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.

In [1]:
%matplotlib inline
In [2]:
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
In [3]:
dta = sm.datasets.webuse('lutkepohl2', 'http://www.stata-press.com/data/r12/')
dta.index = dta.qtr
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.

In [4]:
exog = endog['dln_consump']
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(2,0), trend='nc', exog=exog)
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
/Users/taugspurger/sandbox/statsmodels/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency QS-OCT will be used.
  % freq, ValueWarning)
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                            VARX(2)   Log Likelihood                 348.270
Date:                    Mon, 14 May 2018   AIC                           -670.540
Time:                            21:45:05   BIC                           -640.413
Sample:                        04-01-1960   HQIC                          -658.511
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                58.29, 41.41   Jarque-Bera (JB):         17.34, 10.35
Prob(Q):                        0.03, 0.41   Prob(JB):                   0.00, 0.01
Heteroskedasticity (H):         0.45, 1.13   Skew:                      0.07, -0.55
Prob(H) (two-sided):            0.05, 0.77   Kurtosis:                   5.35, 4.45
                            Results for equation dln_inv                            
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv          -0.2766      0.088     -3.141      0.002      -0.449      -0.104
L1.dln_inc           0.3241      0.621      0.522      0.601      -0.892       1.540
L2.dln_inv          -0.1158      0.157     -0.738      0.461      -0.423       0.192
L2.dln_inc           0.4004      0.387      1.034      0.301      -0.359       1.159
beta.dln_consump     0.5566      0.753      0.739      0.460      -0.919       2.032
                            Results for equation dln_inc                            
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv           0.0339      0.042      0.805      0.421      -0.049       0.117
L1.dln_inc           0.0928      0.132      0.700      0.484      -0.167       0.352
L2.dln_inv           0.0517      0.051      1.018      0.309      -0.048       0.151
L2.dln_inc           0.2731      0.169      1.617      0.106      -0.058       0.604
beta.dln_consump     0.4837      0.199      2.430      0.015       0.094       0.874
                                  Error covariance matrix                                   
============================================================================================
                               coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0441      0.003     14.176      0.000       0.038       0.050
sqrt.cov.dln_inv.dln_inc     0.0013      0.002      0.553      0.581      -0.003       0.006
sqrt.var.dln_inc            -0.0127      0.001    -12.409      0.000      -0.015      -0.011
============================================================================================

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.

In [5]:
ax = res.impulse_responses(10, orthogonalized=True).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.

In [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())
/Users/taugspurger/sandbox/statsmodels/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency QS-OCT will be used.
  % freq, ValueWarning)
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                             VMA(2)   Log Likelihood                 353.887
                              + intercept   AIC                           -683.775
Date:                    Mon, 14 May 2018   BIC                           -655.965
Time:                            21:45:10   HQIC                          -672.670
Sample:                        04-01-1960                                         
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                68.73, 39.27   Jarque-Bera (JB):         12.41, 13.31
Prob(Q):                        0.00, 0.50   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:                   4.99, 4.83
                           Results for equation dln_inv                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0182      0.005      3.794      0.000       0.009       0.028
L1.e(dln_inv)    -0.2621      0.106     -2.482      0.013      -0.469      -0.055
L1.e(dln_inc)     0.5296      0.632      0.839      0.402      -0.708       1.768
L2.e(dln_inv)     0.0343      0.149      0.231      0.818      -0.257       0.326
L2.e(dln_inc)     0.1744      0.478      0.365      0.715      -0.762       1.111
                           Results for equation dln_inc                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0207      0.002     13.120      0.000       0.018       0.024
L1.e(dln_inv)     0.0482      0.042      1.158      0.247      -0.033       0.130
L1.e(dln_inc)    -0.0780      0.139     -0.560      0.576      -0.351       0.195
L2.e(dln_inv)     0.0175      0.042      0.412      0.680      -0.066       0.101
L2.e(dln_inc)     0.1247      0.152      0.820      0.412      -0.173       0.423
                             Error covariance matrix                              
==================================================================================
                     coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------
sigma2.dln_inv     0.0020      0.000      7.353      0.000       0.001       0.003
sigma2.dln_inc     0.0001   2.33e-05      5.833      0.000    9.01e-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.

In [7]:
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
/Users/taugspurger/sandbox/statsmodels/statsmodels/tsa/statespace/varmax.py:152: EstimationWarning: Estimation of VARMA(p,q) models is not generically robust, due especially to identification issues.
  EstimationWarning)
/Users/taugspurger/sandbox/statsmodels/statsmodels/tsa/base/tsa_model.py:171: ValueWarning: No frequency information was provided, so inferred frequency QS-OCT will be used.
  % freq, ValueWarning)
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                         VARMA(1,1)   Log Likelihood                 354.283
                              + intercept   AIC                           -682.567
Date:                    Mon, 14 May 2018   BIC                           -652.439
Time:                            21:45:12   HQIC                          -670.537
Sample:                        04-01-1960                                         
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                68.76, 39.00   Jarque-Bera (JB):         10.81, 14.04
Prob(Q):                        0.00, 0.52   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.43, 0.91   Skew:                      0.00, -0.45
Prob(H) (two-sided):            0.04, 0.81   Kurtosis:                   4.86, 4.91
                           Results for equation dln_inv                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0110      0.067      0.164      0.870      -0.120       0.142
L1.dln_inv       -0.0055      0.722     -0.008      0.994      -1.421       1.410
L1.dln_inc        0.3587      2.788      0.129      0.898      -5.105       5.822
L1.e(dln_inv)    -0.2537      0.734     -0.346      0.729      -1.691       1.184
L1.e(dln_inc)     0.1262      3.028      0.042      0.967      -5.809       6.061
                           Results for equation dln_inc                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0165      0.028      0.584      0.559      -0.039       0.072
L1.dln_inv       -0.0344      0.288     -0.119      0.905      -0.599       0.530
L1.dln_inc        0.2348      1.144      0.205      0.837      -2.008       2.478
L1.e(dln_inv)     0.0904      0.295      0.307      0.759      -0.487       0.668
L1.e(dln_inc)    -0.2390      1.177     -0.203      0.839      -2.545       2.067
                                  Error covariance matrix                                   
============================================================================================
                               coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0449      0.003     14.529      0.000       0.039       0.051
sqrt.cov.dln_inv.dln_inc     0.0017      0.003      0.641      0.522      -0.003       0.007
sqrt.var.dln_inc             0.0116      0.001     11.648      0.000       0.010       0.013
============================================================================================

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