Subject Title: Time Series Analysis and ForecastingSubject Code: AMA332Programme & Dept: 63002 / HD in Mathematics, Statistics and Computing / AMAAcademic Year: 2001 / 2002Term: SecondExamination Session: Examination

Problem 1.

1. The following is the yearly coffee production (in coded units) of a certain country from 2013 to 2021 .
\begin{tabular}{r|ccccccccc}
\hline Year & 2013 & 2014 & 2015 & 2016 & 2017 & 2018 & 2019 & 2020 & 2021 \
\hline$t$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \
\hline$y_{t}$ & 16 & 21 & 27 & 25 & 31 & 29 & 32 & 37 & 43 \
\hline
\end{tabular}
It is felt that the data can be fitted by the linear trend model,
$$Y_{t}-\alpha+\beta t+c_{t} \text {, where } c \sim N I D\left(0, \sigma^{2}\right) \text { and } \beta>0 .$$
Here, $c \sim N I D(0,1)$ means $c_{t} \sim N(0,1)$ for each $t$, and $\left{c_{t}\right}$ are independent of each other at different times.
(a) Find the least square estimates of $\alpha$ and $\beta$, and forecast the value of 2022 . [6 marks]
(b) Instead of using least square estimate, it is suggested to use the Simple Exponential Smoothing (SES) method with exponential smoothing constant $\alpha-0.1$ and the initial forecast $S_{0}-y_{1}$ to construct 1 -step ahead forecasts for the data, i.e.
$$S_{t}=0.1 Y_{t}+0.9 S_{t-1} .$$
i. Calculate the one-step ahead forecast for the year 2015-2018. [8 marks]
ii. Do you think SES overestimates or underestimates the model (1)? Briefly explain.
[4 marks]
iii. What do you suggest to improve the forecast and why?
[2 marks]

Problem 2.

1. For a time series $\left{X_{t}\right}$ with sample size 100 , the sample autocorrelation function $(\mathrm{ACF})$ and sample partial autocorrelation function (PACF) for $\left{X_{t}\right}$ and its differenced sequence $\left{\nabla X_{t}\right}$ are given below.
\begin{tabular}{r|cccccccc}
\hline Lag & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \
\hline \hline Sample ACF of $X_{t}$ & $0.765$ & $0.691$ & $0.606$ & $0.502$ & $0.457$ & $0.419$ & $0.343$ & $0.269$ \
Sample PACF of $X_{t}$ & $*$ &  & $*+$ & $-0.079$ & $0.061$ & $0.069$ & $-0.081$ & $-0.104$ \
\hline Sample ACF of $\nabla X_{t}$ & $0.490$ & $0.077$ & $0.142$ & $0.167$ & $0.020$ & $-0.015$ & $0.085$ & $-0.041$ \
Sample PACF of $\nabla X_{t}$ & $0.490$ & $-0.213$ & $0.277$ & $-0.248$ & $-0.253$ & $0.132$ & $0.070$ & $-0.199$ \
\hline
\end{tabular}
(a) Complete the above table by finding the corresponding sample PACF for $\left{X_{t}\right}$ at lag 1 , lag 2 and lag 3 .
[6 marks]
(b) Propose two possible models for $\left{X_{t}\right}$ and justify them.
[4 marks]
(c) If an ARIMA $(0,1,0)$ model was fitted to $\left{X_{t}\right}$, conduct a Ljung-Box test using values of lag 1 , lag $2, \ldots$,lag 8 , at significance level $\alpha-0.05$ to test whether the residuals are uncorrelated.
[8 marks]

Problem 3.

1. Consider the following process,
$$X_{t}-0.7 X_{t-1}-0.1 X_{t-2}+Z_{t}+2 \theta Z_{t-1}-\theta^{2} Z_{t-2}$$
where $\left{Z_{t}\right}$ is a white noise with mean zero and variance $\sigma^{2}$.
(a) Is $\left{X_{t}\right}$ stationary? Justify your answer.
[3 marks]
(b) Find the range of $\theta$ such that $\left{X_{t}\right}$ is invertible.
[9 marks]
(c) Suppose that $\theta$ is in the range found in (b) so that $Z_{t}$ can be expressed as $Z_{t}-\sum_{j=0}^{\infty} \pi_{j} X_{t-j}$.
i. Find the values of $\pi_{0}, \pi_{1}$ and $\pi_{2}$ in terms of $\theta$.
[4 marks]
ii. Write down a recursive formula of $\pi_{j}$ for $j \geq 3$ in terms of $\theta$.
[2 marks]
(d) Let $\theta-0$. Given that $\operatorname{Var}\left(X_{t}\right)-0.5$, calculate the variance of $Z_{t}, \sigma^{2}$. $\mid 7$ marks $\mid$
(c) Given the observations $X_{2022}-1.6$ and $X_{2021}-0.9$, find the one-step, two-step and three-step ahead forecasts at $t-2022$, when $\theta-0$. Moreover, calculate the variance of these three forecast errors in terms of $\sigma^{2}$.
[10 marks]

Problem 4.

1. Consider the ARCH(2) process
\begin{aligned} &a_{t}-\sigma_{t} c_{t}, \quad c \sim N I D(0,1), \ &\sigma_{t}^{2}-\omega+\alpha_{1} a_{t-1}^{2}+\alpha_{2} a_{t-2}^{2} \end{aligned}
Here, $c \sim N I D(0,1)$ means $c_{\mathrm{t}} \sim N(0,1)$ for each $t$, and $\left{c_{t}\right}$ are independent of each other at different times.
(a) Calculate $\operatorname{Cov}\left(a_{t}, a_{s}\right)$ for $t \neq s$.
[8 marks]
(b) Let $\alpha_{1}-\frac{1}{2}$ and $\alpha_{2}-\frac{1}{4}$. Express $\mathbb{E}\left[a_{t}^{2} a_{t-1}^{2}\right]$ in terms of $\omega$ and $m_{4}-\mathbf{E}\left[a_{t}^{4}\right]$, the fourth moment of $a_{t}$.
[4 marks]
(c) Under the condition in (b), express $\mathbb{E}\left[a_{t}^{2} a_{s}^{2}\right], t+s$, in terms of $\omega$ and $m_{4}-\mathbf{E}\left[a_{t}^{4}\right]$.
[15 marks]

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