#1

data(deere3)

arima(deere3, order=c(1,0,0),method='CSS') # AR(1) using least squares
arima(deere3, order=c(1,0,0),method='ML') # AR(1) using maximum likelihood

arima(deere3, order=c(2,0,0),method='CSS') # AR(2) using least squares
arima(deere3, order=c(2,0,0),method='ML') # AR(2) using maximum likelihood

#2

deere3.ar1.ml <- arima(deere3, order=c(1,0,0),method='ML') # AR(1) using maximum likelihood

plot(rstandard(deere3.ar1.ml), type='o')
qqnorm(residuals(deere3.ar1.ml))

acf(residuals(deere3.ar1.ml),main='Sample ACF of Residuals from AR(1) Model for Deere Data')

Box.test(residuals(deere3.ar1.ml), lag = 8, type = "Ljung-Box", fitdf = 1)

runs(residuals(deere3.ar1.ml))

#3

data(robot)

arima(robot, order=c(1,0,0),method='ML') # AR(1) using maximum likelihood
arima(robot, order=c(0,1,1),method='ML') # IMA(1,1) using maximum likelihood

#4

robot.ar1.ml <- arima(robot, order=c(1,0,0),method='ML') # AR(1) using maximum likelihood

plot(rstandard(robot.ar1.ml), type='o')
qqnorm(residuals(robot.ar1.ml))

acf(residuals(robot.ar1.ml),main='Sample ACF of Residuals from AR(1) Model for Robot Data')

Box.test(residuals(robot.ar1.ml), lag = 20, type = "Ljung-Box", fitdf = 1)

runs(residuals(robot.ar1.ml))

# Overfitting:

arima(robot, order=c(2,0,0),method='ML') # AR(2) using maximum likelihood

arima(robot, order=c(1,0,1),method='ML') # ARMA(1,1) using maximum likelihood

# Repeat with AR(2):

robot.ar2.ml <- arima(robot, order=c(2,0,0),method='ML') # AR(2) using maximum likelihood


acf(residuals(robot.ar2.ml),main='Sample ACF of Residuals from AR(2) Model for Robot Data')

Box.test(residuals(robot.ar2.ml), lag = 30, type = "Ljung-Box", fitdf = 2)

runs(residuals(robot.ar2.ml))