Huron<-scan("https://www.math.mcgill.ca/dstephens/545/Data/huron.txt")
Huron<-Huron+570
n.H<-length(Huron)
t.H<-c(1:n.H)
x<-c(1875:1972)

plot(t.H,Huron,type="l",xlab="Years since 1872",ylab=expression(X[t]))
points(t.H,Huron,pch=19)


plot(t.H,Huron,type="l",xlab="Years since 1872",ylab=expression(X[t]))
points(t.H,Huron,pch=19)
abline(lm(Huron~t.H))
a0<-lm(Huron~t.H)$coeff[1]
a1<-lm(Huron~t.H)$coeff[2]
text(0,577.5,expression(hat(a)[0]==580.204),adj=0,cex=1.5)
text(0,577.1,expression(hat(a)[1]==-0.024),adj=0,cex=1.5)

mt.H<-a0+a1*t.H
yt.H<-Huron-mt.H
plot(t.H,yt.H,type="l",xlab="Years since 1872",ylab=expression(hat(e)[t]))
points(t.H,yt.H,pch=19)
abline(h=0,lty=2)

summary(lm(Huron~t.H))

par(pty="s")
y1<-yt.H[2:98]
y2<-yt.H[1:97]
plot(y1,y2,xlab=expression(hat(e)[t-1]),ylab=expression(hat(e)[t]),pch=19)
par(pty="s")

cor(y1,y2)