# VAR models¶

## Introduction¶

Vector autoregressions (VARs) were introduced in the econometrics literature in the 1980s to allow for (linear) dependencies among multiple variables. For a $$K$$ x $$1$$ vector $$y_{t}$$ we can specify a VAR(p) model as:

$y_{t} = c + A_{1}y_{t-1} + ... + A_{p}y_{t-p} + e_{t}$

These models can be estimated quickly through OLS. But with a large number of dependent variables, the number of parameters to be estimated can grow very quickly. See the notebook on Bayesian VARs for an alternative way to approach these types of model.

## Example¶

We’ll run an VAR model for US banking sector stocks.

import numpy as np
import pyflux as pf
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline

ibm = DataReader(['JPM','GS','BAC','C','WFC','MS'],  'yahoo', datetime(2012,1,1), datetime(2016,6,28))
opening_prices = np.log(ibm['Open'])
plt.figure(figsize=(15,5));
plt.plot(opening_prices.index,opening_prices);
plt.legend(opening_prices.columns.values,loc=3);
plt.title("Logged opening price");


Here we specify an arbitrary VAR(2) model, which we fit via OLS:

model = pf.VAR(data=opening_prices, lags=2, integ=1)


Next we estimate the latent variables. For this example we will use an OLS estimate $$z^{OLS}$$:

x = model.fit()
x.summary()

