#4

set.seed(0)
theta <- -0.5
x <- arima.sim(n = 200,model = list(ma=c(theta)))
y <- acf(x,main = expression(theta[1] == theta))

acf1 = y$acf[2]

In the ACF plot we can see the different values for the sample autocorrelation \(\rho(.)\) as a function of the lag between observations. The first value is \(\rho(0)\) which is equal to 1 by definition. The estimate of \(\rho(1)\) is −0.399, which is very close to the true theoretical value \(\rho(1)\) = −0.4 . Taking into the account of an MA(q) process where \(\rho(h)\) = 0 for all |h| > q, the remaining values of the sample ACF contained between the confidence bound in dotted blue lines are statistically not significantly different from zero.

5

set.seed(5)
theta1 <- 2
theta2 <- -8
x <- arima.sim(n = 300,model = list(ma=c(theta1,theta2)))
y <- acf(x,main = expression(theta[1] == theta1, theta[2] == theta[2]))

rho2 = y$acf[3]

In the ACF plot we can see the different values for the sample autocorrelation \(\rho(·)\) as a function of the lag between observations. \(\rho(0)\) = 1 by definition, \(\rho(1)\) = −0.193 and \(\rho(2)\) = −0.135 . The sample estimate for \(\rho(2)\), -0.135, differs a bit from the true value of \(\rho(2)\), -0.116.