## Theory

Dynamic autoregression models extend autoregressive models by allowing the autoregressive coefficients to vary over time. For a univariate time series $y_{t}$:

$y_{t} = \sum^{p}_{i=1}\phi_{i,t}y_{t-i} + \epsilon_{t}$

$\phi_{i,t}= \phi_{i,t-1} + \eta_{i,t}$

$\epsilon_{t} \sim N\left(0,\sigma^{2}\right)$

$\eta_{i,t} \sim N\left(0,\sigma_{\eta_{i}}^{2}\right)$

In other words, the autoregressive coefficients follow a random walk. This model can be estimated through the Kalman Filter and Smoother.

## PyFlux

We’ll run an Dynamic Autoregressive (DAR) Model for yearly sunspot data:

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

data.index = data['time'].values

plt.figure(figsize=(15,5))
plt.plot(data.index,data['sunspot.year'])
plt.ylabel('Sunspots')
plt.title('Yearly Sunspot Data');


Here we specify an arbitrary DAR(9) model:

model = pf.DAR(data=data,ar=9,integ=0,target='sunspot.year')

Next we estimate the latent variables. For this example we will use a maximum likelihood point mass estimate $z^{MLE}$:

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

DAR(9, integrated=0)
======================================================= =================================================
Dependent Variable: sunspot.year                        Method: MLE
Start Date: 1709                                        Log Likelihood: -1179.097
End Date: 1988                                          AIC: 2380.194
Number of observations: 280                             BIC: 2420.1766
=========================================================================================================
Latent Variable                          Estimate   Std Error  z        P>|z|    95% C.I.
======================================== ========== ========== ======== ======== ========================
Sigma^2 irregular                        0.301
Constant                                 60.0568    23.83      2.5202   0.0117   (13.3499 | 106.7637)
Sigma^2 AR(1)                            0.005
Sigma^2 AR(2)                            0.0
Sigma^2 AR(3)                            0.0005
Sigma^2 AR(4)                            0.0001
Sigma^2 AR(5)                            0.0002
Sigma^2 AR(6)                            0.0011
Sigma^2 AR(7)                            0.0002
Sigma^2 AR(8)                            0.0003
Sigma^2 AR(9)                            0.032
=========================================================================================================


Note we have no standard errors in the results table because it shows the transformed parameters. If we want standard errors, we can call x.summary(transformed=False). Next we will plot the in-sample fit and the dynamic coefficients using plot_fit:

model.plot_fit(figsize=(15,40))


The sharp changes at the beginning reflect the diffuse initialization; together with high initial uncertainty, this leads to stronger updates towards the beginning of the series. We can predict forward using plot_predict:

model.plot_predict(h=50,past_values=40,figsize=(15,5))


The prediction intervals here are unrealistic and reflect the Gaussian distributional assumption we’ve chosen – we can’t have negative sunspots! – but if we are just want the predictions themselves, we can use the predict method:

model.predict(h=50)

sunspot.year
1989 139.637190
1990 166.080697
1991 158.856260
1992 123.442112
1993 84.140754
1994 42.171881
1995 15.495871
1996 4.020670
1997 40.250429
1998 84.376278
1999 136.250978
2000 163.849298
2001 162.628458
2002 134.597983
2003 90.473426
2004 45.883060
2005 9.204294
2006 7.601138
2007 30.750374
2008 80.300545
2009 128.983109
2010 162.943127
2011 166.774942
2012 142.477788
2013 98.476385
2014 47.089022
2015 12.418343
2016 2.323678
2017 26.707932
2018 71.845696
2019 124.183167
2020 160.798035
2021 170.815410
2022 149.830120
2023 104.539250
2024 53.246798
2025 12.534159
2026 0.601882
2027 19.683715
2028 65.149369
2029 117.634307
2030 159.336605
2031 173.882239
2032 155.956235
2033 112.454194
2034 58.413341
2035 15.136978
2036 -2.363851
2037 14.070445
2038 57.020995