[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?

[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?

[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 $\boldsymbol{\theta}^{}$.)

[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]

[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.

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).