VAR(2)
======================================== =================================================
Dependent Variable: Differenced BAC      Method: OLS
Start Date: 2012-01-05 00:00:00          Log Likelihood: 21547.2578
End Date: 2016-06-28 00:00:00            AIC: -42896.5156
Number of observations: 1126             BIC: -42398.8994
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
Diff BAC Constant         0.0007     0.0006     1.2007   0.2299   (-0.0004 | 0.0018)
Diff BAC AR(1)            -0.0525    0.0005     -97.5672 0.0      (-0.0535 | -0.0514)
Diff C to Diff BAC AR(1)  -0.0616    0.0004     -143.365 0.0      (-0.0625 | -0.0608)
Diff GS to Diff BAC AR(1) 0.0595     0.0004     132.6638 0.0      (0.0587 | 0.0604)
Diff JPM to Diff BAC AR(1)0.0296     0.0006     49.3563  0.0      (0.0284 | 0.0308)
Diff MS to Diff BAC AR(1) -0.0231    0.0004     -62.6218 0.0      (-0.0239 | -0.0224)
Diff WFC to Diff BAC AR(1)-0.0417    0.0598     -0.6968  0.4859   (-0.159 | 0.0756)
Diff BAC AR(2)            0.1171     0.0555     2.1087   0.035    (0.0083 | 0.226)
Diff C to Diff BAC AR(2)  -0.1266    0.0444     -2.8528  0.0043   (-0.2136 | -0.0396)
Diff GS to Diff BAC AR(2) 0.1698     0.0464     3.6618   0.0003   (0.0789 | 0.2606)
Diff JPM to Diff BAC AR(2)-0.0959    0.062      -1.5472  0.1218   (-0.2174 | 0.0256)
Diff MS to Diff BAC AR(2) -0.0213    0.0381     -0.557   0.5775   (-0.096 | 0.0535)
Diff WFC to Diff BAC AR(2)-0.001     0.0701     -0.0149  0.9881   (-0.1384 | 0.1363)
Diff C Constant           0.0003     0.065      0.0047   0.9962   (-0.1272 | 0.1278)
Diff C AR(1)              0.0193     0.052      0.3706   0.7109   (-0.0826 | 0.1211)
Diff BAC to Diff C AR(1)  -0.0576    0.0543     -1.0613  0.2886   (-0.164 | 0.0488)
Diff GS to Diff C AR(1)   0.0579     0.0726     0.7979   0.4249   (-0.0844 | 0.2002)
Diff JPM to Diff C AR(1)  0.0831     0.0447     1.8595   0.063    (-0.0045 | 0.1706)
Diff MS to Diff C AR(1)   -0.037     0.0794     -0.4657  0.6414   (-0.1925 | 0.1186)
Diff WFC to Diff C AR(1)  -0.1785    0.0737     -2.4235  0.0154   (-0.3229 | -0.0341)
Diff C AR(2)              0.1612     0.0589     2.7379   0.0062   (0.0458 | 0.2765)
Diff BAC to Diff C AR(2)  -0.1021    0.0615     -1.6598  0.0969   (-0.2226 | 0.0185)
Diff GS to Diff C AR(2)   0.1109     0.0822     1.3483   0.1776   (-0.0503 | 0.272)
Diff JPM to Diff C AR(2)  -0.0453    0.0506     -0.8946  0.371    (-0.1444 | 0.0539)
Diff MS to Diff C AR(2)   0.0127     0.0775     0.1643   0.8695   (-0.1391 | 0.1646)
Diff WFC to Diff C AR(2)  -0.1313    0.0719     -1.8261  0.0678   (-0.2723 | 0.0096)
Diff GS Constant          0.0003     0.0575     0.006    0.9952   (-0.1123 | 0.113)
Diff GS AR(1)             -0.016     0.06       -0.266   0.7903   (-0.1336 | 0.1017)
Diff BAC to Diff GS AR(1) 0.0051     0.0803     0.0633   0.9495   (-0.1523 | 0.1624)
Diff C to Diff GS AR(1)   -0.0785    0.0494     -1.5891  0.112    (-0.1753 | 0.0183)
Diff JPM to Diff GS AR(1) 0.0507     0.0575     0.8814   0.3781   (-0.062 | 0.1633)
Diff MS to Diff GS AR(1)  0.0425     0.0534     0.7961   0.4259   (-0.0621 | 0.1471)
Diff WFC to Diff GS AR(1) -0.0613    0.0426     -1.4376  0.1505   (-0.1449 | 0.0223)
Diff GS AR(2)             0.0865     0.0445     1.9422   0.0521   (-0.0008 | 0.1738)
Diff BAC to Diff GS AR(2) -0.1896    0.0596     -3.1832  0.0015   (-0.3064 | -0.0729)
Diff C to Diff GS AR(2)   0.0423     0.0367     1.1553   0.248    (-0.0295 | 0.1142)
Diff JPM to Diff GS AR(2) 0.0667     0.0769     0.8664   0.3863   (-0.0841 | 0.2174)
