advanced-algorithmic-trading

101
In order to determine whether an AR(p) process is stationary or not we need to solve the
characteristic equation. The characteristic equation is simply the autoregressive model, written
in backward shift form, set to zero:
θ p (B) = 0 (10.7)
We solve this equation for B. In order for the particular autoregressive process to be stationary
we need all of the absolute values of the roots of this equation to exceed unity. This
is an extremely useful property and allows us to quickly calculate whether an AR(p) process is
stationary or not.
The following examples will make this idea concrete:
• Random Walk - The AR(1) process with α 1 = 1 has the characteristic equation θ = 1−B.
Clearly this has root B = 1 and as such is not stationary.
• AR(1) - If we choose α 1 = 1 4 we get x t = 1 4 x t−1 + w t . This gives us a characteristic
equation of 1 − 1 4B = 0, which has a root B = 4 > 1 and so this particular AR(1) process
is stationary.
• AR(2) - If we set α 1 = α 2 = 1 2 then we get x t = 1 2 x t−1 + 1 2 x t−2 + w t . Its characteristic
equation becomes − 1 2
(B − 1)(B + 2) = 0, which gives two roots of B = 1, −2. Since this
has a unit root it is a non-stationary series. However, other AR(2) series can be stationary.
10.4.3 Second Order Properties
The mean of an AR(p) process is zero. However, the autocovariances and autocorrelations are
given by recursive functions, known as the Yule-Walker equations. The full properties are given
below:
µ x = E(x t ) = 0 (10.8)
p∑
γ k = α i γ k−i , k > 0 (10.9)
i=1
p∑
ρ k = α i ρ k−i , k > 0 (10.10)
i=1
Note that it is necessary to know the α i parameter values prior to calculating the autocorrelations.
Now that the second order properties have been states it is possible to simulate various orders
of AR(p) and plot the corresponding correlograms.