## Instructions:

This assignment is worth either 20% or 25% of the final
grade, and is worth a total of 95 points (80 for EMET3007 students). For
questions which ask you to write a program, you must provide the code
you used. The assignment is due by 5pm Friday 22nd 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 27th of October. They will not be accepted after this time. Good Luck.
Clarification on T0: For this assignment, please use T0 = 50 whenever
it comes up.

Problem 1.

[20 marks] The file macrodata.csv contains (amongst other things) US inflation rate data from January 1955 to January 2021 . Use this inflation data for Questions $1,2,4$, and $5 .$
a) Consider the model from Assignment 2 Question $7 ;$ the $\mathrm{AR}(2)$ model with $\mathrm{AR}(1)$ errors. Fix $\phi=0.25 .$ Find the maximum likelihood estimates for $\mu, \rho$, and $\sigma^{2}$.
b) Now we want to estimate $\phi$ instead of fixing it at $0.25 .$ Find the maximum likelihood estimates for $\phi, \mu, \rho_{1}, \rho_{2}$, and $\sigma^{2} .$ [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 $\mathrm{AR}(1)$ errors consider the model with MA(1) errors:
\begin{aligned} &y_{t}=\mu+\rho_{1} y_{t-1}+\rho_{2} y_{t-2}+\epsilon_{t} \ &\epsilon_{t}=u_{t}+\psi u_{t-1}, \quad u_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right) \text { iid white noise } \end{aligned}
where $u_{0}=u_{-1}=0 .$ Derive the log-likelihood function $\ell\left(\psi, \mu, \rho_{1}, \rho_{2}, \sigma^{2} \mid\right.$ $\left.\mathrm{y}, y_{0}\right)$

d) For the inflation data, and using the model in part (c), find the MLE for $\psi, \mu, \rho_{1}, \rho_{2}$, and $\sigma^{2}$. Which model fits the data better (in the MSE sense); the AR(2) errors, or MA(2) errors?

Problem 2.

[15 marks] Consider the AR(2) specification $y_{t}=\mu+$ $\rho_{1} y_{t-1}+\rho_{2} y_{t-2}+\epsilon_{t} .$ Consider three models for the errors:
Model 1: $\epsilon_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right)$ white noise
Model 2: $\epsilon_{t}=\phi \epsilon_{t-1}+u_{t}, \quad u_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right)$ iid white noise Model 3: $\epsilon_{t}=u_{t}+\psi u_{t-1}, \quad u_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right)$ iid white noise
For each model, using the US inflation data in macrodata.csv, compute the MSFE for the one-step-ahead and three-step-ahead forecast. Which model performs best at forecasting?

Problem 3.

[15 marks] This is a theory-only question. Do not attempt to use data in this question.
Consider the ARMA (1,2) process with drift
$y_{t}=\mu+\phi y_{t-1}+u_{t}+\psi_{1} u_{t-1}+\psi_{2} u_{t-2}, \quad u_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right)$ white noise
a) Compute the autocovariance function of $y$. [You may take a maximum of 10 points on this question by setting $\phi=0$ if you wish in this part only. Please be clear and indicate if you do this.]
b) For what values of the parameters is this process covariance stationary?
c) Explain how to produce a two-step-ahead forecast using this model.
d) Consider the poorly-formed model
$y_{t}=\mu+\phi y_{t-1}+\psi_{0} u_{t}+\psi_{1} u_{t-1}+\psi_{2} u_{t-2}, \quad u_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right)$ white noise Suppose $\boldsymbol{\theta}^{}=\left(\mu^{}, \phi^{}, \psi_{0}^{}, \psi_{1}^{}, \psi_{2}^{},\left(\sigma^{2}\right)^{}\right)$ 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 $\boldsymbol{\theta}^{}$.)

Problem 4.

[15 marks] For the inflation data provided, compute the MSFE for the IMA(1,2), IMA(1,3), and IMA(1,4) models. Plot your forecasts. Compare your results. [Recall: The IMA(1,q) model means the differences in the data follow an MA(q) process]

Problem 5.

[15 marks] Consider the following UC model with AR(1) state equation:
$$\begin{gathered} y_{t}=\tau_{t}+\epsilon_{t}, \quad \epsilon_{t} \sim \mathcal{N}\left(0, \sigma^{2}\right) \text { white noise } \ \tau_{t}=\phi \tau_{t-1}+u_{t}, \quad u_{t} \sim \mathcal{N}\left(0, \omega^{2}\right) \text { white noise } \end{gathered}$$
where the unobserved component is initialised by $\tau_{0}=\mathcal{N}(0,1)$ and $\omega^{2}=$ $0.2$. Use the full sample inflation data to find the MLE for $\phi$, and $\sigma^{2}$. Compute the MSFE for the one-step-ahead forecasting exercise for the inflation data under this model.

Problem 6.

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
$$\mathbf{y}{t}=\mathbf{b}+\mathbf{B}{1} \mathbf{y}{t-1}+\mathbf{B}{2} \mathbf{y}{t-2}+\epsilon{t}$$
where $\mathrm{y}_{t}$ is a $3 \times 1$ vector consisting of inflation rate, GDP growth rate, and interest rate.
d) Comment briefly on your results (one or two sentences is sufficient).

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