EMET3007/8012 Assignment 3 Instructions: This assignment is worth either 20% or 25% of the final grade, and is worth a total of 90 points (75 for EMET3007 students). All working must be shown for all questions. For questions which ask you to write a program, you must provide the code you used. The assignment is due by 5pm Friday 18th of October (Friday of Week 11), using Turnitin on Wattle. Late submissions will be accepted without prior written approval and without penalty until 10am Tuesday the 22nd of October. They will not be accepted after this time. Good Luck. Clarification on T0: For this assignment, you may use an appropriate T0 value of your choice. I would encourage you to use T0 = 50. Question 1: [20 marks] The file macrodata.csv contains (amongst other things) US inflation rate data from January 1955 to August 2019. a) Consider the model from Assignment 2 Question 6; the AR(2) model with AR(1) errors. Fix φ = 0.25. Find the maximum likelihood esti- mates for µ, ρ1, ρ2, and σ2. b) Now we want to estimate φ instead of fixing it at 0.25. Find the max- imum likelihood estimates for φ, µ, ρ1, ρ2, and σ2. [There may be more than one ‘local maxima’. Check the plot of log-likelihood to get a feeling for the likelihood function.] c) Instead of AR(1) errors consider the model with MA(1) errors: yt = µ+ ρ1yt−1 + ρ2yt−2 + et, et = ut + ψut−1, ut ∼ N (0, σ2) white noise where e0 = 0. Derive the log-likelihood function `(φ, µ, ρ, σ2 | y, y0). d) For the inflation data, and using the model in part (c), find the MLE for ψ, µ, ρ1, ρ2, and σ2. Which model fits the data better (in the MSE sense); the AR(1) errors, or MA(1) errors? 1 Question 2: [15 marks] Consider the AR(2) specification yt = µ + ρ1yt−1 + ρ2yt−2 + et. Consider three models for the errors: Model 1: et ∼ N (0, σ2)white noise Model 2: et = ut + ψut−1, ut ∼ N (0, σ2) white noise Model 3: et = φet−1 + ut, ut ∼ N (0, σ2) white noise For each model, using the US inflation data in macrodata.csv, compute the MSFE for the one-step-ahead forecast for each model. Which model performs best? Question 3: [10 marks] This is a theory-only question. Do not attempt to use data in this question. Consider the ARMA(1,2) process with drift yt = µ+ φyt−1 + ut + ψ1ut−1 + ψ2ut−2, ut ∼ N (0, σ2) white noise a) Compute the autocovariance function of y. b) For what values of the parameters is this process covariance station- ary? c) Explain how to produce a two-step-ahead forecast using this model. d) Consider the poorly-formed model yt = µ+φyt−1 +ψ0ut+ψ1ut−1 +ψ2ut−2, ut ∼ N (0, σ2) white noise Suppose θ∗ = (µ∗, φ∗,ψ∗0 ,ψ∗1 ,ψ ∗ 2 , (σ 2)∗) is a MLE for this model (given some hypothetical data). Find another MLE for this model. That is, show that the MLE for this model is not unique. (Hint: Your new MLE values will be in terms of the parameters in θ∗.) 2 Question 4: [15 marks] For the inflation data provided, compute the MSFE for the IMA(1,2) model and the IMA(1,3) model. Plot your forecasts. Compare your results. [Recall: The IMA(1,2) model means the differences in the data follow an MA(2) process] Question 5: [15 marks] Consider the following UC model with AR(1) state equation: yt = τt + et, et ∼ N (0, σ2) white noise τt = φτt−1 + ut, ut ∼ N (0,ω2) white noise where the unobserved component is initialised by τ1 = N (0, 1) and ω2 = 0.3. Use the full sample inflation data to find the MLE for φ1, φ2, and σ2. Compute the MSFE for the one-step-ahead forecasting exercise for the in- flation data under this model. Question M: [15 marks] This question is for EMET4312/8012 students only. For the inflation data: a) Compute the one-step ahead MSFE for the naive random walk (as a benchmark). b) Compute the one-step ahead MSFE for the AR(2) model. c) Compute the one-step-ahead MSFE of inflation for the VAR(2) model yt = b + B1yt−1 + B2yt−2 + et where yt is a 3× 1 vector consisting of inflation rate, GDP growth rate, and interest rate. d) Comment briefly on your results (one or two sentence is sufficient). 3
