Question

• Generate data of length n=100 from a Gaussian AR(2) model with phi1=0.5 and phi2=0.2 and input variance sigma^2=1. Afterwards, pretend the values of phi1, phi2 and sigma^2 are unknown.
  i. Use the R function ar() to fit an AR(p) model to your data. As options for the ar() function, first use aic = TRUE. What estimated order did aic minimization give for your data? Try three different fitting methods, method="yule-walker", method="ols"and also method="mle". Do you see significant differences between the methods?
  ii. Repeat the above by setting aic = FALSE and method="yule-walker". First select order p=1; what do you observe in your estimates? Now select order p=3 and then p=5; what do you observe?
  iii. Fit an ARMA(p,q) model to your data using the R function arima(). Specify the orders (p,q) to be fitted as either (2,0) or (1,1) or (2,1). Compare your results under these three specifications. [Note that ARMA(p,q) is the same as ARIMA(p,0,q) so in R function arima() you need to select the middle order to be 0.]

Answer 1

ANSWER: I have attached the below code. I have added the observations in the last.

##################### R CODE ####################################

rm(list=ls(all=TRUE))

set.seed(12345)

X = arima.sim(n=100,model=list(ar=c(0.5,0.2)))
summary(X)

ar(X,aic=TRUE) # AR2 as a result
ar(X,aic=TRUE,method = "mle") # AR2 as a result
ar(X,aic=TRUE,method = "ols") # AR17 as a result
ar(X,aic=TRUE,method = "yule-walker") # AR2 as a result

ar(X,aic=FALSE,order.max = 1)
ar(X,aic=FALSE,method = "mle",order.max = 1)
ar(X,aic=FALSE,method = "ols",order.max = 1)
ar(X,aic=FALSE,method = "yule-walker",order.max = 1)

ar(X,aic=FALSE,order.max = 3)
ar(X,aic=FALSE,method = "mle",order.max = 3)
ar(X,aic=FALSE,method = "ols",order.max = 3)
ar(X,aic=FALSE,method = "yule-walker",order.max = 3)

ar(X,aic=FALSE,order.max = 5)
ar(X,aic=FALSE,method = "mle",order.max = 5)
ar(X,aic=FALSE,method = "ols",order.max = 5)
ar(X,aic=FALSE,method = "yule-walker",order.max = 5)

arima(X,order=c(2,0,0))
arima(X,order=c(1,0,1))
arima(X,order=c(2,0,1))

############## END OF R CODE #############################################

OBSERVATIONS:

(1)

Method Used	Order Selected
Default	2
MLE	2
OLS	17
Yule Walker	2

(3)

Order Selected	AIC
p=2, q=0	312.36
p=1, q=1	312.21
p=2, q=1	314.18