Diff MS to Diff GS AR(2)  0.0433     0.0714     0.6067   0.5441   (-0.0966 | 0.1833)
Diff WFC to Diff GS AR(2) -0.0362    0.0571     -0.6347  0.5256   (-0.1481 | 0.0756)
Diff JPM Constant         0.0005     0.0596     0.0082   0.9934   (-0.1163 | 0.1173)
Diff JPM AR(1)            -0.0304    0.0797     -0.3813  0.703    (-0.1866 | 0.1258)
Diff BAC to Diff JPM AR(1)-0.0281    0.049      -0.5738  0.5661   (-0.1243 | 0.068)
Diff C to Diff JPM AR(1)  0.0695     0.0594     1.1698   0.2421   (-0.047 | 0.186)
Diff GS to Diff JPM AR(1) -0.0106    0.0552     -0.1924  0.8474   (-0.1187 | 0.0975)
Diff MS to Diff JPM AR(1) -0.0338    0.0441     -0.7675  0.4428   (-0.1202 | 0.0526)
Diff WFC to Diff JPM AR(1)-0.0725    0.046      -1.5744  0.1154   (-0.1627 | 0.0178)
Diff JPM AR(2)            0.096      0.0616     1.559    0.119    (-0.0247 | 0.2167)
Diff BAC to Diff JPM AR(2)-0.1246    0.0379     -3.2883  0.001    (-0.1989 | -0.0503)
Diff C to Diff JPM AR(2)  0.0229     0.0696     0.3284   0.7426   (-0.1136 | 0.1593)
Diff GS to Diff JPM AR(2) -0.0084    0.0646     -0.1301  0.8965   (-0.1351 | 0.1182)
Diff MS to Diff JPM AR(2) 0.0319     0.0516     0.6182   0.5364   (-0.0693 | 0.1332)
Diff WFC to Diff JPM AR(2)-0.0117    0.0539     -0.2161  0.8289   (-0.1174 | 0.0941)
Diff MS Constant          0.0004     0.0721     0.005    0.996    (-0.141 | 0.1417)
Diff MS AR(1)             0.0249     0.0444     0.5605   0.5752   (-0.0621 | 0.1119)
Diff BAC to Diff MS AR(1) 0.0456     0.0783     0.5833   0.5597   (-0.1077 | 0.199)
Diff C to Diff MS AR(1)   0.0083     0.0726     0.1148   0.9086   (-0.134 | 0.1507)
Diff GS to Diff MS AR(1)  0.1319     0.0581     2.2717   0.0231   (0.0181 | 0.2457)
Diff JPM to Diff MS AR(1) -0.1771    0.0606     -2.9213  0.0035   (-0.296 | -0.0583)
Diff WFC to Diff MS AR(1) -0.151     0.0811     -1.8629  0.0625   (-0.31 | 0.0079)
Diff MS AR(2)             0.1512     0.0499     3.0308   0.0024   (0.0534 | 0.249)
Diff BAC to Diff MS AR(2) -0.2173    0.0772     -2.8157  0.0049   (-0.3686 | -0.066)
Diff C to Diff MS AR(2)   0.1827     0.0716     2.5499   0.0108   (0.0423 | 0.3231)
Diff GS to Diff MS AR(2)  -0.0107    0.0573     -0.1873  0.8514   (-0.1229 | 0.1015)
Diff JPM to Diff MS AR(2) 0.0004     0.0598     0.0066   0.9947   (-0.1168 | 0.1176)
Diff WFC to Diff MS AR(2) -0.0697    0.08       -0.8711  0.3837   (-0.2264 | 0.0871)
Diff WFC Constant         0.0005     0.0492     0.0095   0.9924   (-0.096 | 0.0969)
Diff WFC AR(1)            0.0092     0.0574     0.1611   0.872    (-0.1032 | 0.1217)
Diff BAC to Diff WFC AR(1)-0.0059    0.0532     -0.1113  0.9114   (-0.1103 | 0.0984)
Diff C to Diff WFC AR(1)  0.0062     0.0425     0.1448   0.8848   (-0.0772 | 0.0896)
Diff GS to Diff WFC AR(1) 0.0525     0.0444     1.1811   0.2376   (-0.0346 | 0.1396)
Diff JPM to Diff WFC AR(1)-0.0047    0.0594     -0.0792  0.9368   (-0.1212 | 0.1118)
Diff MS to Diff WFC AR(1) -0.1996    0.0366     -5.4578  0.0      (-0.2713 | -0.1279)
Diff WFC AR(2)            0.0291     0.0773     0.3759   0.707    (-0.1225 | 0.1806)
Diff BAC to Diff WFC AR(2)-0.0509    0.0718     -0.7087  0.4785   (-0.1915 | 0.0898)
Diff C to Diff WFC AR(2)  0.0255     0.0574     0.4444   0.6567   (-0.0869 | 0.1379)
Diff GS to Diff WFC AR(2) 0.0235     0.0599     0.3922   0.6949   (-0.0939 | 0.1409)
Diff JPM to Diff WFC AR(2)0.015      0.0801     0.1878   0.851    (-0.142 | 0.1721)
Diff MS to Diff WFC AR(2) -0.0556    0.0493     -1.1276  0.2595   (-0.1522 | 0.041)
==========================================================================================


