Vector Autoregressions tsa.vector_ar¶
VAR(p) processes¶
We are interested in modeling a \(T \times K\) multivariate time series \(Y\), where \(T\) denotes the number of observations and \(K\) the number of variables. One way of estimating relationships between the time series and their lagged values is the vector autoregression process:
where \(A_i\) is a \(K \times K\) coefficient matrix.
We follow in large part the methods and notation of Lutkepohl (2005), which we will not develop here.
Model fitting¶
Note
The classes referenced below are accessible via the
statsmodels.tsa.api module.
To estimate a VAR model, one must first create the model using an ndarray of homogeneous or structured dtype. When using a structured or record array, the class will use the passed variable names. Otherwise they can be passed explicitly:
# some example data
In [1]: import numpy as np
In [2]: import pandas
In [3]: import statsmodels.api as sm
In [4]: from statsmodels.tsa.api import VAR, DynamicVAR
In [5]: mdata = sm.datasets.macrodata.load_pandas().data
# prepare the dates index
In [6]: dates = mdata[['year', 'quarter']].astype(int).astype(str)
In [7]: quarterly = dates["year"] + "Q" + dates["quarter"]
In [8]: from statsmodels.tsa.base.datetools import dates_from_str
In [9]: quarterly = dates_from_str(quarterly)
In [10]: mdata = mdata[['realgdp','realcons','realinv']]
In [11]: mdata.index = pandas.DatetimeIndex(quarterly)
In [12]: data = np.log(mdata).diff().dropna()
# make a VAR model
In [13]: model = VAR(data)
Note
The VAR class assumes that the passed time series are
stationary. Non-stationary or trending data can often be transformed to be
stationary by first-differencing or some other method. For direct analysis of
non-stationary time series, a standard stable VAR(p) model is not
appropriate.
To actually do the estimation, call the fit method with the desired lag order. Or you can have the model select a lag order based on a standard information criterion (see below):
In [14]: results = model.fit(2)
In [15]: results.summary()
Out[15]:
Summary of Regression Results
==================================
Model: VAR
Method: OLS
Date: Mon, 14, May, 2018
Time: 21:48:15
--------------------------------------------------------------------
No. of Equations: 3.00000 BIC: -27.5830
Nobs: 200.000 HQIC: -27.7892
Log likelihood: 1962.57 FPE: 7.42129e-13
AIC: -27.9293 Det(Omega_mle): 6.69358e-13
--------------------------------------------------------------------
Results for equation realgdp
==============================================================================
coefficient std. error t-stat prob
------------------------------------------------------------------------------
const 0.001527 0.001119 1.365 0.172
L1.realgdp -0.279435 0.169663 -1.647 0.100
L1.realcons 0.675016 0.131285 5.142 0.000
L1.realinv 0.033219 0.026194 1.268 0.205
L2.realgdp 0.008221 0.173522 0.047 0.962
L2.realcons 0.290458 0.145904 1.991 0.047
L2.realinv -0.007321 0.025786 -0.284 0.776
==============================================================================
Results for equation realcons
==============================================================================
coefficient std. error t-stat prob
------------------------------------------------------------------------------
const 0.005460 0.000969 5.634 0.000
L1.realgdp -0.100468 0.146924 -0.684 0.494
L1.realcons 0.268640 0.113690 2.363 0.018
L1.realinv 0.025739 0.022683 1.135 0.257
L2.realgdp -0.123174 0.150267 -0.820 0.412
L2.realcons 0.232499 0.126350 1.840 0.066
L2.realinv 0.023504 0.022330 1.053 0.293
==============================================================================
Results for equation realinv
==============================================================================
coefficient std. error t-stat prob
------------------------------------------------------------------------------
const -0.023903 0.005863 -4.077 0.000
L1.realgdp -1.970974 0.888892 -2.217 0.027
L1.realcons 4.414162 0.687825 6.418 0.000
L1.realinv 0.225479 0.137234 1.643 0.100
L2.realgdp 0.380786 0.909114 0.419 0.675
L2.realcons 0.800281 0.764416 1.047 0.295
L2.realinv -0.124079 0.135098 -0.918 0.358
==============================================================================
Correlation matrix of residuals
realgdp realcons realinv
realgdp 1.000000 0.603316 0.750722
realcons 0.603316 1.000000 0.131951
realinv 0.750722 0.131951 1.000000
Several ways to visualize the data using matplotlib are available.
