.159      0.246      -0.033       0.130
L1.e(dln_inc)    -0.0745      0.140     -0.533      0.594      -0.348       0.200
L2.e(dln_inv)     0.0178      0.042      0.419      0.675      -0.065       0.101
L2.e(dln_inc)     0.1257      0.153      0.823      0.411      -0.174       0.425
Error covariance matrix
==================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------
sigma2.dln_inv     0.0020      0.000      7.349      0.000       0.001       0.003
sigma2.dln_inc     0.0001   2.33e-05      5.828      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/3.10.13/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                 354.292
+ intercept   AIC                           -682.585
Date:                    Thu, 14 Dec 2023   BIC                           -652.458
Time:                            14:42:19   HQIC                          -670.555
Sample:                        04-01-1960
- 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):             0.00, 0.04   Jarque-Bera (JB):         11.27, 13.92
Prob(Q):                        0.96, 0.84   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:                   4.90, 4.91
Results for equation dln_inv
=================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0105      0.067      0.156      0.876      -0.121       0.142
L1.dln_inv       -0.0001      0.724     -0.000      1.000      -1.420       1.419
L1.dln_inc        0.3753      2.819      0.133      0.894      -5.149       5.900
L1.e(dln_inv)    -0.2513      0.734     -0.342      0.732      -1.690       1.188
L1.e(dln_inc)     0.1206      3.069      0.039      0.969      -5.894       6.135
Results for equation dln_inc
=================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
intercept         0.0164      0.028      0.583      0.560      -0.039       0.071
L1.dln_inv       -0.0334      0.289     -0.116      0.908      -0.601       0.534
L1.dln_inc        0.2389      1.139      0.210      0.834      -1.993       2.471
L1.e(dln_inv)     0.0891      0.296      0.300      0.764      -0.492       0.670
L1.e(dln_inc)    -0.2454      1.174     -0.209      0.834      -2.546       2.055
Error covariance matrix
============================================================================================
coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0449      0.003     14.510      0.000       0.039       0.051
sqrt.cov.dln_inv.dln_inc     0.0017      0.003      0.650      0.515      -0.003       0.007
sqrt.var.dln_inc             0.0116      0.001     11.723      0.000       0.010       0.013
============================================================================================

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


Last update: Dec 14, 2023