a) Plot the inflation data from 1980 to now. Produce a plot of the sample autocovariance function (using a stem plot of xcov or similar.
b) Make an informed judgement as to whether the data is covariance stationary. If it is not, then take appropariate logs and/or differences until you think the data is covariance stationary. (This will likely involve producing additional plot(s) of sample autocovariance.
c) Based on your results from part (b), what kind of ARIMA model do you suspect will be good for modelling the inflation data?

a) Consider the model from Assignment 2 Question 7; the ARX(1) model with $\mathrm{AR}$ (2) errors. Fix $\phi_1=\phi_2=0.25$. Find the maximum likelihood estimates for $\mu, a, \rho$, and $\sigma^2$.
b) Now we want to estimate $\phi_1$ and $\phi_2$ instead of fixing them at $0.25$. Find the maximum likelihood estimates for $\phi_1, \phi_2, \mu, \rho, a$, and $\sigma^2$.
c) Instead of $\mathrm{AR}(2)$ errors consider the model with MA(2) errors:
$$
y_t=\mu+a t+\rho y_{t-1}+\epsilon_t,
$$
$\epsilon_t=u_t+\psi_1 u_{t-1}+\psi_2 u_{t-2,} \quad u_t \sim \mathcal{N}\left(0, \sigma^2\right)$ iid white noise
where $u_0=u_{-1}=0$. Derive the log-likelihood function $\ell\left(\psi_1, \psi_2, \mu, \rho, a, \sigma^2 \mid\right.$ $\left.y, y_0\right)$
d) For the inflation data, and using the model in part (c), find the MLE for $\psi_1, \psi_2, \mu, \rho, a$, and $\sigma^2$. Which model fits the data better (in the MSE sense); the AR(2) errors, or MA(2) errors?

[15 marks] Consider the ARX(1) specification $y_t=\mu+$ $a t+\rho y_{t-1}+\epsilon_t$. Consider three models for the errors:
Model 1: $\epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right)$ white noise
Model 2: $\epsilon_t=\phi_1 \epsilon_{t-1}+\phi_2 \epsilon_{t-2}+u_t, \quad u_t \sim \mathcal{N}\left(0, \sigma^2\right)$ iid white noise
Model 3: $\epsilon_t=u_t+\psi_1 u_{t-1}+\psi_2 u_{t-2}, \quad u_t \sim \mathcal{N}\left(0, \sigma^2\right)$ iid white noise
For each model, using the Australian inflation data in macrodata.csv, compute the MSFE for the one-step-ahead and three-step-ahead forecast. Which model performs best at forecasting?

[15 marks] This is a theory-only question. Do not attempt to use data in this question.
Consider the $\operatorname{ARMA}(2,1)$ process with drift $y_t=\mu+\phi_1 y_{t-1}+\phi_2 y_{t-2}+u_t+\psi u_{t-1}, \quad u_t \sim \mathcal{N}\left(0, \sigma^2\right)$ white noise
a) Compute the autocovariance function of $y$. You may assume the process is covariance stationary. [You may take a maximum of 10 points on this question by setting $\psi=0$ if you wish in this part only. Please be clear and indicate if you do this.]
b) Explain how to produce a two-step-ahead forecast using this model.
c) Consider the poorly-formed model $y_t=\mu+\phi_1 y_{t-1}+\phi_2 y_{t-2}+\psi_0 u_t+\psi_1 u_{t-1}, u_t \sim \mathcal{N}\left(0, \sigma^2\right)$ white noise
Suppose $\theta^=\left(\mu^, \phi_1^, \phi_2^, \psi_0^, \psi_1^,\left(\sigma^2\right)^\right)$ 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 $\theta^$.)

[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.]

Reflect on your results from you IMA analysis with your expected outcome from Question 1.

[15 marks] Consider the following UC model with AR(1) state equation:
$$
y_t=\tau_t+\epsilon_t, \epsilon_t \sim \mathcal{N}\left(0, \sigma^2\right) \text { white noise }
$$
$\tau_t=\phi \tau_{t-1}+u_t, \quad u_t \sim \mathcal{N}\left(0, \omega^2\right)$ white noise
where the unobserved component is initialised by $\tau_0=\mathcal{N}(0,1)$ and $\omega^2=$ $0.04$. Use the 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.

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 $y_t$ 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).