## acf of AR with negative parameters and examples of ACFs and PACFs. 
par(mfrow=c(1,3)) 
n = 1000 
x = arima.sim(n = n, list(ar=c(-.5,-.03,.4)),sd=1) 
plot(1:n,x,xlab="time (days)",ylab="data",type="l",main="AR(3)") 
a = acf(x,plot=F,lag.max=20) 
m = length(a$lag) 
plot(a$lag[2:m],a$acf[2:m],type="h",xlab="lag h (days)", ylab = "r(h)") 
lines(a$lag[2:m],rep(1.96/sqrt(n),m-1),col="red",lty=2)
lines(a$lag[2:m],rep(-1.96/sqrt(n),m-1),col="red",lty=2)
b = pacf(x,plot=F,lag.max=20) 
m = length(b$lag) 
plot(b$lag[1:m],b$acf[1:m],type="h",xlab="lag h (days)", ylab = expression(r[hh])) 
lines(b$lag[1:m],rep(1.96/sqrt(n),m),col="red",lty=2)
lines(b$lag[1:m],rep(-1.96/sqrt(n),m),col="red",lty=2) 

x = arima.sim(n = n, list(ar=c(.5,-.3)),sd=1) 
plot(1:n,x,xlab="time (days)",ylab="data",type="l",main="AR(2)") 
a = acf(x,plot=F,lag.max=20) 
m = length(a$lag) 
plot(a$lag[2:m],a$acf[2:m],type="h",xlab="lag h (days)", ylab = "r(h)",ylim=c(-.1,.7)) 
lines(a$lag[2:m],rep(1.96/sqrt(n),m-1),col="red",lty=2)
lines(a$lag[2:m],rep(-1.96/sqrt(n),m-1),col="red",lty=2)
b = pacf(x,plot=F,lag.max=20) 
plot(b$lag[1:m],b$acf[1:m],type="h",xlab="lag h (days)", ylab = expression(r[hh])) 
lines(a$lag[1:m],rep(1.96/sqrt(n),m),col="red",lty=2)
lines(a$lag[1:m],rep(-1.96/sqrt(n),m),col="red",lty=2) 

x = arima.sim(n = n, list(ma=c(.9,.3,.5,.6,.1,.7)),sd=1) 
plot(1:n,x,xlab="time (days)",ylab="data",type="l",main="MA(6)") 
a = acf(x,plot=F,lag.max=20) 
m = length(a$lag) 
plot(a$lag[2:m],a$acf[2:m],type="h",xlab="lag h (days)", ylab = "r(h)") 
lines(a$lag[2:m],rep(1.96/sqrt(n),m-1),col="red",lty=2)
lines(a$lag[2:m],rep(-1.96/sqrt(n),m-1),col="red",lty=2)
b = pacf(x,plot=F,lag.max=20) 
plot(b$lag[1:m],b$acf[1:m],type="h",xlab="lag h (days)", ylab = expression(r[hh])) 
lines(a$lag[2:m],rep(1.96/sqrt(n),m-1),col="red",lty=2)
lines(a$lag[2:m],rep(-1.96/sqrt(n),m-1),col="red",lty=2)