24.10 Time series models

Typically, we would use the partial autocorrelation plot (above) to determine the order. So, for the lynx data

(p. 800) we would use order 2 or 4, depending on taste. 

Other things being equal, parsimony suggests the use of order 2. The fundamental difference is that a set of random components (εt− j) influences the current value of a MA process, whereas only the current random effect (εt) affects an AR process. Both kinds of effects are at work in an ARMA processes. Ecological models of population dynamics are typically AR models.

For instance, 

windows(7,4)
par(mfrow=c(1,2))
acf(Lynx,main="")

acf(Lynx,type="p",main="")

The order vector specifies the non-seasonal part of the ARIMA model: the three components (p, d, q)
are the AR order, the degree of differencing, and the MA order. We start by investigating the effects of AR

order with no differencing and no moving average terms, comparing models on the basis of the AIC:

model10 <- arima(Lynx,order=c(1,0,0))
model20 <- arima(Lynx,order=c(2,0,0))
model30 <- arima(Lynx,order=c(3,0,0))
model40 <- arima(Lynx,order=c(4,0,0))
model50 <- arima(Lynx,order=c(5,0,0))
model60 <- arima(Lynx,order=c(6,0,0))

AIC(model10,model20,model30,model40,model50,model60)

model01 <- arima(Lynx,order=c(0,0,1))
model02 <- arima(Lynx,order=c(0,0,2))
model03 <- arima(Lynx,order=c(0,0,3))
model04 <- arima(Lynx,order=c(0,0,4))
model05 <- arima(Lynx,order=c(0,0,5))
model06 <- arima(Lynx,order=c(0,0,6))

AIC(model01,model02,model03,model04,model05,model06)

model40 <- arima(Lynx,order=c(4,0,0))
model41 <- arima(Lynx,order=c(4,0,1))
model42 <- arima(Lynx,order=c(4,0,2))
model43 <- arima(Lynx,order=c(4,0,3))

AIC(model40,model41,model42,model43)

model400 <- arima(Lynx,order=c(4,0,0))
model401 <- arima(Lynx,order=c(4,1,0))
model402 <- arima(Lynx,order=c(4,2,0))
model403 <- arima(Lynx,order=c(4,3,0))
AIC(model400,model401,model402,model403)