Unit 12 AR(2) models
Xt=β+ϕ1Xt−1+ϕ2Xt−2+at where β=(1−ϕ1−ϕ2)μ is the moving average constant.
12.1 Notes
It looks like MLR, however the independetn variables are values of the dependent variable at two previous time periods
12.2 Stationarity in AR(2)
E[Xt]=μ
σ2X=σ2a1−ϕ1ρ1−ϕ2ρ2
Instead of depending on positive or negative roots, like AR(1), the behavior of the autocorrelations and spectral density depends on whether or not the roots of the characteristic equation are complex. This means it is already hard to tell from the equation whether or not it is stationary just by looking at the equation.
12.3 AR(2) zero mean form
If μ=0, then the AR(2) model takes the form :
Xt=ϕ1Xt−1+ϕ2Xt−2+at
We can study this much easier and then add the mean back later. A more common way of writing this, and more useful to us is the form: Xt−ϕ1Xt−1−ϕ2Xt−2=at
12.4 More on backshift operator notation
Remember, we defined the backshift operator as B such that BXt=Xt−1
We can extend this and say: BkXt=Xt−k
This means we can rewrite:
Xt−ϕ1BXt−ϕ2B2Xt=at
Xt[1−ϕ1B−ϕ2B2]=at
or, more tersely:
ϕ(B)Xt=at
Where ϕ(B)=1−ϕ1B−ϕ2B2
We can rewrite this ϕ(B) as a characteristic equation now: 1−ϕ1Z−ϕ2Z2=0
Lets quickly expand on our quadratic solver to get the roots of an AR(2) model:
ar2solver <- function(p1, p2) {
res <- as.complex(quad_form(p2, p1, 1))
mag <- Mod(res)
absrecip <- 1/mag
data.frame(roots = res, magnitude = mag, abs.recip = absrecip)
}
ar2solver(0.4, 0.8)
## roots magnitude abs.recip
## 1 -0.25-1.09i 1.118302 0.8942126
## 2 -0.25+1.09i 1.118302 0.8942126
12.5 Characteristic equation
x=−b±√b2−4ac2a 1−ϕ1Z−ϕ2Z2=0
Just solve the quadratic equation using the quadratic formula:
x=−b±√b2−4ac2a
Two cases: * Two real roots * Two complex roots
12.6 Key result
An AR(2) is stationary iff all roots are outside the unit circle
12.6.1 Two real roots
We can factor the AR(2) into two AR(1)s
- one part positive one part negative
- Wandering but oscillating
- Damping ACF but slightly oscillatory
- S(f) has a peak at 0 and at 0.5
- Two Positive roots
- Super wandering
- Super Damped
- S(f) peak at 0
- Two Negative
- super oscillating
12.6.2 Complex conjugate roots
Stationary AR(2)’s with complex conjugate roots have an autocorrelation function which looks more like a damped sinusoid instead of a damped exponential, and a system frequency, f0 defined as+ f0=12πcos−1(ϕ12√−ϕ2). Pretty easy stuff.