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                 361.039
Date:                    Sun, 26 Sep 2021   AIC                           -696.077
Time:                            19:46:48   BIC                           -665.950
Sample:                        04-01-1960   HQIC                          -684.048
                             - 10-01-1978
Covariance Type:                      opg
===================================================================================
Ljung-Box (L1) (Q):            0.05, 10.04   Jarque-Bera (JB):          11.31, 2.39
Prob(Q):                        0.82, 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:                   4.88, 3.44
                            Results for equation dln_inv
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv          -0.2391      0.093     -2.568      0.010      -0.422      -0.057
L1.dln_inc           0.2886      0.449      0.643      0.521      -0.592       1.169
L2.dln_inv          -0.1650      0.155     -1.062      0.288      -0.469       0.139
L2.dln_inc           0.0761      0.422      0.180      0.857      -0.751       0.903
beta.dln_consump     0.9568      0.639      1.498      0.134      -0.295       2.209
                            Results for equation dln_inc
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv           0.0636      0.036      1.782      0.075      -0.006       0.134
L1.dln_inc           0.0821      0.107      0.767      0.443      -0.128       0.292
L2.dln_inv           0.0107      0.033      0.324      0.746      -0.054       0.075
L2.dln_inc           0.0351      0.134      0.262      0.794      -0.228       0.298
beta.dln_consump     0.7710      0.112      6.864      0.000       0.551       0.991
                                  Error covariance matrix
============================================================================================
                               coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0434      0.004     12.287      0.000       0.036       0.050
sqrt.cov.dln_inv.dln_inc  6.088e-05      0.002      0.030      0.976      -0.004       0.004
sqrt.var.dln_inc             0.0109      0.001     11.226      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                 353.888
                              + intercept   AIC                           -683.775
Date:                    Sun, 26 Sep 2021   BIC                           -655.965
Time:                            19:46:54   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())
/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                 354.288
                              + intercept   AIC                           -682.575
Date:                    Sun, 26 Sep 2021   BIC                           -652.448
Time:                            19:46:56   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).