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of our linear regression. The next step is to assume that tomorrow’s intercept and slope are equal
to today’s intercept and slope with the addition of some random system noise. This gives it the
nature of a random walk, the behaviour of which is discussed at length in the previous chapter
on white noise and random walk models.
β t+1 = Iβ t + w t (13.32)
Where the transition matrix is set to the two-dimensional indentify matrix, G t = I. This is
one half of the state space model. The next step is to actually use one of the ETFs in the pair
as the "observations".
13.3.2 Applying the Kalman Filter to a Pair of ETFs
To form the observation equation it is necessary to choose one of the ETF pricing series to be
the "observed" variables, y t , and the other to be given by x t , which provides the linear regression
formulation as above:
y t = F t x t + v t (13.33)
⎛ ⎞
1
= (β 0 , β 1 )
⎜ ⎟
⎝ ⎠ + v t (13.34)
x t
Thus we have the linear regression reformulated as a state space model, which allows us to
estimate the intercept and slope as new price points arrive via the Kalman Filter.
13.3.3 TLT and ETF
We are going to consider two fixed income ETFs, namely the iShares 20+ Year Treasury Bond
ETF (TLT) and the iShares 3-7 Year Treasury Bond ETF (IEI). Both of these ETFs track the
performance of varying duration US Treasury bonds and as such are both exposed to similar
market factors. We will analyse their regression behaviour over the last five years or so.
13.3.4 Scatterplot of ETF Prices
We are now going to use a variety of Python libraries, including NumPy, Matplotlib, Pandas
and PyKalman to to analyse the behaviour of a dynamic linear regression between these two
securities. As with all Python programs the first task is to import the necessary libraries:
from __future__ import print_function
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pandas_datareader as pdr
from pykalman import KalmanFilter