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 pandas_datareader import DataReader
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.