출처 : https://stats.stackexchange.com/questions/118019/a-proof-for-the-stationarity-of-an-ar2

phi1 <- seq(from = -2.5, to = 2.5, length = 51) 
plot(phi1,1+phi1,lty="dashed",type="l",xlab="",ylab="",cex.axis=.8,ylim=c(-1.5,1.5))
abline(a = -1, b = 0, lty="dashed")
abline(a = 1, b = -1, lty="dashed")
title(ylab=expression(phi[2]),xlab=expression(phi[1]),cex.lab=.8)
polygon(x = phi1[6:46], y = 1-abs(phi1[6:46]), col="gray")
lines(phi1,-phi1^2/4)
text(0,-.5,expression(phi[2]<phi[1]^2/4),cex=.7)
text(1.2,.5,expression(phi[2]>1-phi[1]),cex=.7)
text(-1.75,.5,expression(phi[2]>1+phi[1]),cex=.7)

Consider the equation

If z$z$ is a root of the "standard" characteristic equation 1−ϕ1z−ϕ2z2=0$1-{\varphi }_{1}z-{\varphi }_2{z}^2=0$ and setting z−1=λ$z^{-1}=\lambda$, the display obtains from rewriting the standard one as follows:

Hence, an alternative condition for stability of an AR(2)$AR(2)$ is that all roots of the first display are insidethe unit circle, |z|>1⇔|λ|=|z−1|<1$|z|>1\iff |\lambda |=|z^{-1}|<1$.

We use this representation to derive the stationarity triangle of an AR(2)$AR(2)$ process, that is that an AR(2)$AR(2)$ is stable if the following three conditions are met:

1. ϕ2<1+ϕ1${\varphi }_2<1+{\varphi }_1$
2. ϕ2<1−ϕ1${\varphi }_2<1-{\varphi }_1$
3. ϕ2>−1${\varphi }_2>-1$

Recall that you can write the roots of the first display (if real) as

to find the first two conditions.Consider the equation

If z$z$ is a root of the "standard" characteristic equation 1−ϕ1z−ϕ2z2=0$1-{\varphi }_{1}z-{\varphi }_2{z}^2=0$ and setting z−1=λ$z^{-1}=\lambda$, the display obtains from rewriting the standard one as follows:

We use this representation to derive the stationarity triangle of an AR(2)$AR(2)$ process, that is that an AR(2)$AR(2)$ is stable if the following three conditions are met:

1. ϕ2<1+ϕ1${\varphi }_2<1+{\varphi }_1$
2. ϕ2<1−ϕ1${\varphi }_2<1-{\varphi }_1$
3. ϕ2>−1${\varphi }_2>-1$

Recall that you can write the roots of the first display (if real) as

Then, the AR(2)$AR(2)$ is stationary iff |λ|<1$|\lambda |<1$, hence (if the λi${\lambda }_i$ are real):

If λi${\lambda }_i$ is complex, then ϕ21<−4ϕ2${\varphi }_1^2<-4{\varphi }_2$ and so

Plotting the stationarity triangle, also indicating the line that separates complex from real roots, we get

https://blog.naver.com/hafs_snu/220825408168

https://blog.naver.com/hafs_snu/220834890761