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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.ix['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())
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                            VARX(2)   Log Likelihood                 348.269
Date:                    Tue, 28 Feb 2017   AIC                           -670.537
Time:                            21:35:16   BIC                           -640.410
Sample:                        04-01-1960   HQIC                          -658.508
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                59.41, 42.46   Jarque-Bera (JB):         16.41, 13.03
Prob(Q):                        0.02, 0.37   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.46, 1.03   Skew:                      0.05, -0.64
Prob(H) (two-sided):            0.06, 0.95   Kurtosis:                   5.29, 4.59
                            Results for equation dln_inv                            
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv          -0.2773      0.088     -3.149      0.002      -0.450      -0.105
L1.dln_inc           0.3374      0.620      0.544      0.586      -0.877       1.552
L2.dln_inv          -0.1160      0.157     -0.740      0.459      -0.423       0.191
L2.dln_inc           0.3971      0.387      1.025      0.305      -0.362       1.156
beta.dln_consump     0.5448      0.752      0.724      0.469      -0.929       2.019
                            Results for equation dln_inc                            
====================================================================================
                       coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------
L1.dln_inv           0.0335      0.042      0.794      0.427      -0.049       0.116
L1.dln_inc           0.0962      0.133      0.726      0.468      -0.164       0.356
L2.dln_inv           0.0516      0.051      1.017      0.309      -0.048       0.151
L2.dln_inc           0.2734      0.169      1.622      0.105      -0.057       0.604
beta.dln_consump     0.4818      0.198      2.427      0.015       0.093       0.871
                                  Error covariance matrix                                   
============================================================================================
                               coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------------------
sqrt.var.dln_inv             0.0441      0.003     14.164      0.000       0.038       0.050
sqrt.cov.dln_inv.dln_inc     0.0013      0.002      0.549      0.583      -0.003       0.006
sqrt.var.dln_inc            -0.0127      0.001    -12.420      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())
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                             VMA(2)   Log Likelihood                 353.887
                              + intercept   AIC                           -683.775
Date:                    Tue, 28 Feb 2017   BIC                           -655.965
Time:                            21:35:20   HQIC                          -672.670
Sample:                        04-01-1960                                         
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                68.49, 39.14   Jarque-Bera (JB):         12.79, 13.08
Prob(Q):                        0.00, 0.51   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.44, 0.81   Skew:                      0.06, -0.48
Prob(H) (two-sided):            0.04, 0.60   Kurtosis:                   5.02, 4.81
                           Results for equation dln_inv                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0182      0.005      3.800      0.000       0.009       0.028
L1.e(dln_inv)    -0.2572      0.106     -2.430      0.015      -0.465      -0.050
L1.e(dln_inc)     0.5135      0.633      0.812      0.417      -0.726       1.753
L2.e(dln_inv)     0.0295      0.149      0.198      0.843      -0.263       0.322
L2.e(dln_inc)     0.1819      0.475      0.383      0.702      -0.749       1.113
                           Results for equation dln_inc                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0207      0.002     13.087      0.000       0.018       0.024
L1.e(dln_inv)     0.0487      0.042      1.173      0.241      -0.033       0.130
L1.e(dln_inc)    -0.0789      0.139     -0.568      0.570      -0.351       0.193
L2.e(dln_inv)     0.0176      0.042      0.415      0.678      -0.066       0.101
L2.e(dln_inc)     0.1290      0.153      0.844      0.399      -0.170       0.428
                             Error covariance matrix                              
==================================================================================
                     coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------
sigma2.dln_inv     0.0020      0.000      7.351      0.000       0.001       0.003
sigma2.dln_inc     0.0001   2.33e-05      5.822      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.

In [7]:
mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())
/private/tmp/statsmodels/statsmodels/tsa/statespace/varmax.py:153: EstimationWarning: Estimation of VARMA(p,q) models is not generically robust, due especially to identification issues.
  EstimationWarning)
                             Statespace Model Results                             
==================================================================================
Dep. Variable:     ['dln_inv', 'dln_inc']   No. Observations:                   75
Model:                         VARMA(1,1)   Log Likelihood                 354.291
                              + intercept   AIC                           -682.583
Date:                    Tue, 28 Feb 2017   BIC                           -652.455
Time:                            21:35:24   HQIC                          -670.553
Sample:                        04-01-1960                                         
                             - 10-01-1978                                         
Covariance Type:                      opg                                         
===================================================================================
Ljung-Box (Q):                69.15, 40.99   Jarque-Bera (JB):         11.03, 18.14
Prob(Q):                        0.00, 0.43   Prob(JB):                   0.00, 0.00
Heteroskedasticity (H):         0.45, 0.78   Skew:                      0.01, -0.52
Prob(H) (two-sided):            0.05, 0.54   Kurtosis:                   4.88, 5.17
                           Results for equation dln_inv                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0103      0.064      0.161      0.872      -0.115       0.135
L1.dln_inv       -0.0059      0.685     -0.009      0.993      -1.349       1.337
L1.dln_inc        0.3915      2.708      0.145      0.885      -4.916       5.699
L1.e(dln_inv)    -0.2446      0.696     -0.352      0.725      -1.608       1.119
L1.e(dln_inc)     0.1210      2.963      0.041      0.967      -5.686       5.928
                           Results for equation dln_inc                          
=================================================================================
                    coef    std err          z      P>|z|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0164      0.026      0.619      0.536      -0.036       0.068
L1.dln_inv       -0.0318      0.273     -0.116      0.907      -0.568       0.504
L1.dln_inc        0.2372      1.079      0.220      0.826      -1.878       2.352
L1.e(dln_inv)     0.0872      0.280      0.312      0.755      -0.461       0.636
L1.e(dln_inc)    -0.2367      1.115     -0.212      0.832      -2.422       1.949
                                  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      0.654      0.513      -0.003       0.007
sqrt.var.dln_inc             0.0116      0.001     11.775      0.000       0.010       0.013
============================================================================================

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

© 2009–2012 Statsmodels Developers
© 2006–2008 Scipy Developers
© 2006 Jonathan E. Taylor
Licensed under the 3-clause BSD License.
http://www.statsmodels.org/stable/examples/notebooks/generated/statespace_varmax.html