We can plot latent variables with plot_z(): method:

model.plot_z(list(range(0,6)),figsize=(15,5))


We can plot the in-sample fit with plot_fit():

model.plot_fit(figsize=(15,5))


We can make forward predictions with our model using plot_predict():

model.plot_predict(past_values=19, h=5, figsize=(15,5))


How does our model perform? We can get a sense by performing a rolling in-sample prediction – plot_predict_is(): for plotted graphs:

model.plot_predict_is(h=30, figsize=((15,5)))


## Class Description¶

class VAR(data, lags, integ, target, use_ols_covariance)

Vector Autoregression Models (VAR).

Parameter Type Description
data pd.DataFrame or np.ndarray Contains the univariate time series
lags int The number of autoregressive lags
integ int How many times to difference the data (default: 0)
target string or int Which column of DataFrame/array to use.
use_ols_covariance boolean Whether to use fixed OLS covariance

Attributes

latent_variables

A pf.LatentVariables() object containing information on the model latent variables, prior settings. any fitted values, starting values, and other latent variable information. When a model is fitted, this is where the latent variables are updated/stored. Please see the documentation on Latent Variables for information on attributes within this object, as well as methods for accessing the latent variable information.

Methods

adjust_prior(index, prior)

Adjusts the priors for the model latent variables. The latent variables and their indices can be viewed by printing the latent_variables attribute attached to the model instance.

Parameter Type Description
index int Index of the latent variable to change
prior pf.Family instance Prior distribution, e.g. pf.Normal()

Returns: void - changes the model latent_variables attribute

fit(method, **kwargs)

Estimates latent variables for the model. User chooses an inference option and the method returns a results object, as well as updating the model’s latent_variables attribute.

Parameter Type Description
method str Inference option: e.g. ‘M-H’ or ‘MLE’

See Bayesian Inference and Classical Inference sections of the documentation for the full list of inference options. Optional parameters can be entered that are relevant to the particular mode of inference chosen.

Returns: pf.Results instance with information for the estimated latent variables

plot_fit(**kwargs)

Plots the fit of the model against the data. Optional arguments include figsize, the dimensions of the figure to plot.

Returns : void - shows a matplotlib plot

plot_predict(h, past_values, intervals, **kwargs)

Plots predictions of the model, along with intervals.

Parameter Type Description
h int How many steps to forecast ahead
past_values int How many past datapoints to plot
intervals boolean Whether to plot intervals or not

Optional arguments include figsize - the dimensions of the figure to plot. Please note that if you use Maximum Likelihood or Variational Inference, the intervals shown will not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesian prediction intervals. Bayesian intervals with variational inference are not shown because of the limitation of mean-field inference in not accounting for posterior correlations.

Returns : void - shows a matplotlib plot

plot_predict_is(h, fit_once, fit_method, **kwargs)

Plots in-sample rolling predictions for the model. This means that the user pretends a last subsection of data is out-of-sample, and forecasts after each period and assesses how well they did. The user can choose whether to fit parameters once at the beginning or every time step.

Parameter Type Description
h int How many previous timesteps to use
fit_once boolean Whether to fit once, or every timestep
fit_method str Which inference option, e.g. ‘MLE’

Optional arguments include figsize - the dimensions of the figure to plot. h is an int of how many previous steps to simulate performance on.

Returns : void - shows a matplotlib plot

plot_z(indices, figsize)

Returns a plot of the latent variables and their associated uncertainty.

Parameter Type Description
indices int or list Which latent variable indices to plot
figsize tuple Size of the matplotlib figure

Returns : void - shows a matplotlib plot

predict(h)

Returns a DataFrame of model predictions.

Parameter Type Description
h int How many steps to forecast ahead

Returns : pd.DataFrame - the model predictions

predict_is(h, fit_once, fit_method)

Returns DataFrame of in-sample rolling predictions for the model.

Parameter Type Description
h int How many previous timesteps to use
fit_once boolean Whether to fit once, or every timestep
fit_method str Which inference option, e.g. ‘MLE’

Returns : pd.DataFrame - the model predictions

simulation_smoother(beta)

Returns np.ndarray of draws of the data from the Durbin and Koopman (2002) simulation smoother.

Parameter Type Description
beta np.array np.array of latent variables

Recommended just to use model.latent_variables.get_z_values() for the beta input, if you have already fit a model.

Returns : np.ndarray - samples from simulation smoother

## References¶

Lütkepohl, H. & Kraetzig, M. (2004). Applied Time Series Econometrics. Cambridge University Press, Cambridge.