MA585
Homework 3
1. For each of the following scenarios, choose φ1 and φ2 so that the resulting AR(2) process is causal. In each case use R to plot ACF and PACF of the AR(2) process and discuss the characteristics of the plots.
(i) φ 1 > 0 and φ2 > 0
(ii) φ 1 < 0 and φ2 > 0
(iii) φ 1 > 0 and φ2 < 0
(iv) φ 1 < 0 and φ2 < 0
R Note: Theoretical (or population) ACF PACF can be obtained in R using ARMAacf function. However, there is a certain peculiarity in the function that you should be aware of. The function computes ACF starting from lag 0, but PACF starting from lag 1. Since ACF at lag 0 is 1, the two plots are not necessarily on the same scale, so it is hard to compare. Not the end of the world, but a bit annoying. The following code, which fixes the issue, was used to generate ACF and PACF plots for the examples given in the notes.
par(mfrow=c(2,2))
y = ARMAacf(ar = c(1.1,-0.3), lag.max = 20)
y = y[2:21]
plot(y, x = 1:20, type = "h", ylim = c(-1,1), xlab = "h", ylab = "Autocor- relation", main = "AR(2) Population ACF")
abline(h = 0)
y = ARMAacf(ar = c(1.1, -0.3), lag.max = 20,pacf=T)
plot(y, x = 1:20, type = "h", ylim = c(-1,1), xlab = "h",
ylab = "Partial Autocorrelation", main = "AR(2) Population PACF")
abline(h = 0)
y = ARMAacf(ar = c(1,-0.81), lag.max = 20)
y = y[2:21]
plot(y, x = 1:20, type = "h", ylim = c(-1,1), xlab = "h", ylab = "Autocor- relation", main = "AR(2) Population ACF")
abline(h = 0)
y = ARMAacf(ar = c(1,-0.81), lag.max = 20, pacf=T)
plot(y, x = 1:20, type = "h", ylim = c(-1,1), xlab = "h", ylab = "Partial Autocorrelation", main = "AR(2) Population PACF")
abline(h = 0)
2. Do the following problems: 3.1 (use the approach discussed in the class to verify causality and invertibility, no credit for just writing down the answers), 3.2 (only for the processes that are causal and invertible, use R instead of ITSM), 3.4
3. a. Consider a causal AR(2) process
Xt - φ1 Xt - 1 - φ2 Xt -2 = et
where et ~ W N (0; σ2 ): Express the model as an infinite order MA process given by Xt = P ψjet-j and derive the difference equation for ψj .
b. Consider the process
Xt - 1:1Xt - 1 + 0:3Xt -2 = et
Find the first five ψj values and verify your answer using ARMAtoMA function in R.
c. Find the first five ACFs of the model in (b) and verify your answer using
R.
d. Find all PACFs of the model in (b) and verify your answer using R.
4. a. Consider a causal and invertible ARMA(1,1) process
Xt - φXt - 1 = et + θet - 1
where et ~ W N (0; σ2 ): Express the model as an inÖnite order MA process given by Xt = P ψjet-j and show that ψj = φj-1(φ + θ); for j ≥ 1:
b. Suppose the ARMA(1,1) process is given by
Xt - 0:6Xt - 1 = et - 0:2et - 1
Compute the first five ψj values and verify your answer using the AR- MAtoMA function in R.
5. a. Consider the following MA(2) model
Xt = et + 0:8et - 1 - 0:15et -2
a. Compute the ACFs of the model.
b. Use Durbin-Levinson algorithm to compute the first 5 PACFs and verify your answer using R.
6. I generated the following population ACF and PACF plots based on certain ARMA processes. What ARMA processes and the parameter values I used to generate these plots?
a.
b.