library(TSA)
data(lynx)
acf(lynx)
pacf(lynx)
eacf(lynx)
arma.l=arima(lynx,order=c(2,0,2))
arma.l
arma.l2=arima(lynx,order=c(2,0,2),fixed=c(NA,NA,0,NA,NA))
arma.l2
tsdiag(arma.l2,gof.lag=30)
plot(arma.l2$residuals)
qqnorm(arma.l2$residuals);qqline(arma.l2$residuals)
plot(y=as.vector(arma.l2$residuals),x=as.vector(fitted(arma.l2)))
BoxCox.ar(lynx)

# log transformation, then fit an ARMA(3,1)

arma.31=arima(log(lynx),order=c(3,0,1))
arma.31
Rsq=1-.2652/var(log(lynx))
Rsq
tsdiag(arma.31,gof.lag=40)
qqnorm(arma.31$residuals);qqline(arma.31$residuals)
shapiro.test(arma.31$residuals)
plot(y=as.vector(arma.31$residuals)/sqrt(.2652),x=as.vector(fitted(arma.31)))
abline(h=0)

#overfitting for appreciating the goodnesf of fit
# increase either the AR order or the MA order by 1.

arma.41=arima(log(lynx),order=c(4,0,1))
tsdiag(arma.41,gof.lag=40)
arma.51=arima(log(lynx),order=c(5,0,1))
arma.51
arma.32=arima(log(lynx),order=c(3,0,2))
arma.32

ar11=arima(log(lynx),order=c(11,0,0))
ar11
tsdiag(ar11,gof.lag=25)

ar11b=arima(log(lynx),order=c(11,0,0),fixed=c(NA,NA,rep(0,8),NA,NA))
ar11b
tsdiag(ar11b)

#don't overfit both AR and MA, because of possible redundancy.

arma.52=arima(log(lynx),order=c(5,0,2))
arma.52
arma.41
ar(log(lynx))$aic