Problem 1: Consider ARMA(1,1) model Xt = ΦXt-1 + Zt + θZt-1, where |Φ| < 1, θ ∈ R and the random variables Zt are i.i.d. with mean zero and variance σ2Z.
(a) Use the linear representation (see Example 2.9 in Lecture Notes) to obtain the general formula for yx(h), h ≥ 0. Evaluate all infinite series that appear in your formula.
Problem 2: Consider AR(1) model Xt = ΦXt-1+ Zt, where |Φ|< 1 and the random variables Zt are i.i.d. with mean zero and variance σ2Z.
(a) Apply the Durbin-Levinson algorithm to get Φ11, Φ21, Φ22.
(b) Try to prove that Φnn = 0 for all n ≥ 2 (recall that Φnn is partial autocovariance at lag n).
Problem 3: In the following question you will simulate data from AR(1) model and do basic analysis. Please provide relevant codes and relevant numerical answers, but do not include pictures. I give several hints, while the usage of commands ar and arima can be consulted on the course webpage.
(a) Simulate n = 200 observations from AR(1) with Φ = 0.8 and σ2z = 1. You can plot for yourself the ACF and PACF to see that the generated data really come from AR(1), but please do not include plots.
Hint: Use arima.sim(list (order = c (1,0,0) , ar = 0.8) , n = 200)
(b) Use R to estimate the parameters Φ and σ2z using the Yule-Walker method.
Hint: Use ar
(c) Confirm your estimates from (b) using the theoretical formulas.
Hint: Section 5.1 in Lecture Notes.
(d) Use R to estimate the parameters Φ and σ2z using the MLE method.
Hint: Use ar.
(e) Use estimates either from (b) or (d) to predict X201.
Hint: You need to read the last observation from your data set. Remember about the mean!
(f) In (d), why R chose AR(1), not AR(2)?
Hint: Use arima and compare AIC for both models.
Problem 4: In the following question you will use the data set priceNSW.txt available on the course webpage. Please provide relevant codes and relevant numerical answers, but do not include pictures.
(a) Fit an ARMA model.
Hint: You decide on the method
(b) Use R to estimate all model parameters (for example, if your model chosen is AR(2), estimate Φ1, Φ2 and σ2z.
(c) Use your model to predict the next observation.
Hint: You need to read the last observation from your data set. Remember about the mean!
Attachment:- Assignment Dataset.rar