Plotting input time series:
In [16]: results.plot()
Out[16]: <Figure size 1000x1000 with 3 Axes>
Plotting time series autocorrelation function:
In [17]: results.plot_acorr()
Out[17]: <Figure size 1000x1000 with 9 Axes>
Lag order selection¶
Choice of lag order can be a difficult problem. Standard analysis employs
likelihood test or information criteria-based order selection. We have
implemented the latter, accessible through the VAR class:
In [18]: model.select_order(15)
Out[18]: <statsmodels.tsa.vector_ar.var_model.LagOrderResults at 0x10c89fef0>
When calling the fit function, one can pass a maximum number of lags and the order criterion to use for order selection:
In [19]: results = model.fit(maxlags=15, ic='aic')
Forecasting¶
The linear predictor is the optimal h-step ahead forecast in terms of mean-squared error:
We can use the forecast function to produce this forecast. Note that we have to specify the “initial value” for the forecast:
In [20]: lag_order = results.k_ar
In [21]: results.forecast(data.values[-lag_order:], 5)
Out[21]:
array([[ 0.0062, 0.005 , 0.0092],
[ 0.0043, 0.0034, -0.0024],
[ 0.0042, 0.0071, -0.0119],
[ 0.0056, 0.0064, 0.0015],
[ 0.0063, 0.0067, 0.0038]])
The forecast_interval function will produce the above forecast along with asymptotic standard errors. These can be visualized using the plot_forecast function:
In [22]: results.plot_forecast(10)
Out[22]: <Figure size 1000x1000 with 3 Axes>
Impulse Response Analysis¶
Impulse responses are of interest in econometric studies: they are the estimated responses to a unit impulse in one of the variables. They are computed in practice using the MA(\(\infty\)) representation of the VAR(p) process:
We can perform an impulse response analysis by calling the irf function on a VARResults object:
In [23]: irf = results.irf(10)
These can be visualized using the plot function, in either orthogonalized or non-orthogonalized form. Asymptotic standard errors are plotted by default at the 95% significance level, which can be modified by the user.
Note
Orthogonalization is done using the Cholesky decomposition of the estimated error covariance matrix \(\hat \Sigma_u\) and hence interpretations may change depending on variable ordering.
In [24]: irf.plot(orth=False)
Out[24]: <Figure size 1000x1000 with 9 Axes>
Note the plot function is flexible and can plot only variables of interest if so desired:
In [25]: irf.plot(impulse='realgdp')
Out[25]: <Figure size 1000x1000 with 3 Axes>
The cumulative effects \(\Psi_n = \sum_{i=0}^n \Phi_i\) can be plotted with the long run effects as follows:
In [26]: irf.plot_cum_effects(orth=False)
Out[26]: <Figure size 1000x1000 with 9 Axes>
Forecast Error Variance Decomposition (FEVD)¶
Forecast errors of component j on k in an i-step ahead forecast can be decomposed using the orthogonalized impulse responses \(\Theta_i\):
These are computed via the fevd function up through a total number of steps ahead:
In [27]: fevd = results.fevd(5)
In [28]: fevd.summary()
FEVD for realgdp
realgdp realcons realinv
0 1.000000 0.000000 0.000000
1 0.864889 0.129253 0.005858
2 0.816725 0.177898 0.005378
3 0.793647 0.197590 0.008763
4 0.777279 0.208127 0.014594
FEVD for realcons
realgdp realcons realinv
0 0.359877 0.640123 0.000000
1 0.358767 0.635420 0.005813
2 0.348044 0.645138 0.006817
3 0.319913 0.653609 0.026478
4 0.317407 0.652180 0.030414
FEVD for realinv
realgdp realcons realinv
0 0.577021 0.152783 0.270196
1 0.488158 0.293622 0.218220
2 0.478727 0.314398 0.206874
3 0.477182 0.315564 0.207254
4 0.466741 0.324135 0.209124
They can also be visualized through the returned FEVD object:
In [29]: results.fevd(20).plot()
Out[29]: <Figure size 1000x1000 with 3 Axes>
Statistical tests¶
A number of different methods are provided to carry out hypothesis tests about the model results and also the validity of the model assumptions (normality, whiteness / “iid-ness” of errors, etc.).
Granger causality¶
One is often interested in whether a variable or group of variables is “causal”
for another variable, for some definition of “causal”. In the context of VAR
models, one can say that a set of variables are Granger-causal within one of the
VAR equations. We will not detail the mathematics or definition of Granger
causality, but leave it to the reader. The VARResults object has the
test_causality method for performing either a Wald (\(\chi^2\)) test or an
F-test.
In [30]: results.test_causality('realgdp', ['realinv', 'realcons'], kind='f')
Out[30]: <statsmodels.tsa.vector_ar.hypothesis_test_results.CausalityTestResults at 0x10ca15978>
Normality¶
Whiteness of residuals¶
Dynamic Vector Autoregressions¶
Note
To use this functionality, pandas must be installed. See the pandas documentation for more information on the below data structures.
One is often interested in estimating a moving-window regression on time series data for the purposes of making forecasts throughout the data sample. For example, we may wish to produce the series of 2-step-ahead forecasts produced by a VAR(p) model estimated at each point in time.
