%matplotlib inline

import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt

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']]

"""
Explanation: 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.
End of explanation
"""


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())

"""
Explanation: 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.
End of explanation
"""


ax = res.impulse_responses(10, orthogonalized=True).plot(figsize=(13,3))
ax.set(xlabel='t', title='Responses to a shock to `dln_inv`');

"""
Explanation: From the estimated VAR model, we can plot the impulse response functions of the endogenous variables.
End of explanation
"""


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())

"""
Explanation: 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.
End of explanation
"""


mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))
res = mod.fit(maxiter=1000, disp=False)
print(res.summary())

"""
Explanation: 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.
End of explanation
"""