In [31]: np.random.seed(1)
In [32]: import pandas.util.testing as ptest
In [33]: ptest.N = 500
In [34]: data = ptest.makeTimeDataFrame().cumsum(0)
In [35]: data
Out[35]:
A B C D
2000-01-03 1.624345 -1.719394 -0.153236 1.301225
2000-01-04 1.012589 -1.662273 -2.585745 0.988833
2000-01-05 0.484417 -2.461821 -2.077760 0.717604
2000-01-06 -0.588551 -2.753416 -2.401793 2.580517
2000-01-07 0.276856 -3.012398 -3.912869 1.937644
... ... ... ... ...
2001-11-26 29.552085 14.274036 39.222558 -13.243907
2001-11-27 30.080964 11.996738 38.589968 -12.682989
2001-11-28 27.843878 11.927114 38.380121 -13.604648
2001-11-29 26.736165 12.280984 40.277282 -12.957273
2001-11-30 26.718447 12.094029 38.895890 -11.570447
[500 rows x 4 columns]
In [36]: var = DynamicVAR(data, lag_order=2, window_type='expanding')
The estimated coefficients for the dynamic model are returned as a
pandas.Panel object, which can allow you to easily examine, for
example, all of the model coefficients by equation or by date:
In [37]: import datetime as dt
In [38]: var.coefs
Out[38]:
<class 'pandas.core.panel.Panel'>
Dimensions: 9 (items) x 489 (major_axis) x 4 (minor_axis)
Items axis: L1.A to intercept
Major_axis axis: 2000-01-18 00:00:00 to 2001-11-30 00:00:00
Minor_axis axis: A to D
# all estimated coefficients for equation A
In [39]: var.coefs.minor_xs('A').info()
<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 489 entries, 2000-01-18 to 2001-11-30
Freq: B
Data columns (total 9 columns):
L1.A 489 non-null float64
L1.B 489 non-null float64
L1.C 489 non-null float64
L1.D 489 non-null float64
L2.A 489 non-null float64
L2.B 489 non-null float64
L2.C 489 non-null float64
L2.D 489 non-null float64
intercept 489 non-null float64
dtypes: float64(9)
memory usage: 58.2 KB
# coefficients on 11/30/2001
In [40]: var.coefs.major_xs(dt.datetime(2001, 11, 30)).T
Out[40]:
A B C D
L1.A 0.971964 0.045960 0.003883 0.003822
L1.B 0.043951 0.937964 0.000735 0.020823
L1.C 0.038144 0.018260 0.977037 0.129287
L1.D 0.038618 0.036180 0.052855 1.002657
L2.A 0.013588 -0.046791 0.011558 -0.005300
L2.B -0.048885 0.041853 0.012185 -0.048732
L2.C -0.029426 -0.015238 0.011520 -0.119014
L2.D -0.049945 -0.025419 -0.045621 -0.019496
intercept 0.113331 0.248795 -0.058837 -0.089302
Dynamic forecasts for a given number of steps ahead can be produced using the
forecast function and return a pandas.DataMatrix object:
In [41]: var.forecast(2)
Out[41]:
A B C D
2000-01-20 -260.325888 -23.141610 104.930427 -134.489882
2000-01-21 -52.121483 -11.566786 29.383608 -15.099109
2000-01-24 -54.900049 -23.894858 40.470913 -19.199059
2000-01-25 -7.493484 -4.057529 6.682707 4.301623
2000-01-26 -6.866108 -5.065873 5.623590 0.796081
... ... ... ... ...
2001-11-26 31.886126 13.515527 37.618145 -11.464682
2001-11-27 32.314633 14.237672 37.397691 -12.809727
2001-11-28 30.896528 15.488388 38.541596 -13.129524
2001-11-29 30.077228 15.533337 38.734096 -12.900891
2001-11-30 30.510380 13.491615 38.088228 -12.384976
[487 rows x 4 columns]
The forecasts can be visualized using plot_forecast:
In [42]: var.plot_forecast(2)
Class Reference¶
var_model.VAR(endog[, exog, dates, freq, …]) |
Fit VAR(p) process and do lag order selection |
var_model.VARProcess(coefs, coefs_exog, sigma_u) |
Class represents a known VAR(p) process |
var_model.VARResults(endog, endog_lagged, …) |
Estimate VAR(p) process with fixed number of lags |
irf.IRAnalysis(model[, P, periods, order, …]) |
Impulse response analysis class. |
var_model.FEVD(model[, P, periods]) |
Compute and plot Forecast error variance decomposition and asymptotic standard errors |
dynamic.DynamicVAR(data[, lag_order, …]) |
Estimates time-varying vector autoregression (VAR(p)) using equation-by-equation least squares |